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+{"text":"\\section{Introduction}\nIn 1992 Sedykh \\cite{sedykh:originalvertex, sedykh:vertex} generalized the classical four vertex theorem of planar curves by showing that the torsion of any closed space curve  vanishes at least $4$ times, if it lies on a convex surface, see Note \\ref{note:4vertex}. Recently the author \\cite{ghomi:rosenberg} extended Sedykh's theorem to curves which lie on a \\emph{locally} convex simply connected surface. In this work we prove another generalization of Sedykh's theorem which does not require the existence of any underlying surface for the curve. \n\nTo state our result, let us recall that \na set $X$ in Euclidean space $\\mathbf{R}^3$ is \\emph{star-shaped} with respect to a point $o$ if no ray emanating from $o$ intersects $X$ in more than one point. Further let us say that $X$ is \\emph{locally convex} with respect to $o$ if through  every point $p$ of $X$ there passes a plane $H$, not containing $o$, such that a neighborhood of $p$ in $X$ lies on the same side of $H$ as does $o$. For instance, the boundary of any convex body in $\\mathbf{R}^3$ is both star-shaped and locally convex with respect to any of its interior points. In this paper we show:\n\n\\begin{thm}\\label{thm:main}\nLet $\\Gamma\\subset\\mathbf{R}^3$ be a simple closed $\\mathcal{C}^3$ immersed curve with nonvanishing curvature. Suppose that $\\Gamma$ is star-shaped and locally convex with respect to a point  $o$ in the interior of its convex hull. Then the torsion of $\\Gamma$ changes sign at least $4$ times.\n\\end{thm}\n\nIn particular,  if $\\Gamma$ lies on  the boundary of a convex body, then it immediately follows that $\\Gamma$ has at least $4$ points of vanishing torsion, which is Sedykh's result. The above theorem also generalizes a similar result of Thorbergsson and Umehara \\cite[Thm. 0.2]{thorbergsson&umehara}, see Note \\ref{note:TU2}.\n\n\nThe general strategy for proving Theorem \\ref{thm:main} hinges on the fact that at a point of nonvanishing torsion $p$, the torsion $\\tau$ of $\\Gamma$ is positive (resp. negative) if and only if $\\Gamma$ crosses its osculating plane at $p$ in the direction (resp. opposite direction) of the binormal vector $B(p)$. To exploit this phenomenon, we project $\\Gamma$ into a sphere $S$ centered at $o$ to obtain a simple closed  curve $\\overline\\Gamma$ which contains $o$ in its convex hull. By a result of Segre \\cite{segre:tangents, weiner:inflection, ghomi:verticesC}, also known as Arnold's tennis ball theorem, $\\overline\\Gamma$ has at least $4$ inflections $\\overline p_i$. It turns out that  the osculating planes $\\overline\\Pi_{\\overline p_i}$ of $\\overline\\Gamma$ coincide with the osculating planes $\\Pi_{p_i}$ of $\\Gamma$ at $p_i$, where $p_i$ are the preimages of $\\overline p_i$, see Figure \\ref{fig:projection}(a). Further, the local convexity assumption will  ensure that the binormal vectors of these planes are parallel, i.e., $\\overline B(\\overline p_i)=B(p_i)$. So the local position of $\\Gamma$ with respect to $\\Pi_{p_i}$ mirrors that of $\\overline\\Gamma$ with respect to $\\overline\\Pi_{\\overline p_i}$. After a perturbation of $o$, we may assume that $\\tau(p_i)\\neq 0$ and $\\overline p_i$ are \\emph{genuine} inflections, i.e.,  the geodesic curvature of $\\overline\\Gamma$ changes sign at $\\overline p_i$. Further, it is easy to see that at every pair of consecutive genuine inflections, $\\overline\\Gamma$ crosses its osculating planes in opposite directions with respect to $\\overline B$, because $\\overline B$ can never be orthogonal to $S$, see Figure \\ref{fig:projection}(b).\nThus it follows that $\\tau$ changes sign at least once within each of the $4$ intervals of $\\Gamma$ determined by $p_i$.\n \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1.75in]{projection.pdf}\n         \\put(13.5,12.5){\\small $p_i$}\n         \\put(13,20){\\small $B(p_i)$}\n         \\put(22,16){\\small $\\Gamma$}\n          \\put(34.5,19){\\small $\\overline p_i$}\n         \\put(33,27){\\small $\\overline B(\\overline{p}_i)$}\n         \\put(38.5,23){\\small $\\overline\\Gamma$}\n         \\put(30,2){\\small (a)}\n         \\put(81,2){\\small (b)}\n         \\put(13.5,6){$\\Pi_{p_i}=\\overline\\Pi_{\\overline{p}_i}$}\n                    \\put(81,25.5){\\small $\\overline\\Gamma$}\n            \\put(94,23){\\small $\\overline{B}$}\n           \\put(68,14){\\small $\\overline{B}$}\n\n    \\end{overpic}\n    \\caption{}\\label{fig:projection}\n\\end{figure}\n\n\n\nThe above approach for studying the sign of torsion is due to Thorbergsson and Umehara \\cite[p. 240]{thorbergsson&umehara}, who in turn attribute the spherical projection technique to Segre \\cite[p. 258]{segre1968alcune}; however, the method of Thorbergsson and Umehara does not quite lead to a generalization of Sedykh's theorem to a class of nonconvex curves, as we describe in Note \\ref{note:TU} below. \n\nNext we discuss some examples which validate Theorem \\ref{thm:main}.  Figure \\ref{fig:curls}(a) shows a curve which is star-shaped and locally convex, but is not convex. Thus Theorem \\ref{thm:main} is  indeed a nontrivial generalization of Sedykh's result.  Figure \\ref{fig:curls}(b), which shows a torus curve of type $(1, 7)$, demonstrates that the star-shaped assumption by itself would not be sufficient in Theorem \\ref{thm:main}. Indeed all torus curves of type $(1,n)$ are star-shaped with respect to any point on their axis of symmetry, and Costa \\cite{costa:twisted} has shown that, for $n\\geq 2$, they may be realized with nonvanishing torsion if the underlying torus is sufficiently thin. Finally Figure \\ref{fig:curls}(c) shows that the local convexity by itself is not sufficient either. This figure depicts a spherical curve with only two extrema of geodesic curvature and hence only two sign changes of torsion (the torsion of a spherical curve vanishes precisely when its geodesic curvature has a local extremum).\n\n \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1in]{curls.jpg}\n    \\put(10,-2){\\small (a)}\n    \\put(45,-2){\\small (b)}\n    \\put(78,-2){\\small (c)}\n        \\end{overpic}\n    \\caption{}\\label{fig:curls}\n\\end{figure}\n\nFour vertex theorems have had a multifaceted and interesting history, with unexpected applications, since Mukhopadhyaya proved the first version of this phenomenon in 1909. For extensive background and more references see \\cite{ghomi:rosenberg, gluck:notices, ovsienko&tabachnikov}. In particular see \\cite{bray&jauregui} for some recent applications in General Relativity, and \\cite{sedykh1996, sedykh1997} for discrete versions.\n\n\\begin{note}\\label{note:4vertex}\nThe classical four vertex theorem  states that the curvature of any simple closed curve in $\\mathbf{R}^2$ has at least $4$ critical points, which are called vertices.  A point of the curve is a vertex if and only if the osculating circle at that point has contact of order $3$ with the curve. Consequently, the geodesic curvature of simple closed curves on the sphere also satisfies the four vertex theorem, because the stereographic projection preserves circles. Further note that the plane which contains the osculating circle of a spherical curve is actually the osculating plane of the curve. Thus at a vertex, a spherical curve has contact of order $3$ with its osculating plane, which means that the torsion at that point vanishes. Hence all simple closed spherical curves have at least $4$ points of vanishing torsion. Sedykh generalized this result to curves lying on any convex surface. It is in this sense that Sedykh's theorem, and more generally our Theorem \\ref{thm:main}, are extensions of the classical four vertex theorem.\n\\end{note}\n\n\n\n\\begin{note}\\label{note:TU}\nFor convex space curves, a refinement of Sedykh's four vertex theorem is proved by  Thorbegsson and Umehara  in \\cite[Thm. 0.1]{thorbergsson&umehara}. Furthermore,  these authors \\cite[Thm. 0.2]{thorbergsson&umehara} obtain  a $4$ vertex result for space curves $\\Gamma\\subset \\mathbf{R}^3$ which are star-shaped with respect to a point $o$ in the interior of their convex hull, have no tangent line passing through $o$, and further satisfy the property that for all points $p\\in\\Gamma$ the angle between the principal normal $N(p)$ of $\\Gamma$ and the position vector $p-o$ is obtuse, i.e., \n\\begin{equation}\\label{eq:N}\n\\langle p-o, N(p) \\rangle <0.\n\\end{equation}\nThey claim that this result implies Sedykh's theorem, because a ``convex space curve $\\gamma$ satisfies the conditions in Theorem 0.2\" \\cite[p. 230]{thorbergsson&umehara}; however, this is not the case. Indeed there exists a simple closed curve  which lies on the boundary of a convex body,  but does not satisfy the condition \\eqref{eq:N} for any point $o$; see Figure \\ref{fig:apple}.\n  \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1.1in]{apple.jpg}\n    \\put(71.5,26){\\small$p_1$}\n    \\put(80,26){\\small$p_2$}\n     \\put(57,1.5){\\small$H_1^+$}\n      \\put(94,1.5){\\small$H_2^+$}\n    \\end{overpic}\n    \\caption{}\\label{fig:apple}\n\\end{figure}\nThe left diagram here depicts the curve,  and the right diagram shows its projection into its plane of symmetry. Let $p_i$, $i=1$, $2$ be the intersections of the curve with it symmetry plane. Note that at these points the principal normals $N(p_i)$ are antiparallel. Let $H_i$  be the planes orthogonal to $N(p_i)$ which pass through $p_i$, and $H_i^+$ be the corresponding (closed) half-spaces into which $N(p_i)$ point. Note that each $H_i^+$ consists of the set of all points $o$ such that \n$\\l p_i-o, N(p_i) \\r \\leq 0$. But $H_1$ and $H_2$ are disjoint. So there exists no point $o$ with respect to which $\\Gamma$ can satisfy condition \\eqref{eq:N}. Thus Theorem 0.2 in \\cite{thorbergsson&umehara} does not imply Sedykh's theorem.\n\\end{note}\n\n\\begin{note}\\label{note:TU2}\nThe result of Thorbergsson and Umehara  \\cite[Thm. 0.2]{thorbergsson&umehara}  mentioned in the previous note is a special case of Theorem \\ref{thm:main}. Indeed let $H_p$ be the rectifying plane of $\\Gamma$ at $p$, i.e., the plane which passes through $p$ and is orthogonal to the principal normal $N(p)$. Then \\eqref{eq:N} implies that $N(p)$ points to the side of $H$ which contains $o$. Consequently a neighborhood of $p$ in $\\Gamma$ lies on the same side as well, which establishes the local convexity of $\\Gamma$.\n\\begin{comment}\n (let $\\gamma\\colon (-\\epsilon,\\epsilon)\\to\\Gamma$ be a local unit speed parametrization with $\\gamma(0)=p$, and $f(t):=\\langle \\gamma(t)-p,N(p)\\rangle$; by Taylor's theorem, $f(t)=\\|\\gamma''(s)\\|\\langle N(s), N(0)\\rangle t^2\/2$ for some $s\\in(0,t)$, which yields $f\\geq 0$ for small $t$)\n\\end{comment}\n\\end{note}\n\n\n\n\n\n\\section{Basic Notation and Terminology}\nThroughout this work we assume that $\\Gamma$, $o$ are as in Theorem \\ref{thm:main}. In particular $\\Gamma$ has nonvanishing curvature (so that its torsion is well defined). For convenience we also assume that $o$ is the origin of $\\mathbf{R}^3$. Further we let $\\overline\\Gamma$ denote the radial projection of $\\Gamma$ into the unit sphere $\\S^2$ centered at $o$. For every point $p\\in\\Gamma$, $\\overline p:=p\/\\|p\\|$ denotes the corresponding point of $\\overline\\Gamma$. We assume that $\\Gamma$ is oriented, and let $T$, $N$, $B:=T\\times N$, denote the corresponding unit tangent, principal normal, and the binormal vectors of $\\Gamma$ respectively. For every point $p\\in\\Gamma$ there exists a $(\\mathcal{C}^3)$ unit speed parametrization $\\gamma\\colon (-\\epsilon,\\epsilon)\\to \\Gamma$ with $\\gamma(0)=p$ such that $\\gamma'(0)=T(p)$. Then $N(p):=\\gamma''(0)\/\\|\\gamma''(0)\\|$, and  the torsion of $\\Gamma$ is given by \n$$\n\\tau(p):=\\frac{\\langle \\gamma'''(0),B(p)\\rangle}{\\|\\gamma''(0)\\|}.\n$$\n The osculating plane $\\Pi_p$ of $\\Gamma$ at $p$ is the plane which passes through $p$ and is orthogonal to $B(p)$. We let $\\overline\\gamma$, $\\overline T$, $\\overline N$, $\\overline B$,   $\\overline \\Pi$ denote the corresponding quantities for $\\overline\\Gamma$. More specifically, $\\overline\\gamma:=\\gamma\/\\|\\gamma\\|$, $\\overline T:=\\overline\\gamma'\/\\|\\overline\\gamma'\\|$, $\\overline B:=\\overline\\gamma'\\times\\overline\\gamma''\/\\|\\overline\\gamma'\\times\\overline\\gamma''\\|$, and $\\overline N:=\\overline B\\times \\overline T$. In particular note that these quantities are well defined, since by assumption through each point of $\\Gamma$ there passes a local support plane of $H$ not containing $o$. Consequently the tangent lines of $\\Gamma$ do not pass through $o$, since they  lie in $H$. So $\\|\\overline\\gamma'\\|\\neq 0$.\n Further note that $\\overline\\Gamma$ inherits its orientation from $\\Gamma$.\nAn \\emph{inflection point} of $\\overline\\Gamma$ is a point where the geodesic curvature $\\overline k$ of $\\overline\\Gamma$ in $\\S^2$ vanishes. Here we define $\\overline k$ with respect to the conormal vector $\\overline n(\\overline p):=\\overline p\\times \\overline T(\\overline p)$:\n$$\n\\overline k(\\overline p):=\\langle \\overline N(\\overline p),\\overline n(\\overline p)\\rangle.\n$$\nWe say that an inflection point $\\overline p$ is genuine if $\\overline k$ changes sign at $\\overline p$.\n\n\\section{Osculating Planes and Inflections}\nA key part of the proof of Theorem \\ref{thm:main}, which facilitates its reduction to Segre's tennis ball theorem, is the following observation. The first part of this lemma is trivial, the second part goes back to Segre, and the third part is a consequence of our local convexity assumption. \n\\begin{lem}\\label{lem:osculate}\nLet $\\overline p$ be an inflection point of $\\overline\\Gamma$. Then \n\\begin{enumerate}\n\\item[(i)] $o\\in\\overline\\Pi_{\\overline p}$, \n\\item[(ii)] $\\overline\\Pi_{\\overline p}=\\Pi_p$, \n\\item[(iii)] $B(p)= \\overline B(\\overline p)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe argument is presented in three parts corresponding to each of the items enumerated above. Here for any pair of vector $v$, $w$, we use the notation $v\\parallel w$ to indicate that $v=\\lambda w$ for some $\\lambda>0$.\n\n(\\emph{i}) \nThe point $\\overline p$ is an inflection if and only if $\\overline N(\\overline p)$ is orthogonal to $\\S^2$, or\n\\begin{equation}\\label{eq:olN}\n\\overline N(\\overline p)=-\\overline p.\n\\end{equation}\n Thus $o=\\overline p+\\overline N(\\overline p)\\in\\overline \\Pi_{\\overline p}$, as claimed. \n\n(\\emph{ii}) \nSince $\\overline\\Pi_{\\overline p}$ passes through $\\overline p$ and $o$, it contains $p$ as well. So it suffices to check that $\\overline\\Pi_{\\overline p}$ and $\\Pi_p$ are parallel, or that $\\overline B(\\overline p)$ is orthogonal to $\\Pi_p$. To this end let $\\gamma\\colon (-\\epsilon, \\epsilon)\\to\\Gamma$ be a  local parametrization with $\\gamma(0)=p$ and $\\gamma'(0)\\parallel T(p)$. Then $\\overline\\gamma:=\\gamma\/\\|\\gamma\\|$ yields a local parametrization for $\\overline\\Gamma$ with $\\overline\\gamma(0)=\\overline p$ and $\\overline\\gamma'(0)\\parallel \\overline T(\\overline p)$.  We need to check that $\\overline B(\\overline p)$ is orthogonal to both $\\gamma'(0)$ and $\\gamma''(0)$. Simple computations show that\n\\begin{equation}\\label{eq:olgamma1}\n\\overline\\gamma\\times \\overline\\gamma'=\\frac{\\gamma\\times \\gamma'}{\\|\\gamma\\|^2},\n\\end{equation}\n\\begin{equation}\\label{eq:olgamma2}\n\\overline\\gamma\\times\\overline\\gamma''=\\frac{(\\gamma\\times\\gamma'')\\|\\gamma\\|^2-2(\\gamma\\times\\gamma')\\l\\gamma,\\gamma'\\r}{\\|\\gamma\\|^4}.\n\\end{equation}\nUsing \\eqref{eq:olN} and \\eqref{eq:olgamma1}, we have\n\\begin{equation}\\label{eq:olB}\n\\overline B(\\overline p)=\\overline T(\\overline p)\\times \\overline N(\\overline p)=\\overline p\\times \\overline T(\\overline p)=\\overline\\gamma(0)\\times \\overline\\gamma'(0)\n=\\frac{\\gamma(0)\\times \\gamma'(0)}{\\|\\gamma(0)\\|^2}.\n\\end{equation}\nThus\n$\n\\langle \\gamma'(0), \\overline B(\\overline p)\\rangle=0.\n$\nNext, to compute $\\l\\gamma''(0), \\overline B(\\overline p)\\r$, we may assume that $\\overline\\gamma$ has unit speed. Then $\\overline N(\\overline p)\\parallel \\overline\\gamma''(0)$ and \\eqref{eq:olN} yields that \n$$\n\\overline\\gamma(0)\\times \\overline\\gamma''(0)\\parallel  \\overline p\\times \\overline N(\\overline p)\\parallel \\overline p\\times(-\\overline p)=0.\n$$\nConsequently, \\eqref{eq:olgamma2} yields that $\\gamma(0)\\times\\gamma''(0)=\\alpha\\; \\gamma(0)\\times\\gamma'(0)$, for some constant $\\alpha$. Now\nusing \\eqref{eq:olB} we have, \n$$\n\\|\\gamma(0)\\|^2\\l\\gamma''(0), \\overline B(\\overline p)\\r=\\l\\gamma'(0),\\gamma''(0)\\times\\gamma(0)\\r=\n\\alpha\\l\\gamma'(0),\\gamma'(0)\\times\\gamma(0)\\r=0,\n$$\nas desired. \n\n(\\emph{iii})\nWe may assume that $\\gamma$ has unit speed. Then $N(p)\\parallel \\gamma''(0)$.\nConsequently,\n$$\nB(p)=T(p)\\times N(p)\\parallel \\gamma'(0)\\times \\gamma''(0).\n$$\nThis together with \\eqref{eq:olB} shows that $B(p)\\parallel\\overline B(\\overline p)$ if and only if\n$$\n\\gamma'(0)\\times\\gamma''(0)\\parallel \\gamma(0)\\times \\gamma'(0).\n$$\nBy assumption there exists a local support plane $H$ of $\\Gamma$ passing through $p$.   Let $L$ be the line given by $H\\cap \\Pi_p$ ($L$ is well defined because $o\\in \\Pi_p$ by parts $(i)$ and $(ii)$ above, but $o\\not\\in H$ by assumption; thus $H\\neq\\Pi_p$). Let $L^+$ denote the side of $L$ in $\\Pi_p$ which contains $o$. Then $N(p)$ points into $L^+$, see Figure \\ref{fig:planes}; because by assumption $\\Gamma$ lies locally on the side of $H$ which contains $o$, and thus $N(p)$ must point into this side as well. \n  \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1.8in]{planes.pdf}\n    \\put(52.5,16){\\small$o$}\n    \\put(42,14){\\small$\\overline{p}$}\n    \\put(21,14){\\small$p$}\n    \\put(11,3){\\small$L$}\n     \\put(6,23){\\small$H$}\n     \\put(63,11){\\small$L^+$}\n     \\put(76,29){\\small$L$}\n      \\put(19,23.5){\\small$T(p)$}\n       \\put(43,24){\\small$\\overline T(\\overline p)$}\n       \\put(27,11.5){\\small$N(p)$}\n       \\put(50,13){\\small$\\overline N(\\overline p)$}\n       \\put(0,11){\\small$\\Pi_{p}=\\overline\\Pi_{\\overline{p}}$}\n       \\put(89, 6.5){\\small$o$}\n        \\put(89, 16.5){\\small$\\overline p$}\n         \\put(89, 28.5){\\small$p$}\n         \\put(79, 1){\\small$L^+$}\n         \\put(93.5, 26.5){\\small$T(p)$}\n         \\put(94, 16){\\small$\\overline T(p)$}\n         \\put(80.5, 22){\\small$N(p)$}\n         \\put(82, 9){\\small$\\overline N(p)$}\n       \\end{overpic}\n    \\caption{}\\label{fig:planes}\n\\end{figure}\n Further, $L$ is tangent to $\\Gamma$ since $H$ is tangent to $\\Gamma$. Thus $N(p)\\perp L$. Consequently \n $$\n L^+=\\{ x\\in \\Pi_p\\mid \\l\\ x-p,N(p)\\r\\geq 0\\}.\n $$  \n So since $o$ lies in the interior of $L^+$, $\\l o-p, N(p)\\r>0$, or\n $\n\\langle N(p), p\\rangle <0,\n$\n which  yields\n\\begin{equation}\\label{eq:a}\n \\langle \\gamma''(0), \\gamma(0)\\rangle <0.\n\\end{equation}\n Since $\\Pi_p$ passes through $o$,  $\\{\\gamma(0),\\gamma'(0),\\gamma''(0)\\}$ is linearly dependent.\n Further, since we are assuming that $\\gamma$ has unit speed, $\\gamma'\\perp\\gamma''$. Thus \n $$\n \\gamma(0)=a \\gamma'(0)+b\\gamma''(0),\n $$\n where $b=\\langle \\gamma''(0), \\gamma(0)\\rangle\/\\|\\gamma''(0)\\|^2<0$ according to \\eqref{eq:a}.\n So\n $$\n \\gamma(0)\\times\\gamma'(0)=-b\\;\\gamma'(0)\\times\\gamma''(0),\n $$\n which completes the proof.\n\\end{proof}\n\n\\section{Maximum Principles for Torsion and Geodesic Curvature}\n\nHere we collect the facts we need concerning the relation between the sign of torsion of a space curve, and its relative position with respect to its osculating plane. Further we discuss the corresponding facts for the geodesic curvature of spherical curves, which will be used in the proof of our main result in the next section.\n\n\nWe start with torsion. Here by the region \\emph{above} the osculating plane we mean the (closed) half-space into which the binormal vector $B$ points, and the region \\emph{below} will be the other half-space. Since we assume that $\\Gamma$ is oriented, for every pair of points $p$, $q$ of $\\Gamma$, there is a unique choice of a segment with initial point $p$ and final point $q$ which we denote by $[p,q]$. The interior of this segment will be denoted by $(p,q)$.\n\n\\begin{lem}[Lem. 6.12, \\cite{ghomi:rosenberg}]\\label{lem:maxprincipletorsion}\nSuppose that $\\tau\\geq  0$ (resp. $\\tau\\leq 0$) on a segment $[p, q]$ of $\\Gamma$. Then, near $p$,  the segment lies above (resp. below) the osculating plane of $\\Gamma$ at $p$.\\qed\n\\end{lem}\n\nThe above lemma quickly yields the following converse:\n\n\\begin{cor}\\label{cor:maxprincipletorsion}\nSuppose that a segment $[p,q]$ of $\\Gamma$ lies above its osculating  plane at $p$ (resp. $q$) and does not lie completely in the osculating plane. Then $\\tau >0$ (resp. $\\tau<0$) at some point of $[p,q]$.\n\\end{cor}\n\\begin{proof}\nFirst suppose that $[p,q]$ lies above its osculating plane at $p$ and assume, towards a contradiction, that $\\tau\\leq 0$ on $[p,q]$. Then $[p,q]$ lies below $\\Pi_p$ by Lemma \\ref{lem:maxprincipletorsion}. Thus $[p,q]$ must lie entirely in $\\Pi_p$ which is a contradiction. The case where $[p,q]$ lies above its osculating plane at $q$, also follows from Lemma \\ref{lem:maxprincipletorsion}, once we switch the orientation of $[p,q]$ and observe that this switches the direction of $B$, but does not effect the sign of $\\tau$.\n\\end{proof}\n\nSimilarly, here are the  facts concerning geodesic curvature which we need. For every $\\overline p\\in\\overline\\Gamma$, let $C_{\\overline p}$ denote the great circle in $\\S^2$ which is tangent to $\\overline\\Gamma$ at $\\overline p$. By the region \\emph{above} $C_{\\overline p}$ we mean the (closed) hemisphere into which the conormal vector $\\overline n(\\overline p):=\\overline p\\times \\overline T(\\overline p)$ points, and the other hemisphere will be referred to as the region \\emph{below} $C_{\\overline p}$.\n\n\\begin{lem}[Lem. 2.1, \\cite{ghomi:verticesC}]\\label{lem:maxprinciplecurvature}\nSuppose that  a segment $[\\overline p, \\overline q]$ of $\\overline\\Gamma$ lies above (resp. below) the tangent great circle $C_{\\overline p}$. Then either  $[\\overline p, \\overline q]$ is a part of $C_{\\overline p}$, or else $\\overline k > 0$ (resp. $< 0$) at some point of $[\\overline p, \\overline q]$.\\qed\n\\end{lem}\n\nNow we quickly obtain:\n\n\\begin{cor}\\label{cor:maxprinciplecurvature}\nSuppose that $\\overline k>0$ on the interior of a segment $[\\overline p,\\overline q]$ of $\\overline\\Gamma$. Then, near $\\overline p$ and $\\overline q$,  $[\\overline p, \\overline q]$ lies above $C_{\\overline p}$ and $C_{\\overline q}$ respectively, and does not coincide with them.\n\\end{cor}\n\\begin{proof}\nIf $[\\overline p,\\overline q]$ lies locally below  $C_{\\overline p}$, then $[\\overline p, \\overline q]$  coincides with $C_{\\overline p}$ near $\\overline p$, by Lemma \\ref{lem:maxprinciplecurvature}. So $[\\overline p, \\overline q]$ lies locally above $C_{\\overline p}$ as claimed. The same argument may also be applied to $\\overline q$ after switching the orientation of $\\overline\\Gamma$. Finally, since $\\overline k\\neq 0$ on $(p,q)$, these circles cannot coincide with $\\overline\\Gamma$ near $\\overline p$ and $\\overline q$.\n\\end{proof}\n\nNext, we link the last two corollaries together by recording that if $\\overline p$ is an inflection point of $\\overline\\Gamma$, then the region above $C_{\\overline p}$ corresponds to the region above $\\Pi_{p}$. Indeed recall that if $\\overline p$ is an inflection, then $\\overline N(\\overline p)=-\\overline p$ as we discussed in the proof of Lemma \\ref{lem:osculate}. Thus\n$$\n\\overline n(\\overline p)=\\overline p\\times \\overline T(\\overline p)=-\\overline N(\\overline p)\\times \\overline T(\\overline p)=\\overline T(\\overline p)\\times \\overline N(\\overline p)=\\overline B(\\overline p).\n$$\n\nSo Lemma \\ref{lem:osculate} quickly yields:\n\n\n\\begin{lem}\\label{lem:above}\nAt an inflection point  $\\overline p$ of $\\overline\\Gamma$, the region above the great circle $C_{\\overline p}$ in $\\S^2$ coincides with the hemisphere which lies above the osculating plane $\\Pi_p$.\\qed\n\\end{lem}\n\n\n\\section{Proof of the Main Result}\n\nBefore proving our main theorem, we require the following technical fact which shows that the local convexity and star-shaped  properties of $\\Gamma$ are stable under small perturbations of $o$.\n\n\\begin{lem}\\label{lem:oprime}\nThere exists an open neighborhood $U$ of $o$ such that $\\Gamma$ is star-shaped and locally convex with respect to every point $o'$ in $U$.\nFurther, we may choose $o'$ so that only finitely many osculating planes of $\\Gamma$ pass through $o'$.\n \\end{lem}\n\\begin{proof}\nThe local convexity assumption ensures that the tangent lines of $\\Gamma$ do not pass through $o$. Thus $\\overline\\Gamma$ is a $\\mathcal{C}^3$ immersed curve. Further, the star-shaped assumption ensures that $\\overline\\Gamma$ is embedded. Now since embeddings of compact manifold are open in the space of $\\mathcal{C}^1$ mappings, it follows that projections of $\\Gamma$ into unit spheres centered at $o'\\in U$ are embedded  as well, for some open neighborhood $U$ of $o$. So the star-shaped property is preserved under small perturbations of $o$. Next we  check that the local convexity assumption is preserved as well. To this end it suffices to show that the local support planes of $\\Gamma$ may be chosen continuously, for then the support planes cannot get arbitrarily close to $o$, and hence $\\Gamma$ remains locally convex with respect to all points of $U$, assuming $U$ is sufficiently small. To see that the local support planes may be chosen continuously, see \\cite[Sec 3.1]{ghomi:stconvex}.\nFinally, using Sard's theorem, we may choose a point $o'$ in $U$ such that only finitely many osculating planes of $\\Gamma$ pass through $o'$: consider  $f\\colon\\Gamma\\times\\mathbf{R}^2\\to\\mathbf{R}^3$ given by $f(p, t, s)= p+t\\, T(p) +s\\, N(p)$, and let $o'$ be a regular value of $f$. \n\\end{proof}\n\nWe also need the following refinement of the tennis ball theorem. Recall that an inflection point $\\overline p$ of $\\overline\\Gamma$ is \\emph{genuine}, provided that the geodesic curvature $\\overline k$ of $\\overline\\Gamma$ changes sign at $\\overline p$.\n\n\\begin{lem}[Thm. 1.2, \\cite{ghomi:verticesC}]\\label{lem:tennisball}\nSuppose that $\\overline\\Gamma$ has at most  finitely many inflections. Then at least $4$ of these inflections must be genuine.\\qed\n\\end{lem}\n \n\nFinally we are ready to establish our main result:\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nBy Lemma \\ref{lem:oprime}, after replacing $o$ by a nearby point, we may assume that at most finitely many osculating planes of $\\Gamma$ pass through $o$. By Lemma \\ref{lem:osculate}, osculating planes of $\\overline\\Gamma$ at its inflections pass through $o$ and coincide with the osculating planes of $\\Gamma$. Thus  $\\overline\\Gamma$ has now at most finitely many inflections. \nConsequently, by the refinement of the tennis ball theorem, Lemma \\ref{lem:tennisball},  $\\overline\\Gamma$ has at least $4$ genuine inflections. \n\nLet $\\overline p_0$, $\\overline p_1$ be a pair of  genuine inflections of $\\overline\\Gamma$ such that the oriented segment \n$[\\overline p_0, \\overline p_1]$   has no genuine inflections in its interior $(\\overline p_0, \\overline p_1)$. There are at least $4$ such intervals with pairwise disjoint interiors. Thus, to complete the proof, it suffices to show that $\\tau$ changes sign on $(p_0, p_1)$. \n\nWe may assume that $\\overline k\\geq 0$ on $(\\overline p_0,\\overline p_1)$, after switching the orientations of $\\Gamma$ and $\\overline\\Gamma$ if necessary. Then $\\overline k>0$ near $\\overline p_i$ on $(\\overline p_0,\\overline p_1)$, since  $\\overline\\Gamma$ has only finitely many inflections. Consequently,  by the maximum principle for geodesic curvature, Corollary \\ref{cor:maxprinciplecurvature}, $[\\overline p_0, \\overline p_1]$ lies locally above its tangent great circles at $\\overline p_i$ and does not coincide with them near $\\overline p_i$.