diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeuaz" "b/data_all_eng_slimpj/shuffled/split2/finalzzeuaz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeuaz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn a fundamental paper on the coarse geometry of Banach spaces (\\cite{kaltonpropertyq}), N. Kalton introduced a property of metric spaces that he named property $\\mathcal Q$. In particular,\nits absence served as an obstruction to coarse embeddability into reflexive Banach spaces. This property is related to the behavior of Lipschitz maps defined on a particular family of metric graphs that we shall denote $([\\N]^k,d_{\\K}^k)_{k \\in \\N}$. We will recall the precise definitions of these graphs and of property $\\mathcal Q$ in section \\ref{subsection:Q}. Let us just say, vaguely speaking for the moment, that a Banach space $X$ has property $\\mathcal Q$ if for every Lipschitz map $f$ from $([\\N]^k,d_{\\K}^k)$ to $X$, there exists a full subgraph $[\\M]^k$ of $[\\N]^k$, with $\\M$ infinite subset of $\\N$, on which $f$ satisfies a strong concentration phenomenon. It is then easy to see that if a Banach space $X$ has property $\\mathcal Q$, then the family of graphs $([\\N]^k,d_{\\K}^k)_{k \\in \\N}$ does not equi-coarsely embed into $X$ (see the definition in section \\ref{s:coarse}). One of the main results in \\cite{kaltonpropertyq} is that any reflexive Banach space has property $\\mathcal Q$. It then readily follows that a reflexive Banach space cannot contain a coarse copy of all separable metric spaces, or equivalently does not contain a coarse copy of the Banach space $c_0$. In fact, with a sophistication of this argument, Kalton proved an even stronger result in \\cite{kaltonpropertyq}: if a separable Banach space contains a coarse copy of $c_0$, then there is an integer $k$ such that the dual of order $k$ of $X$ is non separable. In particular, a quasi-reflexive Banach space does not contain a coarse copy of $c_0$.\nHowever, Kalton proved that the most famous example of a quasi-reflexive space, namely the James space $\\J$, as well as its dual $\\J^*$, fail property $\\mathcal Q$.\n\nThe main purpose of this paper is to show that, although they do not obey the concentration phenomenon described by property $\\mathcal Q$, neither $\\J$ nor $\\J^*$ equi-coarsely contains the family of graphs $([\\N]^k,d_{\\K}^k)_{k \\in \\N}$ (Corollary \\ref{James}). This provides a coarse invariant, namely ``not containing equi-coarsely the Kalton graphs'', that is very close to but different from property $\\mathcal Q$. This could allow to find obstructions to coarse embeddability between seemingly close Banach spaces. Our result is actually more general. We prove in Theorem \\ref{general} that a quasi-reflexive Banach space $X$ such that both $X$ and $X^*$ admit an equivalent $p$-asymptotically uniformly smooth norm (see the definition in section \\ref{asymptotics}), for some $p$ in $(1,\\infty)$, does not equi-coarsely contain the Kalton graphs.\n\nWe conclude this note by showing that if the James tree space $\\J\\T$ or its predual coarsely embeds into a separable Banach space $X$, then there exists $k\\in \\N$ so that the dual of order $k$ of $X$ is non separable. This extends slightly Theorem 3.5 in \\cite{kaltonpropertyq}.\n\n\\section{Metric notions}\n\n\\subsection{Coarse embeddings}\\label{s:coarse}\n\nLet $M$, $N$ be two metric spaces and $f \\colon M \\to N$ be a map. We define the compression modulus $\\rho_f$ and the expansion modulus $\\omega_f$ as follows. For $t\\in [0,\\infty)$, we set\n\\begin{eqnarray*}\n&&\\rho_f (t) = \\inf \\{ d_N(f(x),f(y)) \\, : \\, d_M(x,y) \\geq t \\},\\\\\n&&\\omega_f (t) = \\sup \\{ d_N(f(x),f(y)) \\, : \\, d_M(x,y) \\leq t \\}.\n\\end{eqnarray*}\nWe adopt the convention $\\sup(\\emptyset)=0$ and $\\inf(\\emptyset)=\\infty$.\nNote that for every $x,y \\in M$,\n$$\\rho_f (d_M(x,y)) \\leq d_N(f(x),f(y)) \\leq \\omega_f (d_M(x,y)).$$\nWe say that $f$ is a \\emph{coarse embedding} if $\\omega_f (t) < \\infty$ for every $t \\in [0,+\\infty)$ and $\\lim_{t \\to \\infty} \\rho_f (t) = \\infty$.\n\nNext, let $(M_i)_{i \\in I}$ be a family of metric spaces. We say that the family $(M_i)_{i \\in I}$ \\emph{equi-coarsely embeds} into a metric space $N$ if there exist two maps $\\rho, \\, \\omega \\colon [0,+\\infty) \\to [0,+\\infty)$ and maps $f_i \\colon M_i \\to N$ for $i \\in I$ such that:\n\\begin{enumerate}[(i)]\n\\item $\\lim_{t \\to \\infty} \\rho(t) = \\infty$,\n\\item $\\omega(t) < \\infty$ for every $t \\in [0,+\\infty)$,\n\\item $\\rho(t) \\leq \\rho_{f_i}(t)$ and $\\omega_{f_i}(t) \\leq \\omega(t)$ for every $i \\in I$ and $t \\in [0,\\infty)$.\n\\end{enumerate}\n\n\n\n\\subsection{The Kalton interlaced graphs and property Q}\n\\label{subsection:Q}\n\nFor $k\\in \\N$ and $\\M$ an infinite subset of $\\N$, we put $[\\M]^{\\le k}=\\{ S\\subset \\M: |S|\\le k\\}$, $[\\M]^{ k}=\\{ S\\subset \\M: |S|=k\\} $, $[\\M]^\\omega=\\{ S\\subset \\M: S\\text{ is infinite}\\} $, and $[\\M]^{< \\omega}=\\{ S\\subset \\M: S\\text{ is finite}\\}$.\nWe always list the elements of some $\\m$ in $[\\N]^{< \\omega}$ or in $[\\N]^{ \\omega}$ in increasing order, meaning that if we write $\\m=(m_1,m_2,\\ldots, m_l)$ or $\\m=(m_1,m_2,m_3, \\ldots )$, we tacitly assume that $m_10$. \nNotice that the sets $\\arg\\max(F)$ and $\\arg\\min(F)$ are disjoint.\nWe select inductively $\\{a_1<\\ldotsa_i\\} \\cap \\arg\\min(F)\\right)$, if this is not empty.\n\\item $a_{i+1}=\\min\\left( \\{n>b_i\\} \\cap \\arg\\max(F)\\right)$, if this set is not empty.\n\\end{itemize}\nNotice that $\\{a_1,\\ldots,a_p\\} \\subset \\n \\setminus \\m$ and $\\{b_1,\\ldots,b_q\\} \\subset \\m\\setminus \\n$.\nNotice also that either $p=q$ or $p=q+1$.\nIn the latter case we define $b_p:=r$ for some $r$ such that $r>a_p$ and $F(r-1)>F(r)$.\nSuch $r$ must exist since $F(\\max\\{n_k,m_k\\})=0$.\nAlso we have $r \\in \\m \\setminus \\n$.\nWe will set\n$$\n\\lbar=\\n \\cup \\{b_1,\\ldots,b_p\\} \\setminus \\{a_1,\\ldots,a_p\\}.\n$$\nIt is clear that $\\lbar \\in [\\M]^k$.\nWe also have $\\max F_{\\lbar,\\m}=\\max F_{\\n,\\m}-1$ and $\\min F_{\\lbar,\\m}=\\min F_{\\n,\\m}$.\nIndeed, the point $\\lbar$ is constructed in such a way that when $F_{\\n,\\m}$ attains its maximum for the first time (going from the left), $F_{\\lbar,\\m}$ is reduced by one and stays reduced by 1 until the next time the minimum of $F_{\\n,\\m}$ is attained (or until the point $r$) where this reduction is corrected back; and so on.\nThus $d(\\lbar,\\m)=d(\\n,\\m)-1$.\nAlso, since the sets $\\set{a_1,\\ldots,a_p}$ and $\\set{b_1,\\ldots,b_p}$ are interlaced we have $F_{\\n,\\m}-1\\leq F_{\\lbar,\\m}\\leq F_{\\n,\\m}$.\nTherefore, since $F_{\\n,\\m}=F_{\\n,\\lbar}+F_{\\lbar,\\m}$, we have that $0\\leq F_{\\n,\\lbar}\\leq 1$ and so finally $d(\\n,\\lbar)=1$, since it is clear that $\\n \\neq \\lbar$.\n\\end{proof}\nNote that if $X$ is a Banach space and $f \\colon ([\\M]^k,d_{\\K}^k) \\to X$ is a map with finite expansion modulus $\\omega_f$, then $\\omega_f(1)$ is actually the Lipschitz constant of $f$ as $d_{\\K}^k$ is a graph distance on $[\\M]^k$.\n\nIn \\cite{kaltonpropertyq} the property $\\mathcal Q$ is defined in the setting of metric spaces. For homogeneity reasons, its definition can be simplified for Banach spaces. Let us recall it here.\n\n\\begin{defi}\nLet $X$ be a Banach space. We say that $X$ has \\emph{property $\\mathcal{Q}$} if there exists $C\\ge 1$ such that for every $k \\in \\N$ and every Lipschitz map $f \\colon ([\\N]^k,d_{\\K}^k) \\to X$, there exists an infinite subset $\\M$ of $\\N$ such that:\n$$\\forall \\, \\overline{n},\\overline{m} \\in [\\M]^k,\\ \\|f(\\overline{n})-f(\\overline{m})\\| \\leq C\\omega_f(1).$$\n\\end{defi}\n\nThe following proposition should be clear from the definitions. We shall however include its short proof.\n\\begin{prop} \\label{QandGk}\nLet $X$ be a Banach space. If $X$ has property $\\mathcal Q$, then the family of graphs $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $X$.\n\\end{prop}\n\n\\begin{proof}\nLet $C\\ge 1$ be given by the definition of property $\\mathcal Q$. Aiming for a contradiction, assume that the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ equi-coarsely embeds into $X$. That is, there are maps $f_k \\colon ([\\N]^k,d_{\\K}^k) \\to X$ and two functions $\\rho, \\omega \\colon [0,+\\infty)\\to[0,+\\infty)$ such that $\\lim_{t \\to \\infty} \\rho (t)=\\infty$ and\n$$\\forall k\\in \\N\\ \\ \\forall t>0\\ \\ \\rho(t) \\leq \\rho_{f_k}(t)\\ \\ \\text{and}\\ \\ \\omega_{f_k}(t) \\leq \\omega(t)<\\infty.$$\nThus, for every $k\\in \\N$, there exists an infinite subset $\\mathbb M_k$ of $\\N$ such that $\\diam(f([\\M_k]^k)))\\le C\\omega(1)$. Since $\\diam([\\M_k]^k)=k$, this implies that for all $k\\in \\N$, $\\rho(k) \\le C\\omega(1)$. This contradicts the fact that $\\lim\\limits_{t \\to \\infty} \\rho(t)=\\infty$.\n\\end{proof}\n\nA concrete bi-Lipschitz copy of the metric spaces $([\\N]^k,d_{\\K}^k)$ in $c_0$ is given by the following proposition.\n\n\\begin{prop}\\label{Kgraphsinc0} Let $(s_n)_{n=1}^\\infty$ be the summing basis of $c_0$, that is\\\\\n$s_n=\\sum_{i=1}^ne_i$, where $(e_i)_{i=1}^\\infty$ is the canonical basis of $c_0$.\\\\\nFor $k\\in \\N$, define $f_k:([\\N]^k,d_{\\K}^k)\\to c_0$ by $f_k(\\n)=\\sum_{i=1}^k s_{n_i}$. Then\n$$\\frac12d_{\\K}^k(\\n,\\m)\\le \\|f_k(\\n)-f_k(\\m)\\|_\\infty\\le d_{\\K}^k(\\n,\\m)$$\nfor all $\\n,\\m \\in [\\N]^k$.\n\\end{prop}\n\n\\begin{proof}\nSince $\\dk=d$, one can show (as in the Fact in the proof of Proposition~\\ref{p:KaltonDistanceFormula}) that\n$\\dk(\\n,\\m)=\\max(f_k(\\n)-f_k(\\m))-\\min(f_k(\\n)-f_k(\\m)).$\nThe result then follows easily since $\\min(f_k(\\n)-f_k(\\m))\\leq 0\\leq \\max(f_k(\\n)-f_k(\\m))$ for all $\\n,\\m\\in [\\N]^k$.\n\\end{proof}\n\n\\begin{remark} We already explained that $c_0$ cannot coarsely embed into any Banach space with property $\\mathcal Q$ (in particular into any reflexive Banach space) and that Kalton even showed with additional arguments that if $c_0$ coarsely embeds into a separable Banach space $X$, then one of the iterated duals of $X$ has to be non separable. An inspection of his proof shows that \nthe uniformly discrete metric spaces\n\\[\n M_k=\\set{\\sum_{i=1}^k s_{n_i} \\times \\indicator{A}: (n_1,\\ldots,n_k) \\in [\\N]^k, A \\in [\\N]^\\omega} \\subset c_0\n\\]\ndo not equi-coarsely embed into any Banach space $X$ such that $X^{(r)}$ is separable for all $r$.\nSee Theorem \\ref{kalton} below for more on this subject.\n\\end{remark}\n\nStudying further the property $\\mathcal Q$ in \\cite{kaltonpropertyq}, Kalton exhibited non reflexive quasi-reflexive spaces with the property $\\mathcal Q$ but showed that $\\J$ and $\\J^*$ fail property $\\mathcal Q$. It is worth noticing that a theorem of Schoenberg \\cite{Schoenberg} implies that $\\ell_1$ coarsely embeds into $\\ell_2$, and therefore $\\ell_1$ provides a simple example of a non-reflexive Banach space with property $\\mathcal Q$.\n\n\\medskip\nWe conclude this section with two propositions that we state here for future reference. We start with a classical version of Ramsey's theorem.\n\\begin{prop}[Corollary 1.2 in \\cite{Gowers}] \\label{ramsey}\nLet $(K,d)$ be a compact metric space, $k \\in \\N$ and $f \\colon [\\N]^k \\to K$. Then for every $\\ep >0$, there exists an infinite subset $\\M$ of $\\N$ such that $d(f(\\overline{n}),f(\\overline{m}))< \\ep$ for every $\\overline{n},\\overline{m} \\in [\\M]^k$.\n\\end{prop}\n\nFor a Banach space $X$, we call \\emph{tree of height $k$} in $X$ any family $(x(\\n))_{\\n\\in[\\N]^{\\le k}}$, with $x(\\n)\\in X$. Then, if $\\M \\in [\\N]^\\omega$, $(x(\\n))_{\\n\\in[\\M]^{\\le k}}$ will be called a \\emph{full subtree} of $(x(\\n))_{\\n\\in[\\N]^{\\le k}}$. A tree $(x^*(\\n))_{\\n\\in[\\M]^{\\le k}}$ in $X^*$ is called \\emph{weak$^*$-null} if for any $\\n \\in [\\M]^{\\le k-1}$, the sequence $(x^*(n_1,\\ldots,n_{k-1},t))_{t>n_{k-1},t\\in \\M}$ is weak$^*$-null.\n\nThe next proposition is\nbased on a weak$^*$-compactness argument and will be crucial for our proofs. Although the distance considered on $[\\N]^k$ is different, the proof follows the same lines as Lemma 4.1 in \\cite{blms}. We therefore state it now without further detail.\n\n\\begin{prop}\\label{nulltree} Let $X$ be a separable Banach space, $k\\in\\N$, and $f:([\\N]^k,d_{\\K}^k)\\to X^*$ a Lipschitz map. Then there exist $\\M \\in [\\N]^\\omega$ and a weak$^*$-null tree $(x^*(\\m))_{\\m\\in[\\M]^{\\le k}}$ in $X^*$ with $\\|x^*_{\\m}\\| \\leq \\omega_f(1)$ for all $\\m\\in [\\M]^{\\leq k}\\setminus\\{\\emptyset\\}$ and so that\n$$\\forall \\n \\in [\\M]^k,\\ f(\\n)=\\sum_{i=0}^k x^*(n_1,\\ldots,n_i)=\\sum_{\\m \\preceq \\n} x^*(\\m).$$\n\\end{prop}\n\n\n\n\n\\section{Uniform asymptotic properties of norms and related estimates}\n\\label{asymptotics}\n\nWe recall the definitions that will be considered in this paper. For a Banach space $(X,\\|\\ \\|)$ we\ndenote by $B_X$ the closed unit ball of $X$ and by $S_X$ its unit\nsphere. The following definitions are due to V. Milman \\cite{Milman} and we adopt the notation from \\cite{JohnsonLindenstraussPreissSchechtman2002}. For $t\\in [0,\\infty)$ we define $$\\overline{\\rho}_X(t)=\\sup_{x\\in S_X}\\inf_{Y}\\sup_{y\\in S_Y}\\big(\\|x+t y\\|-1\\big),$$\nwhere $Y$ runs through all closed subspaces of $X$ of finite codimension. Then, the norm $\\|\\ \\|$ is said to be {\\it asymptotically uniformly smooth} (in short AUS) if\n$$\\lim_{t \\to 0}\\frac{\\overline{\\rho}_X(t)}{t}=0.$$\nFor $p\\in (1,\\infty)$ it is said to be {\\it $p$-asymptotically uniformly smooth} (in short $p$-AUS) if there exists $c>0$ such that for all $t\\in [0,\\infty)$, $\\overline{\\rho}_X(t)\\le ct^p$.\n\nWe will also need the dual modulus defined \nby\n$$ \\overline{\\delta}_X^*(t)=\\inf_{x^*\\in S_{X^*}}\\sup_{E}\\inf_{y^*\\in S_E}\\big(\\|x^*+ty^*\\|-1\\big),$$\nwhere $E$ runs through all finite-codimensional weak$^*$-closed subspaces of~$X^*$. \nThe norm of $X^*$ is said to be {\\it weak$^*$ asymptotically uniformly convex} (in short AUC$^*$) if $\\overline{\\delta}_X^*(t)>0$ for all $t$ in $(0,\\infty)$. If there exists $c>0$ and $q\\in [1,\\infty)$ such that for all $t\\in [0,1]$ $\\overline{\\delta}_X^*(t)\\ge ct^q$, we say that the norm of $X^*$ is $q$-AUC$^*$.\nThe following proposition is elementary.\n\n\\begin{prop}\\label{as-sequences} Let $X$ be a Banach space. For any $t\\in (0,1)$, any weakly null sequence $(x_n)_{n=1}^\\infty$ in $B_{X}$ and any $x \\in S_X$ we have:\n$$ \\limsup_{n \\rightarrow \\infty} \\| x+tx_n \\| \\leq 1 + \\overline{\\rho}_{X}(t).$$\n\nFor any weak$^*$-null sequence $(x^*_n)_{n=1}^\\infty \\subset X^*$ and for any $x^* \\in X^*\\setminus \\set{0}$ we have\n\\[\n \\limsup_{n\\to \\infty} \\norm{x^*+x^*_n} \\geq \\norm{x^*}\\left(1+\\overline{\\delta}_X^*\\left(\\frac{\\limsup \\norm{x_n^*}}{\\norm{x^*}}\\right)\\right).\n\\]\n\n\\end{prop}\n\nWe will also need the following refinement (see Proposition 2.1 in \\cite{LancienRaja2018}).\n\n\\begin{prop}\\label{waus}\nLet $X$ be a Banach space. \nThen the bidual norm on $X^{**}$ has the following property. \nFor any $t\\in (0,1)$, any weak$^*$-null sequence $(x^{**}_n)_{n=1}^\\infty$ in $B_{X^{**}}$ and any $x \\in S_X$ we have:\n$$ \\limsup_{n \\rightarrow \\infty} \\| x+tx^{**}_n \\| \\leq 1 + \\overline{\\rho}_{X}(t).$$\n\\end{prop}\n\n\n\\medskip Let us now recall the following classical duality result concerning these moduli (see for instance \\cite{DKLR} Corollary 2.3 for a precise statement).\n\n\\begin{prop}\\label{duality} Let $X$ be a Banach space. \nThen $\\|\\ \\|_X$ is AUS if and and only if $\\|\\ \\|_{X^*}$ is AUC$^*$.\n\nIf $p,q\\in (1,\\infty)$ are conjugate exponents, then $\\|\\ \\|_X$ is $p$-AUS if and and only if $\\|\\ \\|_{X^*}$ is $q$-AUC$^*$.\n\\end{prop}\n\nWe conclude this section with a list of a few classical properties of Orlicz functions and norms that are related to these moduli.\nA map $\\varphi:[0,\\infty) \\to [0,\\infty)$ is called an \\emph{Orlicz function} if it is continuous, non decreasing, convex and so that $\\varphi(0)=0$ and $\\lim_{t \\to \\infty}\\varphi(t)=\\infty$. \nThe \\emph{Orlicz norm} $\\|\\ \\|_{\\ell_\\varphi}$, associated with $\\varphi$ is defined on $c_{00}$, the space of finitely supported sequences, as follows:\n$$\\forall x=(x_n)_{n=1}^\\infty \\in c_{00},\\ \\ \\|x\\|_{\\ell_\\varphi}=\\inf\\big\\{r>0,\\ \\sum_{n=1}^\\infty \\varphi(x_n\/r)\\le 1\\big\\}.$$\nThe following is immediate from the definition.\n\n\\begin{lem}\\label{Orlicz-lp} Let $\\varphi:[0,\\infty) \\to [0,\\infty)$ be an Orlicz function and $p\\in [1,\\infty)$.\n\\begin{enumerate}[(i)]\n\\item If there exists $C>0$ such that $\\varphi(t)\\le Ct^p$, for all $t\\in [0,1]$, then there exists $A>0$ such that $\\|x\\|_{\\ell_\\varphi} \\le A\\|x\\|_{\\ell_p}$, for all $x\\in c_{00}$.\n\\item If there exists $c>0$ such that $\\varphi(t)\\ge ct^p$, for all $t\\in [0,1]$, then there exists $a>0$ such that $\\|x\\|_{\\ell_\\varphi} \\ge a\\|x\\|_{\\ell_p}$, for all $x\\in c_{00}$.\n\\end{enumerate}\n\\end{lem}\n\nAssume now that $\\varphi:[0,\\infty) \\to [0,\\infty)$ is an Orlicz function which is 1-Lipschitz and such that $\\lim_{t\\to \\infty}\\varphi(t)\/t=1$. \nConsider for $(s,t) \\in \\R^2$, \n\\[\nN_2^\\varphi(s,t)=\n\\begin{cases}|s|+|s|\\varphi(|t|\/|s|) & \\text{ if }s\\neq 0,\\\\\n \\abs{t}& \\text{ if }s=0.\n\\end{cases}\n\\]\nThen define by induction for all $n\\geq 3$:\n$$\\forall (s_1,\\ldots,s_n)\\in \\R^n,\\ N_n^\\varphi(s_1,\\ldots,s_n)=\nN_2^\\varphi\\big(N_{n-1}^\\varphi(s_1,\\ldots,s_{n-1}),s_n\\big).$$\nThe following is proved in \\cite{KaltonTAMS2013} (see Lemma 4.3 and its preparation).\n\n\n\\begin{lemma}\\label{Orlicz-Kalton}\\\n\\begin{enumerate}[(i)]\n\\item For any $n \\ge 2$, the function $N_n^\\varphi$ is an absolute (or lattice) norm on $\\R^n$, meaning that $N_n(s_1,\\ldots,s_n)\\le N_n(t_1,\\ldots,t_n)$, whenever $|s_i|\\le |t_i|$ for all $i\\le n$.\n\\item For any $n\\in \\N$ and any $x\\in \\R^n$:\n$$\\frac12 \\|s\\|_{\\ell_\\varphi} \\le N_n^\\varphi(s) \\le e\\|s\\|_{\\ell_\\varphi}.$$\n\\end{enumerate}\n\\end{lemma}\n\nWhen $X$ is a Banach space, it is easy to see that $\\overline{\\rho}_X$ is a 1-Lipschitz Orlicz function such that $\\lim_{t\\to \\infty}\\rho(t)\/t=1$.\nBut due to its lack of convexity, $\\overline{\\delta}_X^*$ is not an Orlicz function and we need to modify it. Following \\cite{KaltonTAMS2013}, we define\n$$\\delta(t)=\\int_0^t \\frac{\\overline{\\delta}_X^*(s)}{s}\\,ds.$$\nIt is easy to see that $\\overline{\\delta}_X^*(t)\/{t}$ is increasing and tends to $1$ as $t$ tends to $\\infty$. Therefore, $\\delta$ is an Orlicz function which is 1-Lipschitz, such that $\\lim_{t\\to \\infty}\\delta(t)\/t=1$ and satisfying:\n$$\\forall t\\in [0,\\infty),\\ \\ \\overline{\\delta}_X^*(t\/2) \\le \\delta(t) \\le \\overline{\\delta}_X^*(t).$$\nThe following statement is now a direct consequence of Lemmas \\ref{Orlicz-lp} and \\ref{Orlicz-Kalton}.\n\n\\begin{lem}\\label{Nnorm-lp} Let $X$ be a Banach space and $p\\in [1,\\infty)$.\n\\begin{enumerate}[(i)]\n\\item If there exists $C>0$ such that $\\overline{\\rho}_X(x)\\le Ct^p$, for all $t\\in [0,1]$, then there exists $A>0$ such that\n$$\\forall n\\in \\N\\ \\forall x\\in \\R^n,\\ \\ N_n^{\\overline{\\rho}_X}(x)\\le A\\|x\\|_{\\ell_p^n}.$$\n\\item If there exists $c>0$ such that $\\overline{\\delta}_X^*(t)\\ge ct^p$, for all $t\\in [0,1]$, then there exists $a>0$ such that\n$$\\forall n\\in \\N\\ \\forall x\\in \\R^n,\\ \\ N_n^{\\delta}(x)\\ge a\\|x\\|_{\\ell_p^n}.$$\n\\end{enumerate}\n\\end{lem}\n\n\nWe will also use the following reformulation of Propositions~\\ref{as-sequences} and~\\ref{waus} in terms of the norms $N_2^\\delta$ and $N_2^{\\overline{\\rho}_X}$.\n\\begin{lem}\\label{l:asymptotic-Nnorm}\n Let $X$ be a Banach space. \n \\begin{enumerate}[(i)]\n \\item Let $(x_n^*) \\subset X^*$ be weak$^*$-null. Then for any $x^* \\in X^*$ we have \n \\[\n \\limsup_{n\\to \\infty} \\norm{x^*+x_n^*}\\geq N_2^\\delta(\\norm{x^*},\\limsup \\norm{x_n^*}).\n \\]\n \\item Similarly, if $(x_n^{**}) \\subset X^{**}$ is weak$^*$-null and $x\\in X$, then \n$$ \\liminf_{n \\rightarrow \\infty} \\| x+x^{**}_n \\| \\leq N_2^{\\overline{\\rho}_X}(\\norm{x},\\liminf \\norm{x_n^{**}}).$$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n If $x^*=0$ there is nothing to do, so we may assume that $x^*\\neq 0$.\n By application of Proposition~\\ref{as-sequences} we see that \n \\[\n \\begin{aligned}\n \\limsup_{n\\to\\infty}\\norm{x^*+x_n^*}&\\geq \\norm{x^*}\\left(1+\\overline{\\delta}^*_X\\left(\\frac{\\limsup\\norm{x_n^*}}{\\norm{x^*}}\\right)\\right)\\\\\n &\\geq \\norm{x^*}\\left(1+\\delta\\left(\\frac{\\limsup\\norm{x_n^*}}{\\norm{x^*}}\\right)\\right)=N_2^\\delta(\\norm{x^*},\\limsup \\norm{x_n^*})\n \\end{aligned}\n \\]\nThe proof of the second claim is even simpler so we leave it to the reader.\n\\end{proof}\n\n\n\n\n\\section{The general result}\n\nLet us first recall that a Banach space is said to be {\\it quasi-reflexive} if the image of its canonical embedding into its bidual is of finite codimension in its bidual. We can now state our main result.\n\n\\begin{thm}\\label{general} Let $X$ be a quasi-reflexive Banach space, let $p\\in (1,\\infty)$ and denote $q$ its conjugate exponent. Assume that $X$ admits an equivalent $p$-AUS norm and that $X^*$ admits an equivalent $q$-AUS norm. Then the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $X^{**}$.\n\\end{thm}\n\nWe immediately deduce the following.\n\n\\begin{cor}\\label{generalcorollary} \nLet $X$ be a quasi-reflexive Banach space, let $p\\in (1,\\infty)$ and denote $q$ its conjugate exponent. \nAssume that $X$ admits an equivalent $p$-AUS norm and that $X^*$ admits an equivalent $q$-AUS norm. \nThen the family $([\\N]^k,d_{\\K}^k)_{k\\in \\N}$ does not equi-coarsely embed into $X$, nor does it equi-coarsely embed into any iterated dual $X^{(r)}$ ($r\\geq 0$) of $X$.\n\\end{cor}\n\n\\begin{proof} \nSince $X$ is quasi reflexive we infer that $X^{(r)}$ admits an equivalent $p$-AUS norm when $r$ is even and it admits an equivalent $q$-AUS norm when $r$ is odd. \nIndeed, note that when $r$ is even $X^{(r)}$ is isomorphic to $X\\oplus_p F$ where $F$ is finite-dimensional (resp. $X^{(r)}\\simeq X^*\\oplus_q F$ when $r$ is odd).\nNow it is obvious from Theorem~\\ref{general} that $([\\N]^k)_{k\\in \\N}$ do not equi-coarsely embed into $X^{(r)}$ when $r$ is even. \nWhen $r$ is odd, we just exchange the roles of $p$ and $q$.\n\\end{proof}\n\nBefore going into the detailed proof of Theorem~\\ref{general} let us briefly indicate the main idea. We assume that there is an equi-coarse family of embeddings $(f_k)$ of $[\\N]^k$ into $X^{**}$ with moduli $\\rho$ and $\\omega$.\nWe fix $k$ sufficiently large and observe that, up to passing to a subgraph, $f_k$ can be represented as the sum along the branches of a weak$^*$-null countably branching tree of height $k$, say $(z_{\\n})_{\\n\\in [N]^{\\leq k}}$.\nMoreover the norms of the elements of this tree stabilize on each level towards values $(K_i)_{i=1}^k \\subset [0,\\omega(1)]$.\nApplying the existence of a $q-AUS$ norm on $X^*$ one can show that $\\sum_{i=1}^k K_i^p \\leq c^p\\omega(1)^p$ where $c$ is a constant depending only on $X$.\nThe benefit of this observation is twofold.\nOn one hand we will be able to construct two elements $\\n_0,\\m_0 \\in [\\N]^{l}$ (with $l\\leq k$) such that $\\sum_{i=1}^l z_{(n_1,\\ldots,n_i)}-z_{(m_1,\\ldots,m_i)}$ is small in norm (say less than $2c\\omega(1)$) while $d_{\\K}^l(\\n_0,\\m_0)$ is large (say $\\rho(d_{\\K}^l(n_0,\\m_0))> 3c\\omega(1)$). \nOn the other hand the $p-AUS$ renormability of $X$ together with the quasi-reflexivity allows to extend these elements to elements $\\n,\\m \\in [\\N]^k$ such that $\\dk(\\n,\\m)$ is still large and \n\\[\\begin{aligned}\n\\norm{\\sum_{i=l+1}^k z_{(n_1,\\ldots,n_i)}-z_{(m_1,\\ldots,m_i)}} &\\sim \\left(\\sum_{i=l+1}^k\\norm{z_{(n_1,\\ldots,n_i)}-z_{(m_1,\\ldots,m_i)}}^p\\right)^{1\/p}\\\\ &\\sim (\\sum_{i=l+1}^k K_i^p)^{1\/p}\\leq c\\omega(1) .\n\\end{aligned}\n\\]\nEventually, summing the tree from $1$ to $k$ over the branches ending by $\\n$ and $\\m$ we get the desired contradiction\n\\[\n 3c\\omega(1)<\\rho(\\dk(\\n,\\m))\\leq \\norm{f_k(\\n)-f_k(\\m)}\\leq 3c\\omega(1).