\nThis in turn implies, by Lemma \\ref{lem:above},\n that \n$[p_0, p_1]$ lies locally above its osculating planes at $p_i$, and does not coincide with them near $p_i$. Thus, by  the maximum principle for torsion, Corollary \\ref{cor:maxprincipletorsion}, $\\tau$ changes sign on $(p_0, p_1)$ as desired.\n\\end{proof}\n\n\\section*{Acknowledgment}\nThe author is grateful to Slava Sedykh for correcting some computations in the proof of Lemma \\ref{lem:osculate}, and several other comments to improve the exposition of this work. Thanks also to Masaaki Umehara for helpful communications.\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCurrently there is a large body of data coming from cosmological and astrophysical observations that is mostly consistent with the existence of dark matter.  Such observations also suggest that the hypothesized particles that constitute dark matter have very small cross section and travel at speeds much lower than light. These lead to the cold dark matter framework, which is one of the pillars of the current standard cosmological model $\\Lambda$CDM.\n\n\nIt is not only tempting, but mandatory to check if such dark matter particles exist (by detecting them in laboratory based experiments, for instance) and also to check if the gravitational effects that lead to the dark matter hypothesis could follow from a more detailed and complete approach to gravity. The effects of pure classical General Relativity at galaxies have been studied for a long time and, considering galaxy rotation curves, the differences between General Relativity and Newtonian gravity are negligible, since in a galaxy matter moves at speeds much lower than that of light and is typically subject to weak gravitational fields ($\\Phi \\ll c^2$), which leads to the Newtonian limit of General Relativity \\footnote{There are some proposals that consider General Relativity  in the context of galaxies which do not lead to Newtonian gravity, see e.g. \\cite{Cooperstock:2006dt, Vogt:2005hn, Cooperstock:2007sc, Vogt:2005va, Vogt:2005hv}.  It is not impossible that a reasonable explanation for galaxy rotation curves may rely on similar approaches, nevertheless up to now none of such proposals have found a baryonic mass distribution that is in conformity with astrophysical expectations.}. \n\nThere is however a newer approach that may change considerably the role of dark matter, while following standard physical principles. Namely, the investigation of the running of the gravitational coupling parameter $G$ on large scales as induced by the renormalization group framework.\n\nThe running of coupling constants is a well known phenomenon within \nQuantum Field Theory. It is well-known that the renormalization group \nmethod can be extended to quantum field theory on curved space time\nand to some models of quantum gravity (see, e.g., \\cite{Buchbinder:1992rb}), such that  the beta functions can be interpreted in this framework. Concerning \nthe high energy (UV) behavior, there is hope that the running of $G$ \nin quantum gravity may converge to a non-Gaussian fixed point \n(asymptotic safety) \\cite{Niedermaier:2006wt,Weinberg:2009bg}. Our present concern is, however, not about \nthe UV completeness, but with the behavior of $G$ in the far infrared \nregime (IR). In the electromagnetic case the IR behavior corresponds \nto the Appelquist-Carazzone decoupling \\cite{Appelquist:1974tg} (see e.g., \\cite{Goncalves:2009sk} for a recent derivation of this theorem). In the case of gravity the same effect of decoupling has been obtained in \\cite{Gorbar:2002pw,Gorbar:2003yt},\nbut only for the higher derivative terms in the gravitational action. \nIt remains unclear whether such decoupling takes place for the other terms. This possibility was studied on phenomenological grounds a number of times before, e.g. \\cite{Shapiro:2004ch,Reuter:2004nx}.\n\nIn  \\cite{Rodrigues:2009vf} we presented new results on the application of renormalization group (RG) corrections to General Relativity (GR) in the astrophysical domain. Previous attempts to apply this picture to galaxies have considered for simplicity point-like galaxies. We extended previous considerations by identifying the proper renormalization group energy scale $\\mu$ and by evaluating the consequences considering the observational data of disk galaxies. Also we propose a natural choice for the identification of $\\mu$, linking it to the local value of the Newtonian gravitational potential. With this choice, the renormalization group-based approach is capable to mimic dark matter effects with great precision. This picture induces a very small variation on the gravitational coupling parameter $G$, namely a variation of about $10^{-7}$ of its value across $10^5$ light-years. We call our model RGGR, in reference to renormalization group effects in General Relativity.\n\nIn order to evaluate the observational consequences of the RGGR model and to compare it to other proposals, \nrecent high quality observational data \\cite{2008AJ....136.2648D,Gentile:2004tb} from nine regular spiral galaxies were mass-modelled using the standard procedures for the baryonic part, and four different models for the ``dark'' component: i) the RGGR model; ii) one of the most phenomenological successful dark matter profiles, the Isothermal profile \\cite{1991MNRAS.249..523B}; and two alternative models which were built to avoid the need for the dark matter: iii) the Modified Newtonian Dynamics (MOND) \\cite{1983ApJ...270..365M,1983ApJ...270..371M} and iv) the Scalar-Tensor-Vector Gravity (STVG) \\cite{Moffat:2005si}. The latter is a recent relativistic proposal that is capable of dealing with galaxy rotation curves and other phenomena usually attributed to dark matter. For galaxy rotation curves phenomena, STVG becomes equivalent to a similar proposal called MSTG \\cite{Moffat:2004bm, Brownstein:2005zz, Brownstein:2009gz}. The model parameters that we use to fit galaxies in the STVG framework can be found in ref. \\cite{Brownstein:2005zz}. While for MOND we use the $a_0$ value as given in \\cite{Sanders:2002pf}.\n\nThe quality of the rotation curve fits and total stellar mass as inferred from the RGGR model is perfectly satisfactory considering both the general behavior of the model and its results when applied to nine particular galaxies, as analyzed in \\cite{Rodrigues:2009vf}. It is about the same of the Isothermal profile quality, while it seems significantly better than the quality of MOND and STVG. In the case of MOND, we did numerical experiences with $a_0$ as a free parameter and found that, albeit the concordance with the shape of the rotation curve can considerably increase in this case, the concordance with the expected stellar mass-to-light ratios remains unsatisfactory (similar conclusions have also appeared in some recent papers, e.g. \\cite{Gentile:2004tb,Gentile:2010xt,Famaey:2005fd}, and it seems that the concordance can only be improved by adjusting the MOND's $\\mu(x)$ function in an {\\it ad-hoc} way).\n\n\\section{The running of $G$}\n\n\nThe $\\beta$-function for  the gravitational coupling parameter $G$ has been discussed in the framework of different approaches to Quantum Gravity and Quantum Field Theory in curved space-time. In \\cite{Rodrigues:2009vf} we followed the derivation used previously in \\cite{Shapiro:2004ch}.  If  $G$ does not behave as  a constant  in the far  IR limit, it was argued in \\cite{Shapiro:2004ch} (and recently in more details in \\cite{Farina:2011me}) that the logarithmic running of $G$  is a direct consequence of covariance and must hold in all loop orders.  As far as direct derivation of the physical running of $G$ is not available, it is worthwhile to explore the possibility of a logarithmically running $G$ at the phenomenological level.\n\n\nConsider the following infrared $\\beta$-function for General Relativity,\n\\begin{equation}\n\t \\beta_{G^{-1}} \\equiv \\mu \\frac{dG^{-1}}{d \\mu} = 2 \\nu \\,  \\frac{M_{\\mbox{\\tiny Planck}}^{2}}{c \\, \\hbar} = 2 \\nu G_0^{-1}.\n\t\\label{betaG}\n\\end{equation}\n\nEquation (\\ref{betaG}) leads to the logarithmically  varying $G(\\mu)$ function,\n\\begin{equation}\n\t\\label{gmu}\n\tG(\\mu) = \\frac {G_0}{ 1 + \\nu \\,\\mbox{ln}\\,(\\mu^2\/\\mu_0^2)},\n\\end{equation}\nwhere $\\mu_0$ is a reference scale introduced such that $G(\\mu_0) =G_0 $. The constant $G_0$ is the gravitational constant as measured in the Solar System  (actually, there is no need to be very precise on where $G$ assumes the value of $G_0$, due to the smallness of the variation of $G$). The dimensionless constant $\\nu$ is a phenomenological parameter which depends on the details of the quantum theory leading to eq. (\\ref{gmu}). Since we have no means to compute the latter from first principles, its value should be fixed from observations. It will be shown that even a very small $\\nu$ can lead to observational consequences at galactic scales.\n\n\nThe action for this model is simply the Einstein-Hilbert one in which $G$ appears inside the integral, namely,\n\\begin{equation}\n\tS_{\\mbox{\\tiny RGGR}}[g] = \\frac {c^3}{16 \\pi }\\int \\frac {R  } G \\, \\sqrt{-g} \\,  d^4x.\n\t\\label{rggraction}\n\\end{equation}\nIn the above, $G$ should be understood as an external scalar field that satisfies (\\ref{gmu}).  Since for the problem of galaxy rotation curves  the cosmological constant effects are negligible, we have not written the $\\Lambda$ term above. Nevertheless, for a complete cosmological picture, $\\Lambda$ is necessary and it also runs covariantly with the RG flow of $G$ (see e.g.,\\cite{Shapiro:2004ch}). \n\nThere is a simple procedure to map solutions from the Einstein equations with the gravitational constant $G_0$  into RGGR solutions. One need not to follow this route, one may find all the dynamics from the RGGR equations of motion, which can be found by a direct variation of the action (\\ref{rggraction}) in respect to the metric, leading to equations of motion that have the same form of those of a scalar-tensor gravity\\footnote{We stress that it is only the from since RGGR is not a type of scalar-tensor gravity, and $G$ is not a fundamental field of the model.}. In this review, we will proceed to find RGGR solutions via a conformal transformation of the Einstein-Hilbert action, and to this end first we write \n\\begin{equation}\n\tG = G_0 + \\delta G,\n\\end{equation} \nand we assume $\\delta G \/ G_0 \\ll 1$, which will be justified latter. Introducing the conformally related metric\n\\begin{equation}\n\t\\bar g_{\\mu \\nu} \\equiv \\frac {G_0}{G} g_{\\mu \\nu}, \n\t\\label{ct}\n\\end{equation}\nthe RGGR action can be written as\n\\begin{equation}\n\tS_{\\mbox{\\tiny RGGR}}[g] = S_{\\mbox{\\tiny EH}}[\\bar g] + O(\\delta G^2),\n\\end{equation}\nwhere $S_{\\mbox{\\tiny EH}}$ is the Einstein-Hilbert action with $G_0$ as the gravitational constant. The above suggest that the RGGR solutions can be generated from the Einstein equations solutions via the conformal transformation (\\ref{ct}). Indeed, within a good approximation, one can check that this relation persists when comparing the RGGR equations of motion to the Einstein equations even in the presence of matter \\cite{Rodrigues:2009vf}.\n\nIn the context of rotation curves of galaxies, standard General Relativity gives essentially the same predictions of Newtonian gravity. The Newtonian potential $ \\Phi_{\\mbox{\\tiny Newt}}$ is related to the metric by \n\\begin{equation}\n\t\\bar g_{00} = - \\left ( 1  + \\frac {2 \\Phi_{\\mbox{\\tiny Newt}}}{c^2} \\right ).\n\\end{equation}\nHence, using eq. (\\ref{ct}), the effective RGGR potential $\\Phi$ in the non-relativistic limit is given by\n\\begin{equation}\n\t\\Phi = \\Phi_{\\mbox{\\tiny Newt}} + \\frac {c^2}2 \\frac{\\delta G}{G_0}.\n\\end{equation}\nAn equivalent result can also be found from the evaluation of a test particle geodesics \\cite{Rodrigues:2009vf}. In the context of weak gravitational fields $\\Phi_{\\mbox{\\tiny Newt}}\/ c^2 \\ll 1$ (with $\\Phi_{\\mbox{\\tiny Newt}} = 0$ at spatial infinity) holds, and hence the term $\\delta G\/G_0$ should not be neglected.\n\nIn order to derive a test particle acceleration, we have to specify the proper energy scale $\\mu$ for the problem setting in question, which is a time-independent gravitational phenomena in the weak field limit. This is a recent area of exploration of the renormalization group application, where the usual procedures for high energy scattering of particles cannot be applied straightforwardly. Previously to \\cite{Rodrigues:2009vf} the selection of $\\mu \\propto 1\/r$, where $r$ is the distance from a massive point, was repeatedly used, e.g. \\cite{Reuter:2004nv,Dalvit:1994gf,Bertolami:1993mh,Goldman:1992qs, Shapiro:2004ch}. This identification adds a constant velocity proportional to $\\nu$ to any rotation curve. Although it was pointed as an advantage due to the generation of ``flat rotation curves'' for galaxies, it introduced difficulties with the Tully-Fisher law, the Newtonian limit, and the behavior of the galaxy rotation curve close to the galactic center, since there the behavior is closer to the expected one without dark matter. In \\cite{Rodrigues:2009vf} we introduced a $\\mu$ identification that seems better justified both from the theoretical and observational points of view. The characteristic weak-field gravitational energy does not comes from the geometric scaling $1\/r$, but from the Newtonian potential $\\Phi_{\\mbox{\\tiny Newt}}$. However, the straight relation $ \\mu \\propto \\Phi_{\\mbox{\\tiny Newt}}$ leads to $\\mu \\propto 1\/r$ in the large $r$ limit; which is unsatisfactory on observational grounds (bad Newtonian limit and correspondence to the Tully-Fisher law). One way to recover the Newtonian limit is to impose a suitable cut-off, but this does not solves the Tully-Fisher issues \\cite{Shapiro:2004ch}. Another one is to use \\cite{Rodrigues:2009vf}\n\\begin{equation}\n\t\\frac {\\mu}{\\mu_0} =\\left( \\frac{\\Phi_{\\mbox{\\tiny Newt}}}{\\Phi_0} \\right)^\\alpha,\n\t\\label{muphi}\n\\end{equation}\nwhere $\\Phi_0$ and $\\alpha$ are constants. Apart from the condition $0 < \\Phi_0 < c^2$ (i.e., essentially $\\Phi_0$ is a reference Newtonian potential), the precise value of $\\Phi_0$ is largely irrelevant for the problem of rotation curves. The relevant parameter is $\\alpha$. It is a phenomenological parameter that depends on the mass of the system, and it must go to zero when the mass of the system goes to zero. This is necessary to have a good Newtonian limit. From the Tully-Fisher law, it is expected to increase monotonically with the increase of the mass. Such behavior is indeed found from the galaxy fits done in \\cite{Rodrigues:2009vf}. In a recent paper, an upper bound on $\\nu \\alpha$ in the Solar System was derived \\cite{Farina:2011me}. In galaxy systems, $\\nu \\alpha|_{\\mbox{\\tiny Galaxy}} \\sim 10^{-7}$, while for the Solar System, whose mass is about $10^{-10}$ of that of a galaxy,  $\\nu \\alpha|_{\\mbox{\\tiny Solar System}} \\lesssim 10^{-17}$. It shows that a linear decrease on $ \\alpha$ with the mass is sufficient to satisfy both the current upper bound from the Solar System and the results from galaxies.\n\nWe also point that the above energy scale setting (\\ref{muphi}) was recently re-obtained from a more theoretical perspective \\cite{Domazet:2010bk}.\n\nOnce the $\\mu$ identification is set, it is straightforward to find the rotation velocity for a static gravitational system sustained by its centripetal acceleration,\n\\begin{equation}\n\tV^2_{\\mbox{\\tiny RGGR}} \\approx V^2_{\\mbox{\\tiny Newt}} \\left ( 1 - \\frac {\\nu \\, \\alpha  \\, c^2} {\\Phi_{\\mbox{\\tiny Newt}}} \\right ).\n\t\\label{v2rggr}\n\\end{equation}\n\nContrary to Newtonian gravity, the value of the Newtonian \npotential at a given point does play a significant role \nin this approach. This sounds odd from the perspective of \nNewtonian gravity, but this is not so \nfrom the General Relativity viewpoint, since the latter has no \nfree zero point of energy. In particular, the Schwarzschild \nsolution is not invariant under a constant shift of the \npotential.\n\nIn the following, we will comment on the effect of the relation (\\ref{v2rggr}) to galaxy rotation curves. First from a more general perspective, and then to the modeling of individual galaxies.\n\n \n\n\n\n\n\\section{Galaxy rotation curves}\n\nBefore proceeding to specific galaxy rotation curves modeling, it is more instructive to analyze general features of the relation (\\ref{v2rggr}), and to compare it to the standard approach. In \\cite{Rodrigues:2009vf} we analyze some general aspects and scaling laws of the RGGR  model, with no dark matter, in comparison to the isothermal profile; both of them, at this step, without gas and with an exponential stellar disk. In particular, it was pointed that the RGGR rotation curves have a reasonable shape to fit galaxies (i.e., no clear problems like increasing or decreasing too fast, oscillations...),  and that they effectively behave similarly to cored dark matter profiles at inner radii, whose effective core radius scales with the galaxy disk scale length. Further details in our paper.\n\nWe have also extended the previous analysis by adding a gas-like contribution (a re-scaled version of the NGC 3198 gaseous part). In particular, this numerically evaluates how the model behaves on the presence of density perturbations at large radii.  In the first plot of fig. (\\ref{mus}) it is displayed the result for RGGR, which is remarkably good (a similar plot can be found in our original paper), while in the others plots in fig. (\\ref{mus}) (presented at the Conference, but not in \\cite{Rodrigues:2009vf}) one sees the results for the same mass distribution but different choices for the energy scale\\footnote{These other choices are also unsatisfactory from the theoretical perspective, since they have no direct relation to the local energy of the gravitational field in the weak field regime ($\\Phi_{\\mbox{\\tiny Newt}} \\ll c^2 $). } $\\mu$. \n\n\n\n\\begin{figure}[ht]\n\t  \\includegraphics[width=150mm]{IC2Cosmo3mus2.pdf}\n\\caption{The additional circular velocity squared induced by different  choices of $\\mu$. The first plot refers to RGGR, the two others to different identifications of $\\mu$: one depends on the Newtonian acceleration (a variation inspired on MOND) and the other on the (baryonic) matter density. All the plots above display the additional squared velocity of each model divided by $V^2_\\infty \\equiv \\nu \\alpha c^2$ and  as a function of $R\/R_D$, where  $R$ is the the radial cylindrical coordinate and  $R_D$ is the stellar disk scale length. Black lines depict the additional velocity due to a pure exponential stellar disk, while the gray solid lines  take into account the gas mass $M_{\\mbox{gas}}$ for different values of $f \\equiv M_{\\mbox{gas}}\/M_{\\mbox{stars}}$, with $f = 0.2, 0.7, 1.2,..., 9.7$ (i.e., the black lines stand for $f=0$). See \\cite{Rodrigues:2009vf} for further details.}\n\\label{mus}\n\\end{figure}\n\nFrom fig. (\\ref{mus}), the two other proposals different from RGGR are seen to be unsuited as replacements for dark matter. In particular, both are too sensitive to the gas presence, and both eventually add negative contributions to the total circular velocity at large radii.\n\n\nIn \\cite{Rodrigues:2009vf} we used a sample of nine high quality and regular rotation curves of disk galaxies from \\cite{2008AJ....136.2648D, Gentile:2004tb}. In figs. (\\ref{ngc2403}, \\ref{masstolightplot}) we show one of ours results (see \\cite{Rodrigues:2009vf} for the complete set and further details) in comparison to the results of three other models: a cored dark matter profile (Isothermal profile), the  Modified Newtonian Dynamics (MOND) and the recently proposed Scalar-Tensor-Vector Gravity (STVG). \n\n\\begin{figure}[ht]\n\t  \\includegraphics[width=100mm]{NGC2403PlotsIC2Models.pdf}\n\\caption{NGC 2403 rotation curve fits. The red dots and its error bars are the rotation curve observational data, the gray ones close to the abscissa are the residues of the fit. The solid black line for each model is its best fit rotation curve, the dashed yellow curves are the stellar rotation curves from the bulge and disk components,  the dotted purple curve is the gas rotation curve, and the dot-dashed green curve is the resulting Newtonian, with no dark matter, rotation curve. }\n\\label{ngc2403}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\t  \\includegraphics[width=100mm]{ColorVsYIC2Models}\n\\caption{Stellar disk mass-to-light ratio ($Y_{*D}$) in the $3.6  \\mu m$ band  as a function of the color $J - K$.  Each galactic disk is represented above by an open circle, with a {\\it reference} error bar of 50$\\%$ of the $Y_{*D}$ value. The black open squares display the $Y_{*D}$ values and their associated 1$\\sigma$ errors for each galaxy as inferred from the rotation curve fits for each model. The highlighted square and circle correspond to the NGC 2403 galactic disk mass-to-light ratios. See \\cite{Rodrigues:2009vf} for further details.}\n\\label{masstolightplot}\n\\end{figure}\n\n\nDue to the considerably large uncertainty in the total stellar mass of each stellar component (disk and bulge), we first use the total stellar mass as a free parameter for the fittings (achieved from a $\\chi^2$ minimization considering the errors). At a second stage, we compare the resulting value with stellar population expectations, following the  standard approach.\n\nOn the free parameters of each model, we remark that besides the total stellar mass, the Isothermal profile has two additional free parameters, the RGGR model has a single free parameter ($\\alpha$) while MOND and STVG have no free parameters that can vary from galaxy to galaxy. On the other hand, both of the latter depend on constants whose values are calibrated considering its best fit in a large sample of galaxies. We remark that the $\\nu$ parameter in RGGR cannot vary from galaxy to galaxy, but $\\alpha $ can, and galaxy rotation curves are sensible to the combination $\\nu \\alpha$, whose value is about the order of $10^{-7}$. The best fit for NGC 2403 yields $ \\nu \\alpha = (1.66 \\pm0.01) \\times 10^{-7}$.\n\n\n\n\\section{Conclusions}\n\nWe  presented a model, motivated by renormalization group corrections to the Einstein-Hilbert action, that introduces small inhomogeneities in the gravitational coupling across a galaxy (of about 1 part in $10^7$) and can generate galaxy rotation curves in agreement with the observational data, without the introduction of dark matter as a new kind of matter. Both High and Low Surface Brightness galaxies were tested \\cite{Rodrigues:2009vf} . Considering the samples of  galaxies evaluated in \\cite{Rodrigues:2009vf}, the quality of the RGGR rotation curves,  together with the corresponding mass-to-light ratios, is about the same than  the Isothermal profile quality, but with one less free parameter. We expect that similar results would hold in regard to other cored dark matter profiles, while our results seem better than those achieved by the NFW profile \\cite{Rodrigues:2009vf}. We also compared the results of our model with MOND and STVG, and at face value our model yielded clearly better results.   \n\nOur results can be seen as a next step compared to the \nprevious models motivated by renormalization group effects in \ngravity, e.g. \\cite{Shapiro:2004ch, Reuter:2007de}. Their original analyses \ncould only yield a rough estimate on the galaxy rotation curves, \nsince they were restricted to modeling a galaxy as a single \npoint. Trying to extend this approach to real galaxies, we \nhave shown that the proper scale \nfor the renormalization group phenomenology is not of a \ngeometric type, like the inverse of the distance, but is\nrelated to the Newtonian potential with null boundary \ncondition at infinity. \n\n\nThe essential feature for the RGGR rotation curves fits is the formula (\\ref{v2rggr}), which is by itself a  simple formula that provides a very efficient description of galaxy rotation curves. \n\nThere are several tests and implications of this model yet to be evaluated. In particular we are working on applying the RGGR framework to a larger sample of galaxies (including elliptical galaxies) \\cite{FabrisGalaxies} and galaxy-galaxy strong lensing \\cite{rodrigueslentes}. Related work on CMB, BAO and LSS in search for a new cosmological concordance model is also a work in progress \\cite{toribioCMB}.  \n\n\n\n\\bigskip\n\n\\noindent\n{ \\it \\bf Acknowledgements}\n\n\\noindent\nDCR thanks FAPESP and PRPPG-UFES for partial financial support. PSL thanks CNPq and FAPESP for partial financial support. The work of I.Sh. was partially supported by CNPq, FAPEMIG, FAPES and ICTP. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper we establish the structured ring spectra analogs of Goodwillie's widely exploited and powerful cubical diagram results \\cite{Goodwillie_calc2} for spaces. These cubical diagram results are a key ingredient in the authors' homotopic descent results \\cite{Ching_Harper} on a structured ring spectra analog of Quillen-Sullivan theory \\cite{Quillen_rational, Sullivan_MIT_notes, Sullivan_genetics}. They also establish an important part of the foundations for the theory of Goodwillie calculus in the context of structured ring spectra; see, for instance, Arone and Ching \\cite{Arone_Ching}, Bauer, Johnson, and McCarthy \\cite{Bauer_Johnson_McCarthy}, Ching \\cite{Ching_duality}, Harper and Hess \\cite[1.14]{Harper_Hess}, Kuhn \\cite{Kuhn_survey}, and Pereira \\cite{Pereira_general_context, Pereira_spectral_operad}. For example, it follows from our results that the identity functor on a category of structured ring spectra is analytic in the sense of Goodwillie \\cite{Goodwillie_calc2}.\n\n\\begin{assumption}\n\\label{assumption:commutative_ring_spectrum}\nFrom now on in this paper, we assume that ${ \\mathcal{R} }$ is any commutative ring spectrum; i.e., we assume that ${ \\mathcal{R} }$ is any commutative monoid object in the category $({ \\mathsf{Sp}^\\Sigma },{ \\otimes }_S,S)$ of symmetric spectra \\cite{Hovey_Shipley_Smith, Schwede_book_project}. We work mostly in the category of ${ \\mathcal{R} }$-modules which we denote by ${ \\mathsf{Mod}_{ \\mathcal{R} } }$.\n\\end{assumption}\n\n\\begin{rem}\nOur results apply to many different types of algebraic structures on spectra including (i) associative ring spectra, which we simply call ring spectra, (ii) commutative ring spectra, (iii) all of the $E_n$ ring spectra for $1\\leq n\\leq \\infty$ that interpolate  between these two extremes of non-commutativity and commutativity. These structures, and many others, are examples of algebras over operads. We therefore work in the following general context: throughout this paper, ${ \\mathcal{O} }$ is an operad in the category of ${ \\mathcal{R} }$-modules (unless otherwise stated), ${ \\Alg_{ \\mathcal{O} } }$ is the category of ${ \\mathcal{O} }$-algebras, and ${ \\Lt_{ \\mathcal{O} } }$ is the category of left ${ \\mathcal{O} }$-modules. \n\nWhile ${ \\mathcal{O} }$-algebras are the main objects of interest for most readers, our results also apply in the more general case of left modules over the operad ${ \\mathcal{O} }$; that generalization will be needed elsewhere.\n\\end{rem}\n\n\\begin{rem}\nIn this paper, we say that a symmetric sequence $X$ of ${ \\mathcal{R} }$-modules is $n$-connected if each ${ \\mathcal{R} }$-module $X[\\mathbf{t}]$ is an $n$-connected spectrum. We say that an algebra (resp. left module) over an operad is $n$-connected if the underlying ${ \\mathcal{R} }$-module (resp. symmetric sequence of ${ \\mathcal{R} }$-modules) is $n$-connected, and similarly for operads. Similarly, we say that a map $X { \\rightarrow } Y$ of symmetric sequences is $n$-connected if each map $X[\\mathbf{t}] { \\rightarrow } Y[\\mathbf{t}]$ is an $n$-connected map of ${ \\mathcal{R} }$-modules, and a map of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) is $n$-connected if the underlying map of spectra (resp. symmetric sequences) is $n$-connected.\n\\end{rem}\n\nThe main results of this paper are Theorems~\\ref{thm:higher_blakers_massey} and \\ref{thm:higher_dual_blakers_massey}, which are the analogs of Goodwillie's higher Blakers-Massey theorems \\cite[2.4 and 2.6]{Goodwillie_calc2}. These results include various interesting special cases which we now highlight.