\n\\]\n\n\\begin{proof}[Proof of Theorem \\ref{general}] Let us assume that there are two maps $\\rho, \\, \\omega \\colon [0,+\\infty) \\to [0,+\\infty)$ and maps $f_k ([\\N]^k,d_{\\K}^k) \\colon \\to (X^{**},\\|\\ \\|)$ for $k \\in \\N$ such that:\n\\begin{enumerate}[(i)]\n\\item $\\lim_{t \\to \\infty} \\rho(t) = \\infty$,\n\\item $\\omega(t) < \\infty$ for every $t \\in (0,+\\infty)$,\n\\item $\\rho(t) \\leq \\rho_{f_k}(t)$ and $\\omega_{f_k}(t) \\leq \\omega(t)$ for every $k \\in \\N$ and $t \\in (0,\\infty)$.\n\\end{enumerate}\nNote that all $f_k$'s are $\\omega(1)$-Lipschitz for $\\|\\ \\|$ and so $\\omega(1)>0$. Since all the sets $[\\N]^k$ are countable, we may and will assume that $X$ and therefore, by the quasi-reflexivity of $X$, that all its iterated duals are separable.\\\\\nLet us fix $N\\in \\N$. Pick $\\alpha\\in \\N$ such that $\\alpha\\ge \\frac{p}{q}$ and set $k=N^{1+\\alpha} \\in \\N$. We also fix $\\eta>0$. We shall provide at the end of our proof a contradiction if $N$ is chosen large enough and $\\eta$ small enough. We denote $\\|\\ \\|$ the original norm on $X$, as well as its dual and bidual norms. Let us assume, as we may, that $\\|\\ \\|$ is $p$-AUS on $X$. We denote its modulus of asymptotic uniform smoothness $\\overline{\\rho}_{\\|\\ \\|}$ or simply $\\overline{\\rho}_{X}$.\n\n\\medskip\nFor the first step of the proof we shall exploit the existence of an equivalent $q$-AUS norm $|\\ |$ on $X^*$ (we also denote $|\\ |$ its dual norm on $X^{**}$). \nIt is worth mentioning that if $X$ is not reflexive, $|\\ |$ cannot be the dual norm of an equivalent norm on $X$ (see for instance Proposition 2.6 in \\cite{CauseyLancien}). \nAssume also that there exists $b>0$ such that\n\\begin{equation}\\label{equivalence}\n\\forall z \\in X^{**}\\ \\ b\\|z\\|\\le |z|\\le \\|z\\|.\n\\end{equation}\nThen we have that all $f_k$'s are also $\\omega(1)$-Lipschitz for $|\\ |$.\\\\\nBy Proposition \\ref{duality}, we have that there exists $c>0$ such that for all $t\\in [0,1]$, $\\overline{\\delta}_{|\\ |}^*(t)\\ge ct^p$. \nWe denote again\n$$\\delta(t)=\\int_0^t \\frac{\\overline{\\delta}_{|\\ |}^*(s)}{s}\\, ds.$$\nRecall that Lemma \\ref{Nnorm-lp} ensures the existence of $a>0$ such that for all $n\\in \\N$, $N_n^\\delta \\ge 2a\\|\\ \\|_{\\ell_p^n}$.\n\nFirst, using the separability of $X^*$ and Proposition \\ref{nulltree}, we may assume by passing to a full subtree, that there exist a weak$^*$-null tree $(z(\\m))_{\\m\\in[\\N]^{\\le k}}$ in $X^{**}$ with $|z_{\\m}| \\leq \\omega(1)$ for all $\\m\\in [\\N]^{\\leq k}\\setminus\\{\\emptyset\\}$ and so that\n$$\\forall \\n \\in [\\N]^k,\\ f_k(\\n)=\\sum_{i=0}^k z(n_1,\\ldots,n_i)=\\sum_{\\m \\preceq \\n} z(\\m).$$\n\nFor $r\\in \\N$ we denote $E_r=\\{\\m=(m_1,\\ldots,m_j)\\in [\\N]^{\\le k}\\setminus \\{\\emptyset\\},\\ m_j=r\\}$ and $F_r=\\bigcup_{u=1}^r E_u$. \nFix a sequence $(\\lambda_r)_{r=1}^\\infty$ in $(0,1)$ such that $\\prod_{r=1}^\\infty \\lambda_r > \\frac12$. \nWe now use Lemma~\\ref{l:asymptotic-Nnorm}~(i) and the fact that $(z(\\m))_{\\m\\in[\\N]^{\\le k}}$ is a weak$^*$-null tree to build inductively $n_1<\\ldotsj+2N+1$ such that\n\\begin{multline*}\n\\|x_{j+N+1}-x'_{j+N+1}+z_{j+N+2}-z'_{j+N+2}\\| \\\\\n\\le N_2^{\\overline{\\rho}_{X}}\\big(\\|x_{j+N+1}-x'_{j+N+1}\\|,\n\\|z_{j+N+2}-z'_{j+N+2}\\|\\big)+\\eta\n\\end{multline*}\nIt follows from (\\ref{stabilize}) that\n\\begin{multline*}\n\\|z_{j+N+1}-z'_{j+N+1}+z_{j+N+2}-z'_{j+N+2}\\| \\\\\n\\begin{aligned}\n&\\le N_2^{\\overline{\\rho}_{X}}\\big(\\|z_{j+N+1}-z'_{j+N+1}\\|+\\eta,\n\\|z_{j+N+2}-z'_{j+N+2}\\|\\big)+2\\eta\\\\\n&\\le N_2^{\\overline{\\rho}_{X}}\\Big(\\frac{2}{b}\\big(K_{j+N+1}+\\eta\\big)+\\eta,\\frac{2}{b}\\big(K_{j+N+2}+\\eta\\big)\\Big)+2\\eta.\n\\end{aligned}\n\\end{multline*}\nSimilarly, we can inductively find $m_{j+N+2}=u_{j+N+2}<\\cdots0$ such that $N_n^{\\overline{\\rho}_{X}}\\le C\\|\\ \\|_{\\ell_p^n}$ for all $n\\in \\N$ the above inequality yields \n$$\\Big\\|\\sum_{i=j+N+1}^k (z_i-z'_i)\\Big\\| \\le \\frac{2C}{b}\\Big(\\sum_{i=j+N+1}^k K_i^p\\Big)^{1\/p} +\\omega(1)\\le \\Big(\\frac{2CA}{b}+1\\Big)\\omega(1).$$\nFinally, combining the above estimate with (\\ref{samedebut}) and (\\ref{smallblock}), we get that\n$$\\|f(\\m)-f(\\overline{u})\\|\\le \\frac{3A+2CA+b}{b}\\omega(1).$$\nAs announced at the beginning of the proof, this yields a contradiction if $N$ was initially chosen, as it was possible, so that $\\rho(N)>\\frac{3A+2CA+b}{b}\\omega(1).$\n\\end{proof}\n\n\n\n\n\nUnlike reflexivity, quasi-reflexivity itself is not enough to prevent the Kalton graphs from embedding into a Banach space. We thank P. Motakis for showing us the next example.\n\n\\begin{prop}[Motakis]\\label{Pavlos} There exists a quasi-reflexive Banach space $X$ such that the family of graphs $([\\N]^k,d_\\K^k)_{k\\in \\N}$ equi-Lipschitz embeds into $X$.\n\\end{prop}\n\n\\begin{proof} The proof relies on the existence of a quasi-reflexive Banach space $X$ of order one which admits a spreading model, generated by a basis of $X$ that is equivalent to the summing basis $(s_n)_{n=1}^\\infty$ of $c_0$. This is shown in \\cite{FOSZ} (Proposition 3.2) and based on a construction given in \\cite{BHO}. We refer the reader to \\cite{BeauzLap} for the necessary definitions. Consequently, there exists a sequence $(x_n)_{n=1}^\\infty$ in $S_X$ and constants $A,B>0$ such that for all $k\\le n_1<\\cdots0\n\\end{align}\nfor every $s,t\\in[0,1]$ with $s0\n\\end{align}\nfor every $s,t\\in[0,1]$ with $s0,\\\\\n1 & \\text{if } \\kappa =0,\\\\\n\\dfrac{\\sinh(\\!\\sqrt{-\\kappa}\\,\\vartheta)}{\\sqrt{-\\kappa}\\,\\vartheta} & \\text{otherwise},\n\\end{cases}\\\\\n\\sigma_\\kappa^{(t)}(\\vartheta) &:= \\begin{cases} \\infty & \\textnormal{if }\\kappa\\,\\vartheta^2 \\geq \\pi^2,\\\\\nt\\,\\dfrac{\\mathfrak{S}_\\kappa(t\\,\\vartheta)}{\\mathfrak{S}_\\kappa(\\vartheta)} & \\textnormal{otherwise},\n\\end{cases}\n\\end{align*}\n\n\\begin{definition}\\label{Def:Dist coeff} For $K\\in\\R$ and $N\\in[1,\\infty)$, slightly abusing notation we set\n\\begin{align*}\n\\sigma_{K,N}^{(t)}(\\vartheta) &:= \\sigma_{K\/N}^{(t)}(\\vartheta),\\\\\n\\tau_{K,N}^{(t)}(\\vartheta) &:= t^{1\/N}\\,\\sigma_{K,N-1}^{(t)}(\\vartheta)^{1-1\/N}.\n\\end{align*}\n\\end{definition}\n\nNote that for every $t\\in(0,1)$ and every $\\vartheta > 0$, $\\smash{\\sigma_{K,N}^{(t)}(\\vartheta)}$ is continuous in $(K,N)\\in \\R\\times[1,\\infty)$, nondecreasing in $K$, and nonincreasing in $N$ \\cite[Rem.~2.2]{bacher2010}. Analogous claims apply to the quantity $\\smash{\\tau_{K,N}^{(t)}(\\vartheta)}$ \\cite[p.~138]{sturm2006b}. Furthermore, for every $t\\in [0,1]$, every $\\vartheta\\geq 0$, and every $\\kappa\\in (-\\infty,\\pi^2\/\\vartheta^2)$,\n\\begin{align}\\label{Eq:Distortion coeff property}\n\\sigma_\\kappa^{(t)}(\\vartheta) = \\sigma_{\\kappa\\vartheta^2}^{(t)}(1).\n\\end{align}\n\n\\begin{remark}\\label{Re:Lower bounds sigma} We recall the following elementary inequality from \\cite[Rem.~2.3]{cavalletti2017}: for every $K\\in\\R$, every $N\\in[1,\\infty)$, every $t\\in[0,1]$, and every $\\vartheta\\geq 0$,\n\\begin{align*}\n\\sigma_{K,N}^{(t)}(\\vartheta) \\geq t\\,{\\ensuremath{\\mathrm{e}}}^{-(1-t)\\vartheta\\sqrt{K^-\/N}}.\n\\end{align*}\n\\end{remark}\n\nLastly, in view of \\autoref{Th:Equivalence TCD* and TCDe} and \\autoref{Th:Equivalence TMCP* and TMCPe}, given any $t\\in[0,1]$ let us define ${\\ensuremath{\\mathrm{G}}}_t\\colon \\R^2 \\times (-\\infty,\\pi^2)\\to (-\\infty,\\infty]$ and ${\\ensuremath{\\mathrm{H}}}_t\\colon \\R\\times (-\\infty,\\pi^2)\\to (-\\infty,\\infty]$ by\n\\begin{align}\\label{Eq:GtHt}\n\\begin{split}\n{\\ensuremath{\\mathrm{G}}}_t(x,y,\\kappa) &:= \\log\\!\\big[\\sigma_\\kappa^{(1-t)}(1)\\,{\\ensuremath{\\mathrm{e}}}^x + \\sigma_\\kappa^{(t)}(1)\\,{\\ensuremath{\\mathrm{e}}}^y\\big]\\\\\n{\\ensuremath{\\mathrm{H}}}_t(x,\\kappa) &:= \\log\\!\\big[\\sigma_\\kappa^{(1-t)}(1)\\,{\\ensuremath{\\mathrm{e}}}^x\\big] = \\log\\sigma_\\kappa^{(1-t)}(1) + x.\n\\end{split}\n\\end{align}\nThen the functions ${\\ensuremath{\\mathrm{G}}}_t$ and ${\\ensuremath{\\mathrm{H}}}_t$ are jointly convex \\cite[Lem.~2.11]{erbar2015}.\n\n\\subsection{Nonsmooth Lorentzian spaces}\\label{Sub:Lorentzian nonsmooth} We continue with a concise digression on the theory of nonsmooth Lorentzian (pre-length, length, and geodesic) spaces. We refer to \\cite{cavalletti2020, kunzinger2018} for details, proofs, and examples about the corresponding notions.\n\n\\subsubsection{Chronology and causality} Let us fix a preorder $\\leq$ and a transitive relation $\\ll$, contained in $\\leq$, on $\\mms$. The triple $(\\mms,\\ll,\\leq)$ is called \\emph{causal space} \\cite[Def. 2.1]{kunzinger2018}. We say that $x,y\\in\\mms$ are \\emph{timelike} or \\emph{causally} related if $x\\ll y$ or $x\\leq y$, respectively. We write $x0$ if and only if $x\\ll y$, and\n\\item if $x\\leq y\\leq z$ we have the \\emph{reverse triangle inequality}\n\\begin{align}\\label{Eq:Reverse tau}\n\\uptau(x,z) \\geq \\uptau(x,y) + \\uptau(y,z).\n\\end{align}\n\\end{enumerate}\nThe existence of such a $\\uptau$ implies that $\\ll$ is an \\emph{open} relation \\cite[Prop.~2.13]{kunzinger2018}; in particular, the set $\\smash{I^\\pm(A)}$ is open for every $A\\subset\\mms$ \\cite[Lem.~2.12]{kunzinger2018}.\n\n\\begin{definition}\\label{Def:LLSSP} A \\emph{Lorentzian pre-length space} is a quintuple $(\\mms,\\met,\\ll,\\leq,\\uptau)$ which consists of a causal space $(\\mms,\\ll,\\leq)$ endowed with a proper metric $\\met$ and a time separation function $\\uptau$ as introduced above.\n\\end{definition}\n\n\n\\subsubsection{Length of curves}\\label{Sub:Length curves} Let $(\\mms,\\met,\\ll,\\leq,\\uptau)$ be a given Lorentzian pre-length space. A \\emph{path} designates a map $\\gamma\\colon [a,b]\\to\\mms$, where $a,b\\in\\R$ with $a0\n\\end{align}\nfor every $s,t\\in[0,1]$ with $s0$ such that the $\\met$-arclength of any causal curve contained in $C$ is no larger than $c$.\n\n\\begin{definition} The space $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item \\emph{globally hyperbolic} if it is non-totally imprisoning and the causal diamond $J(x,y)$ is compact for every $x,y\\in\\mms$, and\n\\item \\emph{${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic} if it is non-totally imprisoning and the causal diamond $J(C_0,C_1)$ is compact for all compact $C_0,C_1\\subset\\mms$.\n\\end{enumerate}\n\\end{definition}\n\nWe list some properties of (${\\ensuremath{\\mathscr{K}}}$-)globally hyperbolic Lorentzian pre-length spaces which will lead to our standing \\autoref{Ass:ASS} below as well as useful consequences thereof. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is locally causally closed, globally hyperbolic, and $I^\\pm(x) \\neq \\emptyset$ for every $x\\in \\mms$, then ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity holds \\cite[Lem.~1.5]{cavalletti2020}. On the other hand, every locally causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic Lorentzian geodesic space is in fact causally closed \\cite[Lem.~1.6]{cavalletti2020}. By \\cite[Def.~3.25, Thm.~3.26]{kunzinger2018}, global hyperbolicity implies the nonsmooth analogue of the \\emph{strong causality condition} for smooth Lorentzian spacetimes, cf.~e.g.~\\cite[Def.~14.11]{oneill1983}. On every globally hyperbolic Lorentzian length space (see \\cite[Def.~3.22]{kunzinger2018} for the corresponding definition, we will not really need it), the time separation function $\\uptau$ is finite and continuous \\cite[Thm.~3.28]{kunzinger2018}; in particular, we have $\\uptau(x,x) = 0$ for every $x\\in\\mms$. Furthermore, every such space is geodesic by the nonsmooth analogue of Avez--Seifert's theorem \\cite[Thm.~3.30]{kunzinger2018}. In \\cite{burtscher2021}, a singular analogue of Geroch's characterization \\cite{geroch1970} of global hyperbolicity via Cauchy time functions is proven.\n\n\n\\subsection{Optimal transport on Lorentzian spaces}\\label{Sec:OT Lorentzian} Next, let us briefly review the theory of optimal transport on the class of spaces introduced above \\cite{cavalletti2020}. We refer to \\cite{eckstein2017,kellsuhr2020, mccann2020, mondinosuhr2018, suhr2018} for prior developments in the smooth case.\n\n\\subsubsection{Basic probabilistic notation} Let ${\\ensuremath{\\mathscr{P}}}(\\mms)$ denote the set of all Borel probability measures on $\\mms$. Let ${\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ and ${\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ be its subsets consisting of all compactly supported and $\\meas$-absolutely continuous elements, respectively; set $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)} := \\smash{{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$. Given any $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, by $\\mu_\\perp$ we mean the $\\meas$-singular part in the corresponding Lebesgue decomposition of $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$.\n\nFor a Borel map $F\\colon \\mms\\to \\mms'$ into a metric space $(\\mms',\\met')$, given any $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ the measure $F_\\push\\mu \\in{\\ensuremath{\\mathscr{P}}}(\\mms')$ designates the usual \\emph{push-forward} of $\\mu$ under $F$, defined by the formula $F_\\push\\mu[B] := \\smash{\\mu\\big[F^{-1}(B)\\big]}$ for every Borel set $B\\subset\\mms'$.\n\nGiven $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, let $\\Pi(\\mu,\\nu)$ denote the set of all \\emph{couplings} of $\\mu$ and $\\nu$, i.e.~all $\\pi\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ with $\\pi[\\, \\cdot\\times\\mms] = \\mu$ and $\\pi[\\mms\\times\\cdot\\, ] = \\nu$.\n\nWith $\\Cont([0,1];\\mms)$ denoting the set of all curves $\\gamma\\colon [0,1]\\to\\mms$, endowed with the uniform topology, for $t\\in [0,1]$ the so-called \\emph{evaluation map} $\\eval_t\\colon \\Cont([0,1];\\mms) \\to \\mms$ is defined through $\\eval_t(\\gamma) := \\gamma_t$.\n\n\\subsubsection{Chronological and causal couplings} Let $(\\mms,\\met,\\ll,\\leq,\\uptau)$ be a Lorentzian pre-length space. We define the (possibly empty) set $\\Pi_\\ll(\\mu,\\nu)$ of all \\emph{chronological couplings} of two probability measures $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ to consist of all $\\pi\\in\\Pi(\\mu,\\nu)$ such that $\\smash{\\pi[\\mms_\\ll^2]=1}$. The set $\\Pi_\\leq(\\mu,\\nu)$ of all \\emph{causal couplings} of $\\mu$ and $\\nu$ is defined analogously. Under causal closedness, clearly $\\pi\\in\\Pi_\\leq(\\mu,\\nu)$ if and only if $\\pi\\in\\Pi(\\mu,\\nu)$ as well as $\\smash{\\supp\\pi\\subset\\mms_\\leq^2}$; an analogous statement holds for the locally causally closed case if $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$.\n\nA chronological or causal coupling of $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ intuitively describes a way of transporting an infinitesimal mass portion ${\\ensuremath{\\mathrm{d}}}\\mu(x)$ to an infinitesimal mass portion ${\\ensuremath{\\mathrm{d}}}\\nu(y)$ as to guarantee $x\\ll y$ or $x\\leq y$, respectively.\n\nWe call $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ \\emph{chronologically related} if $\\Pi_\\ll(\\mu,\\nu)\\neq\\emptyset$.\n\n\\subsubsection{The $\\smash{\\ell_p}$-optimal transport problem} Given an exponent $p\\in (0,1]$ and following the expositions in \\cite{cavalletti2020,mccann2020} we will adopt the conventions\n\\begin{align*}\n\\sup\\emptyset &:= -\\infty,\\\\\n(-\\infty)^p &:= (-\\infty)^{1\/p} := -\\infty,\\\\\n\\infty -\\infty &:=-\\infty. \\textcolor{white}{a^{1\/p}}\n\\end{align*}\n\nLet the total transport cost function $\\ell_p\\colon{\\ensuremath{\\mathscr{P}}}(\\mms)^2\\to [0,\\infty]\\cup\\{-\\infty\\}$ be given by\n\\begin{align*}\n\\ell_p(\\mu,\\nu) &:= \\sup\\{\\Vert \\uptau\\Vert_{\\Ell^p(\\mms^2,\\pi)} : \\pi \\in \\Pi_\\leq(\\mu,\\nu)\\}\\\\\n&\\textcolor{white}{:}= \\sup\\{\\Vert l\\Vert_{\\Ell^p(\\mms^2,\\pi)} : \\pi\\in\\Pi(\\mu,\\nu)\\},\n\\end{align*}\nwhere $l\\colon \\mms^2 \\to [0,\\infty]\\cup\\{-\\infty\\}$ is defined through\n\\begin{align*}\nl^p(x,y) := \\begin{cases} \\uptau^p(x,y) & \\textnormal{if }x\\leq y,\\\\\n-\\infty & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align*}\n\n\\begin{remark}\nThe sets of maximizers for both suprema defining $\\smash{\\ell_p(\\mu,\\nu)}$ coincide, including the case $\\smash{\\Pi_\\leq(\\mu,\\nu)=\\emptyset}$. One advantage of the formulation via $l$ is that under (local) causal closedness and global hyperbolicity, $l^p$ is upper semicontinuous. In this case, customary optimal transport techniques \\cite{ambrosiogigli2013, villani2009} are applicable to study the second problem, which in turn yields corresponding results for the first \\cite[Rem.~2.2]{cavalletti2020}. Note that the preimages $l^{-1}([0,\\infty))$ and $l^{-1}((0,\\infty))$ encode causality and chronology of points in $\\mms^2$, respectively.\n\\end{remark}\n\nA coupling $\\pi\\in\\Pi(\\mu,\\nu)$ of $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ is called \\emph{$\\ell_p$-optimal} if $\\pi\\in\\Pi_\\leq(\\mu,\\nu)$ and \n\\begin{align*}\n\\ell_p(\\mu,\\nu) = \\Vert\\uptau\\Vert_{\\Ell^p(\\mms^2,\\pi)} = \\Vert l \\Vert_{\\Ell^p(\\mms^2,\\pi)}.\n\\end{align*}\nFor existence of such couplings, for our work it will suffice to keep the following in mind. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is locally causally closed and globally hyperbolic, and if $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ with $\\Pi_\\leq(\\mu,\\nu)\\neq\\emptyset$, existence of an $\\smash{\\ell_p}$-optimal coupling $\\pi$ of $\\mu$ and $\\nu$ holds, and $\\ell_p(\\mu,\\nu) <\\infty$ \\cite[Prop.~2.3]{cavalletti2020}.\n\nA key property of $\\smash{\\ell_p}$ is the \\emph{reverse triangle inequality} \\cite[Prop. 2.5]{cavalletti2020} somewhat reminiscent of \\eqref{Eq:Reverse tau} and of $l$ satisfying the reverse triangle inequality for \\emph{every}, i.e.~not necessarily causally related, $x,y,z\\in\\mms$: for \\emph{every} $\\mu,\\nu,\\sigma\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$,\n\\begin{align}\\label{Eq:Reverse triangle lp}\n\\ell_p(\\mu,\\sigma) \\geq \\ell_p(\\mu,\\nu) + \\ell_p(\\nu,\\sigma).\n\\end{align}\n\n\\subsubsection{Timelike $p$-dualizability}\\label{Sub:Timelike dual} The concept of so-called \\textit{\\textnormal{(}strong\\textnormal{)} timelike $p$-dualiza\\-bi\\-lity}, where $p\\in(0,1]$, of pairs $(\\mu,\\nu)\\in{\\ensuremath{\\mathscr{P}}}(\\mms)^2$ originates in \\cite{cavalletti2020} and generalizes the notion of \\emph{$p$-separation} introduced in \\cite[Def.~4.1]{mccann2020}. Pairs satisfying this condition admit good duality properties \\cite[Prop.~2.19, Prop.~2.21, Thm.~2.26]{cavalletti2020}, which leads to a characterization of $\\smash{\\ell_p}$-geodesics (see \\autoref{Sub:Geodesics} below) in the smooth case \\cite[Thm.~4.3, Thm.~5.8]{mccann2020}, cf.~\\autoref{Th:Equiv}.\n\nIn view of the following \\autoref{Def:TL DUAL}, cf.~\\cite[Def.~2.18, Def.~2.27]{cavalletti2020}, we refer to \\cite[Def.~2.6]{cavalletti2020} for the definition of \\emph{cyclical monotonicity} of a set in $\\smash{\\mms_\\leq^2}$ with respect to $l^p$, see also \\cite[Def.~5.1]{villani2009}. It will not be relevant in our work.\n\nGiven $a,b\\colon\\mms\\to\\R$ we define $a\\oplus b\\colon\\mms^2\\to\\R$ by $(a\\oplus b)(x,y) := a(x) + b(y)$.\n\n\\begin{definition}\\label{Def:TL DUAL} Given any $p\\in (0,1]$, a pair $(\\mu,\\nu)\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ is called \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item \\emph{timelike $p$-dua\\-lizable} by $\\pi\\in \\Pi_\\ll(\\mu,\\nu)$ if $\\pi$ is $\\smash{\\ell_p}$-optimal, and there exist Borel functions $a,b\\colon\\mms\\to\\R$ such that $a\\oplus b\\in\\Ell^1(\\mms^2,\\mu\\otimes\\nu)$ as well as $l^p\\leq a\\oplus b$ on $\\supp\\mu\\times\\supp\\nu$,\n\\item \\emph{strongly timelike $p$-dualizable} by $\\pi\\in\\Pi_\\ll(\\mu,\\nu)$ if the pair $(\\mu,\\nu)$ is timelike $p$-dualizable by $\\pi$, and there is an $l^p$-cyclically monotone Borel set $\\Gamma\\subset \\smash{\\mms_\\ll^2\\cap(\\supp\\mu_0\\times\\supp\\mu_1)}$ such that every given $\\sigma\\in \\Pi_\\leq(\\mu_0,\\mu_1)$ is $\\ell_p$-optimal if and only if $\\sigma[\\Gamma]=1$ \\textnormal{(}in particular, $\\sigma$ is chronological\\textnormal{)}, and\n\\item \\emph{timelike $p$-dualizable} if $(\\mu,\\nu)$ is timelike $p$-dualizable by some coupling $\\pi\\in\\Pi_\\ll(\\mu,\\nu)$; analogously for strong timelike $p$-dualizability.\n\\end{enumerate}\n\nMoreover, any $\\pi$ as in the above items is called \\emph{timelike $p$-dualizing}. \n\\end{definition}\n\n\n\\begin{remark}\\label{Re:Strong timelike} If $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ then the pair $(\\mu,\\nu)$ is timelike $p$-dualizable if and only if there is an $\\ell_p$-optimal coupling $\\smash{\\pi\\in\\Pi_\\leq(\\mu,\\nu)}$ which is concentrated on $\\smash{\\mms_\\ll^2}$.\n\nIf $\\mu,\\nu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ on a causally closed, globally hyperbolic Lorentzian geodesic space $(\\mms,\\met,\\ll,\\leq,\\uptau)$ with $\\smash{\\supp\\mu\\times\\supp\\nu\\subset\\mms_\\ll^2}$, then $(\\mu,\\nu)$ is automatically strongly timelike $p$-dualizable, $p\\in (0,1]$ \\cite[Cor.~2.29]{cavalletti2020}, see also \\cite[Lem.~4.4, Thm.~7.1]{mccann2020}. \n\\end{remark}\n\n\\subsubsection{Geodesics revisited}\\label{Sub:Geodesics} The definitions we consider need the notion of an \\emph{$\\smash{\\ell_p}$-geodesic}, $p\\in (0,1]$, to be made precise. We propose \\autoref{Def:Lorentzian geodesic} as Lorentzian version of Wasserstein geodesics in metric spaces. Precise technical results will be deferred to \\autoref{App:B}.\n\nSolely in this subsection, for simplicity we assume compactness of $\\mms$; the results derived here and in \\autoref{App:B} will mostly be used by replacing $\\mms$ by a causal diamond of two compact sets, which is compact under ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity.\n\nRecall the continuous reparametrization map $\\smash{{\\ensuremath{\\mathsf{r}}}\\colon \\mathrm{TGeo}(\\mms) \\to \\mathrm{TGeo}^\\uptau(\\mms)}$ and the definition $\\mathrm{TGeo}^\\uptau(\\mms) := {\\ensuremath{\\mathsf{r}}}(\\mathrm{TGeo}(\\mms))$ from \\autoref{Sub:GEO}. If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is causally closed, the set $\\Geo(\\mms)$ is Polish \\cite[Prop.~3.17]{kunzinger2018}. Since $\\mathrm{TGeo}(\\mms)$ is open, $\\mathrm{TGeo}^\\uptau(\\mms)$ is a Suslin set, hence universally measurable. \n\nGiven any $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$, we define\n\\begin{align*}\n\\OptGeo_{\\ell_p}(\\mu_0,\\mu_1) &:= \\{{\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}(\\Geo(\\mms)) : (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\Pi_\\leq(\\mu_0,\\mu_1)\\\\\n&\\qquad\\qquad \\textnormal{ is }\\ell_p\\textnormal{-optimal}\\},\\\\\n\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1) &:= \\{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\OptGeo_{\\ell_p}(\\mu_0,\\mu_1) : (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}[\\mms_\\ll^2]=1\\},\\\\\n\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1) &:= {\\ensuremath{\\mathsf{r}}}_\\push\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1).\n\\end{align*}\nWe say that a measure ${\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}(\\Cont([0,1];\\mms))$ \\emph{represents} a curve $(\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$ provided $\\mu_t=(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ for every $t\\in[0,1]$.\n\n\\begin{definition}\\label{Def:Lorentzian geodesic} We term a weakly continuous path $(\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item \\emph{causal $\\ell_p$-geodesic} if it is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\OptGeo_{\\ell_p}(\\mu_0,\\mu_1)}$,\n\\item \\emph{timelike $\\ell_p$-geodesic} if it is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$, and\n\\item\\label{Spans} \\emph{timelike proper-time parametrized $\\ell_p$-geodesic} if it is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$.