\n\nOne such case is given by the homotopy excision result of Theorem~\\ref{thm:homotopy_excision} below. Goerss and Hopkins \\cite[2.3.13]{Goerss_Hopkins_moduli} prove a closely related homotopy excision result in the special case of simplicial algebras over an $E_\\infty$ operad, and remark that it is true more generally for any simplicial operad \\cite[2.3.14]{Goerss_Hopkins_moduli}. In a closely related setting, Baues \\cite[I.C.4]{Baues_combinatorial} proves a homotopy excision result in an algebraic setting that includes simplicial associative algebras, and closely related is a result of  Schwede \\cite[3.6]{Schwede_algebraic} that is very nearly a homotopy excision result in the context of algebras over a simplicial theory. Our result also recovers Dugger and Shipley's \\cite[2.3]{Dugger_Shipley} homotopy excision result for associative ring spectra as a very special case.\n\n\\begin{thm}[Homotopy excision for structured ring spectra]\n\\label{thm:homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a homotopy pushout square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$), then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=k_1+k_2 +1$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=k_1+k_2$.\n\\end{itemize}\n\\end{thm}\n\nRelaxing the assumption in Theorem~\\ref{thm:homotopy_excision} that ${ \\mathcal{X} }$ is a homotopy pushout square, we obtain the following result which is the direct analog for structured ring spectra of the original Blakers-Massey Theorem for spaces.\n\n\\begin{thm}[Blakers-Massey theorem for structured ring spectra]\n\\label{thm:blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$, and  $k_{12}\\in{ \\mathbb{Z} }$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$) and ${ \\mathcal{X} }$ is $k_{12}$-cocartesian, then ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $k_{12}-1$ and $k_{1}+k_{2}$.\n\\end{thm}\n\nThe following higher homotopy excision result lies at the heart of this paper. It can be thought of as a structured ring spectra analog of higher homotopy excision (see Goodwillie \\cite[2.3]{Goodwillie_calc2}) in the context of spaces. This result also implies that the identity functors for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$ are $0$-analytic in the sense of \\cite[4.2]{Goodwillie_calc2}.\n\n\\begin{thm}[Higher homotopy excision for structured ring spectra]\n\\label{thm:higher_homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cocartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i\\in W$),  then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=|W|-1+\\sum_{i\\in W}k_i$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=\\sum_{i\\in W}k_i$.\n\\end{itemize}\n\\end{thm}\n\nThe preceding results are all special cases of the following theorem which relaxes the assumption in Theorem \\ref{thm:higher_homotopy_excision} that ${ \\mathcal{X} }$ is strongly $\\infty$-cocartesian. This result is a structured ring spectra analog of Goodwillie's higher Blakers-Massey theorem for spaces \\cite[2.4]{Goodwillie_calc2}. \n\n\\begin{thm}[Higher Blakers-Massey theorem for structured ring spectra]\n\\label{thm:higher_blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets.\n\\end{thm}\n\nFor instance, when $n=3$, $k$ is the minimum of\n\\begin{align*}\n  k_{\\{1,2,3\\}}-2,\\quad\n  &k_{\\{1,2\\}}+k_{\\{3\\}}-1,\\\\\n  &k_{\\{1,3\\}}+k_{\\{2\\}}-1,\\\\\n  &k_{\\{2,3\\}}+k_{\\{1\\}}-1,\\quad\n  k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}.\n\\end{align*}\n\nOur other results are dual versions of Theorems~\\ref{thm:homotopy_excision}, \\ref{thm:blakers_massey}, \\ref{thm:higher_homotopy_excision} and \\ref{thm:higher_blakers_massey}.\n\n\\begin{thm}[Dual homotopy excision for structured ring spectra]\n\\label{thm:dual_homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a homotopy pullback square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected and ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=k_{1}+k_{2}+2$.\n\\end{thm}\n\nThe following result relaxes the assumption that ${ \\mathcal{X} }$ is a homotopy pullback square.\n\n\\begin{thm}[Dual Blakers-Massey theorem for structured ring spectra]\n\\label{thm:dual_blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2,k_{12}\\geq -1$ with $k_1\\leq k_{12}$ and $k_2\\leq k_{12}$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected, ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, and ${ \\mathcal{X} }$ is $k_{12}$-cartesian, then ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_{12}+1$ and $k_{1}+k_{2}+2$.\n\\end{thm}\n\n\\begin{thm}[Higher dual homotopy excision for structured ring spectra]\n\\label{thm:higher_dual_homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set with $|W|\\geq 2$. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_{W-\\{i\\}}{ \\rightarrow }{ \\mathcal{X} }_W$ is $k_i$-connected ($i\\in W$),  then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=|W|+\\sum_{i\\in W}k_i$.\n\\end{thm}\n\nThe last three results are all special cases of the following theorem which is a structured ring spectra analog of Goodwillie's higher dual Blakers-Massey theorem for spaces \\cite[2.6]{Goodwillie_calc2}. This specializes to the higher dual homotopy excision result (Theorem \\ref{thm:higher_dual_homotopy_excision}) in the special case that ${ \\mathcal{X} }$ is strongly $\\infty$-cartesian, and to Theorem~\\ref{thm:dual_blakers_massey} in the case $|W| = 2$.\n\n\\begin{thm}[Higher dual Blakers-Massey theorem for structured ring spectra]\n\\label{thm:higher_dual_blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_W+|W|-1$ and $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\\end{thm}\n\n\n\\subsection{Organization of the paper}\n\nIn Section \\ref{sec:preliminaries} we recall some preliminaries on algebras and modules over operads. In Section \\ref{sec:cubical_diagrams} we prove our main results. Much of the work is concerned with proving higher homotopy excision (Theorem~\\ref{thm:higher_homotopy_excision}) which we obtain as a special case of a more general result, Theorem \\ref{thm:pushout_cofibration_cube_homotopical_analysis}. We then use an induction argument due to Goodwillie to pass from this to the higher Blakers-Massey result (Theorem~\\ref{thm:higher_blakers_massey}). We can then use higher Blakers-Massey to deduce, first, higher dual homotopy excision (Theorem~\\ref{thm:higher_dual_homotopy_excision}) and then higher dual Blakers-Massey (Theorem~\\ref{thm:higher_dual_blakers_massey}). Finally, in Section \\ref{sec:chain_complexes_over_a_commutative_ring}, we observe that the analogs of the main theorems stated above remain true in the context of unbounded chain complexes over a commutative ring.\n\n\n\\subsection*{Acknowledgments}\n\nThe second author would like to thank Greg Arone, Kristine Bauer, Bjorn Dundas, Bill Dwyer, Brenda Johnson, Nick Kuhn, Ib Madsen, Jim McClure, and Donald Yau for useful remarks. The second author is grateful to Dmitri Pavlov and Jakob Scholbach for helpful comments that directly led to \\cite{Harper_Spectra_Correction}, and to Mark Behrens and Haynes Miller for a stimulating and enjoyable visit to the Department of Mathematics at the Massachusetts Institute of Technology in summer 2011, and for their invitation which made this possible. The first author was partially supported by National Science Foundation Grant DMS-1144149.\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nThe purpose of this section is to recall various preliminaries on algebras and left modules over operads needed in this paper. Define the sets $\\mathbf{n}:=\\{1,\\dots,n\\}$ for each $n\\geq 0$, where $\\mathbf{0}:=\\emptyset$ denotes the empty set. If $W$ is a finite set, we denote by $|W|$ the number of elements in $W$. For a more detailed development of the material in this section, see \\cite{Harper_Spectra, Harper_Modules}.\n\n\\begin{defn}\n\\label{defn:symmetric_sequences}\nLet ${ \\mathsf{M} }$ be a category and $n\\geq 0$.\n\\begin{itemize}\n\\item $\\Sigma$ is the category of finite sets and their bijections.\n\\item $({ \\mathsf{Mod}_{ \\mathcal{R} } },{ \\,\\wedge\\, },{ \\mathcal{R} })$ is the closed symmetric monoidal category of ${ \\mathcal{R} }$-modules.\n\\item A \\emph{symmetric sequence} in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{M} }$) is a functor $\\functor{A}{\\Sigma^{{ \\mathrm{op} }}}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ (resp. $\\functor{A}{\\Sigma^{{ \\mathrm{op} }}}{{ \\mathsf{M} }}$). Denote by ${ \\mathsf{SymSeq} }$ the category of symmetric sequences in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and their natural transformations.\n\\item A symmetric sequence $A$ is \\emph{concentrated at $n$} if $A[\\mathbf{r}]=\\emptyset$ for all $r\\neq n$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{defn} Let $A_1,\\dotsc,A_t\\in{ \\mathsf{SymSeq} }$. Their \\emph{tensor product} $A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} } A_t\\in{ \\mathsf{SymSeq} }$ is the left Kan extension of objectwise smash along coproduct of sets\n\\begin{align*}\n\\xymatrix{\n  (\\Sigma^{{ \\mathrm{op} }})^{\\times t}\n  \\ar[rr]^-{A_1\\times\\dotsb\\times A_t}\\ar[d]^{\\coprod} & &\n  ({ \\mathsf{Mod}_{ \\mathcal{R} } })^{\\times t}\\ar[r]^-{{ \\,\\wedge\\, }} & { \\mathsf{Mod}_{ \\mathcal{R} } } \\\\\n  \\Sigma^{{ \\mathrm{op} }}\\ar[rrr]^{A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} }\n  A_t}_{\\text{left Kan extension}} & & & { \\mathsf{Mod}_{ \\mathcal{R} } }\n}\n\\end{align*}\n\\end{defn}\n\nIf $X$ is a finite set and $A$ is an object in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, we use the usual dot notation $A\\cdot X$ (see Mac Lane \\cite{MacLane_categories} or \\cite[2.3]{Harper_Modules}) to denote the copower $A\\cdot X$ defined by\n$\n  A\\cdot X := \\coprod_X A\n$,\nthe coproduct in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ of $|X|$ copies of $A$. Recall the following useful calculations for tensor products.\n\n\\begin{prop}\nLet $A_1,\\dotsc,A_t\\in{ \\mathsf{SymSeq} }$ and $R\\in\\Sigma$, with $r:=|R|$. There are natural isomorphisms\n\\begin{align}\n  \\notag\n  (A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} } A_t)[R]&{  \\ \\cong \\ }\\\n  \\coprod_{\\substack{\\function{\\pi}{R}{\\mathbf{t}}\\\\ \\text{in ${ \\mathsf{Set} }$}}}\n  A_1[\\pi^{-1}(1)]{ \\,\\wedge\\, }\\dotsb{ \\,\\wedge\\, }\n  A_t[\\pi^{-1}(t)],\\\\\n  \\label{eq:tensor_check_calc}\n  &{  \\ \\cong \\ }\n  \\coprod_{r_1+\\dotsb +r_t=r}A_1[\\mathbf{r_1}]{ \\,\\wedge\\, }\\dotsb{ \\,\\wedge\\, }\n  A_t[\\mathbf{r_t}]\\underset{{\\Sigma_{r_1}\\times\\dotsb\\times\n  \\Sigma_{r_t}}}{\\cdot}\\Sigma_{r}\n\\end{align}\n\\end{prop}\n\nHere, ${ \\mathsf{Set} }$ is the category of sets and their maps, and \\eqref{eq:tensor_check_calc} displays the tensor product $(A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} } A_t)[R]$ as a coproduct of $\\Sigma_{r_1}\\times\\dotsb\\times\\Sigma_{r_t}$-orbits. It will be conceptually useful to extend the definition of tensor powers $A^{{ \\check{\\tensor} } t}$ to situations in which the integers $t$ are replaced by a finite set $T$.\n\n\\begin{defn}\nLet $A\\in{ \\mathsf{SymSeq} }$ and $R,T\\in\\Sigma$. The \\emph{tensor powers} $A^{{ \\check{\\tensor} } T}\\in{ \\mathsf{SymSeq} }$ are defined objectwise by\n\\begin{align*}\n  (A^{{ \\check{\\tensor} }\\emptyset})[R]:=\n  \\coprod_{\\substack{\\function{\\pi}{R}{\\emptyset}\\\\ \\text{in ${ \\mathsf{Set} }$}}}\n  S,\\quad\\quad\n  &(A^{{ \\check{\\tensor} } T})[R]:=\n  \\coprod_{\\substack{\\function{\\pi}{R}{T}\\\\ \\text{in ${ \\mathsf{Set} }$}}}\n  \\bigwedge_{t\\in T} A[\\pi^{-1}(t)]\\quad(T\\neq\\emptyset).\n\\end{align*}\nNote that there are no functions $\\function{\\pi}{R}{\\emptyset}$ in ${ \\mathsf{Set} }$ unless $R=\\emptyset$. We will use the abbreviation $A^{{ \\check{\\tensor} } 0}:=A^{{ \\check{\\tensor} }\\emptyset}$.\n\\end{defn}\n\n\\begin{defn}\\label{defn:circle_product}\nLet $A,B,C\\in{ \\mathsf{SymSeq} }$, and $r,t\\geq 0$. The \\emph{circle product} (or composition product) $A\\circ B\\in{ \\mathsf{SymSeq} }$ is defined objectwise by the coend\n\\begin{align}\n  \\label{eq:circle_product_calc}\n  (A\\circ B)[\\mathbf{r}] := A{ \\,\\wedge\\, }_\\Sigma (B^{{ \\check{\\tensor} }-})[\\mathbf{r}]\n  &{  \\ \\cong \\ }\n  \\coprod_{t\\geq 0}A[\\mathbf{t}]{ \\,\\wedge\\, }_{\\Sigma_t}\n  (B^{{ \\check{\\tensor} } t})[\\mathbf{r}].\n\\end{align}\n\\end{defn}\n\n\n\\begin{prop}\n\\label{prop:closed_monoidal_on_symmetric_sequences}\n\\\n\\begin{itemize}\n\\item [(a)] $({ \\mathsf{SymSeq} },{ \\check{\\tensor} },1)$ has the structure of a closed symmetric monoidal category with all small limits and colimits. The unit for ${ \\check{\\tensor} }$ denoted ``$1$'' is the symmetric sequence concentrated at $0$ with value ${ \\mathcal{R} }$.\n\\item [(b)] $({ \\mathsf{SymSeq} },\\circ,I)$ has the structure of a closed monoidal category with all small limits and colimits. The unit for $\\circ$ denoted ``$I$'' is the symmetric sequence concentrated at $1$ with value ${ \\mathcal{R} }$. Circle product is not symmetric.\n\\end{itemize}\n\\end{prop}\n\n\\begin{defn}\n\\label{defn:hat_construction_embed_at_zero}\nLet $Z\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$. Define $\\hat{Z}\\in{ \\mathsf{SymSeq} }$ to be the symmetric sequence concentrated at $0$ with value $Z$.\n\\end{defn}\n\nThe functor $\\function{\\hat{-}}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{SymSeq} }}$ fits into the adjunction\n\\begin{align*}\n\\xymatrix{\n  { \\mathsf{Mod}_{ \\mathcal{R} } }\\ar@<0.5ex>[r]^-{\\hat{-}} &\n  { \\mathsf{SymSeq} }\\ar@<0.5ex>[l]^-{{ \\mathrm{Ev} }_0}\n}\n\\end{align*}\nwith left adjoint on top and ${ \\mathrm{Ev} }_0$ the \\emph{evaluation} functor defined objectwise by ${ \\mathrm{Ev} }_0(B):=B[\\mathbf{0}]$. Note that $\\hat{-}$ embeds ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ in ${ \\mathsf{SymSeq} }$ as the full subcategory of symmetric sequences concentrated at $0$.\n\n\\begin{defn}\\label{defn:corresponding_functor}\nLet ${ \\mathcal{O} }$ be a symmetric sequence and $Z\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$. The corresponding functor $\\functor{{ \\mathcal{O} }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ is defined objectwise by\n$\n  { \\mathcal{O} }(Z):={ \\mathcal{O} }\\circ(Z):=\\amalg_{t\\geq 0}{ \\mathcal{O} }[\\mathbf{t}]\n  { \\,\\wedge\\, }_{\\Sigma_t}Z^{\\wedge t}.\n$\n\\end{defn}\n\n\\begin{prop}\nLet ${ \\mathcal{O} },A\\in{ \\mathsf{SymSeq} }$ and $Z\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$. There are natural isomorphisms\n\\begin{align}\n\\label{eq:circ_product_and_evaluate_at_zero}\n  \\widehat{{ \\mathcal{O} }(Z)}=\n  \\widehat{{ \\mathcal{O} }\\circ(Z)}{  \\ \\cong \\ }{ \\mathcal{O} }\\circ\\hat{Z},\\quad\\quad\n  { \\mathrm{Ev} }_0({ \\mathcal{O} }\\circ A){  \\ \\cong \\ } { \\mathcal{O} }\\circ\\bigl({ \\mathrm{Ev} }_0(A)\\bigr).\n\\end{align}\n\\end{prop}\n\n\\begin{proof}\nThis follows from \\eqref{eq:circle_product_calc} and \\eqref{eq:tensor_check_calc}.\n\\end{proof}\n\n\\begin{defn}\n\\label{defn:operad}\nAn \\emph{operad} in ${ \\mathcal{R} }$-modules is a monoid object in $({ \\mathsf{SymSeq} },\\circ,I)$ and a \\emph{morphism of operads} is a morphism of monoid objects in $({ \\mathsf{SymSeq} },\\circ,I)$.\n\\end{defn}\n\n\\begin{rem} If ${ \\mathcal{O} }$ is an operad, then the associated functor $\\function{{ \\mathcal{O} }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ is a monad.\n\\end{rem}\n\n\\begin{defn}\n\\label{defn:algebras_and_modules}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules.\n\\begin{itemize}\n\\item A \\emph{left ${ \\mathcal{O} }$-module} is an object in $({ \\mathsf{SymSeq} },\\circ,I)$ with a left action of ${ \\mathcal{O} }$ and a \\emph{morphism of left ${ \\mathcal{O} }$-modules} is a map that respects the left ${ \\mathcal{O} }$-module structure. Denote by ${ \\Lt_{ \\mathcal{O} } }$ the category of left ${ \\mathcal{O} }$-modules and their morphisms.\n\\item An \\emph{${ \\mathcal{O} }$-algebra} is an algebra for the monad $\\functor{{ \\mathcal{O} }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ and a \\emph{morphism of ${ \\mathcal{O} }$-algebras} is a map in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ that respects the ${ \\mathcal{O} }$-algebra structure. Denote by ${ \\Alg_{ \\mathcal{O} } }$ the category of ${ \\mathcal{O} }$-algebras and their morphisms.\n\\end{itemize}\n\\end{defn}\n\nIt follows easily from \\eqref{eq:circ_product_and_evaluate_at_zero} that an ${ \\mathcal{O} }$-algebra is the same as an ${ \\mathcal{R} }$-module $Z$ with a left ${ \\mathcal{O} }$-module structure on $\\hat{Z}$, and if $Z$ and $Z'$ are ${ \\mathcal{O} }$-algebras, then a morphism of ${ \\mathcal{O} }$-algebras is the same as a map $\\function{f}{Z}{Z'}$ in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ such that $\\function{\\hat{f}}{\\hat{Z}}{\\hat{Z'}}$ is a morphism of left ${ \\mathcal{O} }$-modules. In other words, an algebra over an operad ${ \\mathcal{O} }$ is the same as a left ${ \\mathcal{O} }$-module that is concentrated at $0$, and ${ \\Alg_{ \\mathcal{O} } }$ embeds in ${ \\Lt_{ \\mathcal{O} } }$ as the full subcategory of left ${ \\mathcal{O} }$-modules concentrated at $0$, via the functor $\\function{\\hat{-}}{{ \\Alg_{ \\mathcal{O} } }}{{ \\Lt_{ \\mathcal{O} } }}$, $Z\\longmapsto \\hat{Z}$. Define the \\emph{evaluation} functor $\\function{{ \\mathrm{Ev} }_0}{{ \\Lt_{ \\mathcal{O} } }}{{ \\Alg_{ \\mathcal{O} } }}$ objectwise by ${ \\mathrm{Ev} }_0(B):=B[\\mathbf{0}]$.\n\n\\begin{prop}\n\\label{prop:basic_properties_LTO}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{C} }$. There are adjunctions\n\\begin{align}\n\\label{eq:free_forgetful_adjunction}\n\\xymatrix{\n  { \\mathsf{Mod}_{ \\mathcal{R} } }\\ar@<0.5ex>[r]^-{{ \\mathcal{O} }\\circ(-)} & { \\Alg_{ \\mathcal{O} } },\\ar@<0.5ex>[l]^-{U}\n}\\quad\\quad\n\\xymatrix{\n  { \\mathsf{SymSeq} }\\ar@<0.5ex>[r]^-{{ \\mathcal{O} }\\circ-} & { \\Lt_{ \\mathcal{O} } },\\ar@<0.5ex>[l]^-{U}\n}\\quad\\quad\n\\xymatrix{\n  { \\Alg_{ \\mathcal{O} } }\\ar@<0.5ex>[r]^-{\\hat{-}} & { \\Lt_{ \\mathcal{O} } },\\ar@<0.5ex>[l]^-{{ \\mathrm{Ev} }_0}\n}\n\\end{align}\nwith left adjoints on top and $U$ the forgetful functor. All small colimits exist in ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$, and both reflexive coequalizers and filtered colimits are preserved (and created) by the forgetful functors. All small limits exist in ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$, and are preserved (and created) by the forgetful functors.\n\\end{prop}\n\nThroughout this paper, we use the following model structures on the categories of ${ \\mathcal{O} }$-algebras and left ${ \\mathcal{O} }$-modules.\n\n\\begin{defn}\n\\label{defn:stable_flat_positive_model_structures}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. The \\emph{positive flat stable model structure} on ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) has as weak equivalences the stable equivalences (resp. objectwise stable equivalences) and  as fibrations the positive flat stable fibrations (resp. objectwise positive flat stable fibrations).\n\\end{defn}\n\nThe model structures in Definition \\ref{defn:stable_flat_positive_model_structures} are established in \\cite{Harper_Spectra, Harper_Spectra_Correction, Harper_Hess}. For a description of the cofibrations, see \\cite[Section 4]{Harper_Spectra} and \\cite[Section 7]{Harper_Hess}. For ease of notation, we have followed Schwede \\cite{Schwede_book_project} in using the term \\emph{flat} (e.g., flat stable model structure) for what is called $S$ (e.g., stable $S$-model structure) in \\cite{Hovey_Shipley_Smith, Schwede, Shipley_comm_ring}. For some of the good properties of the flat stable model structure, see \\cite[5.3.7 and 5.3.10]{Hovey_Shipley_Smith}.\n\n\n\\section{Homotopical Analysis of Cubical Diagrams}\n\\label{sec:cubical_diagrams}\n\nIn this section we prove the main results of the paper. The following definitions and constructions appear in Goodwillie \\cite{Goodwillie_calc2} in the context of spaces, and will also be useful in our context of structured ring spectra.\n\n\\begin{defn}[Indexing categories for cubical diagrams]\nLet $W$ be a finite set and ${ \\mathsf{M} }$ a category.\n\\begin{itemize}\n\\item Denote by $\\powerset(W)$ the poset of all subsets of $W$, ordered by inclusion $\\subset$ of sets. We will often regard $\\powerset(W)$ as the category associated to this partial order in the usual way; the objects are the elements of $\\powerset(W)$, and there is a morphism $U{ \\rightarrow } V$ if and only if $U\\subset V$.\n\\item Denote by $\\powerset_0(W)\\subset\\powerset(W)$ the poset of all nonempty subsets of $W$; it is the full subcategory of $\\powerset(W)$ containing all objects except the initial object $\\emptyset$.\n\\item Denote by $\\powerset_1(W)\\subset\\powerset(W)$ the poset of all subsets of $W$ not equal to $W$; it is the full subcategory of $\\powerset(W)$ containing all objects except the terminal object $W$.\n\\item A \\emph{$W$-cube} ${ \\mathcal{X} }$ in ${ \\mathsf{M} }$ is a $\\powerset(W)$-shaped diagram ${ \\mathcal{X} }$ in ${ \\mathsf{M} }$; in other words, a functor $\\function{{ \\mathcal{X} }}{\\powerset(W)}{{ \\mathsf{M} }}$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{rem}\nIf $n=|W|$ and ${ \\mathcal{X} }$ is a $W$-cube in ${ \\mathsf{M} }$, we will sometimes refer to ${ \\mathcal{X} }$ simply as an \\emph{$n$-cube} in ${ \\mathsf{M} }$. In particular, a $0$-cube is an object in ${ \\mathsf{M} }$ and a $1$-cube is a morphism in ${ \\mathsf{M} }$.\n\\end{rem}\n\n\\begin{defn}[Faces of cubical diagrams]\nLet $W$ be a finite set and ${ \\mathsf{M} }$ a category. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\mathsf{M} }$ and consider any subsets $U\\subset V\\subset W$. Denote by $\\partial_U^V{ \\mathcal{X} }$ the $(V-U)$-cube defined objectwise by\n\\begin{align*}\n  T\\mapsto(\\partial_U^V{ \\mathcal{X} })_T:={ \\mathcal{X} }_{T\\cup U},\\quad\\quad T\\subset V-U.\n\\end{align*}\nIn other words, $\\partial_U^V{ \\mathcal{X} }$ is the $(V-U)$-cube formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_U$ and ${ \\mathcal{X} }_V$. We say that $\\partial_U^V{ \\mathcal{X} }$ is a \\emph{face} of ${ \\mathcal{X} }$ of \\emph{dimension} $|V-U|$.\n\\end{defn}\n\n\\begin{defn}\n\\label{defn:cofibration_cubes_etc}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) and $k\\in{ \\mathbb{Z} }$.\n\\begin{itemize}\n\\item ${ \\mathcal{X} }$ is a \\emph{cofibration cube} if the map\n$\\colim_{\\powerset_1(V)}{ \\mathcal{X} }{ \\rightarrow }\\colim_{\\powerset(V)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_V$ is a cofibration for each $V\\subset W$; in particular, each ${ \\mathcal{X} }_V$ is cofibrant.\n\\item ${ \\mathcal{X} }$ is \\emph{$k$-cocartesian} if the map\n$\\hocolim_{\\powerset_1(W)}{ \\mathcal{X} }{ \\rightarrow }\\hocolim_{\\powerset(W)}{ \\mathcal{X} }{ \\ \\simeq \\ }{ \\mathcal{X} }_W$ is $k$-connected.\n\\item ${ \\mathcal{X} }$ is \\emph{$\\infty$-cocartesian} if the map\n$\\hocolim_{\\powerset_1(W)}{ \\mathcal{X} }{ \\rightarrow }\\hocolim_{\\powerset(W)}{ \\mathcal{X} }{ \\ \\simeq \\ }{ \\mathcal{X} }_W$ is a weak equivalence.\n\\item ${ \\mathcal{X} }$ is \\emph{strongly $\\infty$-cocartesian} if each face of dimension $\\geq 2$ is $\\infty$-cocartesian.\n\\item ${ \\mathcal{X} }$ is a \\emph{pushout cube} if the map\n$\\colim_{\\powerset_1(V)}{ \\mathcal{X} }{ \\rightarrow }\\colim_{\\powerset(V)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_V$  is an isomorphism for each $V\\subset W$ with $|V| \\geq 2$; i.e., if it is built by colimits in the usual way out of the maps ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_V$, $V\\subset W$, $|V|=1$.\n\\end{itemize}\n\\end{defn}\n\nThese definitions and constructions dualize as follows. Note that when looking for the appropriate dual construction, it is useful to observe that ${ \\mathcal{X} }=\\partial_\\emptyset^V{ \\mathcal{X} }$ when restricted to $\\powerset(V)$; for instance, $\\colim_{\\powerset_1(V)}{ \\mathcal{X} }=\\colim_{\\powerset_1(V)}\\partial_\\emptyset^V{ \\mathcal{X} }$.\n\n\\begin{defn}\n\\label{defn:fibration_cubes_etc}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) and $k\\in{ \\mathbb{Z} }$.\n\\begin{itemize}\n\\item ${ \\mathcal{X} }$ is a \\emph{fibration cube} if the map ${ \\mathcal{X} }_V{  \\ \\cong \\ }\\lim_{\\powerset(W-V)}\\partial_V^W{ \\mathcal{X} }{ \\rightarrow }\\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }$ is a fibration for each $V\\subset W$; in particular, each ${ \\mathcal{X} }_V$ is fibrant.\n\\item ${ \\mathcal{X} }$ is \\emph{$k$-cartesian} if the map ${ \\mathcal{X} }_\\emptyset{ \\ \\simeq \\ }\\holim_{\\powerset(W)}{ \\mathcal{X} }{ \\rightarrow }\\holim_{\\powerset_0(W)}{ \\mathcal{X} }$ is $k$-connected.\n\\item ${ \\mathcal{X} }$ is \\emph{$\\infty$-cartesian} if the map ${ \\mathcal{X} }_\\emptyset{ \\ \\simeq \\ }\\holim_{\\powerset(W)}{ \\mathcal{X} }{ \\rightarrow }\\holim_{\\powerset_0(W)}{ \\mathcal{X} }$ is a weak equivalence.\n\\item ${ \\mathcal{X} }$ is \\emph{strongly $\\infty$-cartesian} if each face of dimension $\\geq 2$ is $\\infty$-cartesian.\n\\item ${ \\mathcal{X} }$ is a \\emph{pullback cube} if the map ${ \\mathcal{X} }_V{  \\ \\cong \\ }\\lim_{\\powerset(W-V)}\\partial_V^W{ \\mathcal{X} }{ \\rightarrow }\\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }$ is an isomorphism for each $V\\subset W$ with $|W-V| \\geq 2$; i.e., if it is built by limits in the usual way out of the maps ${ \\mathcal{X} }_V{ \\rightarrow }{ \\mathcal{X} }_W$, $V\\subset W$, $|W-V|=1$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{rem}\nIt is important to note that every $1$-cube in ${ \\Alg_{ \\mathcal{O} } }$, ${ \\Lt_{ \\mathcal{O} } }$, ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, or ${ \\mathsf{SymSeq} }$ is strongly $\\infty$-cocartesian (resp. strongly $\\infty$-cartesian), since there are no faces of dimension $\\geq 2$, but only the $1$-cubes that are weak equivalences are $\\infty$-cocartesian (resp. $\\infty$-cartesian).\n\\end{rem}\n\nThe following is an exercise left to the reader.\n\\begin{prop}\n\\label{prop:connectivity_estimates_for_composition_of_maps}\nLet $k\\in{ \\mathbb{Z} }$. Consider any maps $X{ \\rightarrow } Y{ \\rightarrow } Z$ in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] If $X{ \\rightarrow } Y$ and $Y{ \\rightarrow } Z$ are $k$-connected, then $X{ \\rightarrow } Z$ is $k$-connected.\n\\item[(b)] If $X{ \\rightarrow } Y$ is $(k-1)$-connected and $X{ \\rightarrow } Z$ is $k$-connected, then $Y{ \\rightarrow } Z$ is $k$-connected.\n\\item[(c)] If $X{ \\rightarrow } Z$ is $k$-connected and $Y{ \\rightarrow } Z$ is $(k+1)$-connected, then $X{ \\rightarrow } Y$ is $k$-connected.\n\n\\end{itemize}\n\\end{prop}\n\nVersions of the following connectivity estimates are proved in Goodwillie \\cite[1.6--1.8]{Goodwillie_calc2} in the context of spaces, and exactly the same arguments give a proof of Propositions \\ref{prop:map_of_cubical_diagrams} and \\ref{prop:composed_map_of_cubical_diagrams} below in the context of structured ring spectra; this is an exercise left to the reader.\n\n\\begin{prop}\n\\label{prop:map_of_cubical_diagrams}\nLet $W$ be a finite set and $k\\in{ \\mathbb{Z} }$. Consider any map ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ of $W$-cubes in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{X} }$ are $k$-cocartesian, then ${ \\mathcal{Y} }$ is $k$-cocartesian.\n\\item[(b)] If ${ \\mathcal{X} }$ is $(k-1)$-cocartesian and ${ \\mathcal{Y} }$ is $k$-cocartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $k$-cocartesian.\n\\item[(c)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{Y} }$ are $k$-cartesian, then ${ \\mathcal{X} }$ is $k$-cartesian.\n\\item[(d)] If ${ \\mathcal{X} }$ is $k$-cartesian and ${ \\mathcal{Y} }$ is $(k+1)$-cartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $k$-cartesian.\n\n\\end{itemize}\n\\end{prop}\n\n\\begin{prop}\n\\label{prop:composed_map_of_cubical_diagrams}\nLet $W$ be a finite set and $k\\in{ \\mathbb{Z} }$. Consider any map ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ of $W$-cubes in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ are $k$-cocartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cocartesian.\n\\item[(b)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $(k-1)$-cocartesian and ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cocartesian, then ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cocartesian.\n\\item[(c)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ are $k$-cartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cartesian.