\n\\end{enumerate}\n\nA measure ${\\ensuremath{\\boldsymbol{\\pi}}}$ as in the last item will be called \\emph{$\\smash{\\ell_p}$-optimal geodesic plan}.\n\\end{definition}\n\n\\begin{remark}\\label{Re:Lip} If $(\\mms,\\met,\\ll,\\leq,\\uptau)$ is ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, every causal or timelike $\\smash{\\ell_p}$-geodesic connecting compactly supported $\\mu_0$ and $\\mu_1$ is Lipschitz continuous with respect to the $q$-Wasserstein metric $W_q$ induced by $(\\mms,\\met)$ for every $q\\in[1,\\infty]$.\n\\end{remark}\n\nBy construction, we have ${\\ensuremath{\\boldsymbol{\\pi}}}[\\mathrm{TGeo}(\\mms)]=1$ for every $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}(\\mu_0,\\mu_1)}$. Consequently, ${\\ensuremath{\\boldsymbol{\\pi}}}[\\mathrm{TGeo}^\\uptau(\\mms)]=1$ for every $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, and therefore ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\Cont([0,1];\\mms)$ obeys $\\uptau(\\gamma_s,\\gamma_t) = (t-s)\\,\\uptau(\\gamma_0,\\gamma_1)>0$ for every $s,t\\in [0,1]$ with $s 0\n\\end{align*}\nfor every timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ and every $s,t\\in[0,1]$ with $s0$, the rescaled measured Lorentzian pre-length space $(\\mms, a\\met, b\\meas,\\ll,\\leq, \\theta\\uptau)$ satisfies $\\smash{\\mathrm{TCD}_p^*(K\/\\theta^2,N)}$.\n\\end{enumerate}\n\nAnalogous statements are obeyed by the conditions $\\smash{\\mathrm{wTCD}_p^*(K,N)}$, $\\smash{\\mathrm{TCD}_p(K,N)}$, and $\\smash{\\mathrm{wTCD}_p(K,N)}$. \n\\end{proposition}\n\n\\begin{proof} Concerning \\ref{La:DieEINS}, for all four conditions consistency in $K$ follows by nondecreasingness of $\\smash{\\sigma_{K,N}^{(r)}(\\vartheta)}$ and $\\smash{\\tau_{K,N}^{{(r)}}(\\vartheta)}$ in $K\\in\\R$ for fixed $r\\in[0,1]$, $N\\in [1,\\infty)$, and $\\vartheta\\geq 0$. Consistency in $N$ is clear since the defining inequality, in either case, is asked to hold for every $N''\\geq N$, so is particularly satisfied for every $N''\\geq N'$.\n\nItem \\ref{La:DieZWEI} follows from the scaling properties \n\\begin{align*}\n\\sigma_{K\/\\theta^2,N'}^{(r)}(\\theta\\,\\vartheta) &= \\sigma_{K,N'}^{(r)}(\\vartheta),\\\\\n\\tau_{K\/\\theta^2,N'}^{(r)}(\\theta\\,\\vartheta) &= \\tau_{K,N'}^{(r)}(\\vartheta).\\qedhere\n\\end{align*}\n\\end{proof}\n\nLastly, recall that the $N$-R\u00e9nyi entropy ${\\ensuremath{\\mathscr{S}}}_N(\\mu)$, $N\\in[1,\\infty)$, at $\\mu\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ only depends on the $\\meas$-absolutely continuous part of $\\mu$, which might be trivial. Along those timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics in \\autoref{Def:TCD*} or \\autoref{Def:TCD}, this does not happen; in fact, under mild additional hypotheses, these always consist of $\\meas$-absolutely continuous measures. This is addressed in \\autoref{Le:Stu} and will be useful at various occasions. (We also use variants of it without explicit notice occasionally, e.g.~in \\autoref{Sub:Local global}.) Its proof uses \\eqref{Eq:Ent SN limit} and is analogous to the one of \\cite[Prop.~1.6]{sturm2006b}, hence omitted. \n\n\\begin{lemma}\\label{Le:Stu} Fix $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\mu_1 \\,\\meas)\\in({\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap \\Dom(\\Ent_\\meas))^2$, let $\\smash{\\mu\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)}$, and let $\\smash{\\pi\\in\\Pi(\\mu_0,\\mu_1)}$ be a coupling of $\\mu_0$ and $\\mu_1$ such that for every $N'\\geq N$, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu)&\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad - \\int_{\\mms^2} \\sigma_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N'}\\d\\pi(x^0,x^1).\n\\end{align*}\nThen $\\smash{\\mu = \\rho\\,\\meas\\in\\Dom(\\Ent_\\meas)}$ with\n\\begin{align*}\n\\Ent_\\meas(\\mu) \\leq (1-t)\\Ent_\\meas(\\mu_0) + t\\Ent_\\meas(\\mu_1) - \\frac{K}{2}\\,t(1-t)\\,\\big\\Vert \\uptau\\big\\Vert_{\\Ell^2(\\mms^2,\\pi)}^2.\n\\end{align*}\n\\end{lemma}\n\n\\begin{remark} Using \\autoref{Le:Villani lemma for geodesic}, the assumptions in \\autoref{Le:Stu} are satisfied for $\\mu := \\mu_t$, $t\\in[0,1]$, and $\\pi$ if these objects are coming from a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic as well as a timelike $p$-dualizing coupling, respectively, witnessing the inequality defining $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{remark}\n\n\\begin{corollary} For every $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$, the $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition implies $\\smash{\\mathrm{wTCD}_p(K,\\infty)}$ in the sense of \\textnormal{\\cite[Def.~4.1]{braun2022}}.\n\\end{corollary}\n\n\n\\subsection{Geometric inequalities} Now we derive fundamental geometric inequalities from the conditions introduced in \\autoref{Ch:TCD conditions}. We start with the more restrictive $\\smash{\\mathrm{wTCD}_p(K,N)}$-case which implies sharp versions of these facts, cf.~\\autoref{Cor:HD}; \\autoref{Sub:Versions reduced} provides nonsharp extensions for $\\smash{\\mathrm{wTCD}_p^*(K,N)}$-spaces. All proofs are variations of standard arguments \\cite{sturm2006b}, see also \\cite{bacher2010,cavalletti2020}.\n\nSome of the geometric inequalities to follow hold in fact under weaker timelike measure-contraction properties, cf.~\\autoref{Pr:TMCP to TCD} and \\autoref{Re:Geom inequ TMCP}.\n\nMoreover, all results in this section also hold if ${\\ensuremath{\\mathscr{X}}}$ is a locally causally closed and globally hyperbolic measured Lorentzian geodesic space.\n\n\\subsubsection{Sharp timelike Brunn--Minkowski inequality}\n\nTo introduce the \\emph{timelike Brunn--Minkowski inequality} in the next \\autoref{Pr:Brunn-Minkowski}, given $A_0,A_1\\subset \\mms$ and $t\\in [0,1]$, define the set of timelike $t$-intermediate points between $A_0$ and $A_1$ as\n\\begin{align*}\nA_t := \\{\\gamma_t : \\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms),\\, \\gamma_0 \\in A_0,\\, \\gamma_1\\in A_1\\}.\n\\end{align*}\nFurthermore, define \n\\begin{align}\\label{Eq:THETA}\n\\Theta := \\begin{cases} \\sup\\uptau(A_0\\times A_1) & \\textnormal{if }K<0,\\\\\n\\inf\\uptau(A_0\\times A_1) & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align}\n\n\\begin{proposition}\\label{Pr:Brunn-Minkowski} Assume $\\smash{\\mathrm{wTCD}_p(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in [1,\\infty)$. Let $A_0,A_1\\subset\\mms$ be relatively compact Borel sets with $\\meas[A_0]\\,\\meas[A_1] >0$, and assume strong timelike $p$-dualizability of \n\\begin{align*}\n(\\mu_0,\\mu_1) := (\\meas[A_0]^{-1}\\,\\meas\\mres A_0,\\meas[A_1]^{-1}\\, \\meas\\mres A_1).\n\\end{align*}\nThen for every $t\\in [0,1]$ and every $N'\\geq N$, \n\\begin{align}\\label{Eq:BM}\n\\meas[A_t]^{1\/N'} \\geq \\tau_{K,N'}^{(1-t)}(\\Theta)\\,\\meas[A_0]^{1\/N'} + \\tau_{K,N'}^{(t)}(\\Theta)\\,\\meas[A_1]^{1\/N'}.\n\\end{align}\n\nAssuming $\\smash{\\mathrm{TCD}_p(K,N)}$ in place of $\\smash{\\mathrm{wTCD}_p(K,N)}$, the same conclusion holds if $(\\mu_0,\\mu_1)$ is merely timelike $p$-dualizable.\n\\end{proposition}\n\n\\begin{proof} Without loss of generality, we assume $N' > 1$ and $\\meas[A_t] < \\infty$. Let a time\\-like proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ to $\\mu_1$ and a timelike $p$-dualizing $\\pi\\in\\smash{\\Pi_\\ll(\\mu_0,\\mu_1)}$ witness the semiconvexity inequality defining $\\smash{\\mathrm{wTCD}_p(K,N)}$. Note that $\\supp\\mu_t\\subset A_t$ and, by $\\smash{\\mathrm{wTCD}_p(K,N)}$, that $\\meas[A_t]>0$. Let $\\rho_t$ denote the density of the absolutely continuous part of $\\mu_t$ with respect to $\\meas$. Employing Jensen's inequality, and $\\smash{\\mathrm{wTCD}_p(K,N)}$ again we obtain\n\\begin{align*}\n&\\tau_{K,N}^{(1-t)}(\\Theta)\\,\\meas[A_0]^{1\/N'} + \\tau_{K,N}^{(t)}(\\Theta)\\,\\meas[A_1]^{1\/N'}\\\\\n&\\qquad\\qquad\\leq \\int_{A_t}\\rho_t^{1-1\/N'}\\d\\meas \\leq \\meas[A_t]\\,\\Big[\\!\\fint_{A_t}\\rho_t\\d\\meas\\Big]^{1-1\/N'} \\leq \\meas[A_t]^{-1\/N'},\n\\end{align*}\nwhich is the desired claim.\n\nThe proof of the last statement is completely analogous.\n\\end{proof}\n\n\n\\begin{remark} Note that in the case $K\\geq 0$, \\eqref{Eq:BM} implies\n\\begin{align*}\n\\meas[A_t]^{1\/N'} \\geq (1-t)\\,\\meas[A_0]^{1\/N'} + t\\,\\meas[A_1]^{1\/N'}.\n\\end{align*}\n\\end{remark}\n\n\\begin{remark} Recall from \\autoref{Re:Strong timelike} that the strong timelike $p$-dualizability hy\\-pothesis for $(\\mu_0,\\mu_1)$ in \\autoref{Pr:Brunn-Minkowski} is satisfied if $\\smash{A_0 \\times A_1 \\subset \\mms_\\ll^2}$ provided the space ${\\ensuremath{\\mathscr{X}}}$ is locally causally closed, globally hyperbolic, and geodesic.\n\\end{remark}\n\n\n\\subsubsection{Sharp timelike Bonnet--Myers inequality} Next, an immediate consequence of \\autoref{Pr:Brunn-Minkowski} is the subsequent \\emph{timelike Bonnet--Myers inequality}.\n\n\\begin{corollary}\\label{Cor:Bonnet-Myers} Assume the $\\smash{\\mathrm{wTCD}_p(K,N)}$ condition for some $p\\in(0,1)$, $K>0$, and $N\\in [1,\\infty)$. Then \n\\begin{align*}\n\\sup\\uptau(\\mms^2) \\leq \\pi\\sqrt{\\frac{N-1}{K}}.\n\\end{align*}\n\\end{corollary}\n\n\\begin{proof} Suppose to the contrapositive that for some $\\varepsilon > 0$, we have\n\\begin{align*}\n\\uptau(z_0,z_1) \\geq \\pi\\sqrt{\\frac{N-1}{K}}+4\\varepsilon\n\\end{align*}\nfor two given points $z_0,z_1\\in\\mms$. By continuity of $\\uptau$, we fix $\\delta > 0$ and $x_0,x_1\\in\\mms$ such that $\\smash{A_0 := {\\ensuremath{\\mathsf{B}}}^\\met(x_0,\\delta) \\subset I^+(z_0)}$, $\\smash{A_1 := {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta) \\subset I^-(z_1)}$, and \n\\begin{align*}\n\\inf\\uptau({\\ensuremath{\\mathsf{B}}}^\\met(x_0,\\delta) \\times {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta)) \\geq \\pi\\sqrt{\\frac{N-1}{K}}+\\varepsilon.\n\\end{align*}\nThis implies strong timelike $p$-dualizability of $\\smash{(\\meas[A_0]^{-1}\\,\\meas\\mres{A_0},\\meas[A_1]^{-1}\\,\\meas\\mres{A_1})}$ by \\autoref{Re:Strong timelike}. Hence \\autoref{Pr:Brunn-Minkowski} is applicable, and the inherent inequality $\\smash{\\Theta \\geq \\pi\\sqrt{(N-1)\/K}+\\varepsilon}$ together with the definition of $\\smash{\\tau_{K,N}^{(r)}}$ gives $\\smash{\\meas[A_{1\/2}]=\\infty}$. On the other hand, the set $\\smash{A_{1\/2}}$ is relatively compact since $\\smash{A_{1\/2} \\subset J(z_0,z_1)}$ by global hy\\-per\\-bolicity, which makes the situation $\\meas[A_{1\/2}]=\\infty$ impossible.\n\\end{proof}\n\n\\begin{remark} \\autoref{Cor:Bonnet-Myers} implies that, under its assumptions, the $\\uptau$-length of every causal curve in $\\mms$ is no larger than $\\smash{\\pi\\sqrt{(N-1)\/K}}$. Moreover, if $K>0$ and $N=1$ then $\\mms$ does not contain any chronologically related pair of points.\n\\end{remark}\n\n\\subsubsection{Sharp timelike Bishop--Gromov inequality}\\label{Sub:Sharp BG inequ} To prove volume growth estimates \u00e0 la \\emph{Bishop--Gromov}, cf.~\\autoref{Th:BG} below, we recall the notion of $\\uptau$-star-shaped sets from \\cite[Sec.~3.1]{cavalletti2020}; this is used to localize the volume of $\\uptau$-balls \n\\begin{align}\\label{Eq:TAU BALLS}\n{\\ensuremath{\\mathsf{B}}}^\\uptau(x,r) := \\{y\\in I^+(x) : \\uptau(x,y)< r\\}\\cup\\{x\\},\n\\end{align}\nwith $x\\in\\mms$ and $r>0$, which typically have infinite volume. \n\nA set $\\smash{E\\subset I^+(x)\\cup\\{x\\}}$ is termed \\emph{$\\uptau$-star-shaped} with respect to $x\\in\\mms$ if for every $\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$ with $\\gamma_0 = x$ and $\\gamma_1 \\in E$, we have $\\gamma_t\\in E$ for every $t\\in (0,1)$. For such $E$ and $x$, and given $r>0$, we define the quantities\n\\begin{align*}\n{\\ensuremath{\\mathrm{v}}}_r &:= \\meas\\big[\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,r)\\cap E\\big],\\\\\n{\\ensuremath{\\mathrm{s}}}_r &:= \\limsup_{\\delta \\to 0} \\delta^{-1}\\,\\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,r+\\delta)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,r))\\cap E\\big].