\n\\item[(d)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cartesian and ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ is $(k+1)$-cartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $k$-cartesian.\n\\end{itemize}\n\\end{prop}\n\nThe following results depend on the fact that the model structures on ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and ${ \\mathsf{SymSeq} }$ are stable, so that fibration and cofibration sequences coincide. Note that these do not hold, in general, for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$.\n\n\\begin{prop}\n\\label{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR}\nLet $W$ be a finite set and $k\\in{ \\mathbb{Z} }$. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $k$-cocartesian if and only if ${ \\mathcal{X} }$ is $(k-|W|+1)$-cartesian.\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian if and only if ${ \\mathcal{X} }$ is $(k+|W|-1)$-cocartesian.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nThis is because the total homotopy cofiber of ${ \\mathcal{X} }$ (see Goodwillie \\cite[1.4]{Goodwillie_calc2}) is weakly equivalent to the $|W|$-th suspension, usually denoted $\\Sigma^{|W|}$, of the total homotopy fiber of ${ \\mathcal{X} }$ (see \\cite[1.1a]{Goodwillie_calc2}).\n\\end{proof}\n\n\n\\subsection{Proof of higher homotopy excision for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nThe purpose of this section is to prove Theorem~\\ref{thm:higher_homotopy_excision}. At the heart of our proof is a homotopical analysis of the construction ${ \\mathcal{O} }_A$ described in Proposition~\\ref{prop:coproduct_modules}. We deduce Theorem~\\ref{thm:higher_homotopy_excision} from a more general result about the effect of the construction $A \\mapsto { \\mathcal{O} }_A$ on strongly $\\infty$-cocartesian cubes. \n\n\\begin{defn}\\label{def:symmetric_array}\nConsider symmetric sequences in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$. A \\emph{symmetric array} in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ is a symmetric sequence in ${ \\mathsf{SymSeq} }$; i.e., a functor $\\functor{A}{\\Sigma^{ \\mathrm{op} }}{{ \\mathsf{SymSeq} }}$. Denote by ${ \\mathsf{SymArray} }:={ \\mathsf{SymSeq} }^{\\Sigma^{ \\mathrm{op} }}$ the category of symmetric arrays in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and their natural transformations.\n\\end{defn}\n\nThe following ${ \\mathcal{O} }_A$ construction is crucial to our arguments; a proof of the following proposition is given in \\cite[4.7]{Harper_Spectra}.\n\n\\begin{prop}\n\\label{prop:coproduct_modules}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, $A\\in{ \\Alg_{ \\mathcal{O} } }$ (resp. $A\\in{ \\Lt_{ \\mathcal{O} } }$), and $Y\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. $Y\\in{ \\mathsf{SymSeq} }$). Consider any coproduct in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) of the form $A\\amalg{ \\mathcal{O} }\\circ(Y)$ (resp. $A\\amalg({ \\mathcal{O} }\\circ Y)$). There exists a symmetric sequence ${ \\mathcal{O} }_A$ (resp. symmetric array ${ \\mathcal{O} }_A$) and natural isomorphisms\n\\begin{align*}\n  A\\amalg{ \\mathcal{O} }\\circ(Y) {  \\ \\cong \\ }\n  \\coprod\\limits_{q\\geq 0}{ \\mathcal{O} }_A[\\mathbf{q}]\n  { \\,\\wedge\\, }_{\\Sigma_q}Y^{\\wedge q}\\quad\n  \\Bigl(\\text{resp.}\\quad\n  A\\amalg({ \\mathcal{O} }\\circ Y) {  \\ \\cong \\ }\n  \\coprod\\limits_{q\\geq 0}{ \\mathcal{O} }_A[\\mathbf{q}]\n  { \\check{\\tensor} }_{\\Sigma_q}Y^{{ \\check{\\tensor} } q}\n    \\Bigr)\n\\end{align*}\nin the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$). If $q\\geq 0$, then ${ \\mathcal{O} }_A[\\mathbf{q}]$ is naturally isomorphic to a colimit of the form\n\\begin{align*}\n  { \\mathcal{O} }_A[\\mathbf{q}]&{  \\ \\cong \\ }\n  \\colim\\biggl(\n  \\xymatrix{\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}A^{\\wedge p} &\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}({ \\mathcal{O} }\\circ (A))^{\\wedge p}\\ar@<-0.5ex>[l]^-{d_1}\n    \\ar@<-1.5ex>[l]_-{d_0}\n  }\n  \\biggl),\\\\\n  \\text{resp.}\\quad\n  { \\mathcal{O} }_A[\\mathbf{q}]&{  \\ \\cong \\ }\n  \\colim\\biggl(\n  \\xymatrix{\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}A^{{ \\check{\\tensor} } p} &\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}({ \\mathcal{O} }\\circ A)^{{ \\check{\\tensor} } p}\\ar@<-0.5ex>[l]^-{d_1}\n    \\ar@<-1.5ex>[l]_-{d_0}\n  }\n  \\biggl),\n\\end{align*}\nin ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_q^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_q^{ \\mathrm{op} }}$), with $d_0$ induced by operad multiplication and $d_1$ induced by the left ${ \\mathcal{O} }$-action map $\\function{m}{{ \\mathcal{O} }\\circ (A)}{A}$ (resp. $\\function{m}{{ \\mathcal{O} }\\circ A}{A}$).\n\\end{prop}\n\nRecall from \\cite{Harper_Hess} the following proposition.\n\n\\begin{prop}\n\\label{prop:OA_commutes_with_certain_colimits}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and let $q\\geq 0$. Then the functor\n$\n  \\function{{ \\mathcal{O} }_{(-)}[\\mathbf{q}]}{{ \\Alg_{ \\mathcal{O} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_q^{ \\mathrm{op} }}}\n$ (resp.\n$\n  \\function{{ \\mathcal{O} }_{(-)}[\\mathbf{q}]}{{ \\Lt_{ \\mathcal{O} } }}{{ \\mathsf{SymSeq} }^{\\Sigma_q^{ \\mathrm{op} }}}\n$) preserves reflexive coequalizers and filtered colimits.\n\\end{prop}\n\n\\begin{defn}\\label{def:filtration_setup_modules}\nLet $\\function{i}{X}{Y}$ be a morphism in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) and $t\\geq 1$. Define $Q_0^t:=X^{\\wedge t}$ (resp. $Q_0^t:=X^{{ \\check{\\tensor} } t}$) and $Q_t^t:=Y^{\\wedge t}$ (resp. $Q_t^t:=Y^{{ \\check{\\tensor} } t}$). For $0<q<t$ define $Q_q^t$ inductively by the left-hand (resp. right-hand) pushout diagrams\n\\begin{align*}\n\\xymatrix{\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{\\wedge(t-q)}\n  { \\,\\wedge\\, } Q_{q-1}^q\\ar[d]^{i_*}\\ar[r]^-{{ \\mathrm{pr} }_*} & Q_{q-1}^t\\ar[d]\\\\\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{\\wedge(t-q)}\n  { \\,\\wedge\\, } Y^{\\wedge q}\\ar[r] & Q_q^t\n}\\quad\n\\xymatrix{\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{{ \\check{\\tensor} }(t-q)}\n  { \\check{\\tensor} } Q_{q-1}^q\\ar[d]^{i_*}\\ar[r]^-{{ \\mathrm{pr} }_*} & Q_{q-1}^t\\ar[d]\\\\\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{{ \\check{\\tensor} }(t-q)}\n  { \\check{\\tensor} } Y^{{ \\check{\\tensor} } q}\\ar[r] & Q_q^t\n}\n\\end{align*}\nin ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_t}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_t}$). We sometimes denote $Q_q^t$ by $Q_q^t(i)$ to emphasize in the notation the map $\\function{i}{X}{Y}$. The maps ${ \\mathrm{pr} }_*$ and $i_*$ are the obvious maps induced by $i$ and the appropriate projection maps.\n\\end{defn}\n\nRecall from \\cite{Harper_Hess} the following proposition.\n\n\\begin{prop}\n\\label{prop:filtering_OA}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, $A\\in{ \\Alg_{ \\mathcal{O} } }$ (resp. $A\\in{ \\Lt_{ \\mathcal{O} } }$), and $\\function{i}{X}{Y}$ in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$). Consider any pushout diagram in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) of the form\n\\begin{align}\n\\label{eq:small_arg_pushout_modules}\n\\xymatrix{\n  { \\mathcal{O} }\\circ (X)\\ar[r]^-{f}\\ar[d]^{{ \\mathrm{id} }\\circ (i)} & A\\ar[d]^{j}\\\\\n  { \\mathcal{O} }\\circ (Y)\\ar[r] & B\n}\\quad\\quad\\text{resp.}\\quad\n\\xymatrix{\n  { \\mathcal{O} }\\circ X\\ar[r]^-{f}\\ar[d]^{{ \\mathrm{id} }\\circ i} & A\\ar[d]^{j}\\\\\n  { \\mathcal{O} }\\circ Y\\ar[r] & B\n}\n\\end{align}\nFor each $r\\geq 0$, ${ \\mathcal{O} }_B[\\mathbf{r}]$ is naturally isomorphic to a filtered colimit of the form\n\\begin{align}\n\\label{eq:filtered_colimit_modules_refined}\n  { \\mathcal{O} }_B[\\mathbf{r}]{  \\ \\cong \\ }\n  \\colim\\Bigl(\n  \\xymatrix{\n    { \\mathcal{O} }_A^0[\\mathbf{r}]\\ar[r]^{j_1} &\n    { \\mathcal{O} }_A^1[\\mathbf{r}]\\ar[r]^{j_2} &\n    { \\mathcal{O} }_A^2[\\mathbf{r}]\\ar[r]^{j_3} & \\dotsb\n  }\n  \\Bigr)\n\\end{align}\nin ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_r^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$), with ${ \\mathcal{O} }_A^0[\\mathbf{r}]:={ \\mathcal{O} }_A[\\mathbf{r}]$ and ${ \\mathcal{O} }_A^t[\\mathbf{r}]$ defined inductively by pushout diagrams in ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_r^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$) of the form\n\\begin{align}\n\\label{eq:good_filtration_modules_refined}\n\\xymatrix{\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\,\\wedge\\, }_{\\Sigma_t}Q_{t-1}^t\\ar[d]^{{ \\mathrm{id} }\\wedge_{\\Sigma_t}i_*}\n  \\ar[r]^-{f_*} & { \\mathcal{O} }_A^{t-1}[\\mathbf{r}]\\ar[d]^{j_t}\\\\\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\,\\wedge\\, }_{\\Sigma_t}Y^{\\wedge t}\\ar[r]^-{\\xi_t} & { \\mathcal{O} }_A^t[\\mathbf{r}]\n}\\quad\\quad\n\\text{resp.}\\quad\n\\xymatrix{\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\check{\\tensor} }_{\\Sigma_t}Q_{t-1}^t\\ar[d]^{{ \\mathrm{id} }{ \\check{\\tensor} }_{\\Sigma_t}i_*}\n  \\ar[r]^-{f_*} & { \\mathcal{O} }_A^{t-1}[\\mathbf{r}]\\ar[d]^{j_t}\\\\\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\check{\\tensor} }_{\\Sigma_t}Y^{{ \\check{\\tensor} } t}\\ar[r]^-{\\xi_t} & { \\mathcal{O} }_A^t[\\mathbf{r}]\n}\n\\end{align}\n\\end{prop}\n\n\\begin{rem}\nIt is important to note (see \\cite{Harper_Hess}) that for $r=0$ the filtration \\eqref{eq:filtered_colimit_modules_refined} specializes to a filtered colimit of the pushout in \\eqref{eq:small_arg_pushout_modules} of the form\n\\begin{align}\n\\label{eq:good_filtration_modules}\n  B{  \\ \\cong \\ }\n  { \\mathcal{O} }_B[\\mathbf{0}]{  \\ \\cong \\ }\n  \\colim\\bigl(\n  \\xymatrix@1{\n    A_0\\ar[r]^{j_1} & A_1\\ar[r]^{j_2} & A_2\\ar[r]^{j_3} & \\dotsb\n  }\n  \\bigr)\n\\end{align}\nin the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$), with $A_0:={ \\mathcal{O} }_A[\\mathbf{0}]{  \\ \\cong \\ } A$ and $A_t:={ \\mathcal{O} }_A^t[\\mathbf{0}]$.\n\\end{rem}\n\n\\begin{prop}\\label{prop:pushout_of_n_connected_cofibration}\nLet $n\\geq -1$. If the map $\\function{i}{X}{Y}$ in Proposition \\ref{prop:filtering_OA} is an $n$-connected generating cofibration or generating acyclic cofibration in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) with the positive flat stable model structure, and ${ \\mathcal{R} },{ \\mathcal{O} }_A$ are $(-1)$-connected, then each map $j_t$ in \\eqref{eq:good_filtration_modules} and \\eqref{eq:good_filtration_modules_refined} is an $n$-connected monomorphism. In particular, the map $j$ in \\eqref{eq:small_arg_pushout_modules} is an $n$-connected monomorphism in the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\end{prop}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The generating cofibrations and acyclic cofibrations in ${ \\mathsf{SymSeq} }$ have cofibrant domains. Hence by \\cite[4.28*]{Harper_Spectra_Correction}, each $j_t$ in \\eqref{eq:good_filtration_modules} is a monomorphism. We know that $A_t\/A_{t-1}{  \\ \\cong \\ } { \\mathcal{O} }_A[\\mathbf{t}]{ \\check{\\tensor} }_{\\Sigma_t}(Y\/X)^{{ \\check{\\tensor} } t}$ and $*{ \\rightarrow } Y\/X$ is an $n$-connected cofibration in ${ \\mathsf{SymSeq} }$. It follows from \\cite[4.40]{Harper_Hess} that each $j_t$ in \\eqref{eq:good_filtration_modules} is $n$-connected. The case for each map $j_t$ in \\eqref{eq:good_filtration_modules_refined} is similar.\n\\end{proof}\n\nThe following proposition is closely related to Dugger-Shipley \\cite[A.3, A.4]{Dugger_Shipley}.\n\n\\begin{prop}\n\\label{prop:factorization_into_n_connected_cofibration}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $n\\geq -1$. If $\\function{f}{A}{C}$ is an $n$-connected map in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$), $C$ is fibrant,  and ${ \\mathcal{R} },{ \\mathcal{O} }_A$ are $(-1)$-connected, then $f$ factors in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) as\n\\begin{align}\n\\label{eq:factoring_n_connected_maps}\n\\xymatrix{\n  A\\ar[r]^-{j} & B\\ar[r]^-{p} & C\n}\n\\end{align}\na nice $n$-connected cofibration followed by an acyclic fibration. Here, ``nice'' means that $j$ is a (possibly transfinite) composition of pushouts of $n$-connected generating cofibrations and generating acyclic cofibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$). This factorization is functorial in all such $f$.\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:factorization_into_n_connected_cofibration}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $\\function{i}{X}{Y}$ be an $n$-connected generating cofibration or generating acyclic cofibration in ${ \\mathsf{SymSeq} }$ with the positive flat stable model structure, and consider the pushout diagram\n\\begin{align}\n\\label{eq:gluing_on_cells}\n\\xymatrix{\n  { \\mathcal{O} }\\circ X\\ar[r]\\ar[d] & Z_0\\ar[d]^{i_0}\\\\\n  { \\mathcal{O} }\\circ Y\\ar[r] & Z_1\n}\n\\end{align}\nin ${ \\Lt_{ \\mathcal{O} } }$. Assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected; let's verify that ${ \\mathcal{O} }_{Z_1}$ is $(-1)$-connected and $i_0$ is an $n$-connected monomorphism in ${ \\mathsf{SymSeq} }$. Let $A:=Z_0$. By Proposition \\ref{prop:filtering_OA}, we know ${ \\mathcal{O} }_{Z_1}[\\mathbf{r}]$ is naturally isomorphic to a filtered colimit of the form\n\\begin{align*}\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]&{  \\ \\cong \\ }\n  \\colim\\bigl(\n  \\xymatrix{\n    { \\mathcal{O} }_A^0[\\mathbf{r}]\\ar[r]^{j_1} &\n    { \\mathcal{O} }_A^1[\\mathbf{r}]\\ar[r]^{j_2} &\n    { \\mathcal{O} }_A^2[\\mathbf{r}]\\ar[r]^{j_3} & \\dotsb\n  }\n  \\bigr)\n\\end{align*}\nin ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$, and Proposition \\ref{prop:pushout_of_n_connected_cofibration} verifies that each $j_t$ is an $n$-connected monomorphism. Since ${ \\mathcal{O} }_A^0={ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected by assumption, it follows that ${ \\mathcal{O} }_{Z_1}$ is $(-1)$-connected, and taking $r=0$ (or using Proposition \\ref{prop:pushout_of_n_connected_cofibration} again) finishes the argument that $i_0$ is an $n$-connected monomorphism in ${ \\mathsf{SymSeq} }$.\n\nConsider a sequence\n$\n\\xymatrix@1{\n  Z_0\\ar[r] & Z_1\\ar[r] & Z_2\\ar[r] & \\dotsb\n}\n$\nof pushouts of maps as in \\eqref{eq:gluing_on_cells}, and let $Z_\\infty:=\\colim_k Z_k$. Consider the naturally occurring map $Z_0{ \\rightarrow } Z_\\infty$, and assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected. By the argument above we know that\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_2}[\\mathbf{r}]\\ar[r] & \\dotsb\n}\n\\end{align*}\nis a sequence of $n$-connected monomorphisms, hence ${ \\mathcal{O} }_{Z_\\infty}$ is $(-1)$-connected, and taking $r=0$ verifies that $Z_0{ \\rightarrow } Z_\\infty$ is an $n$-connected monomorphism in ${ \\mathsf{SymSeq} }$.\n\nThe small object argument (see \\cite[7.12]{Dwyer_Spalinski} for a useful introduction) produces a factorization \\eqref{eq:factoring_n_connected_maps} of $f$ such that $p$ has the right lifting property with respect to the $n$-connected generating cofibrations and generating acyclic cofibrations in ${ \\Lt_{ \\mathcal{O} } }$, and $j$ is a (possibly transfinite) composition of pushouts of maps as in \\eqref{eq:gluing_on_cells}, starting with $Z_0=A$. Since $C$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$ by assumption, it follows from the latter lifting property that $p$ is a fibration between fibrant objects in ${ \\Lt_{ \\mathcal{O} } }$. By the argument above, it follows that $j$ is $n$-connected. Since $f$ is $n$-connected by assumption, it follows that $p$ is $n$-connected, and since $p$ furthermore has the right lifting property with respect to the $n$-connected generating cofibrations, it follows that $p$ is a weak equivalence which completes the proof.\n\\end{proof}\n\n\\begin{rem}\nIt is important to note that an $n$-connected fibration between fibrant objects in symmetric spectra with the positive flat stable model structure (resp. positive stable model structure) is a positive levelwise $n$-connected fibration; this property has been exploited in the argument above.\n\\end{rem}\n\n\\begin{rem}\nTo keep the statement of Proposition \\ref{prop:factorization_into_n_connected_cofibration} as simple and non-technical as possible, we have been conservative in our choice for the set of maps used in the small object argument. In other words, running the small object argument with the set of $n$-connected generating cofibrations and generating acyclic cofibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) is sufficient for our purposes and makes for an attractive and simple statement, but one can obtain the desired factorizations using a smaller set of maps; this is an exercise left to the reader.\n\\end{rem}\n\n\\begin{prop}\n\\label{prop:OA_construction_on_n_connected_cofibrations}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $n\\geq -1$.  Let $\\function{j}{A}{B}$ be a cofibration in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$). Assume that ${ \\mathcal{R} },{ \\mathcal{O} }_A$ are $(-1)$-connected. If $j$ is $n$-connected, then ${ \\mathcal{O} }_A[\\mathbf{r}]{ \\rightarrow }{ \\mathcal{O} }_B[\\mathbf{r}]$ is an $n$-connected monomorphism for each $r\\geq 0$.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The proof is in two parts; in part (a) we assume that $B$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$, and in part (b) we do not assume that $B$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$. Consider part (a). Proceed exactly as in the proof of Proposition \\ref{prop:factorization_into_n_connected_cofibration}, and consider the pushout diagram \\eqref{eq:gluing_on_cells}. Assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected; let's verify that ${ \\mathcal{O} }_{Z_0}[\\mathbf{r}]{ \\rightarrow }{ \\mathcal{O} }_{Z_1}[\\mathbf{r}]$ is an $n$-connected monomorphism for each $r\\geq 0$. This follows by arguing exactly as in the proof of Proposition \\ref{prop:factorization_into_n_connected_cofibration}. Consider a sequence\n\\begin{align*}\n\\xymatrix@1{\n  Z_0\\ar[r] & Z_1\\ar[r] & Z_2\\ar[r] & \\dotsb\n}\n\\end{align*}\nof pushouts of maps as in \\eqref{eq:gluing_on_cells}, and let $Z_\\infty:=\\colim_k Z_k$. Consider the naturally occurring map $Z_0{ \\rightarrow } Z_\\infty$, and assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected. By the argument above we know that\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_2}[\\mathbf{r}]\\ar[r] & \\dotsb\n}\n\\end{align*}\nis a sequence of $n$-connected monomorphisms, hence ${ \\mathcal{O} }_{Z_0}[\\mathbf{r}]{ \\rightarrow } { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]$ is an $n$-connected monomorphism. Noting that every $n$-connected cofibration of the form $A{ \\rightarrow } B$ in ${ \\Lt_{ \\mathcal{O} } }$ is a retract of a (possibly transfinite) composition of pushouts of maps as in \\eqref{eq:gluing_on_cells}, starting with $Z_0=A$, finishes the proof of part (a).\n\nConsider part (b). The map $B{ \\rightarrow } *$ in ${ \\Lt_{ \\mathcal{O} } }$ factors as $B{ \\rightarrow } C{ \\rightarrow } *$, an acyclic cofibration followed by a fibration. Since the map $B{ \\rightarrow } C$ is an acyclic cofibration and the composite map $A{ \\rightarrow } C$ is an $n$-connected cofibration, we know from part (b) that the composite map\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{O} }_{A}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{B}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{C}[\\mathbf{r}]\n}\n\\end{align*}\nis an $n$-connected monomorphism and the right-hand map is a monomorphism and a weak equivalence. Hence the left-hand map is an $n$-connected monomorphism which finishes the proof of part (b).\n\\end{proof}\n\n\\begin{prop}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. If $A$ is a cofibrant ${ \\mathcal{O} }$-algebra (resp. left ${ \\mathcal{O} }$-module) and ${ \\mathcal{R} },{ \\mathcal{O} },A$ are $(-1)$-connected, then ${ \\mathcal{O} }_A$ is $(-1)$-connected.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. This follows from Proposition \\ref{prop:OA_construction_on_n_connected_cofibrations} by considering the map ${ \\mathcal{O} }\\circ\\emptyset{ \\rightarrow } A$ in ${ \\Lt_{ \\mathcal{O} } }$, together with the natural isomorphisms ${ \\mathcal{O} }\\circ\\emptyset{  \\ \\cong \\ }\\widehat{{ \\mathcal{O} }[\\mathbf{0}]}$ and ${ \\mathcal{O} }_{{ \\mathcal{O} }\\circ\\emptyset}[\\mathbf{r}]{  \\ \\cong \\ }\\widehat{{ \\mathcal{O} }[\\mathbf{r}]}$ for each $r\\geq 0$ (see \\cite[5.31]{Harper_Hess}).\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:map_of_n_cubes}\nAny $3$-cube $\\mathcal{X}$ of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) may be regarded as a map of $2$-cubes $A{ \\rightarrow } B$ with $A=\\partial_\\emptyset^{\\{1,2\\}}{ \\mathcal{X} }$ (the top face of ${ \\mathcal{X} }$) and $B=\\partial_{\\{3\\}}^{\\{1,2,3\\}}{ \\mathcal{X} }$ (the bottom face of ${ \\mathcal{X} }$) as follows:\n\\begin{align*}\n\\xymatrix@!0{\n{ \\mathcal{X} }_\\emptyset\\ar[rr]\\ar[dd]\\ar[dr] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar[dr]\\ar'[d][dd]\\\\\n&{ \\mathcal{X} }_{\\{2\\}}\\ar[rr]\\ar[dd] &&\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rr] &&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\\\\n&{ \\mathcal{X} }_{\\{2,3\\}}\\ar[rr] &&\n{ \\mathcal{X} }_{\\{1,2,3\\}}\n}\n\\quad\\quad\n\\xymatrix@!0{\nA_\\emptyset\\ar[rr]\\ar[dd]\\ar[dr] &&\nA_{\\{1\\}}\\ar[dr]\\ar'[d][dd]\\\\\n&A_{\\{2\\}}\\ar[rr]\\ar[dd] &&\nA_{\\{1,2\\}}\\ar[dd]\\\\\nB_\\emptyset\\ar[dr]\\ar'[r][rr] &&\nB_{\\{1\\}}\\ar[dr]\\\\\n&B_{\\{2\\}}\\ar[rr] &&\nB_{\\{1,2\\}}\n}\n\\end{align*}\nMore generally, we may regard an $(n+1)$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) as a map of $n$-cubes $A{ \\rightarrow } B$ with $A=\\partial_\\emptyset^{\\{1,\\dots,n\\}}{ \\mathcal{X} }$ and $B=\\partial_{\\{n+1\\}}^{\\{1,\\dots,n+1\\}}{ \\mathcal{X} }$, for each $n\\geq 0$. In particular, the map ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_{\\{n+1\\}}$ in ${ \\mathcal{X} }$ is the map $A_\\emptyset{ \\rightarrow } B_\\emptyset$ in $A{ \\rightarrow } B$.\n\\end{rem}\n\nWe now prove the following result of which Theorem~\\ref{thm:higher_homotopy_excision} is the special case $r = 0$. \n\n\\begin{thm}[Homotopical analysis of ${ \\mathcal{O} }_{ \\mathcal{X} }$ for a pushout cofibration cube ${ \\mathcal{X} }$]\n\\label{thm:pushout_cofibration_cube_homotopical_analysis}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $n\\geq 1$. Let ${ \\mathcal{X} }$ be a pushout $(\\mathbf{n}\\boldsymbol{+}\\mathbf{1})$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) regarded as a map of pushout $\\mathbf{n}$-cubes $A{ \\rightarrow } B$ as in Remark \\ref{rem:map_of_n_cubes}. Assume that ${ \\mathcal{R} },{ \\mathcal{O} }_{A_\\emptyset}$ are $(-1)$-connected. Let $k_1,\\dots,k_{n+1}\\geq -1$. Assume that each $A_\\emptyset{ \\rightarrow } A_{\\{i\\}}$ and $A_\\emptyset{ \\rightarrow } B_\\emptyset$ are cofibrations between cofibrant objects in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) $(1\\leq i\\leq n)$. Consider the associated left-hand diagram of the form\n\\begin{align}\n\\label{eq:colim_punctured_cube_associated_diagrams}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}A\\ar[d]\\ar[r] & \\tilde{A}\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}B\\ar[r] & \\tilde{B}\n}\\quad\\quad\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A[\\mathbf{r}]\\ar[d]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{r}]\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_B[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]\n}\n\\end{align}\nin the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$), and more generally, the associated right-hand diagrams $(r\\geq 0)$ in ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_r^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$). If each $A_\\emptyset{ \\rightarrow } A_{\\{i\\}}$ is $k_i$-connected $(1\\leq i\\leq n)$ and $A_\\emptyset{ \\rightarrow } B_\\emptyset$ is $k_{n+1}$-connected, then the diagrams \\eqref{eq:colim_punctured_cube_associated_diagrams} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian; here, $\\tilde{A}:=A_\\mathbf{n}$ and $\\tilde{B}:=B_{\\mathbf{n}}$.\n\\end{thm}\n\n\\begin{rem}\nIn other words, this theorem shows that the $(\\mathbf{n}\\boldsymbol{+}\\mathbf{1})$-cube ${ \\mathcal{O} }_{ \\mathcal{X} }[\\mathbf{r}]$ is $(k_1+\\dots + k_{n+1}+n)$-cocartesian for each $r\\geq 0$, or equivalently, it shows that the $(\\mathbf{n}\\boldsymbol{+}\\mathbf{1})$-cube ${ \\mathcal{O} }_{ \\mathcal{X} }$ is $(k_1+\\dots + k_{n+1}+n)$-cocartesian. The left-hand diagram in (\\ref{eq:colim_punctured_cube_associated_diagrams}) is the case $r = 0$ and the result here is precisely that needed for Theorem~\\ref{thm:higher_homotopy_excision}.\n\\end{rem}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The proof is in two parts; in part (a) we assume that $B_\\emptyset$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$, and in part (b) we do not assume that $B_\\emptyset$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$. The argument is by induction on $n$. It is convenient to start the induction at $n=0$ in which case the diagrams in \\eqref{eq:colim_punctured_cube_associated_diagrams} are maps (i.e., $1$-cubes) of the form $\\tilde{A}{ \\rightarrow }\\tilde{B}$ and ${ \\mathcal{O} }_{\\tilde{A}}[\\mathbf{r}]{ \\rightarrow }{ \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]$. Hence the case $n=0$ is verified by Proposition \\ref{prop:OA_construction_on_n_connected_cofibrations}. Let $N\\geq 1$ and assume the proposition is true for each $0\\leq n\\leq N-1$. Consider part (a); let's verify it remains true for $n=N$. Let $\\function{i}{X}{Y}$ be a $k_{n+1}$-connected generating cofibration or generating acyclic cofibration in ${ \\mathsf{SymSeq} }$ with the positive flat stable model structure, $Z_0$ a pushout $n$-cube in ${ \\Lt_{ \\mathcal{O} } }$, and consider any left-hand pushout diagram of the\nform\n\\begin{align}\n\\label{eq:gluing_on_cells_higher_cube}\n\\xymatrix{\n  { \\mathcal{O} }\\circ X\\ar[d]_-{{ \\mathrm{id} }\\circ i}\\ar[r] &\n  {Z_0}_\\emptyset\\ar[d]\\\\\n  { \\mathcal{O} }\\circ Y\\ar[r] &\n  {Z_1}_\\emptyset\n}\\quad\\quad\n\\xymatrix{\n  Z_0\\ar[d] \\\\\n  Z_1\n}\\quad\\quad\n\\xymatrix{\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[d]\\\\\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\n}\n\\end{align}\nin ${ \\Lt_{ \\mathcal{O} } }$ with the middle map of pushout $n$-cubes the associated pushout $(n+1)$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Assume each ${Z_0}_\\emptyset{ \\rightarrow } {Z_0}_{\\{i\\}}$ is a $k_i$-connected cofibration between cofibrant objects in ${ \\Lt_{ \\mathcal{O} } }$ $(1\\leq i\\leq n)$ and ${ \\mathcal{O} }_{{Z_0}_\\emptyset}$ is $(-1)$-connected; let's verify that the associated right-hand maps  of $n$-cubes $(r\\geq 0)$, each regarded as an $(n+1)$-cube in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$, are $(k_1+\\dots+k_{n+1}+n)$-cocartesian. If $A:=Z_0$ and $\\tilde{A}:=A_{\\{1,\\dots,n\\}}$, then by Proposition \\ref{prop:filtering_OA} there are corresponding filtrations\n\\begin{align}\n\\label{eq:induced_filtration_diagram_for_studying_induced_map_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^0[\\mathbf{r}]\\ar[dd]^{\\xi_0}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^1[\\mathbf{r}]\\ar@{.>}[d]^{\\xi_1}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^2[\\mathbf{r}]\\ar@{.>}[d]^{\\xi_2}\\ar[r] &\n  \\dots\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^\\infty[\\mathbf{r}]\\ar[d]^{\\xi_\\infty}\\\\\n  &\n  \\cdot\\ar@{.>}[d]^{(*)_1}\\ar[r] &\n  \\cdot\\ar@{.>}[d]^{(*)_2}\\ar[r] &\n  \\dots\\ar[r] &\n  \\cdot\\ar[d]^{(*)_\\infty}\\\\\n  { \\mathcal{O} }_{\\tilde{A}}^0[\\mathbf{r}]\\ar[r]\\ar@\/_0.5pc\/[ur] &\n  { \\mathcal{O} }_{\\tilde{A}}^1[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}^2[\\mathbf{r}]\\ar[r] &\n  \\dots\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}^\\infty[\\mathbf{r}]\n}\n\\end{align}\ntogether with induced maps $\\xi_t$ and $(*)_t$ ($t\\geq 1$) that make the diagram in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$ commute; here, the upper diagrams are pushout diagrams and $\\xi_\\infty:=\\colim_t\\xi_t$, the maps $(*)_t$ are the obvious induced maps and $(*)_\\infty:=\\colim_t(*)_t$, the left-hand vertical map is naturally isomorphic to\n\\begin{align*}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\longrightarrow\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}],\n\\end{align*}\nand the right-hand vertical maps are naturally isomorphic to the diagram\n\\begin{align}\n\\label{eq:right_hand_vertical_maps_higher_cubes}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\longrightarrow\n  \\bigl(\\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\bigr)\\cup\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\longrightarrow\n  { \\mathcal{O} }_{\\tilde{Z}_1}[\\mathbf{r}];\n\\end{align}\nhere, $\\tilde{Z}_0:={Z_0}_{\\{1,\\dots,n\\}}$ and $\\tilde{Z}_1:={Z_1}_{\\{1,\\dots,n\\}}$. We want to show that the right-hand map in \\eqref{eq:right_hand_vertical_maps_higher_cubes} is $(k_1+\\dots+k_{n+1}+n)$-connected; since the horizontal maps in \\eqref{eq:induced_filtration_diagram_for_studying_induced_map_higher_cubes} are monomorphisms, it suffices to verify each map $(*)_t$ is $(k_1+\\dots+k_{n+1}+n)$-connected. The argument is by induction on $t$. The map $\\xi_0$ factors as\n\\begin{align*}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^0[\\mathbf{r}]\\longrightarrow\n  \\bigl(\\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^0[\\mathbf{r}]\\bigr)\\cup{ \\mathcal{O} }_{\\tilde{A}}^0[\\mathbf{r}]\\xrightarrow[{  \\ \\cong \\ }]{(*)_0}\n  { \\mathcal{O} }_{\\tilde{A}}^0[\\mathbf{r}]\n\\end{align*}\nand since the right-hand map $(*)_0$ is an isomorphism, it is $(k_1+\\dots+k_{n+1}+n)$-connected. Consider the commutative diagram\n\\begin{align}\n\\label{eq:filtration_quotients_diagram_for_analyzing_connectivity_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A^{t-1}[\\mathbf{r}]\\ar[d]^{\\xi_{t-1}}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A^t[\\mathbf{r}]\\ar[d]^{\\xi_t}\\ar[r] &\n  \\bigl(\\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A[\\mathbf{t+r}]\\bigr){ \\check{\\tensor} }_{\\Sigma_t}(Y\/X)^{{ \\check{\\tensor} } t}\\ar[d]_{{  \\ \\cong \\ }}\\ar@\/^2pc\/[dd]^{(\\#)}\\\\\n  \\cdot\\ar[d]^{(*)_{t-1}}\\ar[r] &\n  \\cdot\\ar[d]^{(*)_t}\\ar[r] &\n  \\cdot\\ar[d]_{(\\#\\#)}\\\\\n  { \\mathcal{O} }_{\\tilde{A}}^{t-1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}^t[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{t+r}]{ \\check{\\tensor} }_{\\Sigma_t}(Y\/X)^{{ \\check{\\tensor} } t}\n}\n\\end{align}\nwith rows cofiber sequences. Since we know $(Y\/X)^{{ \\check{\\tensor} } t}$ is at least $k_{n+1}$-connected and\n$\n  \\colim_{\\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A[\\mathbf{t+r}]\\longrightarrow\n  { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{t+r}]\n$\nis $(k_1+\\dots+k_{n}+n-1)$-connected by the induction hypothesis, it follows that $(\\#)$ is $(k_1+\\dots+k_{n+1}+n)$-connected, and hence $(\\#\\#)$ is also. Since the rows in \\eqref{eq:filtration_quotients_diagram_for_analyzing_connectivity_higher_cubes} are cofiber sequences, it follows by induction on $t$ that $(*)_t$ is $(k_1+\\dots+k_{n+1}+n)$-connected for each $t\\geq 1$. This finishes the argument that the the right-hand maps of $n$-cubes $(r\\geq 0)$ in \\eqref{eq:gluing_on_cells_higher_cube}, each regarded as an $(n+1)$-cube in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$, are $(k_1+\\dots+k_{n+1}+n)$-cocartesian.\n\nConsider a sequence $Z_0{ \\rightarrow } Z_1{ \\rightarrow } Z_2{ \\rightarrow }\\cdots$ of pushout $n$-cubes in ${ \\Lt_{ \\mathcal{O} } }$ as in \\eqref{eq:gluing_on_cells_higher_cube}, define $\\tilde{Z}_n:={Z_n}_{\\{1,\\dots,n\\}}$, $Z_\\infty:=\\colim_nZ_n$, and $\\tilde{Z}_\\infty:=\\colim_n\\tilde{Z}_n$, and consider the naturally occurring map $Z_0{ \\rightarrow } Z_\\infty$ of pushout $n$-cubes, regarded as a pushout $(n+1)$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the associated left-hand diagram of the form\n\\begin{align}\n\\label{eq:gluing_on_cells_filtration_sequence_O_construction_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}Z_0\\ar[d]\\ar[r] & \\tilde{Z}_0\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}Z_\\infty\\ar[r] & \\tilde{Z}_\\infty\n}\\quad\\quad\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[d]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_\\infty}[\\mathbf{r}]\n}\n\\end{align}\nin the underlying category ${ \\mathsf{SymSeq} }$ and the associated right-hand diagrams ($r\\geq 0$) in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$. Assume each ${Z_0}_\\emptyset{ \\rightarrow } {Z_0}_{\\{i\\}}$ is a $k_i$-connected cofibration between cofibrant objects in ${ \\Lt_{ \\mathcal{O} } }$ $(1\\leq i\\leq n)$ and ${ \\mathcal{O} }_{{Z_0}_\\emptyset}$ is $(-1)$-connected. We want to show that the right-hand diagrams in \\eqref{eq:gluing_on_cells_filtration_sequence_O_construction_higher_cubes} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian. Consider the associated commutative diagram\n\\begin{align}\n\\label{eq:filtration_quotients_diagram_for_analyzing_connectivity_for_Z_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[dd]\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\ar@{.>}[d]^{\\eta_1}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_2}[\\mathbf{r}]\\ar@{.>}[d]^{\\eta_2}\\ar[r] &\n  \\cdots\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\\ar[d]^{\\eta_\\infty}\\\\\n  &\n  \\cdot\n  \\ar@{.>}[d]^{(\\#)_1}\\ar[r] &\n  \\cdot\n  \\ar@{.>}[d]^{(\\#)_2}\\ar[r] &\n  \\cdots\\ar[r] &\n  \\cdot\\ar[d]^{(\\#)_\\infty}\\\\\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[r]\\ar@\/_0.5pc\/[ur] &\n  { \\mathcal{O} }_{\\tilde{Z}_1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_2}[\\mathbf{r}]\\ar[r] &\n  \\cdots\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_\\infty}[\\mathbf{r}]\n}\n\\end{align}\nin ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$ and induced maps $\\eta_t$ and $(\\#)_t$ $(t\\geq 1)$; here, the upper diagrams are pushout diagrams and $\\eta_\\infty:=\\colim_t\\eta_t$, the maps $(\\#)_t$ are the obvious induced maps and $(\\#)_\\infty:=\\colim_t(\\#)_t$, and the right-hand vertical maps are naturally isomorphic to the diagram\n\\begin{align}\n\\label{eq:right_hand_vertical_maps_sequential_diagram_higher_cubes}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\n  \\longrightarrow\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\n  \\longrightarrow\n  { \\mathcal{O} }_{\\tilde{Z}_\\infty}[\\mathbf{r}]\n\\end{align}\nWe want to show that the right-hand map in \\eqref{eq:right_hand_vertical_maps_sequential_diagram_higher_cubes} is $(k_1+\\dots + k_{n+1}+n)$-connected; since the horizontal maps in \\eqref{eq:filtration_quotients_diagram_for_analyzing_connectivity_for_Z_higher_cubes} are monomorphisms, it suffices to verify each map $(\\#)_t$ is $(k_1+\\dots + k_{n+1}+n)$-connected. The argument is by induction on $t$. The map $(\\#)_t$ factors as\n\\begin{align}\n\\label{eq:factorization_of_maps_to_analyze_higher_cubes}\n\\xymatrix{\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_t}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[r] &\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_t}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_{t-1}}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_t}[\\mathbf{r}]\n}\n\\end{align}\nWe know from above that $(\\#)_1$ and the right-hand map in \\eqref{eq:factorization_of_maps_to_analyze_higher_cubes} are $(k_1+\\dots + k_{n+1}+n)$-connected for each $t\\geq 1$, hence it follows by induction on $t$ that $(\\#)_t$ is $(k_1+\\dots + k_{n+1}+n)$-connected for each $t\\geq 1$. This finishes the argument that the right-hand diagrams $(r\\geq 0)$ in \\eqref{eq:gluing_on_cells_filtration_sequence_O_construction_higher_cubes} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian in  ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$.\n\nIt follows from Proposition \\ref{prop:factorization_into_n_connected_cofibration} that the pushout $(n+1)$-cube $A{ \\rightarrow } B$ factors as $Z_0\\xrightarrow{i_\\lambda} Z_\\lambda \\xrightarrow{p} B$, a composition of pushout $(n+1)$-cubes in ${ \\Lt_{ \\mathcal{O} } }$, starting with $Z_0=A$, where $i_\\lambda$ is a (possibly transfinite) composition of pushout $n$-cubes as in \\eqref{eq:gluing_on_cells_higher_cube} and $p$ is an objectwise weak equivalence. Consider the associated diagram\n\\begin{align*}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[d]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[d]\\ar@\/^0.5pc\/[dr]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\lambda}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\ar[r] &\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\lambda}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\ar[r]^-{(**)} &\n  { \\mathcal{O} }_{\\tilde{Z}_\\lambda}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_B[\\mathbf{r}]\\ar[r] &\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_B[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[r]^-{(*)} &\n  { \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]\n}\n\\end{align*}\nNoting that the bottom vertical arrows are weak equivalences, it follows that $(*)$ has the same connectivity as $(**)$, which finishes the proof of part (a) that the right-hand diagrams $(r\\geq 0)$ in \\eqref{eq:colim_punctured_cube_associated_diagrams} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$.  In particular, taking $r=0$ verifies that the left-hand diagram in \\eqref{eq:colim_punctured_cube_associated_diagrams} is $(k_1+\\dots + k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }$.\n\nConsider part (b). The map $B_\\emptyset{ \\rightarrow } *$ in ${ \\Lt_{ \\mathcal{O} } }$ factors as $B_\\emptyset{ \\rightarrow } C_\\emptyset{ \\rightarrow } *$, an acyclic cofibration followed by a fibration. Consider the associated pushout $(n+1)$-cube $B{ \\rightarrow } C$ in ${ \\Lt_{ \\mathcal{O} } }$ and the associated diagram of pushout squares of the form\n\\begin{align*}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A[\\mathbf{r}]\\ar[d]\\ar[r] & { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{r}]\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_B[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_C[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{C}}[\\mathbf{r}]\n}\n\\end{align*}\nin ${ \\Lt_{ \\mathcal{O} } }$. Since the map $B_\\emptyset{ \\rightarrow } C_\\emptyset$ is an acyclic cofibration and the composite map $A_\\emptyset{ \\rightarrow } C_\\emptyset$ is a $k_{n+1}$-connected cofibration, we know from part (a) that the outer diagram is $(k_1+\\dots+k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$. Since the vertical maps in the bottom square are weak equivalences, it follows that the upper square is $(k_1+\\dots+k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$ which finishes the proof of part (b).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_homotopy_excision}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. It is enough to treat the special case where ${ \\mathcal{X} }$ is a pushout cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. The case $|W|=1$ is trivial and the case $|W|\\geq 2$ follows from Theorem \\ref{thm:pushout_cofibration_cube_homotopical_analysis}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:homotopy_excision}]\nThis is the special case $|W| = 2$ of Theorem \\ref{thm:higher_homotopy_excision}.\n\\end{proof}\n\n\n\\subsection{Proof of the higher Blakers-Massey theorem for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nThe purpose of this section is to prove the Blakers-Massey theorems \\ref{thm:blakers_massey} and \\ref{thm:higher_blakers_massey}. We first show that Blakers-Massey for square diagrams (Theorem \\ref{thm:blakers_massey}) follows fairly easily from the higher homotopy excision result proved in the previous section.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $W:=\\{1,2\\}$. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align*}\nWe know that $(*)$ is $(k_1+k_2+1)$-connected by homotopy excision (Theorem \\ref{thm:homotopy_excision}) and $(**)$ is $k_{12}$-connected by assumption. Hence by Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(a) the composition is $l$-connected, where $l$ is the minimum of $k_1+k_2+1$ and $k_{12}$; in other words, we have verified that ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{SymSeq} }$, and Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR}(a) finishes the proof.\n\n\\end{proof}\n\nWe now turn to the proof of the higher Blakers-Massey result (Theorem~\\ref{thm:higher_blakers_massey}). Our approach follows that used by Goodwillie at the corresponding point in \\cite{Goodwillie_calc2}.\n\nThe following is an important warm-up calculation for Proposition \\ref{prop:cocartesian_estimates_for_induction_argument}.\n\n\\begin{prop}\n\\label{prop:warmup_cocartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for every $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }$ is $k_{V-U}$-cocartesian.\n\\end{prop}\n\n\\begin{proof}\nThe argument is by induction on $|U|$. The case $|U|=0$ is true by assumption. Let $n\\geq 1$ and assume the proposition is true for each $0\\leq|U|<n$. Let's verify it remains true for $|U|=n$. Let $u\\in U$ and note that $\\partial_{U-\\{u\\}}^V{ \\mathcal{X} }$ can be written as the composition of cubes\n\\begin{align*}\n\\xymatrix{\n  \\partial_{U-\\{u\\}}^{V-\\{u\\}}{ \\mathcal{X} }\\ar[r] &\n  \\partial_U^V{ \\mathcal{X} }\n}\n\\end{align*}\nWe know by the induction assumption that the composition of cubes is $k_{(V-U)\\cup\\{u\\}}$-cocartesian and the left-hand cube is $k_{V-U}$-cocartesian. Since $k_{V-U}\\leq k_{(V-U)\\cup\\{u\\}}$ by assumption, it follows from Proposition \\ref{prop:map_of_cubical_diagrams}(a) that the right-hand cube is $k_{V-U}$-cocartesian which finishes the proof.\n\\end{proof}\n\n\\begin{defn}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any subset ${ \\mathcal{B} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item A subset ${ \\mathcal{A} }\\subset{ \\mathcal{B} }$ is \\emph{convex} if every element of ${ \\mathcal{B} }$ which is less than an element of ${ \\mathcal{A} }$ is in ${ \\mathcal{A} }$.\n\\item Define ${ \\mathcal{A} }^W_\\mathrm{min}:=\\{V\\subset W: |V|\\leq 1\\}$ and ${ \\mathcal{A} }^W_\\mathrm{max}:=\\powerset(W)$.\n\\item For each convex subset ${ \\mathcal{A} }\\subset\\powerset(W)$, the $W$-cube ${ \\mathcal{X} }_{ \\mathcal{A} }$ is defined objectwise by $({ \\mathcal{X} }_{ \\mathcal{A} })_U := \\colim_{{ \\mathcal{A} }\\cap\\powerset(U)}{ \\mathcal{X} }{  \\ \\cong \\ }\\colim_{T\\in{ \\mathcal{A} },\\, T\\subset U}{ \\mathcal{X} }_T$.\n\\end{itemize}\n\\end{defn}\n\nThe following proposition will be needed in the proof of Proposition \\ref{prop:properties_of_the_X_sub_A_construction} below.\n\n\\begin{prop}\n\\label{prop:colim_over_punctured_cube_of_X_sub_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }\\subset\\powerset(W)$. Then for each nonempty subset $V\\subset W$, there is a natural isomorphism\n\\begin{align*}\n  \\colim_{\\powerset_1(V)}{ \\mathcal{X} }_{ \\mathcal{A} }{  \\ \\cong \\ }\n  \\colim_{{ \\mathcal{A} }\\cap\\powerset_1(V)}{ \\mathcal{X} }=\n  \\colim_{T\\in{ \\mathcal{A} },\\,T\\subsetneqq V}{ \\mathcal{X} }_T\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}\nThis is because the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\subset U,\\, U\\subsetneqq V\\}$ and $\\{T\\in{ \\mathcal{A} }:T\\subsetneqq V\\}$ are the same.\n\\end{proof}\n\nThe following proposition explains the key properties of the ${ \\mathcal{X} }_{ \\mathcal{A} }$ construction and its relationship to ${ \\mathcal{X} }$; it is through these properties that the ${ \\mathcal{X} }_{ \\mathcal{A} }$ construction is useful and meaningful.\n\n\\begin{prop}\n\\label{prop:properties_of_the_X_sub_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }^W_\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }^W_\\mathrm{max}$.\n\\begin{itemize}\n\\item[(a)] There are natural isomorphisms $\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }{  \\ \\cong \\ }\\partial_\\emptyset^V{ \\mathcal{X} }$ if $V\\in{ \\mathcal{A} }$,\n\\item[(b)] The $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }$ is $\\infty$-cocartesian if $V\\notin{ \\mathcal{A} }$ and ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Alg_{ \\mathcal{O} } }$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nConsider part (a). Let $V\\in{ \\mathcal{A} }$. Then we know that $\\powerset(V)\\subset{ \\mathcal{A} }$, by ${ \\mathcal{A} }$ convex, and hence ${ \\mathcal{A} }\\cap\\powerset(V)=\\powerset(V)$. It follows that $({ \\mathcal{X} }_{ \\mathcal{A} })_V=\\colim_{\\powerset(V)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_V$. The same argument shows that $({ \\mathcal{X} }_{ \\mathcal{A} })_{V'}{  \\ \\cong \\ }{ \\mathcal{X} }_{V'}$, for each $V'\\subset V$. Consider part (b). Let $V\\subset W$ and $V\\notin{ \\mathcal{A} }$. Then ${ \\mathcal{A} }\\cap\\powerset(V)={ \\mathcal{A} }\\cap\\powerset_1(V)$ and hence the composition\n\\begin{align*}\n  \\colim_{\\powerset_1(V)}\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }=\n  \\colim_{\\powerset_1(V)}{ \\mathcal{X} }_{ \\mathcal{A} }{  \\ \\cong \\ }\n  \\colim_{{ \\mathcal{A} }\\cap\\powerset_1(V)}{ \\mathcal{X} }=\n  \\colim_{{ \\mathcal{A} }\\cap\\powerset(V)}{ \\mathcal{X} }=\n  ({ \\mathcal{X} }_{ \\mathcal{A} })_V=(\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} })_V\n\\end{align*}\nis an isomorphism which finishes the proof of part (b).\n\\end{proof}\n\nThe following proposition shows that ${ \\mathcal{X} }_{ \\mathcal{A} }$ inherits several of the homotopical properties of ${ \\mathcal{X} }$.\n\n\\begin{prop}\n\\label{prop:cocartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }^W_\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }^W_\\mathrm{max}$. Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for each $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }_{ \\mathcal{A} }$ is $k_{V-U}$-cocartesian.\n\\end{prop}\n\n\\begin{proof}\nBy the induction argument in the proof of Proposition \\ref{prop:warmup_cocartesian_estimates_for_induction_argument}, it suffices to verify that the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }$ is $k_{V}$-cocartesian, for each nonempty subset $V\\subset W$, and this follows immediately from Proposition \\ref{prop:properties_of_the_X_sub_A_construction}.\n\\end{proof}\n\nThe following proposition is proved in Goodwillie \\cite[2.8]{Goodwillie_calc2} in the context of spaces, and exactly the same argument gives a proof in the context of structured ring spectra; this is an exercise left to the reader.\n\n\\begin{prop}\n\\label{prop:cofibration_cube_properties}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item[(a)] For each inclusion ${ \\mathcal{A} }'\\subset{ \\mathcal{A} }$ of convex subsets of $\\powerset(W)$, the induced map $\\colim_{{ \\mathcal{A} }'}{ \\mathcal{X} }{ \\rightarrow }\\colim_{{ \\mathcal{A} }}{ \\mathcal{X} }$ is a cofibration in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$),\n\\item[(b)] For any convex subsets ${ \\mathcal{A} },{ \\mathcal{B} }$ of $\\powerset(W)$, the diagram\n\\begin{align*}\n\\xymatrix{\n  \\colim_{{ \\mathcal{A} }\\cap{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[r]\\ar[d] & \\colim_{{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[d]\\\\\n  \\colim_{{ \\mathcal{A} }}{ \\mathcal{X} }\\ar[r] & \\colim_{{ \\mathcal{A} }\\cup{ \\mathcal{B} }}{ \\mathcal{X} }\n}\n\\end{align*}\nis a pushout diagram of cofibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$).\n\\item[(c)] If $A\\in{ \\mathcal{A} }$ is maximal and the cofibration\n$\n  \\colim_{\\powerset_1(A)}{ \\mathcal{X} }{ \\rightarrow }\\colim_{\\powerset(A)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_A\n$\nis $k_A$-connected, then the cofibration\n$$\n  \\colim_{{ \\mathcal{A} }-\\{A\\}}{ \\mathcal{X} }{ \\rightarrow }\\colim_{{ \\mathcal{A} }}{ \\mathcal{X} }\n$$\nis $k_A$-connected.\n\\end{itemize}\n\\end{prop}\n\nThe purpose of the following induction argument is to leverage the higher homotopy excision result (Theorem \\ref{thm:higher_homotopy_excision}) for structured ring spectra into a proof of the first main theorem in this paper (Theorem \\ref{thm:higher_blakers_massey})---the higher Blakers-Massey theorem for structured ring spectra. Proposition \\ref{prop:cubical_induction_argument} is motivated by Goodwillie \\cite[2.12]{Goodwillie_calc2}; it is essentially Goodwillie's cubical induction argument, appropriately modified to our situation.\n\n\\begin{prop}[Cubical induction argument]\n\\label{prop:cubical_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Suppose ${ \\mathcal{A} }\\subset{ \\mathcal{A} }_\\mathrm{max}^W$ is convex, $A\\in{ \\mathcal{A} }$ maximal, $|A|\\geq 2$, and ${ \\mathcal{A} }':={ \\mathcal{A} }-\\{A\\}$. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nIf $X_{{ \\mathcal{A} }'}$ is $k$-cartesian, then ${ \\mathcal{X} }_{ \\mathcal{A} }$ is $k$-cartesian, where $k$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets.\n\\end{prop}\n\n\\begin{rem}\n\\label{rem:3_cube_example_of_cubical_induction}\nIn the case that ${ \\mathcal{X} }$ is a $3$-cube, the following diagram illustrates one of the cubical decompositions covered by Proposition \\ref{prop:cubical_induction_argument}. It corresponds to the sequence of maximal elements: $\\{1,2\\}$, $\\{1,3\\}$, $\\{2,3\\}$, $\\{1,2,3\\}$; we benefitted from the discussion in Munson-Voli\\'c \\cite{Munson_Volic_book_project} where this cubical decomposition is used, in the context of spaces, to illustrate Goodwillie's cubical induction argument.\n\n\\begin{align*}\n\\xymatrix@!0{\n{ \\mathcal{X} }_\\emptyset\\ar[rr]\\ar[dd]\\ar[dr] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar[dr]\\ar'[d][dd]\\ar@{=}[rr] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar@{=}[rr]\\ar[dr]\\ar'[d][dd] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar[dr]\\ar[dd]|(0.5)\\hole|(0.53)\\hole\\\\\n&{ \\mathcal{X} }_{\\{2\\}}\\ar[rr]\\ar[dd] &&\n\\cdot\\ar[dd]\\ar[rr] &&\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar@{=}[rr]\\ar[dd] &&\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rr]\\ar@{=}[dd] &&\n\\cdot\\ar[dr]\\ar@{=}'[r][rr] &&\n\\cdot\\ar[dr]\\ar'[r][rr] &&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\ar@{=}'[d][dd]\\\\\n&\\cdot\\ar[rr]\\ar[dd] &&\n\\cdot\\ar[rr] &&\n\\cdot\\ar[rr] &&\n\\cdot\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rrrrrr]\\ar@{=}[dd]&&&&&&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\ar@{=}'[d][dd]\\\\\n&{ \\mathcal{X} }_{\\{2,3\\}}\\ar[rrrrrr]\\ar@{=}[dd]&&&&&&\n\\cdot\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rrrrrr] &&&&&&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\\\\n&{ \\mathcal{X} }_{\\{2,3\\}}\\ar[rrrrrr]&&&&&&{ \\mathcal{X} }_{\\{1,2,3\\}}\n}\n\\end{align*}\n\\end{rem}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:cubical_induction_argument}]\nThe argument is by induction on $|W|$. The case $|W|=2$ is verified by the proof of Theorem \\ref{thm:blakers_massey}. Let $n\\geq 3$ and assume the proposition is true for each $2\\leq |W|<n$. Let's verify it remains true for $|W|=n$.\n\nWe know by assumption that ${ \\mathcal{X} }_{{ \\mathcal{A} }'}$ is $k$-cartesian. We want to verify that ${ \\mathcal{X} }_{ \\mathcal{A} }$ is $k$-cartesian (it might be helpful at this point to look ahead to  \\eqref{eq:final_decomposition_for_induction_argument} for the decomposition of ${ \\mathcal{X} }_{ \\mathcal{A} }$ that we will use to finish the proof). Consider the induced map of $W$-cubes ${ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }{ \\mathcal{X} }_{ \\mathcal{A} }$. Note that if $V\\not\\supset A$, then $\\powerset(V)\\not\\ni A$ and hence ${ \\mathcal{A} }'\\cap\\powerset(V)={ \\mathcal{A} }\\cap\\powerset(V)$; in particular, each of the maps\n\\begin{align}\n\\label{eq:subdiagram_of_identity_maps}\n\\xymatrix{\n  ({ \\mathcal{X} }_{{ \\mathcal{A} }'})_V\\ar[r]^-{{ \\mathrm{id} }} & ({ \\mathcal{X} }_{ \\mathcal{A} })_V,\n  & V\\not\\supset A\n}\n\\end{align}\nin ${ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }{ \\mathcal{X} }_{ \\mathcal{A} }$ is the identity map. Note that if $V\\supset U\\supset A$, then the diagram\n\\begin{align}\n\\label{eq:pushout_diagram_in_induction_step}\n\\xymatrix{\n  ({ \\mathcal{X} }_{{ \\mathcal{A} }'})_U\\ar[r]\\ar[d] & ({ \\mathcal{X} }_{ \\mathcal{A} })_U\\ar[d]\\\\\n  ({ \\mathcal{X} }_{{ \\mathcal{A} }'})_V\\ar[r] & ({ \\mathcal{X} }_{ \\mathcal{A} })_V,&\n  V\\supset U\\supset A\n}\n\\end{align}\nis a pushout diagram by Proposition \\ref{prop:cofibration_cube_properties}(b); in particular, focusing on this case is the same as focusing on the subdiagram $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$ of ${ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }{ \\mathcal{X} }_{ \\mathcal{A} }$.\n\nLet's first verify that $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$ is $(k+|A|-1)$-cartesian. Let ${ \\mathcal{Y} }$ denote $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$ regarded as a $((W-A)\\cup\\{*\\})$-cube as follows:\n\\begin{align*}\n  { \\mathcal{Y} }_V&:=(\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'})_V=({ \\mathcal{X} }_{{ \\mathcal{A} }'})_{V\\cup A},\\quad\\ V\\subset W-A,\\\\\n  { \\mathcal{Y} }_V&:=(\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} })_U=({ \\mathcal{X} }_{ \\mathcal{A} })_{U\\cup A},\\quad\\quad U\\subset W-A,\\quad V=U\\cup\\{*\\}.\n\\end{align*}\nSince $|(W-A)\\cup\\{*\\}|<|W|$, our induction assumption can be applied to ${ \\mathcal{Y} }$, provided that the appropriate $k'_V$-cocartesian estimates are satisfied. We claim that with the following definitions\n\\begin{align*}\n  k'_V&:=k_V,\\quad\\quad \\emptyset\\neq V\\subset W-A,\\\\\n  k'_V&:=k_A,\\quad\\quad V=\\{*\\}\\\\\n  k'_V&:=\\infty,\\quad\\quad \\emptyset\\neq U\\subset W-A,\\quad V=U\\cup\\{*\\},\n\\end{align*}\nthe cube $\\partial_\\emptyset^V{ \\mathcal{Y} }$ (formed by all maps in ${ \\mathcal{Y} }$ between ${ \\mathcal{Y} }_\\emptyset$ and ${ \\mathcal{Y} }_V$) is $k'_V$-cocartesian, for each nonempty subset $V\\subset (W-A)\\cup\\{*\\}$. Let's verify this now: If $V\\subset W-A$ is nonempty, then $\\partial_\\emptyset^V{ \\mathcal{Y} }$ is the cube $\\partial_A^{A\\cup V}{ \\mathcal{X} }_{{ \\mathcal{A} }'}$ which is $k_V$-cocartesian by Proposition \\ref{prop:cocartesian_estimates_for_induction_argument}. If $V=\\{*\\}$, then $\\partial_\\emptyset^V{ \\mathcal{Y} }$ is the map $({ \\mathcal{X} }_{{ \\mathcal{A} }'})_A{ \\rightarrow }({ \\mathcal{X} }_{ \\mathcal{A} })_A$ which is $k_A$-connected by Proposition \\ref{prop:cofibration_cube_properties}(c). Finally, if $V=U\\cup\\{*\\}$ for any nonempty $U\\subset W-A$, then $\\partial_\\emptyset^V{ \\mathcal{Y} }$ is the cube $\\partial_A^{A\\cup U}{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^{A\\cup U}{ \\mathcal{X} }_{ \\mathcal{A} }$ which is $\\infty$-cocartesian by\n\\eqref{eq:pushout_diagram_in_induction_step} and Proposition \\ref{prop:map_of_cubical_diagrams}(b). Hence by our induction hypothesis applied to ${ \\mathcal{Y} }$: since the sum $\\sum_{V\\in\\lambda'}(k'_V+1)$ for a partition $\\lambda'$ of $(W-A)\\cup\\{*\\}$ by nonempty sets is always either $\\infty$, or the sum $\\sum_{U\\in\\lambda}(k_U+1)$ for a partition $\\lambda$ of $W$ by nonempty sets in which some $U$ is $A$, we know that ${ \\mathcal{Y} }$ is $k'$-cartesian, where $k'$ is the minimum of $-|(W-A)\\cup\\{*\\}|+\\sum_{U\\in\\lambda}(k_U+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets in which some $U$ is $A$. In particular, this implies that ${ \\mathcal{Y} }$, and hence $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$, is $(k+|A|-1)$-cartesian.\n\nWe next want to verify that\n\\begin{align}\n\\label{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams}\n  \\partial_{A'}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }\n  \\quad \\text{is $(k+|A'|-1)$-cartesian}\n\\end{align}\nfor each $A'\\subset A$. We know that \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} is true for $A'=A$ by above. We will argue by downward induction on $|A'|$. Suppose that \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} is true for some nonempty $A'\\subset A$. Let $a\\in A'$ and note that the cube $\\partial_{A'-\\{a\\}}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_{A'-\\{a\\}}^W{ \\mathcal{X} }_{ \\mathcal{A} }$ can be written as the diagram of cubes\n\\begin{align}\n\\label{eq:downward_induction_cube_factorization}\n\\xymatrix{\n  \\partial_{A'-\\{a\\}}^{W-\\{a\\}}{ \\mathcal{X} }_{{ \\mathcal{A} }'}\\ar[r]^{{ \\mathrm{id} }}\\ar[d] &\n  \\partial_{A'-\\{a\\}}^{W-\\{a\\}}{ \\mathcal{X} }_{ \\mathcal{A} }\\ar[d]\\\\\n  \\partial_{A'}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}\\ar[r] &\n  \\partial_{A'}^W{ \\mathcal{X} }_{{ \\mathcal{A} }}\n}\n\\end{align}\nWe know the top arrow is the identity by \\eqref{eq:subdiagram_of_identity_maps}, and the bottom arrow is $(k+|A'|-1)$-cartesian by assumption. It follows from  Proposition \\ref{prop:map_of_cubical_diagrams}(d) that  \\eqref{eq:downward_induction_cube_factorization} is $(k+|A'|-2)$-cartesian, which finishes the argument that \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} is true for each $A'\\subset A$.