\n\\end{align*}\nWhenever confusion is excluded, we avoid making notationally explicit the dependency of ${\\ensuremath{\\mathrm{v}}}_r$ and ${\\ensuremath{\\mathrm{s}}}_r$ on $E$ and $x$. Lastly, we set\n\\begin{align}\\label{Eq:integral def}\n\\mathfrak{v}_{K,N}(r) := \\Big[\\!\\int_0^r \\mathfrak{s}_{K,N-1}(r)^{N-1}\\d r\\Big]^{1\/N}.\n\\end{align}\n\nIn the next result, we employ the convention $1\/0 := \\infty$.\n\n\\begin{theorem}\\label{Th:BG} Assume $\\smash{\\mathrm{wTCD}_p(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in (1,\\infty)$. Let $\\smash{E\\subset I^+(x)\\cup\\{x\\}}$ be a compact set which is $\\uptau$-star-shaped with respect to $x\\in\\mms$. Then for every $r,R > 0$ with $\\smash{r < R \\leq \\pi\\sqrt{(N-1)\/\\max\\{K,0\\}}}$,\n\\begin{align*}\n\\frac{{\\ensuremath{\\mathrm{s}}}_r}{{\\ensuremath{\\mathrm{s}}}_R} &\\geq \\Big[\\frac{\\mathfrak{s}_{K,N-1}(r)}{\\mathfrak{s}_{K,N-1}(R)}\\Big]^{N-1},\\\\\n\\frac{{\\ensuremath{\\mathrm{v}}}_r}{{\\ensuremath{\\mathrm{v}}}_R} &\\geq \\Big[\\frac{\\mathfrak{v}_{K,N}(r)}{\\mathfrak{v}_{K,N}(R)}\\Big]^N.\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof} We assume $K>0$, the rest is argued similarly. Define $t := r\/R$, let $\\delta > 0$, and let $\\varepsilon > 0$ be small enough such that $\\smash{A_0 \\times A_1 \\subset \\mms_\\ll^2}$, where $\\smash{A_0 := {\\ensuremath{\\mathsf{B}}}^\\uptau(x,\\varepsilon)\\cap E}$ and $A_1 := (\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,R+\\delta R) \\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,R))\\cap E$. Thus $(\\meas[A_0]^{-1}\\,\\meas\\mres{A_0},\\meas[A_1]^{-1}\\,\\meas\\mres{A_1})$ is strongly time\\-like $p$-dualizable according to \\autoref{Re:Strong timelike}. Now we observe that $\\smash{A_t \\subset ({\\ensuremath{\\mathsf{B}}}^\\uptau(x,r+\\delta r)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,r))\\cap E}$. By the reverse triangle inequality for $\\uptau$, we get $\\Theta \\leq R+\\delta R$. Employing \\autoref{Pr:Brunn-Minkowski}, we therefore obtain\n\\begin{align*}\n&\\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,r+\\delta r)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,r))\\cap E\\big]^{1\/N}\\\\\n&\\qquad\\qquad \\geq \\tau_{K,N}^{(1-r\/R)}(R+\\delta R)\\,\\meas\\big[{\\ensuremath{\\mathsf{B}}}^\\uptau(x,\\varepsilon)\\cap E\\big]^{1\/N}\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\tau_{K,N}^{(r\/R)}(R+\\delta R)\\, \\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,R+\\delta R)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,R))\\cap E\\big]^{1\/N}\\\\\n&\\qquad\\qquad \\geq \\tau_{K,N}^{(r\/R)}(R+\\delta R)\\, \\meas\\big[(\\bar{{\\ensuremath{\\mathsf{B}}}}^\\uptau(x,R+\\delta R)\\setminus {\\ensuremath{\\mathsf{B}}}^\\uptau(x,R))\\cap E\\big]^{1\/N}.\n\\end{align*}\nWriting out the expression $\\smash{\\tau_{K,N}^{(r\/R)}(R+\\delta R)^N}$ and finally sending $\\delta\\to 0$ implies the claimed lower bound for ${\\ensuremath{\\mathrm{s}}}_r\/{\\ensuremath{\\mathrm{s}}}_R$.\n\nAs in the proof of \\cite[Thm.~2.3]{sturm2006b}, we argue that ${\\ensuremath{\\mathrm{v}}}$ is locally Lipschitz continuous on $(0,\\infty)$. Hence, it is differentiable $\\smash{\\Leb^1}$-a.e.~on $(0,\\infty)$, and $\\smash{{\\ensuremath{\\mathrm{v}}}_r = \\int_0^r \\dot{{\\ensuremath{\\mathrm{v}}}}_s\\d s}$ for every $r > 0$. Therefore, the claimed lower bound on ${\\ensuremath{\\mathrm{v}}}_r\/{\\ensuremath{\\mathrm{v}}}_R$ follows from the previous part in com\\-bination with \\cite[Lem.~18.9]{villani2009}.\n\\end{proof}\n\n\\begin{remark} If $K=0$, the estimates from \\autoref{Th:BG} become\n\\begin{align*}\n\\frac{{\\ensuremath{\\mathrm{s}}}_r}{{\\ensuremath{\\mathrm{s}}}_R} &\\geq \\Big[\\frac{r}{R}\\Big]^{N-1},\\\\\n\\frac{{\\ensuremath{\\mathrm{v}}}_r}{{\\ensuremath{\\mathrm{v}}}_R} &\\geq \\Big[\\frac{r}{R}\\Big]^N.\n\\end{align*}\nIn fact, these estimates hold for $N=1$ as well.\n\\end{remark}\n\nIncidentally, from \\autoref{Th:BG} we deduce the following natural and sharp upper bound on the $\\uptau$-Hausdorff dimension as introduced and studied in \\cite{mccann2021}. Our results improve \\cite[Thm.~5.2, Cor.~5.3]{mccann2021}. We refer to \\cite{mccann2021} for details about the notions presented in the subsequent \\autoref{Cor:HD}.\n\n\\begin{corollary}\\label{Cor:HD} Let $(\\mms,\\langle\\cdot,\\cdot\\rangle)$ be a continuous, globally hyperbolic, causally plain spacetime of dimension $n\\in\\N$ \\textnormal{(}which is thus causally closed \\cite[Prop.~3.5]{kunzinger2018} and geodesic \\cite[Thm.~3.30, Thm.~5.12]{kunzinger2018}\\textnormal{)}. Assume its induced Lorentzian geodesic space, with $\\meas$ being the Lorentzian volume measure induced by $\\langle\\cdot,\\cdot\\rangle$, to obey $\\smash{\\mathrm{wTCD}_p(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Then the geometric dimension $\\smash{\\dim^\\uptau\\mms}$ in the sense of \\textnormal{\\cite[Def.~3.1]{mccann2021}} satisfies\n\\begin{align*}\nn=\\dim^\\uptau\\mms \\leq N.\n\\end{align*}\n\\end{corollary}\n\n\\begin{remark}\\label{Re:HD2} As in \\cite[Thm.~5.2]{mccann2021}, which is proven under $\\smash{\\mathrm{wTCD}_p^e(K,N)}$, assuming $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ instead of $\\smash{\\mathrm{wTCD}_p(K,N)}$ in \\autoref{Cor:HD} would only lead to the conclusion $n = \\dim^\\uptau\\mms \\leq N+1$ by \\autoref{Th:Reduced BG}.\n\\end{remark}\n\n\\subsubsection{Versions in the reduced case}\\label{Sub:Versions reduced} With evident modifications, the following nonsharp versions of \\autoref{Pr:Brunn-Minkowski}, \\autoref{Cor:Bonnet-Myers}, \\autoref{Th:BG}, and \\autoref{Cor:HD} are readily verified under the more general $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition. The deduced inequalities are analogous to their counterparts drawn from the $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ condition in \\cite[Prop.~3.4, Prop.~3.5, Prop.~3.6]{cavalletti2020}.\n\nIn view of \\autoref{Th:Reduced BG} below, recall the definition of $\\mathfrak{v}_{K,N}$ from \\eqref{Eq:integral def}.\n\n\\begin{proposition}\\label{Propos} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $A_0,A_1\\subset\\mms$ be relatively compact Borel sets which satisfy $\\meas[A_0]\\,\\meas[A_1]>0$, and assume strong timelike $p$-dualizability of \n\\begin{align*}\n(\\mu_0,\\mu_1) := (\\meas[A_0]^{-1}\\,\\meas\\mres A_0, \\meas[A_1]^{-1}\\,\\meas\\mres A_1).\n\\end{align*}\nLet $\\Theta$ be as in \\eqref{Eq:THETA}. Then for every $t\\in [0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n\\meas[A_t]^{1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\Theta)\\,\\meas[A_0]^{1\/N'} + \\sigma_{K,N'}^{(t)}(\\Theta)\\,\\meas[A_1]^{1\/N'}.\n\\end{align*}\n\nAssuming $\\smash{\\mathrm{TCD}_p^*(K,N)}$ in place of $\\smash{\\mathrm{wTCD}_p^*(K,N)}$, the same conclusion holds if $(\\mu_0,\\mu_1)$ is merely timelike $p$-dualizable.\n\\end{proposition}\n\n\\begin{corollary}\\label{Cor:Reduced BM} Assume the $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition for some $p\\in(0,1)$, $K>0$, and $N\\in [1,\\infty)$. Then \n\\begin{align*}\n\\sup\\uptau(\\mms^2) \\leq \\pi\\sqrt{\\frac{N}{K}}.\n\\end{align*}\n\\end{corollary}\n\n\\begin{theorem}\\label{Th:Reduced BG} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in (1,\\infty)$. Let $\\smash{E\\subset I^+(x)\\cup\\{x\\}}$ be a compact set which is $\\uptau$-star-shaped with respect to $x\\in\\mms$. Then for every $r,R > 0$ with $\\smash{r < R \\leq \\pi\\sqrt{N\/\\max\\{K,0\\}}}$,\n\\begin{align*}\n\\frac{{\\ensuremath{\\mathrm{s}}}_r}{{\\ensuremath{\\mathrm{s}}}_R} &\\geq \\Big[\\frac{\\mathfrak{s}_{K,N}(r)}{\\mathfrak{s}_{K,N}(R)}\\Big]^N,\\\\\n\\frac{{\\ensuremath{\\mathrm{v}}}_r}{{\\ensuremath{\\mathrm{v}}}_R} &\\geq \\Big[\\frac{\\mathfrak{v}_{K,N+1}(r)}{\\mathfrak{v}_{K,N+1}(R)}\\Big]^{N+1}.\n\\end{align*}\n\\end{theorem}\n\n\\subsection{Stability}\\label{Sec:Stability TCD} In this section, we prove a key property of our timelike curvature-dimension conditions, namely the weak stability of the notions from \\autoref{Def:TCD*} and \\autoref{Def:TCD}, cf.~\\autoref{Th:Stability TCD}. The relevant notion of convergence of measured Lorentzian pre-length spaces, see \\autoref{Def:Convergence}, is due to \\cite[Sec. 3.3]{cavalletti2020}.\n\n\\begin{definition}\\label{Def:Lor isometry} For Lorentzian pre-length spaces $\\smash{(\\mms^i,\\met^i,\\ll^i,\\leq^i,\\uptau^i)}$, $i\\in\\{0,1\\}$, we term a map $\\smash{\\iota\\colon \\mms^0\\to\\mms^1}$ a \\emph{Lorentzian isometric embedding} if $\\iota$ is a topolo\\-gical embedding such that for every $x,y\\in\\mms^0$,\n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item $x\\leq^0 y$ if and only if $\\iota(x) \\leq^1\\iota(y)$, and\n\\item $\\tau^1(\\iota(x),\\iota(y)) = \\tau^0(x,y)$.\n\\end{enumerate}\n\\end{definition}\n\nGiven any $k\\in\\N_\\infty := \\N\\cup\\{\\infty\\}$ let $(\\mms_k,\\met_k,\\meas_k,\\ll_k,\\leq_k,\\uptau_k)$ be a fixed measured Lorentzian pre-length space which, as in \\eqref{Eq:X}, we will abbreviate by ${\\ensuremath{\\mathscr{X}}}_k$.\n\n\\begin{definition}\\label{Def:Convergence} We define $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ to converge to ${\\ensuremath{\\mathscr{X}}}_\\infty$ if the following holds.\n\\begin{enumerate}[label=\\textnormal{\\alph*\\textcolor{black}{.}}]\n\\item\\label{La:Blurr1reg} There exists a causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic measured Lorentzian geodesic space $\\smash{(\\mms,\\met,\\ll,\\leq,\\uptau)}$ such that for every $k\\in\\N_\\infty$ there exists a Lorentzian isometric embedding $\\smash{\\iota_k\\colon \\mms_k\\to \\mms}$.\n\\item In terms of the maps $\\iota_k$ from \\ref{La:Blurr1reg}, the sequence $((\\iota_k)_\\push\\meas_k)_{k\\in\\N}$ converges to $(\\iota_\\infty)_\\push\\meas_\\infty$ in duality with $\\Cont_\\comp(\\mms)$, i.e.~for every $\\varphi\\in\\Cont_\\comp(\\mms)$ we have\n\\begin{align*}\n\\lim_{k\\to\\infty}\\int_\\mms \\varphi\\d(\\iota_k)_\\push\\meas_k = \\int_\\mms \\varphi\\d(\\iota_\\infty)_\\push\\meas_\\infty.\n\\end{align*}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} If $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ converges to ${\\ensuremath{\\mathscr{X}}}_\\infty$ according to \\autoref{Def:Convergence}, ${\\ensuremath{\\mathscr{X}}}_k$ is automatically causally closed, ${\\ensuremath{\\mathscr{K}}}$-globally hyperbolic, and geodesic for every $k\\in\\N_\\infty$.\n\\end{remark}\n\n\n\nThe following \\autoref{Le:CM lemma} is an implicit consequence from the proof of \\cite[Lem. 3.5]{cavalletti2020} and will be useful for the proof of \\autoref{Th:Stability TCD}. \n\n\\begin{lemma}\\label{Le:CM lemma} Assume that ${\\ensuremath{\\mathscr{X}}}$ is causally closed, globally hyperbolic, and geodesic. Let the pair $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$ admit some $\\smash{\\ell_p}$-optimal coupling $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$, $p\\in (0,1]$. Then there exist sequences $(\\pi^n)_{n\\in\\N}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms^2)$ and $(a_n)_{n\\in\\N}$ in $[1,\\infty)$ with the properties \n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item $(a_n)_{n\\in\\N}$ converges to $1$,\n\\item $\\pi^n[\\mms_\\ll^2]=1$ for every $n\\in\\N$,\n\\item $\\smash{\\pi^n = \\rho^n\\,\\meas^{\\otimes 2}\\in {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2,\\meas^{\\otimes 2})}$ with $\\smash{\\rho^n\\in\\Ell^\\infty(\\mms^2,\\meas^{\\otimes 2})}$ for every $n\\in\\N$,\n\\item $(\\pi^n)_{n\\in\\N}$ converges weakly to $\\pi$, \n\\item\\label{DENS} the density $\\smash{\\rho_0^n}$ and $\\smash{\\rho_0^n}$ of the first and second marginal of $\\pi_n$ with respect to $\\meas$ is no larger than $a_n\\,\\rho_0$ and $a_n\\,\\rho_1$, respectively, for every $n\\in\\N$, and\n\\item $\\smash{\\rho_0^n\\to \\rho_0}$ and $\\smash{\\rho_1^n\\to\\rho_1}$ in $\\Ell^1(\\mms,\\meas)$ as $n\\to\\infty$.