\n\nTo finish off the proof, let $a\\in A$ and note by \\eqref{eq:subdiagram_of_identity_maps} that ${ \\mathcal{X} }_{ \\mathcal{A} }$ can be written as the composition of cubes\n\\begin{align}\n\\label{eq:final_decomposition_for_induction_argument}\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }_{ \\mathcal{A} }.\n\\end{align}\nThe right-hand arrow is $k$-cartesian by \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} and the left-hand arrow is ${ \\mathcal{X} }_{{ \\mathcal{A} }'}$ which is $k$-cartesian by assumption, hence by Proposition \\ref{prop:composed_map_of_cubical_diagrams}(c) it follows that ${ \\mathcal{X} }_{ \\mathcal{A} }$ is $k$-cartesian which finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Let ${ \\mathcal{A} }:={ \\mathcal{A} }_{\\mathrm{max}}^W$, ${ \\mathcal{A} }':={ \\mathcal{A} }_{\\mathrm{max}}^W-\\{W\\}$, and note that ${ \\mathcal{X} }_{ \\mathcal{A} }$ is equal to ${ \\mathcal{X} }$. Then it follows by induction from Proposition \\ref{prop:cubical_induction_argument}, together with Theorem \\ref{thm:higher_homotopy_excision} (to start the induction using ${ \\mathcal{A} }'={ \\mathcal{A} }^W_{\\mathrm{min}}$) that ${ \\mathcal{X} }$ is $k$-cartesian.\n\\end{proof}\n\n\n\\subsection{Proof of higher dual homotopy excision for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nWe now turn to the dual versions of our main results. In this section we prove the dual homotopy excision results (Theorems \\ref{thm:dual_homotopy_excision} and \\ref{thm:higher_dual_homotopy_excision}). Notice that here we are leveraging the fact that cartesian-ness in the categories ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$ is detected in the underlying categories of ${ \\mathcal{R} }$-modules and symmetric sequences, and that, in those underlying categories, there is a close relationship between cartesian-ness and cocartesian-ness, given by Proposition~\\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:dual_homotopy_excision}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $W:=\\{1,2\\}$. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align*}\nWe know that $(*)$ is $(k_1+k_2+1)$-connected by homotopy excision (Theorem \\ref{thm:homotopy_excision}), and since ${ \\mathcal{X} }$ is $\\infty$-cocartesian in the underlying category ${ \\mathsf{SymSeq} }$, the composition is $\\infty$-connected. Hence by Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) the map $(**)$ is $(k_1+k_2+2)$-connected which finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_dual_homotopy_excision}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The argument is by induction on $|W|$. The case $|W|=2$ is verified by Theorem \\ref{thm:dual_homotopy_excision}. Let $n\\geq 3$ and assume the theorem is true for each $2\\leq |W|< n$. Let's verify it remains true for $|W|=n$.\n\nIt suffices to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align}\n\\label{eq:composition_of_maps_for_determining_cocartesian_estimate}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align}\nWe want to show that $(**)$ is $k$-connected. Consider the $W$-cube ${ \\mathcal{X} }':={ \\mathcal{X} }_{{ \\mathcal{A} }_\\mathrm{max}-\\{W\\}}$ in ${ \\Lt_{ \\mathcal{O} } }$ and note that $(*)$ is isomorphic to the map\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }'\\ar[r]^-{(*)'} &\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset(W)}{ \\mathcal{X} }'{  \\ \\cong \\ }{ \\mathcal{X} }'_W\n}\n\\end{align*}\nWe know that ${ \\mathcal{X} }'$ is $\\infty$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$, hence by higher Blakers-Massey (Theorem \\ref{thm:higher_blakers_massey}) applied to ${ \\mathcal{X} }'$, together with the induction hypothesis, it follows that ${ \\mathcal{X} }'$ is $k'$-cartesian with $k'=\\sum_{i\\in W}k_i$. Hence by Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR} we know that ${ \\mathcal{X} }'$ is $(k'+|W|-1)$-cocartesian in ${ \\mathsf{SymSeq} }$, and therefore $(*)$ is $(k'+|W|-1)$-connected. Since the composition in \\eqref{eq:composition_of_maps_for_determining_cocartesian_estimate} is $\\infty$-connected, it follows from Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) that $(**)$ is $(k'+|W|)$-connected which finishes the proof.\n\\end{proof}\n\n\\subsection{Proof of the higher dual Blakers-Massey theorem for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nThe purpose of this section is to prove Theorems \\ref{thm:dual_blakers_massey} and \\ref{thm:higher_dual_blakers_massey}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:dual_blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $W:=\\{1,2\\}$. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align}\n\\label{eq:composition_of_maps_in_dual_blakers_massey}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align}\nSince ${ \\mathcal{X} }$ is $k_{12}$-cartesian, we know by Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR} that ${ \\mathcal{X} }$ is $(k_{12}+1)$-cocartesian in ${ \\mathsf{SymSeq} }$, and hence the composition in \\eqref{eq:composition_of_maps_in_dual_blakers_massey} is $(k_{12}+1)$-connected. Since we know the map $(*)$ is $(k_1+k_2+1)$-connected by homotopy excision (Theorem \\ref{thm:homotopy_excision}), it follows that $(**)$ is $k$-connected by Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) which finishes the proof.\n\\end{proof}\n\n\n\n\\begin{defn}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any subset ${ \\mathcal{B} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item A subset ${ \\mathcal{A} }\\subset{ \\mathcal{B} }$ is \\emph{concave} if every element of ${ \\mathcal{B} }$ which is greater than an element of ${ \\mathcal{A} }$ is in ${ \\mathcal{A} }$.\n\\item Define ${ \\mathcal{A} }_W^\\mathrm{min}:=\\{V\\subset W: |W-V|\\leq 1\\}$ and ${ \\mathcal{A} }_W^\\mathrm{max}:=\\powerset(W)$.\n\\item For each concave subset ${ \\mathcal{A} }\\subset\\powerset(W)$, the $W$-cube ${ \\mathcal{X} }^{ \\mathcal{A} }$ is defined objectwise by $({ \\mathcal{X} }^{ \\mathcal{A} })_U := \\lim_{T\\in{ \\mathcal{A} },\\, T\\supset U}{ \\mathcal{X} }_T$.\n\\end{itemize}\n\\end{defn}\n\n\n\\begin{prop}\n\\label{prop:warmup_cartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$  (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for every $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }$ is $k_{V-U}$-cartesian.\n\\end{prop}\n\n\\begin{proof}\nIt will be useful to note that assumptions (i) and (ii) are equivalent to the following assumptions:\n\\begin{itemize}\n\\item[(1)] for each subset $U\\subsetneqq W$, the $(W-U)$-cube $\\partial_{U}^W{ \\mathcal{X} }$  (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_U$ and ${ \\mathcal{X} }_W$) is $k_{W-U}$-cartesian,\n\\item[(2)] $k_{W-V}\\leq k_{W-U}$ for each $U\\subset V$.\n\\end{itemize}\nWe want to verify that $\\partial_U^V{ \\mathcal{X} }$ is $k_{V-U}$-cartesian for each $U\\subsetneqq V\\subset W$. The argument is by downward induction on $|V|$. The case $|V|=|W|$ is true by assumption. Assume the proposition is true for some nonempty $V\\subset W$ and consider any $U\\subsetneqq V$. Let $v\\in V$ and note that $\\partial_{U-\\{u\\}}^V{ \\mathcal{X} }$ can be written as the composition of cubes\n\\begin{align*}\n\\xymatrix{\n  \\partial_{U-\\{u\\}}^{V-\\{u\\}}{ \\mathcal{X} }\\ar[r] &\n  \\partial_U^V{ \\mathcal{X} }\n}\n\\end{align*}\nWe know by the induction assumption that the composition of cubes is $k_{(V-U)\\cup\\{u\\}}$-cartesian and the right-hand cube is $k_{V-U}$-cartesian. Since $k_{V-U}\\leq k_{(V-U)\\cup\\{u\\}}$ by assumption, it follows from Proposition \\ref{prop:map_of_cubical_diagrams}(c) that the left-hand cube is $k_{V-U}$-cartesian which finishes the proof; note that the sets $(V-\\{u\\})-(U-\\{u\\})$ and $V-U$ are the same.\n\\end{proof}\n\nThe following proposition will be needed in the proof of Proposition \\ref{prop:properties_of_the_X_sup_A_construction} below.\n\n\\begin{prop}\n\\label{prop:lim_over_punctured_cube_of_X_sup_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }\\subset\\powerset(W)$. Then for each nonempty subset $V\\subset W$, there is a natural isomorphism\n\\begin{align*}\n  \\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }{  \\ \\cong \\ }\n  \\lim_{T\\in{ \\mathcal{A} },\\,T\\supsetneqq V}{ \\mathcal{X} }_T\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}\nThis is because the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\supset U,\\, U\\supsetneqq V\\}$ and $\\{T\\in{ \\mathcal{A} }:T\\supsetneqq V\\}$ are the same.\n\\end{proof}\n\nThe following proposition explains the key properties of the ${ \\mathcal{X} }^{ \\mathcal{A} }$ construction and its relationship to ${ \\mathcal{X} }$; it is through these properties that the ${ \\mathcal{X} }^{ \\mathcal{A} }$ construction is useful and meaningful.\n\n\\begin{prop}\n\\label{prop:properties_of_the_X_sup_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }_W^\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }_W^\\mathrm{max}$.\n\\begin{itemize}\n\\item[(a)] There are natural isomorphisms $\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }{  \\ \\cong \\ }\\partial_V^W{ \\mathcal{X} }$ if $V\\in{ \\mathcal{A} }$,\n\\item[(b)] The $(W-V)$-cube $\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }$ is $\\infty$-cartesian if $V\\notin{ \\mathcal{A} }$ and ${ \\mathcal{X} }$ is a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules).\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nConsider part (a). Let $V\\in{ \\mathcal{A} }$. Then we know that the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\supset V\\}$ and $\\{T:W\\supset T\\supset V\\}$ are the same, by ${ \\mathcal{A} }\\subset{ \\mathcal{A} }_W^\\mathrm{max}$ concave. It follows that ${ \\mathcal{X} }_V{  \\ \\cong \\ }\\lim_{W\\supset T\\supset V}{ \\mathcal{X} }_T=\\lim_{T\\in{ \\mathcal{A} },\\,T\\supset V}=({ \\mathcal{X} }^{ \\mathcal{A} })_V$. The same argument shows that ${ \\mathcal{X} }_{V'}{  \\ \\cong \\ }({ \\mathcal{X} }^{ \\mathcal{A} })_{V'}$, for each $V'\\supset V$. Consider part (b). Let $V\\subset W$ and $V\\notin{ \\mathcal{A} }$. Then the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\supset V\\}$ and $\\{T\\in{ \\mathcal{A} }:T\\supsetneqq V\\}$ are the same, and hence the composition\n\\begin{align*}\n  (\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} })_\\emptyset=({ \\mathcal{X} }^{ \\mathcal{A} })_V=\\lim_{T\\in{ \\mathcal{A} },\\,T\\supset V}X_T=\\lim_{T\\in{ \\mathcal{A} },\\,T\\supsetneqq V}X_T{  \\ \\cong \\ }\\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }\n\\end{align*}\nis an isomorphism which finishes the proof of part (b).\n\\end{proof}\n\nThe following proposition shows that ${ \\mathcal{X} }^{ \\mathcal{A} }$ inherits several of the homotopical properties of ${ \\mathcal{X} }$.\n\n\n\\begin{prop}\n\\label{prop:cartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }_W^\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }_W^\\mathrm{max}$. Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for each $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }^{ \\mathcal{A} }$ is $k_{V-U}$-cartesian.\n\\end{prop}\n\n\\begin{proof}\nBy the downward induction argument in the proof of Proposition \\ref{prop:warmup_cartesian_estimates_for_induction_argument}, it suffices to verify that the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }^{ \\mathcal{A} }$ is $k_V$-cartesian, for each nonempty subset $V\\subset W$, and this follows immediately from Proposition \\ref{prop:properties_of_the_X_sup_A_construction}.\n\\end{proof}\n\nThe following proposition appears in Goodwillie \\cite[2.8]{Goodwillie_calc2} in the context of spaces, and exactly the same argument gives a proof in the context of structured ring spectra; this is an exercise left to the reader.\n\n\\begin{prop}\n\\label{prop:fibration_cube_properties}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item[(a)] For each inclusion ${ \\mathcal{A} }'\\subset{ \\mathcal{A} }$ of concave subsets of $\\powerset(W)$, the induced map $\\lim_{{ \\mathcal{A} }}{ \\mathcal{X} }{ \\rightarrow }\\lim_{{ \\mathcal{A} }'}{ \\mathcal{X} }$ is a fibration in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$),\n\\item[(b)] For any concave subsets ${ \\mathcal{A} },{ \\mathcal{B} }$ of $\\powerset(W)$, the diagram\n\\begin{align*}\n\\xymatrix{\n  \\lim_{{ \\mathcal{A} }\\cup{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[r]\\ar[d] & \\lim_{{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[d]\\\\\n  \\lim_{{ \\mathcal{A} }}{ \\mathcal{X} }\\ar[r] & \\lim_{{ \\mathcal{A} }\\cap{ \\mathcal{B} }}{ \\mathcal{X} }\n}\n\\end{align*}\nis a pullback diagram of fibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$).\n\\item[(c)] If $A\\in{ \\mathcal{A} }$ is minimal and the fibration $${ \\mathcal{X} }_A{  \\ \\cong \\ }\\lim_{\\powerset(W-A)}\\partial_A^W{ \\mathcal{X} }{ \\rightarrow }\\lim_{\\powerset_0(W-A)}\\partial_A^W{ \\mathcal{X} }$$\nis $k_{W-A}$-connected, then the fibration $$\\lim_{ \\mathcal{A} }{ \\mathcal{X} }{ \\rightarrow }\\lim_{{ \\mathcal{A} }-\\{A\\}}{ \\mathcal{X} }$$ is $k_{W-A}$-connected.\n\\end{itemize}\n\\end{prop}\n\nThe purpose of the following induction argument is to leverage the higher dual homotopy excision result (Theorem \\ref{thm:higher_dual_homotopy_excision}) for structured ring spectra into a conceptual proof of the second main theorem in this paper (Theorem \\ref{thm:higher_dual_blakers_massey})---the higher dual Blakers-Massey theorem for structured ring spectra. Proposition \\ref{prop:dual_cubical_induction_argument} is motivated by Goodwillie \\cite[proof of (2.6)]{Goodwillie_calc2}; it is essentially Goodwillie's dual cubical induction argument, appropriately modified to our situation. The reader who is interested in an alternate proof of Theorem \\ref{thm:higher_dual_blakers_massey}, which is more efficient, but requires a little extra calculation at the end, may skip directly to Remark \\ref{rem:alt_proof_of_higher_dual_blakers_massey}.\n\n\\begin{prop}[Dual cubical induction argument]\n\\label{prop:dual_cubical_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Suppose ${ \\mathcal{A} }\\subsetneqq{ \\mathcal{A} }_W^\\mathrm{max}$ is concave, $A\\in{ \\mathcal{A} }$ minimal, $|W-A|\\geq 2$, and ${ \\mathcal{A} }':={ \\mathcal{A} }-\\{A\\}$. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nIf $X^{{ \\mathcal{A} }'}$ is $j$-cocartesian, then ${ \\mathcal{X} }^{ \\mathcal{A} }$ is $j$-cocartesian, where $j$ is the minimum of $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\\end{prop}\n\n\n\\begin{rem}\nIn the case that ${ \\mathcal{X} }$ is a $3$-cube, the following diagram illustrates one of the cubical decompositions covered by Proposition \\ref{prop:dual_cubical_induction_argument}. It corresponds to the sequence of minimal elements: $\\{3\\}$, $\\{2\\}$, $\\{1\\}$, $\\emptyset$.\n\n\\begin{align*}\n\\xymatrix@!0{\n{ \\mathcal{X} }_{\\{1,2,3\\}} &&\n{ \\mathcal{X} }_{\\{2,3\\}}\\ar[ll]\\ar@{=}[rr] &&\n{ \\mathcal{X} }_{\\{2,3\\}}\\ar@{=}[rr] &&\n{ \\mathcal{X} }_{\\{2,3\\}}\\\\\n&{ \\mathcal{X} }_{\\{1,3\\}}\\ar[ul] &&\n\\cdot\\ar[ll]\\ar[ul] &&\n{ \\mathcal{X} }_{\\{3\\}}\\ar@{=}[rr]\\ar[ll]\\ar[ul] &&\n{ \\mathcal{X} }_{\\{3\\}}\\ar[ul]\\\\\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar@{=}[dd]\\ar[uu] &&\n\\cdot\\ar@{=}'[r][rr]\\ar'[l][ll]\\ar'[u][uu] &&\n\\cdot\\ar'[u][uu] &&\n{ \\mathcal{X} }_{\\{2\\}}\\ar@{=}'[d][dd]\\ar'[l][ll]\\ar[uu]|(0.47)\\hole|(0.5)\\hole\\\\\n&\\cdot\\ar[dd]\\ar[ul]\\ar[uu] &&\n\\cdot\\ar[ll]\\ar[ul]\\ar[uu] &&\n\\cdot\\ar[ll]\\ar[ul]\\ar[uu] &&\n\\cdot\\ar[ll]\\ar[ul]\\ar[uu]\\\\\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar@{=}[dd]&&&&&&\n{ \\mathcal{X} }_{\\{2\\}}\\ar@{=}'[d][dd]\\ar'[lllll][llllll]\\\\\n&{ \\mathcal{X} }_{\\{1\\}}\\ar@{=}[dd]\\ar[ul]\\ar[uu]&&&&&&\n\\cdot\\ar[llllll]\\ar[ul]\\ar[uu]\\\\\n{ \\mathcal{X} }_{\\{1,2\\}} &&&&&&\n{ \\mathcal{X} }_{\\{2\\}}\\ar'[lllll][llllll]\\\\\n&{ \\mathcal{X} }_{\\{1\\}}\\ar[ul]&&&&&&{ \\mathcal{X} }_\\emptyset\\ar[llllll]\\ar[ul]\\ar[uu]\n}\n\\end{align*}\n\\end{rem}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:dual_cubical_induction_argument}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The argument is by induction on $|W|$. The case $|W|=2$ is verified by the proof of Theorem \\ref{thm:dual_blakers_massey}. Let $n\\geq 3$ and assume the proposition is true for each $2\\leq|W|<n$. Let's verify it remains true for $|W|=n$.\n\nWe know by assumption that ${ \\mathcal{X} }^{{ \\mathcal{A} }'}$ is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$. We want to verify that ${ \\mathcal{X} }^{ \\mathcal{A} }$ is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ (it might be helpful at this point to look ahead to \\eqref{eq:dual_final_decomposition_for_induction_argument} for the decomposition of ${ \\mathcal{X} }^{ \\mathcal{A} }$ that we will use to finish the proof). Consider the induced map of $W$-cubes ${ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }{ \\mathcal{X} }^{{ \\mathcal{A} }'}$. Note that if $U\\not\\subset A$, then $\\{T\\in{ \\mathcal{A} }:T\\supset U\\}=\\{T\\in{ \\mathcal{A} }':T\\supset U\\}$; in particular, each of the maps\n\\begin{align}\n\\label{eq:dual_subdiagram_of_identity_maps}\n\\xymatrix{\n  ({ \\mathcal{X} }^{{ \\mathcal{A} }})_U\\ar[r]^-{{ \\mathrm{id} }} & ({ \\mathcal{X} }^{{ \\mathcal{A} }'})_U,\n  & U\\not\\subset A\n}\n\\end{align}\nin ${ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ is the identity map. Note that if $U\\subset V\\subset A$, then the diagram\n\\begin{align}\n\\label{eq:pullback_diagram_in_induction_step}\n\\xymatrix{\n  ({ \\mathcal{X} }^{{ \\mathcal{A} }})_U\\ar[r]\\ar[d] & ({ \\mathcal{X} }^{{ \\mathcal{A} }'})_U\\ar[d]\\\\\n  ({ \\mathcal{X} }^{{ \\mathcal{A} }})_V\\ar[r] & ({ \\mathcal{X} }^{{ \\mathcal{A} }'})_V, &\n  U\\subset V\\subset A\n}\n\\end{align}\nis a pullback diagram by Proposition \\ref{prop:fibration_cube_properties}(b); in particular, focusing on this case is the same as focusing on the subdiagram $\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ of ${ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }{ \\mathcal{X} }^{{ \\mathcal{A} }'}$.\n\nLet's first verify that $\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ is $(j+|A|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$. Let ${ \\mathcal{Y} }$ denote $\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ regarded as an $(A\\cup\\{*\\})$-cube as follows:\n\\begin{align*}\n  { \\mathcal{Y} }_V&:=(\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} })_V=({ \\mathcal{X} }^{ \\mathcal{A} })_V,\\quad\\ V\\subset A,\\\\\n  { \\mathcal{Y} }_V&:=(\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'})_U=({ \\mathcal{X} }^{{ \\mathcal{A} }'})_U,\\quad U\\subset A,\\quad V=U\\cup\\{*\\}.\n\\end{align*}\nSince $|A\\cup\\{*\\}|<|W|$, our induction assumption can be applied to ${ \\mathcal{Y} }$, provided that the appropriate $k'_V$-cartesian estimates are satisfied. We claim that with the following definitions\n\\begin{align*}\n  k'_V&:=k_V,\\quad\\quad \\emptyset\\neq V\\subset A,\\\\\n  k'_V&:=k_{W-A},\\quad\\quad V=\\{*\\}\\\\\n  k'_V&:=\\infty,\\quad\\quad \\emptyset\\neq U\\subset A,\\quad V=U\\cup\\{*\\},\n\\end{align*}\nthe cube $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ (formed by all maps in ${ \\mathcal{Y} }$ between ${ \\mathcal{Y} }_{(A\\cup\\{*\\})-V}$ and ${ \\mathcal{Y} }_{A\\cup\\{*\\}}$) is $k'_V$-cartesian, for each nonempty subset $V\\subset A\\cup\\{*\\}$. Let's verify this now: If $V\\subset A$ is nonempty, then $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ is the cube $\\partial_{A-V}^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ which is $k_V$-cartesian by Proposition \\ref{prop:cartesian_estimates_for_induction_argument}. If $V=\\{*\\}$, then $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ is the map $({ \\mathcal{X} }^{ \\mathcal{A} })_A{ \\rightarrow }({ \\mathcal{X} }^{{ \\mathcal{A} }'})_A$ which is $k_{W-A}$-connected by Proposition \\ref{prop:fibration_cube_properties}(c). Finally, if $V=U\\cup\\{*\\}$ for any nonempty $U\\subset A$, then $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ is the cube $\\partial_{A-U}^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_{A-U}^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ which is $\\infty$-cartesian by\n\\eqref{eq:pullback_diagram_in_induction_step} and Proposition \\ref{prop:map_of_cubical_diagrams}(d). Hence by our induction hypothesis applied to ${ \\mathcal{Y} }$: since the sum $\\sum_{V\\in\\lambda'}k'_V$ for a partition $\\lambda'$ of $A\\cup\\{*\\}$ by nonempty sets is always either $\\infty$, or the sum $\\sum_{U\\in\\lambda}k_U$ for a partition $\\lambda$ of $W$ by nonempty sets in which some $U$ is $W-A$, we know that ${ \\mathcal{Y} }$ is $j'$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$, where $j'$ is the minimum of $|A\\cup\\{*\\}|+\\sum_{U\\in\\lambda}k_U$ over all partitions $\\lambda$ of $W$ by nonempty sets in which some $U$ is $W-A$. In particular, this implies that ${ \\mathcal{Y} }$, and hence $\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$, is $(j+|A|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$.\n\n\nWe next want to verify that\n\\begin{align}\n\\label{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams}\n  \\partial_\\emptyset^{A'}{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^{A'}{ \\mathcal{X} }^{{ \\mathcal{A} }'}\n  \\quad \\text{is $(j+|A'|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$}\n\\end{align}\nfor each $A'\\supset A$. We know that \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} is true for $A'=A$ by above. We will argue by upward induction on $|A'|$. Suppose that \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} is true for some $A'\\supset A$. Let $a\\in W-A'$ and note that the cube $\\partial_\\emptyset^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }\\partial_\\emptyset^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ can be written as the diagram of cubes\n\\begin{align}\n\\label{eq:upward_induction_cube_factorization}\n\\xymatrix{\n  \\partial_\\emptyset^{A'}{ \\mathcal{X} }^{ \\mathcal{A} }\\ar[r]\\ar[d] &\n  \\partial_\\emptyset^{A'}{ \\mathcal{X} }^{{ \\mathcal{A} }'}\\ar[d]\\\\\n  \\partial_{\\{a\\}}^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{ \\mathcal{A} }\\ar[r]^-{{ \\mathrm{id} }} &\n  \\partial_{\\{a\\}}^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}\n}\n\\end{align}\nWe know the bottom arrow is the identity by \\eqref{eq:dual_subdiagram_of_identity_maps}, and the top arrow is $(j+|A'|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ by assumption. It follows from  Proposition \\ref{prop:map_of_cubical_diagrams}(b) that  \\eqref{eq:upward_induction_cube_factorization} is $(j+|A'|+2-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$, which finishes the argument that \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} is true for each $A'\\supset A$.\n\n\nTo finish off the proof, let $a\\in W-A$ and note by \\eqref{eq:dual_subdiagram_of_identity_maps} that ${ \\mathcal{X} }^{ \\mathcal{A} }$ can be written as the composition of cubes\n\\begin{align}\n\\label{eq:dual_final_decomposition_for_induction_argument}\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }^{{ \\mathcal{A} }'}.\n\\end{align}\nThe left-hand arrow is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ by \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} and the right-hand arrow is ${ \\mathcal{X} }^{{ \\mathcal{A} }'}$ which is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ by assumption, hence by Proposition \\ref{prop:composed_map_of_cubical_diagrams}(a) it follows that ${ \\mathcal{X} }^{ \\mathcal{A} }$ is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ which finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_dual_blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. It is enough to consider the special case where ${ \\mathcal{X} }$ is a fibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Let ${ \\mathcal{A} }:={ \\mathcal{A} }^{\\mathrm{max}}_W$, ${ \\mathcal{A} }':={ \\mathcal{A} }^{\\mathrm{max}}_W-\\emptyset$, and note by \\eqref{eq:dual_subdiagram_of_identity_maps} that ${ \\mathcal{X} }^{ \\mathcal{A} }$, which is equal to ${ \\mathcal{X} }$, can be written as the composition of cubes\n\\begin{align}\n\\label{eq:dual_final_decomposition_dual_higher_homotopy_excision}\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }^{{ \\mathcal{A} }'}.\n\\end{align}\nWe know by Proposition \\ref{prop:dual_cubical_induction_argument}, together with Theorem \\ref{thm:higher_dual_homotopy_excision} (to start the induction using ${ \\mathcal{A} }'={ \\mathcal{A} }_W^{\\mathrm{min}}$) that the right-hand arrow, which is ${ \\mathcal{X} }^{{ \\mathcal{A} }'}$, is $j$-cocartesian in ${ \\Alg_{ \\mathcal{O} } }$, where $j$ is the minimum of $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$. We claim that the left-hand arrow is $(k_W+|W|-1)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$; this follows from upward induction by arguing exactly as in \\eqref{eq:upward_induction_cube_factorization}, but by starting with the observation (see the $k'_V$ estimates in the proof of Proposition \\ref{prop:dual_cubical_induction_argument}) that the map $\\partial_\\emptyset^\\emptyset{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^\\emptyset{ \\mathcal{X} }^{{ \\mathcal{A} }'}$, which is the map $({ \\mathcal{X} }^{ \\mathcal{A} })_\\emptyset{ \\rightarrow }({ \\mathcal{X} }^{{ \\mathcal{A} }'})_\\emptyset$, is $k_W$-connected. Hence it follows from Proposition \\ref{prop:composed_map_of_cubical_diagrams} that the composition,\nwhich is ${ \\mathcal{X} }$, is $k$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ which finishes the proof.\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:alt_proof_of_higher_dual_blakers_massey}\nAn alternate proof of Theorem \\ref{thm:higher_dual_blakers_massey} can be obtained from the higher Blakers-Massey theorem (Theorem \\ref{thm:higher_blakers_massey}) by arguing as in the proof of Theorem \\ref{thm:higher_dual_homotopy_excision}. Since the dual cubical induction argument is conceptually very useful and will be needed elsewhere, we include both approaches for the interested reader, with only a few details of the alternate proof below left to the reader.\n\\end{rem}\n\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The argument is by induction on $|W|$. The case $|W|$=1 is trivial and the case $|W|=2$ is verified by Theorem \\ref{thm:dual_blakers_massey}. Let $n\\geq 3$ and assume the theorem is true for each $2\\leq |W|< n$. Let's verify it remains true for $|W|=n$.\n\nIt suffices to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align}\n\\label{eq:alt_composition_of_maps_for_determining_cocartesian_estimate}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align}\nWe want to show that $(**)$ is $k$-connected. Consider the $W$-cube ${ \\mathcal{X} }':={ \\mathcal{X} }_{{ \\mathcal{A} }^W_\\mathrm{max}-\\{W\\}}$ in ${ \\Lt_{ \\mathcal{O} } }$ and note that $(*)$ is isomorphic to the map\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }'\\ar[r]^-{(*)'} &\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset(W)}{ \\mathcal{X} }'{  \\ \\cong \\ }{ \\mathcal{X} }'_W\n}\n\\end{align*}\nWe know that ${ \\mathcal{X} }'$ is $\\infty$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$. By Proposition \\ref{prop:warmup_cartesian_estimates_for_induction_argument} we know that the $V$-cube $\\partial_U^{U\\cup V}{ \\mathcal{X} }$ is $k_V$-cartesian for each disjoint $U$ and $V$. Hence by the induction hypothesis, for each $V\\subsetneqq W$ the cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ is $k'_V$-cocartesian, where $k'_V$ is the minimum of $k_V+|V|-1$ and $|V|+\\sum_{U\\in\\lambda'}k_U$ over all partitions $\\lambda'$ of $V$ by nonempty sets not equal to $V$. In particular, the $W$-cube ${ \\mathcal{X} }'$ satisfies the conditions of the higher Blakers-Massey theorem (Theorem \\ref{thm:higher_blakers_massey}) with $k'_W=\\infty$ and the $k'_V$ above, and hence it follows that ${ \\mathcal{X} }'$ is $k'$-cartesian, where $k'$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k'_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\nHence by Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR} we know that ${ \\mathcal{X} }'$ is $(k'+|W|-1)$-cocartesian in ${ \\mathsf{SymSeq} }$, and therefore $(*)$ is $(k'+|W|-1)$-connected. Since the composition in \\eqref{eq:alt_composition_of_maps_for_determining_cocartesian_estimate} is $(k_W+|W|-1)$-connected, it follows from Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) that $(**)$ is $k$-connected, where $k$ is the minimum of $k_W+|W|-1$ and $\\sum_{V\\in\\lambda}(k'_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$; it is an exercise left to the reader to verify that this description of $k$ agrees with Theorem \\ref{thm:higher_dual_blakers_massey}.\n\\end{proof}\n\n\n\n\\section{Operads in chain complexes over a commutative ring}\n\\label{sec:chain_complexes_over_a_commutative_ring}\n\nThe purpose of this section is to observe that the main results of this paper remain true in the context of unbounded chain complexes over a commutative ring, provided essentially that the desired model category structures exist on algebras (resp. left modules) over the operad ${ \\mathcal{O} }$. Since the constructions and proofs of the theorems are essentially identical to the arguments above in the context of ${ \\mathcal{R} }$-modules, modulo the obvious changes, the arguments are left to the reader; see \\cite[5.17]{Harper_Hess} and Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes} below.\n\n\\begin{assumption}\n\\label{assumption:commutative_ring_notation}\nFrom now on in this section, we assume that ${ {\\mathcal{K}} }$ is any commutative ring.\\end{assumption}\n\nDenote by $({ \\mathsf{Ch} }_{ {\\mathcal{K}} },{ \\otimes },{ {\\mathcal{K}} })$ the closed symmetric monoidal category of unbounded chain complexes over ${ {\\mathcal{K}} }$ \\cite{Hovey, MacLane_homology}.\n\n\\begin{homotopical_assumption}\n\\label{HomotopicalAssumption_chain_complexes}\nIf ${ \\mathcal{O} }$ is an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$, assume that the following model structure exists on ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$): the model structure on ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) has weak equivalences the homology isomorphisms (resp. objectwise homology isomorphisms) and fibrations the maps that are dimensionwise surjections (resp. objectwise dimensionwise surjections). Furthermore, assume that ${ \\mathcal{O} }[\\mathbf{r}]$ is cofibrant in $({ \\mathsf{Ch} }_{ {\\mathcal{K}} })^{\\Sigma_r^{ \\mathrm{op} }}$ with respect to the projective model structure for each $r\\geq 0$.\n\\end{homotopical_assumption}\n\nIf ${ {\\mathcal{K}} }$ is any field of characteristic zero, then Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes} is satisfied by every operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ (see, for instance, \\cite{Harper_Modules}).\n\nThe following is a commutative rings version of Theorem \\ref{thm:homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a homotopy pushout square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$), then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=k_1+k_2 +1$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=k_1+k_2$.\n\\end{itemize}\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:blakers_massey}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$, and  $k_{12}\\in{ \\mathbb{Z} }$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$) and ${ \\mathcal{X} }$ is $k_{12}$-cocartesian, then ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $k_{12}-1$ and $k_{1}+k_{2}$.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:higher_homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a nonempty finite set. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cocartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i\\in W$),  then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=|W|-1+\\sum_{i\\in W}k_i$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=\\sum_{i\\in W}k_i$.\n\\end{itemize}\n\\end{thm}\n\nThe following is a commutative rings version of the first main theorem of this paper (Theorem \\ref{thm:higher_blakers_massey}).\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a nonempty finite set. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:dual_homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a homotopy pullback square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected and ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=k_{1}+k_{2}+2$.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:dual_blakers_massey}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2,k_{12}\\geq -1$ with $k_1\\leq k_{12}$ and $k_2\\leq k_{12}$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected, ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, and ${ \\mathcal{X} }$ is $k_{12}$-cartesian, then ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_{12}+1$ and $k_{1}+k_{2}+2$.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:higher_dual_homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a finite set with $|W|\\geq 2$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_{W-\\{i\\}}{ \\rightarrow }{ \\mathcal{X} }_W$ is $k_i$-connected ($i\\in W$),  then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=|W|+\\sum_{i\\in W}k_i$.\n\\end{thm}\n\nThe following is a commutative rings version of the second main theorem of this paper (Theorem \\ref{thm:higher_dual_blakers_massey}).\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a nonempty finite set. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_W+|W|-1$ and $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\\end{thm}\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe optical properties of thin metal films secure their wide use in different fields \u2013 from mirrors to supersensitive detectors \\cite{Maier} capable of detecting refraction coefficient changes as small as ${10^{ - 7}}$ \\cite{Mikaelian}. These properties are considered to be stable to external actions and changes of the absorption spectrum are detected only in going from macro samples to nano samples. The latter effect is caused by collective oscillations of electrons (so-called plasmon oscillations) in nanoparticles  with  sizes smaller than  the electron mean free  path  in massive samples. It was revealed that the static polarizability of metal clusters containing  several tens of atoms, which determines the frequency of collective oscillations, differs sufficiently from  macroscopic polarizability \\cite{Knight}. Besides, calculations show that in $Ag$ clusters $1 - 2 nm$ in diameter an extra charge of $1$ to $3$ electron charges should shift the plasmon absorption line by about $5 nm$, while for even $10 - nm$ clusters this spectral shift is negligibly small \\cite{Quinten}. It sounds reasonable because the spectral shift is caused by a change of the charge density, which is noticeable in small clusters and negligibly small in large clusters. It is however known that plasmon frequencies are strongly dependent on the form and size of nanoparticles. The examples show that noticeable changes of optical properties of metal nano-size samples are hard to cause without changing their geometry.\n\n\tIn the present communication we give the experimental data that clearly show a considerable change of the effective relative permittivity of $Ag$ layers in the MDM nanostructure.\n\n\\section{ Experiment}\nHigh-vacuum electron-beam deposition was used to make plasmon 4-nanolayer MDM samples ($A{l_2}{O_3}\/Ag\/A{l_2}{O_3}\/Ag$). Figure~\\ref{fig:fig01} illustrates a 4-layer nanostructure whose optical properties are studied in the experiment. The layers of the structure are $12,\\,49,\\,177\\;{\\textrm{and}}\\;36\\,nm$ thick. A quartz substrate was kept at a relatively low temperature (about ${150^ \\circ }C$) in the process of layer deposition which allows obtaining smaller crystal grains in the layers  and was necessary to ensure the surface smothness of the film. The outer $A{l_2}{O_3}$ layer ($d = 12nm$) serves as antioxidation coating. No adhesive sublayers were used. A laser with wavelength $\\lambda  = 632.8nm$ and $p$-polarization was used to generate SPR. It should be noted that this structure can be considered as a metal-dielectric plasmonic waveguide with two SPR angles ${\\theta _1} = {43^0}$ and ${\\theta _2} = {56.3^0}$.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig01}\n\\caption{\\label{fig:fig01}MDM plasmonic structure}\n\\end{figure}\n\n\tIn deposition we used masks to form a thin-film capacitor structure. The diameter of the outer silver electrode was 3 mm. A constant electric field was applied to the capacitor. The structures could stand up to $30V$ without electric breakdown. This corresponds to the field strength of $1.7 \\cdot {10^6}V\/cm$ in the dielectric layer, given the corundum permittivity $\\varepsilon  = 10$. However, when the air humidity was high, a lower breakdown voltage and greater leakage currents were observed.\n\n\tIn the experiment we  measured the SPR spectrum (the angular reflection spectrum of the MDM structure) using the Kretschmann configuration \\cite{Kretschmann}. We used an automatic semicylinder $\\theta - 2\\theta $ goniometer with additional plane-wavefront-forming optics to measure the angular spectrum of total reflection.\n\n\tThe experimental results are given in Figure~\\ref{fig:fig02} which shows how the reflection spectrum changes with the electric field changing from $0$ to $30$ volts. The curves allow us to assert that the reflection coefficient changes noticeably when the voltage increases above $4V$, at $U > 16V$ it reaches  saturation. The resonance peak width grows considerably with $U$, yet the shift from the resonance angles ${\\theta _1} = {43^0}$ and ${\\theta _2} = {56.3^0}$ does not occur. The reflection coefficient changes significantly: when $U$ approaches $20 V$, it decreases fourfold near the first resonance angle and increases fivefold near the second SPR angle. In Figure~\\ref{fig:fig02} we can select three groups of curves: the first group corresponds to $U \\in [0,\\;6V]$, the second to $U \\in [8 V,\\;16 V]$, and the third to $U \\in [18 V,\\;32 V]$. The grouping will become clear in the next paragraph where we examine the voltage dependence of optical parameters. Besides, there are two points where the reflection coefficient is independent of the applied voltage: $R \\approx 11.6\\% $ at ${\\theta} = {53.95^0}$ and  $R \\approx 11.8\\% $ at  ${\\theta} = {57.39^0}$ for any $U$. It is hard to explain the existence of these points, however they can be used for testing the results: the theoretical curves should intersect at the points.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig02}\n\\caption{\\label{fig:fig02}The reflection spectrum versus the voltage $U$ across the MDM structure. The upper curves corresponds to $U$changing from $0 V$ to $28 V$ in $4 - V$ steps. The lowest curve corresponds to near-breakdown voltage $U = 30 V$.}\n\\end{figure}\n\n\tThe curves of reflection coefficient $R$  versus angle of incidence $\\theta $ allowed us to compute the optical parameters of all layers of the MDM structure (optical thickness, refraction coefficients $n$ and absorption coefficients $k$) as function of voltage $U \\in [0,\\;30 V]$. The effective optical parameters of the layers were determined by the best agreement between experimental and theoretical curves.\n\n\tThe conventional methods and software were used in computation of theoretical angular spectrum (the method is described in \\cite{Palagushkin1,Palagushkin2} in detail). The optical constants of silver and corundum from the SOPRA database and layer thicknesses measured during the deposition process were taken as initial values for the theoretical model in the absence of the external electric field. Then we computed the best agreement between the theoretical and experimental curves $R = R(\\theta )$ to correct the layer thicknesses to use them for further computations. The results of simulation are shown in Figure~\\ref{fig:fig03} and in Table~\\ref{table:table1} for $U = 0$.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig03}\n\\caption{\\label{fig:fig03}SPR spectrum when there is no electric field (the solid line corresponds to the theory, marks to the experiments).}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{table:table1} The initial optical parameters of the layers ($U=0$)}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nNo & material \t\t& $n$ \t& $k$ \t& thickness, ($nm$) \\\\\n1  & ${Al_2}{O_3}$\t& 1.659\t& 0\t\t& 12.02\\\\\n2  & $Ag$\t\t\t& 0.136\t& 4.011\t& 49.20\\\\\n3  & ${Al_2}{O_3}$\t& 1.659\t& 0\t\t& 177.36\\\\\n4  & $Ag$\t\t\t& 0.136\t& 4.011\t& 36.46\\\\\n5  & substrate\t\t& 1.525\t& 0\t\t& -\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\tGiven non-zero external electric field ($U > 0$), the thicknesses of the layers were considered fixed (Table~\\ref{table:table1}) in computations and effective optical parameters of the $Ag$ and $A{l_2}{O_3}$ layers (refraction $n$ and absorption $k$ coefficients) were varied to reach the best agreement with the experimental spectrum. All parameters of the outer protective layer ($d = 12nm$) were also fixed according to Table~\\ref{table:table1} because it should have no electric field in it.\n\n\tComparing the experimental and theoretical curves allows the following conclusions. With no external field (Fig.~\\ref{fig:fig03}) the model is nearly perfect: the original values of optical parameters of the layers are identical to those given in the SOPRA data base. If there is an external field, the simulation needs closer consideration. If we assume that the optical properties of both $Ag$ layers keep the same in the presence of the electric field, the difference between theoretical and experimental data will increase with the field strength and we can't eliminate it by varying $n$ and $k$. Indeed, the theoretical curves in Fig.~\\ref{fig:fig04} built from this assumption agrees with the experiment poorly. It is clearly caused by the fact that the computations do not take into account nonlinear effects which can take place when the electric field strength is high. Surface charges caused by dielectric polarization and changes of  the numbers of electrons  in the metal layers should also be taken into consideration. The theory and experiment agree better if we assume that the properties of $Ag$ of the cathode and anode differ dramatically. Validation of such approach and corresponding simulation results are given in the next section.\n\n\\begin{figure*}\n\\includegraphics{fig04}\n\\caption{\\label{fig:fig04}SPR spectra for $U = 16 V$ and $U = 30 V$ (the marks are the experimental data, the solid line are the computation results). The theoretic curve  are built under assumption that the optical parameters of the both $Ag$ layers are the same. It is seen that these curves do not agree well with the experiment.}\n\\end{figure*}\n\n\\section{ The results of the simulation}\nThe difference between the theory and experiment can be eliminated if we suppose that the optical parameters of the upper (anode) and lower (cathode) $Ag$ layers can be different and must be optimized  separately. The separate optimization is possible under the assumption that charges accumulate at the dielectric-metal interface and the electron plasma density changes across the metal layers in a strong electric field.\n\n\tIndeed, when the dielectric layer is about $d_0\\sim200\\,nm$ thick and the applied voltage is nearly $30\\,V$, the surface charge density is $\\sigma  = U\/4\\pi \\varepsilon {d_0} \\sim 40CGSE$ (the permittivity of corundum $\\varepsilon \\sim 10$). If the layer area is $1\\;c{m^2}$, the number of excess electrons on one of the $Ag$ layer (and deficient ones on the other) is  $\\sim{10^{11}}$, while the full number of conduction electrons is $\\sim{10^{18}}$.\n\n\tLet us ask ourselves what the deficiency of electrons in the layer can result in. It is clear that the depth of the potential well for all electrons of the anode increases by $30\\;eV$. At the same time, we should take into account the changed band structure. The matter is that the additional attraction force from the uncompensated positive charge brings about a shift of the conduction band and, of course, the Fermi level ${E_F}$ down to a lower energy. Correspondingly, this results in the interband  absorption  threshold  ($\\sim4\\,eV$)  lowering. However, the shift of the d-electron band is much smaller because of  stronger binding to the nuclei.\n\nThe change of the interband absorption level changes the dielectric permittivity of the metal and, therefore, parameters $n$ and $k$. For instance, the model function $\\varepsilon (\\omega )$  is built for gold (see \\cite{Maier}) which allows for two interband transitions. The eleven  fitting parameters (resonance frequencies, resonance widths, complex amplitudes, asymptotic parameter, etc.) of the function allowed to reach a good agreement with the data \\cite{Etchegoin}. In the case of silver a similar function has not been built so far, so we limit ourselves to the rough estimation using formulae for gold \\cite{Maier}. A $0.1 - eV$ change of threshold $\\Delta $ results in an about unit change of the dielectric permittivity, which can be detected experimentally. Note, that the formula presented  in \\cite{Etchegoin2} describes well the permittivity of $Ag$ for the photon energies less than $3\\,eV$, which does not allow to analyze the influence of  the shift of  the interband  transition threshold on the permittivity value at $\\lambda  = 632.8nm$ ($1.96\\,eV$).\n\n\tAt the same time, the excess of electrons on the cathode raises the interband absorption level. Excess electrons do not leave the metal as long as the attraction caused by image charges exceeds the repulsion caused  by excess electrons. The condition of equilibrium of  these forces can be written as ${{e}^{2}}\/{{l}^{2}}=eU\/2d_0$, where $U\/2d_0$ is the field strength on the cathode (it is twice greater in the capacitor itself), $l$ is the electron-cathode distance. Then the saturation voltage is ${U_s} = 6 \\times {10^{ - 12}}\/{l^2}$. Let us suppose that slow electrons are scattered by defects whose concentration is about ${10^{18}}\\;c{m^{ - 3}}$. The average free path is about $10nm$ and saturation voltage is a few volts in this case. Further increase of $U$ leads to dynamic equilibrium, that is, the number of electrons arriving to the cathode in unit time is equal to the number of electrons leaking through the dielectric per unit time.\n\n\tThe above considerations suggest the necessity to optimize optical parameters of the upper (anode) and lower (cathode) $Ag$ layer independently. Indeed, the modelling using independent optimization of anode and cathode layers gives good agreement between theoretic and experimental curves. Figure~\\ref{fig:fig05} gives the results of simulation of an SPR angular spectrum for $U = 16\\,V$ and $U = 30\\,V$. Comparing Figure~\\ref{fig:fig04} and Figure~\\ref{fig:fig05} shows that the independent optimization allows a significantly better theory-experiment agreement. The computed values of optical properties for this kind of modelling are given in Table~\\ref{table:table2}. The first thing that draws attention is that the refractive index of the anode layer proves to be smaller than that of the cathode. This is an expected observation which agrees with measurements made during the deposition process and is explained by the fact that the deposition on a smooth substrate always gives a better quality of the $Ag$ layer than in deposition on a relatively loose intermediate layer. Secondly, with the growing voltage the $A{l_2}{O_3}$ layer starts exhibiting slight absorption and its refractive index decreases. Probably, it is caused by electrons arriving to this layer from the cathode. To avoid misunderstanding, we should point out that in modelling it is impossible to take into account surface irregularities, interpenetration of layers and non-linear effects caused by a strong electric field. That is why the values of $n$ and $k$ given in Table~\\ref{table:table2} should be regarded as certain effective values that allows for the drawbacks of this theoretical approach in a way.\n\n\\begin{figure*}\n\\includegraphics{fig05}\n\\caption{\\label{fig:fig05}SPR spectra for $U = 16 V$  and  $U = 30 V$ (marks \u2013 experiment, line \u2013 theory). The theoretic curves are built given separate optimizations of $Ag$  parameters on the anode and cathode.}\n\\end{figure*}\n\n\\begin{table*}\n\\caption{\\label{table:table2} The optical parameters of the MDM structure}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc}\n   & \n\\multicolumn{2}{c}{$Ag$ (anode), $d=49 nm$} &\n\\multicolumn{2}{c}{${Al_2}{O_3}$, $d=177 nm$} &\n\\multicolumn{2}{c}{$Ag$ (cathode), $d=36 nm$} \\\\\n$U(V)$ \t& $n$ \t\t& $k$ \t& $n$ \t& $k$ \t& $n$ \t& $k$ \\\\\n0  \t\t& 0.1360\t&4.0110\t&1.6908\t&0.0000\t&0.1341\t&4.0100\\\\\t\n4  \t\t& 0.1341\t&4.0109\t&1.6592\t&0.0022\t&0.1340\t&4.0100\\\\\n8  \t\t& 0.5928\t&4.2967\t&1.6593\t&0.0087\t&0.0881\t&3.4853\\\\\n12  \t& 0.5931\t&4.2969\t&1.6533\t&0.0116\t&0.0882\t&3.4852\\\\\n16  \t& 0.5932\t&4.2970\t&1.6531\t&0.0113\t&0.0882\t&3.4852\\\\\n20  \t& 0.3752\t&4.6909\t&1.5059\t&0.0583\t&0.0000\t&1.0263\\\\\n24  \t& 0.3717\t&4.6840\t&1.5064\t&0.0589\t&0.0000\t&1.0308\\\\\n28  \t& 0.3828\t&4.6885\t&1.5122\t&0.0622\t&0.0000\t&1.0619\\\\\n30  \t& 0.3778\t&4.7022\t&1.5161\t&0.0642\t&0.0000\t&1.0816\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\tThe voltage dependence of the optical parameters of MDM layers are presented in Figure~\\ref{fig:fig06}. It is seen that all the optical parameters of the $Ag$ layers experience noticeable changes at $U\\sim8V$ \u0438 and $U\\sim16V$. Similar irregular behaviour of the parameters is observed for the $A{l_2}{O_3}$ layers. Between these two voltage points parameters $n$ and $k$ keep fairly stable for all layers. The exclusion is the absorption coefficient of the $A{l_2}{O_3}$ layer which grows slightly on the interval  $U =20-30V$. It is interesting that the refractive index of the cathode $Ag$ layer falls to zero when $U > 16V$, which means that the dielectric permittivity becomes strictly negative and loses its imaginary component.\n\t\n\t\t\\begin{figure*}\n\\includegraphics{fig06}\n\\caption{\\label{fig:fig06}The refractive indices and absorptions coefficients versus voltage $U$. The top panels show the behaviour of the parameters of the cathode and anode $Ag$ layers, the bottom ones refer to the $A{l_2}{O_3}$ layers.}\n\\end{figure*}\n\n\tFollowing the changes of parameters $n$ and $k$, the reflection coefficient also changes sharply.  The most pronounced changes in the MDM structure reflection occur when the voltage exceeds $16V$. An experiment on this MDM structure showed  that the voltage increase  up to $U =20-30V$ results in a $4.2$-times decrease of the reflection coefficient at ${\\theta} = {42.9^0}$ and $5.4$-times increase at ${\\theta} = {55.8^0}$. Calculations show that the contrast of the reflectivity can be increased dramatically by varying the layer thicknesses slightly. For instance, if we increase the anode layer thickness to $54\\,nm$, we get a structure which at fixed angle ${\\theta} = {60.5^0}$ may be considered as a switcher: with zero reflection at $U=0$ and  with $20\\% $ reflectivity when $U$ increases to $30V$.\n\n\\section{Conclusions}\nWe have discovered significant changes of the dielectric permittivity of silver when a constant voltage of up to $30V$ is applied to a MDM-structure waveguide. It is shown that we can modulate the reflection coefficient widely by varying the voltage. Looking forward, the effect may be used in development of electrically controlled optical valves.\n\n\tWe could not so far find an acceptable theory that can explain the abrupt change of optical parameters of the MDM structure. This will be the goal of later research.\n\n\\begin{acknowledgments} \nThe work was supported by the project 2.1 of the RAS Presidium and RFBR grant 11-07-92470.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\pagestyle{plain}\n\nDuring the last two years new connections between integrable\nmodels and quantum groups were established. The representation\ntheory of quantum groups at roots of unit provides new solutions\nto the start--triangle equation increasing in this way the\nfamily of lattice integrable models. Most of these new solutions\ncan be obtained by applying the ``descendent''--procedure [1] to\nwell known models as for instance the 6--vertex model.\n\nGiven an $R$ matrix satisfying the Yang--Baxter equation the first\nstep in the descendent procedure is to find all possible\nsolutions to:\n\n\\begin{equation}\nR (\\lambda, \\mu) \\left[ L (\\lambda) \\otimes L (\\mu) \\right]  =\n\\left[ L (\\mu) \\otimes L (\\lambda) \\right] R (\\lambda, \\mu)\n\\label{1}\n\\end{equation}\n\nIf the $R$--matrix is, for instance, the quantum $R$ matrix of\n$\\widehat{SU (2)}_q$ in the 1\/2--representation then we should\nexpect the solutions of (1) to be in one to one correspondence with the\ndifferent irreps of the Hopf algebra $\\widehat{SU (2)}_q$.\nIt is in this way that the representation theory plays an\nimportant role in the discovery of new solutions to thestar--triangle equations\nby the descendent procedure [2]. The second\nstep of this method is to find for two given solutions\n$L^{\\rho_1}, L^{\\rho_2}$ of (1) a new $R$--matrix, solution to\nthe Yang--Baxter equation, and satisfying:\n\n\\begin{equation}\n{\\cal R}^{\\rho_1 \\rho_2} \\left( L^{\\rho_1} \\otimes L^{\\rho_2}\n\\right) = \\left( L^{\\rho_2} \\otimes L^{\\rho_1} \\right) {\\cal\nR}^{\\rho_1 \\rho_2}\n\\label{2}\n\\end{equation}\n\nIf we interpret $L^{\\rho_1}, L^{\\rho_2}$ as associated with two\ndifferent irreps $(\\rho_1, \\rho_2)$ of $\\widehat{SU (2)}_q$ the\nnew ${\\cal R}$--matrix will define the intertwiner between these\ntwo representations. The two steps of the descendent--procedure\nare graphically represented in fig. 1.\n\n\\vspace{4cm}\n\\begin{center}\nFig. 1. \\underline{The descendent--equations}\n\\end{center}\n\n\\vspace{0.3cm}\nThe simplest example of this procedure is the one known as\nfusion where starting, for instance, with the $R^{1\/2,\n1\/2}$--matrix of $\\widehat{SU (2)}_q$ we obtain the intertwiner\nfor regular representations with higher spins [3].\n\nThe case of quantum groups at roots of unity is a very\ninteresting example for the application of this technique. In\nthis case, the number of different finite dimensional irreps is\nmuch bigger as a consequence of the existence of new central\nelements in the Hopf algebra [4]. Hence for the $SU(2)_q$ case, the\ncentral Hopf subalgebra for $q = \\varepsilon$ an Nth--root of unit,\ncontains, in addition to the standard casimir, the new elements\n$E^N, F^N, K^N$. The different irreps can be divided into two\nmain sets: i) regular representations and ii)\ngeneric--representation. The last ones are classified into\ncyclic, with non vanishing eigenvalue for $E^N$ and $F^N$,\nsemicyclic with vanishing eigenvalue only for $E^N$ or $F^N$ but\nnot both and nilpotent for which $E^N = F^N = 0$ but $K^N$ having\ngeneric values. Starting with the $R$--matrix of $\\widehat{SU\n(2)}_q$ it is possible to get solutions to the descendent\nequations (1) and (2) associated with cyclic, semicyclic and\nnilpotent representations. The first case i.e. cyclic, was first\ndiscovered by Bazhanov and Stroganov [1] and the Kyoto group [5]. Surprisingly\nenough, the solution obtained coincides with the one\ncorresponding to the chiral Potts model [6]. These kinds of models are\nspecially important from a mathematical point of view. Their\nspectral manifold is a higher genus curve:\n\n\\begin{equation}\n\\Gamma_k \\; : \\; x^N + y^N = k (1 + x^N y^N)\n\\label{3}\\end{equation}\n\n\\noindent\nwith $k$ a parameter of the model (for instance $k=0$\ncorresponds to Fateev--Zamolodchikov model). Given two irreps\n$(\\xi_1, \\xi_2)$ of $\\widehat{SU (2)}_q$ with $q^N =1$ and\n$\\xi \\in Spec Z_q$, where $Z_q$ is the central Hopf subalgebra,\nthen the intertwining condition defines a submanifold in $Spec Z_q$\nwhich can be shown to be isomorphic to the product of two copies\nof the spectral curve $\\Gamma_k$ of the chiral Potts model.