\n\\end{enumerate}\n\\end{lemma}\n\nFurthermore, in the notation of \\autoref{Le:CM lemma}, given any $K\\in\\R$, $N\\in[1,\\infty)$, and $t\\in [0,1]$, as well as a measure $\\pi\\in\\Pi(\\mu_0,\\mu_1)$ with marginals $\\mu_0 = \\rho_0\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ and $\\mu_1 = \\rho_1\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, for brevity we define\n\\begin{align}\\label{Eq:TDdef}\n\\begin{split}\n{\\ensuremath{\\mathscr{T}}}_{K,N}^{(t)}(\\pi) &:= -\\int_{\\mms^2} \\tau_{K,N}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\tau_{K,N}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N}\\d\\pi(x^0,x^1).\n\\end{split}\n\\end{align}\n\nWe note the following variant of \\cite[Lem.~3.2]{sturm2006b}. The latter proof easily carries over to the setting of \\autoref{Le:Const perturb}, where the marginal densities, instead of being assumed to be the same throughout the sequence as in \\cite{sturm2006b}, are allowed to be perturbed by constants vanishing in the limit. Compare with \\autoref{Le:USC lemma}.\n\n\\begin{lemma}\\label{Le:Const perturb} Suppose $\\meas\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$. Let $(a_k)_{k\\in\\N}$ be a sequence in $[1,\\infty)$ converging to $1$, and let $(b_k)_{k\\in\\N}$ be a sequence of nonnegative real numbers converging to $0$. Define $\\smash{\\nu_{k,0},\\nu_{k,1}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, $k\\in\\N$, by\n\\begin{align*}\n\\nu_{k,0} &:= (a_k+b_k)^{-1}\\,(a_k\\,\\mu_0 + b_k\\,\\meas) = (a_k+b_k)^{-1}\\,(a_k\\,\\rho_0 + b_k)\\,\\meas,\\\\\n\\nu_{k,1} &:= (a_k+b_k)^{-1}\\,(a_k\\,\\mu_1 + b_k\\,\\meas) = (a_k+b_k)^{-1}\\,(a_k\\,\\rho_1 + b_k)\\,\\meas.\n\\end{align*}\nThen for every sequence $(\\pi_k)_{k\\in\\N}$ of measures $\\pi_k\\in \\Pi(\\nu_{k,0},\\nu_{k,1})$ which converges weakly to $\\pi\\in\\Pi(\\mu_0,\\mu_1)$, every $K\\in\\R$, every $N\\in[1,\\infty)$, and every $t\\in[0,1]$, \n\\begin{align*}\n\\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N}^{(t)}(\\pi_k) \\leq {\\ensuremath{\\mathscr{T}}}_{K,N}^{(t)}(\\pi).\n\\end{align*}\n\\end{lemma}\n\n\\begin{remark} Of course, the assertion from \\autoref{Le:Const perturb} remains valid if $\\smash{\\tau_{K,N}^{(1-t)}}$ and $\\smash{\\tau_{K,N}^{(t)}}$ are replaced by $\\smash{\\sigma_{K,N}^{(1-t)}}$ and $\\smash{\\sigma_{K,N}^{(t)}}$ in \\eqref{Eq:TDdef}, respectively.\n\\end{remark}\n\nThe subsequent proof of \\autoref{Th:Stability TCD} below follows \\cite[Thm.~3.1]{sturm2006b}. However, by the nature of our timelike curvature-dimension conditions, we additionally have to ensure chronological relations of many measures under consideration. Hence, the proof becomes much longer and quite technical. It may be skipped at first reading; alternatively, the reader may consult the proof of the counterpart \\autoref{Th:Stability TMCP} of \\autoref{Th:Stability TCD} below for a somewhat easier argument in which one does not have to trace chronological $\\smash{\\ell_p}$-optimal couplings.\n\n\n\n\n\\begin{theorem}\\label{Th:Stability TCD} Assume the convergence of $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ to ${\\ensuremath{\\mathscr{X}}}_\\infty$ as in \\autoref{Def:Convergence}. Moreover, let $(K_k,N_k)_{k\\in\\N}$ be a sequence in $\\R\\times [1,\\infty)$ converging to $(K_\\infty,N_\\infty)\\in\\R\\times[1,\\infty)$. Suppose the existence of $p\\in (0,1)$ such that ${\\ensuremath{\\mathscr{X}}}_k$ obeys $\\mathrm{TCD}_p(K_k,N_k)$ for every $k\\in\\N$. \nThen $\\smash{{\\ensuremath{\\mathscr{X}}}_\\infty}$ satisfies $\\smash{\\mathrm{wTCD}_p(K_\\infty,N_\\infty)}$. \n\nThe analogous statement in which $\\smash{\\mathrm{TCD}_p(K_k,N_k)}$ and $\\smash{\\mathrm{wTCD}_p(K_\\infty,N_\\infty)}$ are replaced by $\\smash{\\mathrm{TCD}_p^*(K_k,N_k)}$ and $\\smash{\\mathrm{wTCD}_p^*(K_\\infty,N_\\infty)}$, $k\\in\\N$, respectively, holds too.\n\\end{theorem}\n\n\\begin{proof} We only prove the first implication, the second is similar. To relax notation, without restriction and further notice we identify $\\mms_k$ with its image $\\iota_k(\\mms_k)$ in $\\mms$, and the measure $\\meas_k$ with its push-forward $(\\iota_k)_\\push\\meas_k$ for every $k\\in\\N_\\infty$.\n\n\\textbf{Step 1.} \\textit{Reduction to compact $\\mms$.} Fix any strongly timelike $p$-dualizable pair $(\\mu_{\\infty,0},\\mu_{\\infty,1}) = (\\rho_{\\infty,0}\\,\\meas_\\infty,\\rho_{\\infty,1}\\,\\meas_\\infty)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas_\\infty)$. By compactness of $\\supp\\mu_{\\infty,0}$ and $\\supp\\mu_{\\infty,1}$, and by ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity of $\\smash{\\mms}$, we restrict all arguments below to a compact subset $\\smash{C\\subset \\mms}$ with $\\smash{\\meas_k[C]^{-1}\\,\\meas_k\\mres C \\to \\meas_\\infty[C]^{-1}\\,\\meas_\\infty\\mres C}$ weakly as $k\\to\\infty$. To relax notation, we will thus assume with no loss of generality that $\\smash{\\mms}$ itself is compact, and that $\\smash{\\meas_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms)}$ for every $k\\in\\N_\\infty$. In particular, without notationally expressing this property all the time, all measures will henceforth be compactly supported.\n\n\\textbf{Step 2.} \\textit{Restriction of the assumptions on $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$.} We will first assume that $\\rho_{\\infty,0},\\rho_{\\infty,1}\\in\\Ell^\\infty(\\mms,\\meas_\\infty)$. The general case is discussed in Step 8 below; we note for now that this conclusion will not conflict with our reductions from Step 1.\n\n\\textbf{Step 3.} \\textit{Construction of a chronological recovery sequence.} The goal of this step is the construction of a sequence $\\smash{(\\mu_{k,0},\\mu_{k,1})_{k\\in\\N}}$ of timelike $p$-dualizable pairs $\\smash{(\\mu_{k,0},\\mu_{k,1})\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)^2}$, $k\\in\\N$, such that $\\smash{\\mu_{k,0}\\to \\mu_{\\infty,0}}$ and $\\smash{\\mu_{k,1}\\to \\mu_{\\infty,1}}$ weakly as $k\\to\\infty$, up to a subsequence. The highly nontrivial part we address here is that such a pair can in fact be constructed to be chronological.\n\n\\textbf{Step 3.1.} Let $W_2$ denote the $2$-Wasserstein distance on $\\smash{(\\mms,\\met)}$. Since $(\\meas_k)_{k\\in\\N}$ converges weakly to $\\meas_\\infty$ and since $\\mms$ is compact, we have $W_2(\\meas_k,\\meas_\\infty)\\to 0$ as $k\\to\\infty$. Let $\\mathfrak{q}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ be a $W_2$-optimal coupling of $\\meas_k$ and $\\meas_\\infty$, $k\\in\\N$. For later use, we disintegrate $\\mathfrak{q}_k$ with respect to $\\pr_1$ and $\\pr_2$, respectively, writing\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}\\mathfrak{q}_k(x, y) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{p}_{x}^k(y)\\d\\meas_k(x) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{p}_{y}^\\infty(x)\\d\\meas_\\infty(y)\n\\end{align*}\nfor certain Borel maps $\\smash{\\mathfrak{p}^k \\colon \\mms\\to {\\ensuremath{\\mathscr{P}}}(\\mms)}$ and $\\smash{\\mathfrak{p}^\\infty \\colon \\mms\\to {\\ensuremath{\\mathscr{P}}}(\\mms)}$. (Although $\\mathfrak{p}^\\infty$ depends on $k$ as well, we do not make this explicit to not overload our notation.) These disin\\-te\\-grations induce nonrelabeled Borel maps $\\smash{\\mathfrak{p}^k\\colon {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)\\to {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ and $\\smash{\\mathfrak{p}^\\infty}\\colon {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)\\to {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)$ defined by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}\\mathfrak{p}^k(f\\,\\meas_\\infty)(x) &:= \\Big[\\!\\int_\\mms f(y)\\d\\mathfrak{p}_{x}^k(y) \\Big]\\d\\meas_k(x),\\\\\n{\\ensuremath{\\mathrm{d}}}\\mathfrak{p}^\\infty(g\\,\\meas_k)(y) &:= \\Big[\\!\\int_\\mms g(x)\\d\\mathfrak{p}_{y}^\\infty(x) \\Big]\\d\\meas_\\infty(y).\n\\end{align*} \n\n\\textbf{Step 3.2.} Let $\\pi_\\infty\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})$ be timelike $p$-dualizing, and let $(\\pi^n_\\infty)_{n\\in\\N}$ be an asso\\-ciated sequence in $\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2,\\meas_\\infty^{\\otimes 2})}$ as in \\autoref{Le:CM lemma}, where we write\n\\begin{align*}\n\\pi^n_\\infty &= \\rho_\\infty^n\\,\\meas_\\infty^{\\otimes 2},\\\\\n\\mu_{\\infty,0}^n := (\\pr_1)_\\push\\pi_\\infty^n &= \\rho_{\\infty,0}^n\\,\\meas_\\infty,\\\\\n\\mu_{\\infty,1}^n := (\\pr_2)_\\push\\pi_\\infty^n &= \\rho_{\\infty,1}^n\\,\\meas_\\infty.\n\\end{align*}\nFor $k\\in\\N$, define $\\smash{\\mu_{k,0}^n,\\mu_{k,1}^n\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ by\n\\begin{align*}\n\\mu_{k,0}^n &:= \\mathfrak{p}^k(\\mu_{\\infty,0}^n) = \\rho_{k,0}^n\\,\\meas_k,\\\\\n\\mu_{k,1}^n &:= \\mathfrak{p}^k(\\mu_{\\infty,1}^n) = \\rho_{k,1}^n\\,\\meas_k\n\\end{align*}\nas well as $\\smash{\\pi_k^n\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2,\\meas_k^{\\otimes 2})}$ by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}} \\pi_k^n(x^0,x^1) &:= \\Big[\\!\\int_{\\mms^2}\\rho_\\infty^n(y^0,y^1)\\d(\\mathfrak{p}_{x^0}^k\\otimes\\mathfrak{p}_{x^1}^k)(y^0,y^1)\\Big]\\d\\meas_k^{\\otimes 2}(x^0,x^1),\n\\end{align*}\nor in other words,\n\\begin{align*}\n\\pi_k^n = (\\pr_1,\\pr_3)_\\push\\big[(\\rho_\\infty^n \\circ (\\pr_2,\\pr_4))\\,\\mathfrak{q}_k\\otimes\\mathfrak{q}_k\\big].\n\\end{align*}\nA straightforward computation entails $\\smash{\\pi_k^n\\in\\Pi(\\mu_{k,0}^n,\\mu_{k,1}^n)}$. Also, for given $n\\in\\N$, weak convergence of $(\\mathfrak{q}_k)_{k\\in\\N}$ to the unique $W_2$-optimal (diagonal) coupling of $\\meas_\\infty$ and $\\meas_\\infty$ by tightness \\cite[Lem.~4.3, Lem.~4.4]{villani2009}, up to a nonrelabeled subse\\-quence, imply weak convergence of $\\smash{(\\pi_k^n)_{k\\in\\N}}$ to $\\smash{\\pi_\\infty^n}$ up to a nonrelabeled subsequence.\n\n\\textbf{Step 3.3.} By \\autoref{Le:CM lemma}, a tightness and a diagonal argument, passage to a subsequence, and relabeling of indices, we find $n\\colon\\N\\to\\N$ such that, setting\n\\begin{align*}\n\\tilde{\\pi}_k := \\pi_k^{n_k},\n\\end{align*}\nthe sequence $\\smash{(\\tilde{\\pi}_k)_{k\\in\\N}}$ converges weakly to $\\pi_\\infty$. By Portmanteau's theorem,\n\\begin{align}\\label{Eq:Portmanteau in stability}\n1 = \\pi_\\infty[\\mms_\\ll^2] \\leq \\liminf_{k\\to\\infty} \\tilde{\\pi}_k[\\mms_\\ll^2].\n\\end{align}\nHence, up passing to a subsequence, we henceforth assume that $\\tilde{\\pi}_k[\\mms_\\ll^2] > 0$ for every $k\\in\\N$. We write the mar\\-ginals $\\smash{\\tilde{\\mu}_{k,0},\\tilde{\\mu}_{k,1}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ of $\\tilde{\\pi}_k$ as\n\\begin{align*}\n\\tilde{\\mu}_{k,0} &= \\tilde{\\rho}_{k,0}\\,\\meas_k = \\rho_{k,0}^{n_k}\\,\\meas_k,\\\\\n\\tilde{\\mu}_{k,1} &= \\tilde{\\rho}_{k,1}\\,\\meas_k = \\rho_{k,1}^{n_k}\\,\\meas_k.\n\\end{align*}\n\n\n\n\\textbf{Step 3.4.} Now we \ndefine $\\smash{\\hat{\\pi}_k\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms^2)}$ by\n\\begin{align*}\n\\hat{\\pi}_k := \\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k\\mres \\mms^2_\\ll.\n\\end{align*}\nNote that $\\smash{(\\hat{\\pi}_k)_{k\\in\\N}}$ converges weakly to $\\smash{\\pi_\\infty}$, whence\n\\begin{align}\\label{Eq:Der Limes}\n\\lim_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p\\d\\hat{\\pi}_k = \\int_{\\mms^2}\\uptau^p\\d\\pi_\\infty.\n\\end{align}\nLet $\\smash{\\hat{\\mu}_{k,0},\\hat{\\mu}_{k,0}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ denote the marginals of $\\smash{\\hat{\\pi}_k}$, with\n\\begin{align*}\n\\hat{\\mu}_{k,0} &= \\hat{\\rho}_{k,0}\\,\\meas_k,\\\\\n\\hat{\\mu}_{k,1} &= \\hat{\\rho}_{k,1}\\,\\meas_k.\n\\end{align*}\n\nThese measures are chronologically related, but we do not know whether they have a chronological \\emph{$\\smash{\\ell_p}$-optimal} coupling, i.e.~whether they are timelike $p$-dualizable. To achieve this, note that $\\smash{\\ell_p(\\hat{\\mu}_{k,0},\\hat{\\mu}_{k,1})} \\in (0,\\infty)$ by chronology of $\\smash{\\hat{\\pi}_k}$ as well as compactness of $\\mms^2$. Let $\\smash{\\check{\\pi}_k\\in\\Pi_\\leq(\\hat{\\mu}_{k,0},\\hat{\\mu}_{k,1})}$ be an $\\smash{\\ell_p}$-optimal coupling. By weak convergence of $\\smash{(\\hat{\\pi}_k)_{k\\in\\N}}$, its marginal sequences $\\smash{(\\hat{\\mu}_{k,0})_{k\\in\\N}}$ and $\\smash{(\\hat{\\mu}_{k,1})_{k\\in\\N}}$ are tight, and so is $\\smash{(\\check{\\pi}_k)_{k\\in\\N}}$ \\cite[Lem.