\nMoreover the solution $R(\\xi_1 \\xi_2)$ to equation (2)\nfactorizes into four pieces which can be represented in terms of\nthe Boltzman weights $W, \\bar{W}$ of the chiral Potts model [5].\n\nThe new solution which we would like to discuss in this lecture\ncorresponds to the case of semicyclic and nilpotent\nrepresentations. The model defined in this way shares some\nsimilarities with chiral Potts and with the Heisenberg--Ising models with\nhigher spin. However, it presents some new features which we now move\non to present.\n\n\\section{The semi--cyclic descendent}\n\nIn a previous paper [7] we have presented the $R$ matrix solution\nto the Yang--Baxter and the intertwiner conditions:\n\n$$\n(1 \\otimes R (\\xi_1 \\xi_2)) (R (\\xi_1 \\xi_3) \\otimes 1) (1\n\\otimes R (\\xi_2 \\xi_3)) =\n\\eqno{(3.a)}\n$$\n\n\\[\n= (R (\\xi_2 \\xi_3) \\otimes 1) (1 \\otimes R(\\xi_1 \\xi_3)) (R\n(\\xi_1 \\xi_2) \\otimes 1)\n\\]\n\n$$\nR (\\xi_1 \\xi_2) \\Delta_{\\xi_1 \\xi_2} (a) = \\Delta_{\\xi_2 \\xi_1}\n(a) R (\\xi_1 \\xi_2)\n\\eqno{(3.b)}\n$$\n\n\\[\na \\in SU (2)_q \\; \\; \\; q^N = 1\n\\]\n\n\\noindent\nfor $\\xi_1, \\xi_2$ semicyclic representations of $SU(2)_q$.\nDenoting by $x, y, \\lambda^N$ the eigenvalues of $E^N, F^N$ and\n$K^N$ respectively the semicyclic representations are\nparametrized by $(y_1, \\lambda_1)$, $(y_2, \\lambda_2)$. Using (3.b)\nfor $a$ in the center, we get the intertwining constraints on\nthe values of $\\xi_1, \\xi_2$:\n\\begin{equation}\n\\frac{y_1}{1 - \\lambda^N_1} = \\frac{y_2}{1 - \\lambda^N_2} = \\chi\n\\label{4}\n\\end{equation}\n\n\\noindent\nwith $\\chi$ an arbitrary complex number. The solution $R (\\xi_1\n\\xi_2)$ satisfies:\n\n\\begin{equation}\n\\begin{array}{rl}\ni) & R (\\xi \\xi) = 1 \\\\\nii) & R (\\xi_1 \\xi_2) R (\\xi_2 \\xi_1) = 1 \\otimes 1 \\\\\niii) & R (\\xi_1 \\xi_2) = P R (\\xi_2 \\xi_1) P\n\\end{array}\n\\label{5}\n\\end{equation}\n\n\\noindent\nwhere $P$ is the permutation operator.\n\nFor more details on this solution see reference [7].\n\nFollowing the spirit of the descendent--technology we proceed\nnow to find solutions to equation (1), for $R$ the quantum\n$R$--matrix of $\\widehat{SU(2)}_q$ in the 1\/2--representation\ni.e. the six vertex model, which can be associated with\nsemicyclic representations of $\\widehat{SU(2)}_q$ at $q =\n\\varepsilon$ an Nth--root of unit (we shall consider $N$ to be\nan odd integer). We will first consider the simplest\nnon trivial case:  $x = y = 0$ and $\\lambda$ generic (the\nregular representations correspond to $\\lambda =\n\\varepsilon^{2s}$ with $s$ integer or half integer spin).\n\nThe 6--vertex $R$--matrix is given by [3]:\n\n\\begin{equation}\nR^{1\/2, 1\/2} (u) = sh \\left[ u + i\\gamma \\frac{1}{2} (1 + \\sigma^3\n\\otimes \\sigma^3) \\right] + i \\; {\\rm sin\\gamma} \\; (\\sigma^+ \\otimes\n\\sigma^- + \\sigma^- \\otimes \\sigma^+)\n\\label{6}\n\\end{equation}\n\nEquation (1) reads now:\n\n\\begin{equation}\nR^{1\/2, 1\/2} (u-v) (L (u) \\otimes L(v)) = (L (v) \\otimes L(u))\nR^{1\/2, 1\/2} (u-v)\n\\label{7}\n\\end{equation}\n\nThe solutions, we are interested in, are $L^{(\\lambda)} (u)$--matrices\nacting on $V^{1\/2} \\otimes V^\\lambda$ such that:\n\n\\begin{equation}\nL^{(\\lambda)} (u) : V^{1\/2} \\otimes V^\\lambda \\rightarrowV^{1\/2} \\otimes\nV^{\\lambda}\n\\label{8}\n\\end{equation}\n\n\\noindent\nwhere $V^\\lambda$ is the irrep of $\\widehat{SU\n(2)}_\\varepsilon$ with eigenvalue of $K^N = \\lambda^N$ and where\nthe tensor product in (7) is defined with respect to the\n$V^{1\/2}$--indices. Notice that the $L^{(\\lambda)} (u)$--matrices we\nare looking for define the intertwiners between the irreps 1\/2\nand $\\lambda$. In matrix notation the solution is given by:\n\n$$\nL^{(\\lambda)} (u) = \\frac{1}{e^u \\varepsilon^{1\/2} \\lambda^{1\/2}\n- e^{-u} \\varepsilon^{-1\/2} \\lambda^{-1\/2}} \\times\n$$\n\n\\begin{equation}\n\\times \\left( \\begin{array}{cc}\ne^u \\varepsilon^{1\/2} K^{1\/2}\n- e^{-u} \\varepsilon^{-1\/2} K^{-1\/2} & (\\varepsilon -\n\\varepsilon^{-1}) \\varepsilon^{-1\/2} FK^{1\/2} \\\\\n(\\varepsilon- \\varepsilon^{-1}) \\varepsilon^{-1\/2} E K^{-1\/2} &\ne^u \\varepsilon^{1\/2} K^{-1\/2}\n- e^{-u} \\varepsilon^{-1\/2} K^{1\/2} \\end{array} \\right)\n\\label{9}\n\\end{equation}\n\n\n\\noindent\nwith  $E, K, F$ the generators of $SU(2)_q$ which satisfy the relations:\n\n\\[\nEK = \\varepsilon^2 K E \\; \\; , \\; \\; FK = \\varepsilon^{-2} KF\n\\]\n\n\\begin{equation}\n[E, K] = \\frac{K - K^{-1}}{\\varepsilon - \\varepsilon^{-1}}\n\\label{10}\n\\end{equation}\n\nThe representation $V^\\lambda$ is defined in the basis $\\{ |r> \\}_{r=0}^{N-1}$\n\n\\begin{equation}\n\\begin{array}{l}\nE | r> = d_{r-1} | r-1> \\\\\nF | r> = d_{r} | r+1> \\\\\nK | r> = \\lambda \\varepsilon^{-2r} | r>\n\\end{array}\n\\label{11}\n\\end{equation}\n\n\\[\nd^2_j (\\lambda) = [j+1] \\frac{\\lambda \\varepsilon^{-j} -\n\\lambda^{-1} \\varepsilon^j}{\\varepsilon - \\varepsilon^{-1}}\n\\]\n\\[\n[ x] = \\frac{\\varepsilon^x - \\varepsilon^{-x}}{\\varepsilon - \\varepsilon^{-1}}\n\\]\n\nFor a graphic representation of the matrix $L^{(\\lambda)}(u)$ see\nFig. 2. Notice that each entry in the 2 x 2 matrix (9)\nrepresents an operator acting on the space $V^\\lambda$.\n\n\\vspace{5cm}\n\\begin{center}\nFig. 2.  Graphic representation of the $L^{(\\lambda)}(u)$--matrix\n\\end{center}\n\n\\vspace{1cm}\nThe second step in the descendent procedure corresponds to\nfinding the $R$--matrix solution to the equation (2):\n\n\\begin{equation}\n{\\cal R}^{\\lambda_1 \\lambda_2} (u-v) \\left( L^{(\\lambda_1)} (u)\n\\otimes L^{(\\lambda_2)} (v) \\right) = \\left( L^{(\\lambda_2)} (v)\n\\otimes L^{(\\lambda_1)} (u) \\right) R^{\\lambda_1 \\lambda_2} (u-v)\n\\label{12}\n\\end{equation}\n\n\\noindent\nwhere now the tensor product is defined with respect\nto the \\underline{$V^{\\lambda_1}$ - $V^{\\lambda_2}$} \\underline{indices} of the\n$L^{(\\lambda)}$ (u) - matrices (see fig. 1 B). The solution to\n(12) defines the intertwiner between the representations\n$\\lambda_1, \\lambda_2$ and it is given by:\n\n\\[\nR^{\\lambda_1 \\lambda_2} (u)^{l, r_1 + r_2 - l}_{r_1 r_2} =\n\\frac{\\epsilon^{(r_1 + r_2 - l) l - r_1\nr_2}}{\\prod^{r_1+r_2-1}_{j=0} (e^u \\lambda_1 \\lambda_2\n\\epsilon^{-j} - e^{-u} \\epsilon^j)} \\times\n\\]\n\n\\[\n\\times {\\sum^{r_1}_{l_1 = 0}\n\\sum^{r_2}_{l_2 = 0}} \\left[ \\begin{array}{c} r_1\n\\\\ l_1 \\end{array} \\right] \\left[ \\begin{array}{c} r_2 \\\\ l_2\n\\end{array} \\right] \\frac{[l] ! [r_2 - l_2] !}{[r_1 + l_2] !\n[r_2]!} (\\epsilon - \\epsilon^{-1})^{r_1 - l_1 +l_2}\n\\]\n\n\\[\n\\times \\prod^{r_1 + l_2 -1}_{j= r_1} d_j (\\lambda_1) \\prod^{r_1\n+l_2-1}_{j=l_1 +l_2} d_j (\\lambda_1) \\prod^{r_2\n- 1}_{j=r_2 - l_2} d_j (\\lambda_2) \\prod^{r_1+r_2\n-l-1}_{j=r_2 -l_2}\n\\]\n\n\\begin{equation}\\times \\lambda^{l_2}_1 \\lambda^{r_1-l_1}_2 \\prod^{r_2 -l_2\n-1}_{j=0} (e^u \\lambda_2 \\epsilon^{-j} - e^{-u} \\lambda_1\n\\epsilon^j) \\prod^{l_1 -1}_{j=0} (e^u \\lambda_1 \\epsilon^{-j+r_2-l_2}\n-e^{-u} \\lambda_2\n\\epsilon^{j+l_2 -r_2})\n\\label{13}\n\\end{equation}\n\n\\noindent\nwith the following conventions: a) whenever in above products the upper index\nis less than the lower index the result is one.\nb) the constraint $l_1 + l_2 = l$ must be used to carry out summation.\n\nIt is easy to check that this solution coincides in the case of\n$u=0$ with the $R^{\\xi_1 \\xi_2}$--matrix of reference [7] for\n$\\xi_1, \\xi_2$ two semi--cyclic irreps with $x=y=0$ and $K^N =\n\\lambda^N_1, \\lambda^N_2$. Moreover  for $\\lambda_1 = \\lambda_2\n= \\epsilon^{2s}$ eqn. (13) gives us the spin $s - R$ matrix.\n\nSummarizing, we have obtained starting with the $R$--matrix of\nthe six vertex model for $q$ an Nth-root of unit a class of\ndescendent models characterized by the quantum $R$--matrix (13).\nThe transfer matrix of these models can be defined for periodic\nboundary conditions as follows (see fig. 3):\n\n\\begin{equation}\nT_{\\lambda_0} (u, \\lambda) : \\otimes^L V^{\\lambda_0}\n\\rightarrow\n\\otimes^L V^{\\lambda_0}\n\\label{14}\n\\end{equation}\n\n\\begin{equation}\n\\langle r'_1 \\cdots r'_N | T_{\\lambda_0} (u, \\lambda) | r_1,\nr_2 \\cdots r_N \\rangle = \\left( \\sum_{l's} R^{\\lambda,\n\\lambda_0} (u)_{l_1 r_1}^{l_2 r'_1} R^{\\lambda \\lambda_0}\n(u)^{l_3 r'_2}_{l_2 r_2} \\cdots \\right)\n\\label{15}\n\\end{equation}\n\n\\noindent\nwhere $\\lambda_0$ appears as a parameter characterizing the model\nand $u$ and $\\lambda$ as spectral variables. The Yang-Baxter\nrelation for the $R$--matrix (eq. 13) implies the integrability equation:\n\n\\begin{equation}\n\\left[ T_{\\lambda_0} (u, \\lambda'), T_{\\lambda_0} (u', \\lambda'') \\right]\n= 0\n\\label{16}\n\\end{equation}\n\nEquation (16) is the standard integrability condition for\nsoluble models with the important peculiarity that the spectral\nvariables are now living on a manifold of complex dimension\nequal 2. Notice that for the kind of irreps we are considering\n(i.e. with vanishing eigenvalues of $E^N$ and $F^N$ and generic\neigenvalue of K) the intertwining condition is not imposing any\nconstraint on the allowed values of $\\lambda$.\n\n\\[(\\lambda'_1 u) \\begin{array}{cc|cc|cc|cc|cc|ccc}\n& & r'_1 & & r'_2 & & r'_3 & & r'_4 \\cdots & & r'_L & & \\\\\n\\hline\nl_1 & & &l_2  & & l_3  & & l_4  & & l_5 \\cdots  & e 1  & \\\\\n& & r_1  & & r_2  & &  r_3 & & r_4 \\cdots & & r_L & | r_1 \\cdots\nr_L> \\\\\n\\end{array}\n\\overbrace{\\otimes^L V^{\\lambda_0}}\n\\]\n\n\\begin{center}\n$L = \\#$ sites in the row.\n\nFig. 3: The transfer matrix $T_{\\lambda_0} (\\lambda, u)$\n\\end{center}\n\n\\noindent\n\n\\section*{3. The associated 1D--chain}\n\nGiven the transfer matrix (14-15) we can define a local 1D\nhamiltonian as follows:\n\n\\begin{equation}\n\\left. H^{(\\lambda_0)} = i \\frac{\\partial}{\\partial u}  l n\nT_{\\lambda_0} (u, \\lambda_0) \\right|_{u= 0}\n\\label{17}\n\\end{equation}\n\n\\noindent\nwhich will be hermitian for certain allowed regions of\n$\\lambda_0$. This hamiltonian defines a one dimensional\nspin chain with the Hilbert space given by:\n\n\\begin{equation}\n{\\cal H} = \\otimes^L V^{\\lambda_0}\n\\label{18}\n\\end{equation}\n\nThe most natural interpretation of this spin--chain is as a\ngeneralization to continuous ``spin\" of the higher spin quantum\nchains defined in references [8]. In fact, as we have\nalready mentioned for the special case $\\lambda_0 =\n\\epsilon^{2S_{max}} (2S_{max} +1 = N)$ the hamiltonian (16)\ncoincides with the hamiltonian $H^{S_{max}}_\\epsilon$ of spin\n$S_{max}$ and anisotropy depending of $\\epsilon$.\n\nThe two main new features of the model defined by the transfer\nmatrix (14) are:\n\n\\begin{itemize}\n\\item[i)] All the $V^{\\lambda_0}$ representations are of the\nsame dimensions $N$. This is specific of working with the\nquantum group at roots of unit.\n\\item[ii)] The integrability condition (16) with two independent\nspectral parameters.\n\\end{itemize}\n\nThe main consequence of (16) for the spin chain defined by (16)\nis the existence of a local operator $Q^{(\\lambda_0)}$ defined by:\n\n\n\\begin{equation}\nQ^{(\\lambda_0)} = \\left. 2 \\lambda \\frac{\\partial}{\\partial \\lambda} l n\nT_{\\lambda_0} (u = 0,\\lambda) \\right|_{\\lambda = \\lambda_0}\n\\label{19}\n\\end{equation}\n\n\\noindent\nsuch that:\n\n\\begin{equation}\n\\left[ H^{\\lambda_0}, Q^{\\lambda_0} \\right] = 0\n\\label{20}\n\\end{equation}\n\nDenoting $\\xi \\equiv (u, \\lambda)$, the transfer matrix (14) can\nbe written as:\n\n\\begin{equation}\nT_{\\xi, \\xi'} = T_{(u, \\lambda)|(u'\\lambda_0)} \\equiv\nT_{\\lambda_0} (u - u', \\lambda)\n\\label{21}\n\\end{equation}\n\n\\noindent\nwhere we have used the difference property with respect to the\nspectral variable $u$. The most general local hamiltonian we can\ndefine is:\n\n\\begin{equation}\n{\\cal H} = \\frac{d}{dt} \\left. ln T_{\\xi, \\xi'(t)}\n\\right|_{t= 0}\n\\label{22}\n\\end{equation}\n\n\\noindent\nwith $\\lim_{t\n\\rightarrow 0} \\xi'(t) =  \\xi$. This definition of the hamiltonian would\nbe, in spirit, very close to the way a 1D hamiltonian is defined\nfor the chiral Potts model. However, it is clear that in our case\nany Hamiltonian obtained as indicated by eqn. (22) will be a\nlinear combination of $H^{(\\lambda_0)}$ and $Q^{(\\lambda_0)}$.\nBefore entering into the detailed study of the 1D hamiltonians, it\nis worthwhile to discuss the differences between the models we\nare defining and the chiral Potts. The model which we can compare\nwith is the Fateev--Zamolodchikov model. The way the cyclicity\nof the irreps, entering into the definition of this model, is reflected\nin the physics is through the $Z(N)$--invariance. In fact, the spectrum of\nthe hamiltonian decomposes in different sectors with well\ndefined $Z(N)$--charge.\n\nIn the noncyclic case, we are presenting here, there exists a well\ndefined ``reference state\" $| \\Omega_0 > \\in \\otimes^L V^{\\lambda_0}$:\n\n\\begin{equation}\n| \\Omega_0 > = | 0,0 \\cdots 0 >\n\\label{23}\n\\end{equation}\n\nRecall that the states in $\\otimes^L V^\\lambda$ are given by $|r_1\nr_2 \\cdots r_L>$ with $r_i = 0,\\cdots,N-1$. The reference state\nis the natural generalization of the ferromagnetic state with\nall the spins up. The conserved number, as in the case of the\nHeisenberg chains, is the total number of spins ``down'' which in our\ncase is given by:\n\n\\begin{equation}\n\\sum^L_{i=1} r_i = \\rho.\n\\label{24}\n\\end{equation}\n\nDifferent sectors correspond to different values $\\rho$. In the\nchiral Potts model, i.e. cyclic case, there is no good\ndefinition of ``reference state\" and the conserved quantum\nnumber is given by the $Z(N)$ charge defined by:\n\n\\begin{equation}\nX = e^{2 i\\pi Q\/N}.\n\\label{25}\n\\end{equation}\n\n\\noindent\nwhere\n\n\\begin{equation}\nX | r_1 \\cdots r_L >= |r_1 +1 \\cdots r_2 +1>.\n\\label{26}\n\\end{equation}\n\nIn the next section and using the fact that we have a good\nreference state we will proceed to diagonalize the hamiltonian\n$H^{\\lambda_0}$ by using the Bethe Ansatz technique.\n\n\\section*{4. Bethe Ansatz Equations}\n\nIn terms of the $L^{\\lambda_0} (u)$--matrices we define the\nmonodromy matrix by:\n\n\\begin{equation}\nt_{\\lambda_0} (u) = L^{\\lambda_0}_L (u) \\cdots L^{\\lambda_0}_2\nL^{\\lambda_0}_1 (u)\\label{27}\n\\end{equation}\n\n\\noindent\nwith $L$ the length of the row in lattice units. The monodromy\nmatrix $t_{\\lambda_0}(u)$ satisfies:\n\n\\begin{equation}\nR^{1\/2, 1\/2} (u - v) t_{\\lambda_0} (u) t_{\\lambda_0} (v)\n= t_{\\lambda_0} (v) t_{\\lambda_0} (u) R^{1\/2, 1\/2} (u - v)\n\\label{28}\n\\end{equation}\n\nRepresenting it as a 2x2 matrix:\n\n\\begin{equation}\nt_{\\lambda_0} (u) = \\left[ \\begin{array}{cc} A_{\\lambda_0} (u) &\nB_{\\lambda_0} (u) \\\\\nC_{\\lambda_0} (u) & D_{\\lambda_0} (u) \\end{array} \\right]\n\\label{29}\n\\end{equation}\n\n\\noindent\nwe obtain the operators $B_{\\lambda_0} (u) : \\otimes^L\nV^{\\lambda_0} \\rightarrow \\otimes^L V^{\\lambda_0}$ which can be\ninterpreted as creating on the reference state (23) an\n``elementary excitation\". Notice that:\n\n\\begin{equation}\nB_{\\lambda_0} (u_1) B_{\\lambda_0} (u_2) = B_{\\lambda_0} (u_2)\nB_{\\lambda_0} (u_1)\n\\label{30}\n\\end{equation}\n\n$$\nC_{\\lambda_0} (u) | \\Omega_0 > = 0.\n\\eqno{(30;b)}\n$$\n\nTo diagonalize the transfer matrix (13) we use the standard\nalgebraic Bethe Ansatz:\n\n\\begin{equation}\n|\\psi> = \\prod^M_{i=1} B_{\\lambda_0} (u_i) | \\Omega_0 >\n\\label{31}\n\\end{equation}\n\nThe number $M$ of ``excitations\" is for our model a conserved\nquantum number and therefore we can diagonalize the transfer\nmatrix in each sector.\n\nUsing equation (9) it is easy to find the eigenvalues $\\Lambda\n(\\lambda_0, \\lambda, u; \\{ u_i \\}_{i=1, \\cdots, M})$ of the\ntransfer matrix T in each sector:\n\n\\begin{equation}T_{\\lambda_0} (u, \\lambda) |\\psi> = \\Lambda (\\lambda_0,\n\\lambda,\nu; \\{u_i \\}) |\\psi > +\\;unwanted\\;terms.\n\\label{32}\n\\end{equation}\n\n\\noindent\nwith\n\n\\begin{equation}\n\\Lambda (\\lambda_0, \\lambda,\nu; \\{u_i \\}) = \\sum^{N-1}_{r=0} (R^{r0}_{r0} (\\lambda,\n\\lambda', u))^L \\prod^M_{i=1} {\\cal L}^{\\lambda'}_r (u - u_i)\n\\label{33}\n\\end{equation}\n\n\\noindent\nwhere the $R$ matrix is the one given in (12) and\n${\\cal L}^{\\lambda'}_r (u-u_i)$ is:\n\n\\begin{equation}\n{\\cal L}^{\\lambda'}_r (u)= \\frac{(\\epsilon^u \\epsilon^{1\/2}\n\\lambda^{\\prime 1\/2} - e^{-u} \\epsilon^{-1\/2} \\lambda'^{-1\/2}) (e^u\n\\epsilon^{-1\/2} \\lambda'^{-1\/2} - e^{-u} \\epsilon^{1\/2}\n\\lambda'^{1\/2})}{(e^u \\epsilon^{1\/2 - r} \\lambda'^{1\/2} - e^{-u}\n\\epsilon^{r-1\/2} \\lambda'^{-1\/2}) (e^u \\epsilon^{-r-1\/2}\n\\lambda'^{1\/2} - e^{-u} \\epsilon^{r+1\/2} \\lambda'^{-1\/2})}\n\\label{34}\n\\end{equation}\n\nTo fix the values of the ``rapidities'' $u_i$ we must eliminate\nthe unwanted terms in (32). This can be done using equation (28)\nor more easily imposing that the residues at the poles both in\n$u$ and $\\lambda$ of (33) vanish. The result is:\n\n\\begin{equation}\n\\left( \\frac{e^{u_j} \\epsilon^{1\/2} \\lambda^{1\/2}_0 - e^{-u_j}\n\\epsilon^{-1\/2} \\lambda^{-1\/2}_0}{e^{u_j} \\epsilon^{1\/2}\n\\lambda^{-1\/2}_0 - e^{-u_j} \\epsilon^{-1\/2} \\lambda^{1\/2}_0}\n\\right)^L = \\prod^{M}_{\\begin{array}{c} k=1 \\\\ k \\neq j\n\\end{array}} \\frac{sh (u_j - u_k + i \\gamma)}{sh (u_j - u_k - i \\gamma)}\n\\label{35}\n\\end{equation}\n\n\\noindent\nwhere $\\epsilon = e^{i \\gamma}$. In the case $\\lambda_0 =\n\\epsilon^{2s}$ with 2s being an integer, i.e. regular representations of spin\ns, equations (35) become the Bethe Ansatz equations for the higher spin\nHeisenberg--Ising chains of references [8].\n\n{}From (33) we can find the eigenvalues of the operators\n$H^{\\lambda_0}$ and $Q^{\\lambda_0}$ defined in the previous section:\n\n\\begin{equation}\nE^{\\lambda_0} = 2i \\sum^M_{j=1} \\frac{\\lambda_0 -\n\\lambda^{-1}_0}{(e^{u_j} \\varepsilon^{1\/2} \\lambda^{1\/2}_0 -e^{-u_j}\n\\varepsilon^{-1\/2} \\lambda^{-1\/2}_0 ) (e^{u_j}\n\\varepsilon^{1\/2} \\lambda^{-1\/2}_0 - e^{-u_j} \\varepsilon^{-1\/2}\n\\lambda^{1\/2}_0)}\n\\label{36}\n\\end{equation}\n\n\\begin{equation}\nQ^{\\lambda_0} = 2 \\sum^M_{j=1}\n\\frac{e^{2u_j} \\varepsilon - e^{-2u_j} \\varepsilon^{-1}}{(e^{u_j}\n\\varepsilon^{1\/2} \\lambda^{1\/2}_0 -\ne^{-u_j} \\varepsilon^{-1\/2} \\lambda^{-1\/2}_0 ) (e^{u_j}\n\\varepsilon^{1\/2} \\lambda^{-1\/2}_0 - e^{-u_j} \\varepsilon^{-1\/2}\n\\lambda^{1\/2}_0)}\n\\label{37}\n\\end{equation}\n\nFor the intermediate case corresponding to $\\lambda_0 =\n\\varepsilon^{2s}$ with arbitrary $s$, the equations (36) (37) become:\n\n\\begin{equation}\nE^s = - \\sum^M_{j=1} \\frac{\\sin (2 \\gamma s)}{sh [\n\\frac{\\gamma}{2} (\\alpha_j + 2 is)] sh [ \\frac{\\gamma}{2} (\\alpha_j -\n2 is)]}\n\\label{38}\n\\end{equation}\n\n\\begin{equation}\nQ^s =  \\sum^M_{j=1} \\frac{s h ( \\gamma \\alpha_j)}{sh [\n\\frac{\\gamma}{2} (\\alpha_j + 2 is)] sh [ \\frac{\\gamma}{2} (d_j -\n2 is)]}\n\\label{39}\n\\end{equation}\n\n\\noindent\nwhere we have introduced the new variable $\\alpha$ defined by\n\n\\begin{equation}\nu + \\frac{i}{2} \\gamma = \\frac{\\gamma}{2} \\alpha\n\\label{40}\n\\end{equation}\n\nNow we briefly discuss the case of the nonvanishing $\\chi$, defined by eqn.(4).\nThe first observation to be made is that the number of excitations\n``$M$'', introduced in (31), is no longer ``good quantum number\".\nWhat replaces it is an eigenvalue of $\\Delta K$ (which is $\\lambda^L_0\ne^{i\\frac{2 \\pi Q}{N}}$ for $Q=0, 1, \\cdots N-1)$ since this\noperator commutes with both transfer matrices, namely\n\n\\[\n\\Delta K = K^{\\lambda_0}_L \\cdots K_2^{\\lambda_0} K_1^{\\lambda_0}\n\\]\n\n\\begin{equation}[ \\Delta K, tr t_{\\lambda_0} (u)] = [\\Delta K, T_{\\lambda_0}\n(u,\n\\lambda) ] =0\n\\label{41}\n\\end{equation}\n\nSurprisingly enough, some nice features of $\\chi =0$ case\nsurvive. In particular inspecting formula (13), one concludes\nthat R--matrix preserves its low--triangular form while acting\non the ``reference state\" (23). Therefore, this state remains an\neigenvector of $T_{\\lambda_0} (u_1 \\lambda)$. More precisely\n\n\\begin{equation}\nT_{\\lambda_0} (u, \\lambda) | \\Omega_0 > = \\sum^{N-1}_{r=0} (R^{r0}_{r0}\n(\\lambda_0, \\lambda, u))^L | \\Omega_0 >\n\\label{42}\n\\end{equation}\n\n\\noindent\nWhat makes $\\chi \\neq 0$ situation somewhat more complicated is\nthe fact that the commutation relations for $A_{\\lambda_0},\nB_{\\lambda_0}, C_{\\lambda_0}, D_{\\lambda_0}$ and the elements of\n$T_{\\lambda_0}(u,\\lambda)$ no longer have simple form and\ntherefore, one does not expect for the state (31) to be an\neigenvector of $T_{\\lambda_0} (u, \\lambda)$. Consequently,\nthe appropriate generalization of Algebraic Bethe Ansatz is\ncalled for in this case.\n\nTo get the feeling of what transpires, one should note that the\nstate (31) is still an eigenvector of $tr t_{\\lambda_0} (u)$\n\n\\begin{equation}\ntr t_{\\lambda_0} (u) | \\psi_M > = \\sum^1_{r=0} \\left( L^{\\lambda_0}\n(u)^{r0}_{r0} \\right)^L\n\\prod^M_{i=1} R^{1\/2, 1\/2} ( (-1)^r (u_i - u) )^{01}_{10}\n| \\psi_M >\n\\label{43}\n\\end{equation}\n\nIndeed, in deriving eqn.(43) above, one only uses 6--vertex\ncommutation relations (28) and the degeneracy eqn.(30;b) for the\noperator $C_{\\lambda_0} (u)$. Since neither eqn.(28) nor eqn.(30;b)\nare affected by turning on nonzero $\\chi$, one could infer that\neqn.(43) should hold true. Thus, it appears that going from $\\chi\n=0$ to $\\chi \\neq0$ case produces no visible effect on the\nspectrum of the transfer matrix $tr t_{\\lambda_0} (u)$. This\nconclusion, however, may be a bit premature. To see that, let us\nrecall, that not all the solutions of (35), generally speaking,\ncorrespond to non-zero vectors of the form (31).\n\nQuite frequently an additional investigation (involving delicate\nlimiting procedures) is required to determine the fate of the\nparticular solution. In general, the answer to {\\it ``To Be Or\nNot To Be}\" question may depend on whether or not $\\chi$ takes\non the zero value.\nLet us now generate \\underline{finite--dimensional} vector space\n$V_M (Q = M - int [ \\frac{M}{N}])$ by premultiplying non-zero\nvector $| \\psi_M >$ by any sum of products of operators\n$T_{\\lambda_0} (u, \\lambda), T_{\\lambda_0} (u', \\lambda')\n\\cdots$ for any $u, \\lambda; u', \\lambda'; \\cdots$ (but all\nwith the same $\\lambda_0$ and $\\chi$). It is clear, that in this\nspace one can diagonalize simultaneously the whole family of\ncommuting transfer--matrices $T_{\\lambda_0} (u, \\lambda)$.\nRecalling that $tr t_{\\lambda_0} (u)$ commutes with\n$T_{\\lambda_0} (u, \\lambda)$, one immediately arrives to the conclusion\nthat the eigenvectors of family $T_{\\lambda_0} (u, \\lambda)$\ncan be constructed as linear combinations of vectors (31), all\nof the same length (modulo N) and eigenvalue of transfer--matrix\n$tr t_{\\lambda_0} (u)$. Apparently, what seems to be going on is\nas follows: The eigenvalues of transfer matrices $T_{\\lambda_0}\n(u, \\lambda)$ and  $tr t_{\\lambda_0} (u)$ are highly degenerate\nwhen $\\chi = 0$. Turning on finite $\\chi$ results in lifting this\ndegeneracy for $T_{\\lambda_0} (u, \\lambda)$ and changing\nmultiplicities of eigenvalues of $tr t_{\\lambda_0} (u)$. This is\nvery intriguing phenomena and certainly warrants further\ninvestigations. To summarize, we can still characterize\neigenstates of commuting family $T_{\\lambda_0} (u, \\lambda)$\nby the set of Bethe Ansatz roots (35), abandoning, however, the\nsimple representation for Bethe vector given by formula (31). The\ngeneralization of algebraic Bethe Ansatz, briefly sketched\nabove, is similar in spirit to Tarasov's proposal for\nsuperintegrable chiral Potts Model [9].\n\nThis similarity is, by no means, accidental. Indeed, chiral Potts\nmodel on superintegrable line shares an important property of\nsemi--cyclic case: The L--matrix which intertwines spin 1\/2 and\ncyclic repr. has ``top\" reference state but no bottom state.\nWhether this similitude has a deeper implication remains to be seen.\n\nWhen algebraic Bethe Ansatz does not work (or does not provide\nthe shortest route), one may resort to the alternative procedure\nin order to solve for eigenvalues of Hamiltonian and transfer\nmatrix. This procedure, known as ``Functional Relations\"\nmethod, was exploited quite successfully in recent years to\nfind the spectrum of RSOS [10] and chiral Potts models [11].\nThe gist of this approach can be described in a few words as\nfollows: One keeps on fusing various R--matrices (related to the\nmodel under investigation) throwing in some obvious (and not so\nobvious) symmetries along the way, until one comes back to where\njourney began. The result is the system of functional equations\nfor the transfer matrices. In a sense, one can regard this\nprocedure as a generalization of Zamolodchikov--Karowski\nBootstrap program [12] to determine S-matrices of exactly\nintegrable models.\n\nIn order to apply this technique to our model, let us recall\nthat the tensor product of a spin $j$ representation with a\nsemi--cyclic one is completely reducible. In particular,\nfor $j= \\frac{1}{2}$, we have (see reference [14])\n\\begin{equation}\n( \\frac{1}{2} ) \\otimes (y, \\lambda) = ( y, \\epsilon \\lambda)_+\n+ (y, \\epsilon^{-1} \\lambda)_-\n\\label{44}\n\\end{equation}\n\nIn each subspace one can find the highest weight vector $V^+_0$\nand $V^-_0$ such that\n\n\\begin{equation}\n\\Delta(E) V^{\\pm}_0 = 0 \\;\\;and\\;\\;\\Delta (K) V^\\pm_0 =\n\\epsilon^{\\pm1} \\lambda V^\\pm_0\n\\label{45}\n\\end{equation}\n\nMaking use of (11) we obtain for each sign a semi--cyclic\nrepresentation with the basis $[ V^\\pm_0 , \\cdots, V^\\pm_{N-1} ].$\n\nLet us now define two projection points $u_\\pm$ as follows:\n\n\\begin{equation}\nL^{\\lambda_0} (u_\\mp) V^\\pm_i = 0\n\\label{46}\n\\end{equation}\n\nAt these points the operators $L^{\\lambda_0}$ becomes\nessentially the projector $P^\\pm$ onto corresponding $\\pm$ subspace.\n\n\\begin{equation}\nL^{\\lambda_0} (u_{\\pm}) \\sim P^\\pm;\\;\\;P^+ + P^- = 1\\;\\;and\\;\\;P^+P^-\n= P^- P^+ = 0\n\\label{47}\n\\end{equation}\n\nIt follows then from eqn. (12) that\n\n\\[\nP^- [ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u\n- u_-)] = \\cdots [ R^{\\lambda_1, \\lambda_0} (u - u_-) \\otimes\nL^{\\lambda_0} (u)] \\cdots P^-\n\\]\n\n\\noindent\nand\n\n\\begin{equation}\nP^- [ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u - u_-)\n] P^+ = 0\n\\label{48}\n\\end{equation}\n\nThe relation (48) above reveals a block--triangular structure of\nthe product\n\n\\[O^{-1} [ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u -\nu_-) ] O =\n\\left(\n\\begin{array}{cc}\nP^+ LRP^+ & * \\\\\n0 & P^-LRP^-\n\\end{array}\n\\right)\n\\]\n\nRemarkably, $P^+LRP^+$ and $P^-LRP^-$ turn out to be\n\n\\[\nP^+ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u - u_-)\nP^+ =\n\\]\n\n\\begin{equation}\n= L^{\\lambda_0} (u)^{00}_{00} R^{\\epsilon \\lambda_1, \\lambda_0}\n(u - u_- + i \\frac{2\\pi}{N})\n\\label{49}\n\\end{equation}\n\n\\[\nP^- L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u - u_-)\nP^- =\n\\]\n\\begin{equation}\n= L^{\\lambda_0} (u)^{10}_{10} R^{\\epsilon^{-1} \\lambda_1,\n\\lambda_0} (u - u_- - i \\frac{2\\pi}{N})\n\\label{50}\n\\end{equation}\n\nIn the product of block--triangular matrices the diagonal blocks\nare multiplied independently. Thus, we have for transfer--matrices\n\n\\[\ntr t_{\\lambda_0} (u) T_{\\lambda_0} (u - u_-, \\lambda_1) =\n[L^{\\lambda_0} (u)^{00}_{00}]^L T_{\\lambda_0} (u-u_- + i\n\\frac{2\\pi}{N}, \\epsilon^{\\lambda_1}) +\n\\]\n\n\\begin{equation}\n+ (L^{\\lambda_0} (u)^{10}_{10} )^L T_{\\lambda_0} (u - u_- - i\n\\frac{2\\pi}{N}, \\epsilon^{-1} \\lambda_1)\n\\label{51}\n\\end{equation}\n\n\nThe use of the second projection point $u_+$ leads to the\nsimilar equation. Note, that $T_{\\lambda_0} (u, \\lambda)$\ndepends essentially on two parameters $u$ and $\\lambda$.\nTherefore, another functional relation is needed in order to\ndetermine eigenvalues of $T_{\\lambda_0} (u, \\lambda)$ interms of known\neigenvalues of $tr t_{\\lambda_0} (u)$. This\nadditional relation (along with eigenvalues of $T_{\\lambda_0}\n(u, \\lambda))$ will be presented in the forthcoming publication [13].\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}