~4.4]{villani2009}. By Prokhorov's theorem, the latter converges weakly to some $\\smash{\\check{\\pi}_\\infty\\in\\Pi(\\mu_{\\infty,0},\\mu_{\\infty,1})}$; causal closedness of $\\mms$ and Portmanteau's theorem imply $\\smash{\\check{\\pi}_\\infty[\\mms_\\leq^2]=1}$. Moreover, \\eqref{Eq:Der Limes} gives\n\\begin{align}\\label{Eq:Lp calc}\n\\begin{split}\n\\ell_p(\\mu_{\\infty,0},\\mu_{\\infty,1})^p &= \\int_\\mms \\uptau^p\\d\\pi_\\infty = \\lim_{k\\to\\infty}\\int_\\mms \\uptau^p\\d\\hat{\\pi}_k\\\\ &\\leq \\lim_{k\\to\\infty} \\int_\\mms\\uptau^p\\d\\check{\\pi}_k = \\int_\\mms\\uptau^p\\d\\check{\\pi}_\\infty.\n\\end{split}\n\\end{align}\nHere the inequality follows from $\\smash{\\ell_p}$-optimality of $\\smash{\\check{\\pi}_k}$. Thus, $\\smash{\\check{\\pi}_\\infty}$ is an $\\smash{\\ell_p}$-optimal coupling of $\\smash{\\mu_{\\infty,0}}$ and $\\smash{\\mu_{\\infty,1}}$. Since the latter pair is strongly timelike $p$-dualizable, every such coupling is chronological, whence $\\smash{\\check{\\pi}_\\infty[\\mms_\\ll^2]=1}$; thus,\n\\begin{align*}\n1 = \\check{\\pi}_\\infty[\\mms_\\ll^2] \\leq \\liminf_{k\\to\\infty} \\check{\\pi}_k[\\mms_\\ll^2].\n\\end{align*}\nAs usual, we may and will assume $\\smash{\\check{\\pi}_k[\\mms_\\ll^2] > 0}$ for every $k\\in\\N$.\n\n\\textbf{Step 3.5.} Lastly, we define $\\smash{\\bar{\\pi}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)}$ by\n\\begin{align*}\n\\bar{\\pi}_k := \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\check{\\pi}_k\\mres \\mms_\\ll^2.\n\\end{align*}\nSince the restriction of an $\\smash{\\ell_p}$-optimal coupling is $\\smash{\\ell_p}$-optimal \\cite[Lem.~2.10]{cavalletti2020}, $\\smash{\\bar{\\pi}_k}$ is a chronological $\\smash{\\ell_p}$-optimal coupling of its marginals $\\smash{\\mu_{k,0},\\mu_{k,1}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ with\n\\begin{align*}\n\\mu_{k,0} &= \\rho_{k,0}\\,\\meas_k,\\\\\n\\mu_{k,1} &= \\rho_{k,1}\\,\\meas_k.\n\\end{align*}\nThe pair $\\smash{(\\mu_{k,0},\\mu_{k,1})}$ is timelike $p$-dualizable by $\\bar{\\pi}_k$ for every $k\\in\\N$, as desired.\n\n\\textbf{Step 4.} \\textit{Invoking the $\\mathrm{TCD}$ condition.} Fix $K\\in\\R$ and $N\\in (1,\\infty)$ with $K < K_\\infty$ and $N > N_\\infty$. Then $K < K_k$ and $N > N_k$ for large enough $k\\in\\N$. Hence, again up to a subsequence, we may and will assume that the previous strict inequalities hold for every $k\\in\\N$, respectively.\n\nBy \\autoref{Pr:Consistency TCD}, for every $k\\in\\N$ there are a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{k,t})_{t\\in[0,1]}$ con\\-necting $\\smash{\\mu_{k,0}}$ to $\\mu_{k,1}$ as well as a timelike $p$-dualizing coupling $\\smash{\\pi_k\\in\\Pi_\\ll(\\mu_{k,0},\\mu_{k,1})}$ such that for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:TCD COND INVOKING}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t}) \\leq {\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k).\n\\end{align}\n\n\\textbf{Step 5.} \\textit{Estimating $\\smash{-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k)}$ from below.} Tracing back the construction from Step 2 and Step 3, we infer the inequalities\n\\begin{align}\\label{Eq:Inequ rhok0}\n\\begin{split}\n\\rho_{k,0} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\rho}_{k,0}\\quad \\meas\\textnormal{-a.e.},\\\\\n\\rho_{k,1} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\rho}_{k,1} \\quad\\meas\\textnormal{-a.e.}\n\\end{split}\n\\end{align}\nHowever, when using these estimates to bound $\\smash{-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k)}$ from below, we change the involved densities in the integrands, but not the coupling with respect to which it is integrated; we thus have to make up for this in the coupling as well. In turn, this requires us to take into account that both $\\smash{\\tilde{\\pi}_k}$ and $\\smash{\\check{\\pi}_k}$ are not chro\\-nological, but only close to being so for large $k\\in\\N$.\n\n\\textbf{Step 5.1.} We modify $\\pi_k$ into a measure with marginals \n\\begin{align*}\n\\nu_{k,0} &:= (1+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(\\tilde{\\rho}_{k,0}+ \\delta_k + \\varepsilon_k+\\zeta_k)\\,\\meas_k = \\varrho_{k,0}\\,\\meas_k,\\\\\n\\nu_{k,1} &:= (1+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(\\tilde{\\rho}_{k,1}+ \\delta_k + \\varepsilon_k+\\zeta_k)\\,\\meas_k = \\varrho_{k,1}\\,\\meas_k,\n\\end{align*}\nwhere $\\delta_k,\\varepsilon_k \\in[0,1]$ and $\\zeta_k \\geq 0$ are defined by\n\\begin{align}\\label{Eq:DEFS}\n\\begin{split}\n\\delta_k &:= \\tilde{\\pi}_k[\\{\\uptau=0\\}],\\\\\n\\varepsilon_k &:= \\check{\\pi}_k[\\{\\uptau=0\\}],\\\\\n\\zeta_k &:= W_2(\\meas_k,\\meas_\\infty).\n\\end{split}\n\\end{align}\nNote that $(\\delta_k)_{k\\in\\N}$, $(\\varepsilon_k)_{k\\in\\N}$, and $(\\zeta_k)_{k\\in\\N}$ converge to $0$. Define $\\varpi_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ by\n\\begin{align*}\n\\varpi_k &:= (1+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,\\big[\\tilde{\\pi}_k[\\mms_\\ll^2]\\,\\check{\\pi}_k[\\mms_\\ll^2]\\,\\pi_k + \\tilde{\\pi}_k\\mres \\{\\uptau=0\\}\\\\\n&\\qquad\\qquad + (\\delta_k + \\varepsilon_k+\\zeta_k)\\,\\meas_k^{\\otimes 2} + \\tilde{\\pi}_k[\\mms_\\ll^2]\\,\\check{\\pi}_k\\mres \\{\\uptau=0\\}\\big].\n\\end{align*}\nBy tracing back the definitions of $\\tilde{\\pi}_k$ and $\\check{\\pi}_k$ in Step 2, one verifies that $\\varpi_k$ is a coupling, not necessarily $\\smash{\\ell_p}$-optimal, of $\\nu_{k,0}$ and $\\nu_{k,1}$. \n\n\\textbf{Step 5.2.} By \\eqref{Eq:Inequ rhok0}, we have \n\\begin{alignat*}{3}\n\\rho_{k,0} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,(1+\\delta_k + \\varepsilon_k+\\zeta_k)\\,\\varrho_{k,0} & \\quad & \\meas\\textnormal{-a.e.},\\\\\n\\rho_{k,1} &\\leq \\check{\\pi}_k[\\mms_\\ll^2]^{-1}\\,\\tilde{\\pi}_k[\\mms_\\ll^2]^{-1}\\,(1+\\delta_k + \\varepsilon_k+\\zeta_k)\\,\\varrho_{k,1} & \\quad & \\meas\\textnormal{-a.e.}\n\\end{alignat*}\nIn the sequel, to clear up notation, given $a,b\\in\\R$ we write $a\\geq_k b$ if there exists a sequence $(c_k)_{k\\in\\N}$ of positive real numbers converging to $1$ with $a\\geq c_k\\,b$ for every $k\\in\\N$. By definition of the inherent distortion coefficients, we obtain\n\\begin{align*}\n-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) &= \\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\qquad\\qquad +\\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_{k,1}(x^1)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\geq_k \\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\qquad\\qquad + \\int_{\\mms^2} \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,1}(x^1)^{-1\/N'}\\d\\pi_k(x^0,x^1)\\\\\n&\\geq_k\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad + \\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad - (1-t)\\int_{\\{\\uptau=0\\}}\\varrho_{k,0}(x^0)^{-1\/N'} \\d\\tilde{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - t\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^1)^{-1\/N'}\\d\\tilde{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - (1-t)\\int_{\\{\\uptau=0\\}}\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - t\\int_{\\{\\uptau=0\\}}\\varrho_{k,1}(x^1)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad - c\\,(\\delta_k + \\varepsilon_k+\\zeta_k)\\int_{\\mms} \\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0)\\\\\n&\\qquad\\qquad - c\\,(\\delta_k + \\varepsilon_k+\\zeta_k)\\int_{\\mms} \\varrho_{k,1}(x^1)^{-1\/N'}\\d\\meas_k(x^1),\n\\end{align*}\nwhere, recalling our choice $K < K_k$ for large enough $k\\in\\N$ and \\autoref{Cor:Bonnet-Myers},\n\\begin{align}\\label{Eq:c value}\nc := \\max\\{\\sup\\tau_{K,N'}^{(1-t)}\\circ\\uptau(\\mms^2),\\sup\\tau_{K,N'}^{(t)}\\circ\\uptau(\\mms^2)\\}.\n\\end{align}\nSince $\\varrho_{k,0} \\geq_k \\delta_k + \\varepsilon_k+\\zeta_k$ $\\meas_k$-a.e.~and $\\varrho_{k,1} \\geq_k \\delta_k+\\varepsilon_k+\\zeta_k$ $\\meas_k$-a.e., by definition of $\\delta_k$ and $\\varepsilon_k$ we obtain the estimates\n\\begin{align*}\n(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'}&\\geq_k (1-t)\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\tilde{\\pi}_k(x^0,x^1) \\\\\n&\\qquad\\qquad + t\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\tilde{\\pi}_k(x^0,x^1),\\\\\n(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'} &\\geq_k (1-t)\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1)\\\\\n&\\qquad\\qquad + t\\int_{\\{\\uptau=0\\}} \\varrho_{k,1}(x^0)^{-1\/N'}\\d\\check{\\pi}_k(x^0,x^1),\\\\\n2\\,(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'} &\\geq_k (\\delta_k+\\varepsilon_k+\\zeta_k)\\int_{\\mms}\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0)\\\\\n&\\qquad\\qquad + (\\delta_k+\\varepsilon_k+\\zeta_k)\\int_{\\mms}\\varrho_{k,1}(x^1)^{-1\/N'}\\d\\meas_k(x^1),\n\\end{align*}\nand therefore\n\\begin{align*}\n-{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) &\\geq_k\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad + \\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad -2\\,(c+1)\\,(\\delta_k + \\varepsilon_k+\\zeta_k)^{1-1\/N'}.\\textcolor{white}{\\int}\n\\end{align*}\n\n\\textbf{Step 5.3.} Let $\\smash{\\mathfrak{r}^0,\\mathfrak{r}^1\\colon \\mms\\to {\\ensuremath{\\mathscr{P}}}(\\mms)}$ denote the ($k$-dependent) disintegrations of $\\varpi_k$ with respect to $\\pr_1$ and $\\pr_2$, respectively, i.e.\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}} \\varpi_k(x^0,x^1) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{r}^0_{x^0}(x^1)\\d\\nu_{k,0}(x^0) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{r}^1_{x^1}(x^0)\\d\\nu_{k,1}(x^1),\n\\end{align*}\nand define\n\\begin{align}\\label{Eq:Disint tau}\n\\begin{split}\nv_0(x^0) &:= \\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\d\\mathfrak{r}_{x^0}^0(x^1),\\\\\nv_1(x^1) &:= \\int_{\\mms} \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\d\\mathfrak{r}_{x^1}^1(x^0).\n\\end{split}\n\\end{align}\nMoreover, define $\\smash{\\nu_{\\infty,0}^k,\\nu_{\\infty,1}^k\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ by\n\\begin{align*}\n\\nu_{\\infty,0}^k &:= (1+\\delta_k + \\varepsilon_k+\\zeta_k)^{-1}\\,(\\rho_{\\infty,0}^{n_k} + \\delta_k +\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\varrho_{\\infty,0}^k\\,\\meas_\\infty,\\\\\n\\nu_{\\infty,1}^k &:= (1+\\delta_k + \\varepsilon_k+\\zeta_k)^{-1}\\,(\\rho_{\\infty,1}^{n_k} + \\delta_k +\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\varrho_{\\infty,1}^k\\,\\meas_\\infty,\n\\end{align*}\nand observe that\n\\begin{align}\\label{Eq:Observation!}\n\\begin{split}\n\\mathfrak{p}^k(\\nu_{\\infty,0}^k) &= \\nu_{k,0} = \\varrho_{k,0}\\,\\meas_k,\\\\\n\\mathfrak{p}^k(\\nu_{\\infty,1}^k) &= \\nu_{k,1} = \\varrho_{k,1}\\,\\meas_k.\n\\end{split}\n\\end{align}\nThen by Jensen's inequality,\n\\begin{align*}\n&\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\textcolor{white}{\\sum_0^1}\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mms^2}\\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\varpi_k(x^0,x^1)\\\\\n&\\qquad\\qquad = \\sum_{i=0}^1 \\int_{\\mms} \\varrho_{k,i}(x^i)^{1-1\/N'}\\,v_i(x^i)\\d\\meas_k(x^i)\\\\\n&\\qquad\\qquad = \\sum_{i=0}^1 \\int_{\\mms}\\Big[\\!\\int_{\\mms} \\varrho_{\\infty,i}^k(y^i)\\d\\mathfrak{p}_{x^i}^k(y^i)\\Big]^{1-1\/N'}v_i(x^i)\\d\\meas_k(x^i)\\\\\n&\\qquad\\qquad \\geq \\sum_{i=0}^1 \\int_{\\mms^2} \\varrho_{\\infty,i}^k(y^i)^{1-1\/N'}\\,v_i(x^i)\\d\\mathfrak{p}_{x^i}^k(y^i)\\d\\meas_k(x^i)\\\\\n&\\qquad\\qquad = \\sum_{i=0}^1 \\int_{\\mms^2} \\varrho_{\\infty,i}^k(y^i)^{1-1\/N'}\\,v_i(x^i)\\d\\mathfrak{p}_{y^i}^\\infty(x^i)\\d\\meas_\\infty(y^i).\n\\end{align*}\n\n\\textbf{Step 5.4.} Next, we estimate the latter sum from below. Using \\eqref{Eq:Disint tau} and \\eqref{Eq:Observation!},\n\\begin{align*}\n&\\int_{\\mms^2} \\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,v_0(x^0)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad =\\int_{\\mms^4}\\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big].\n\\end{align*}\n\nLet $\\varepsilon > 0$. Since $\\smash{\\tau_{K,N'}^{(t)}\\circ\\uptau}$ is uniformly continuous by compactness of $\\smash{\\mms^2}$, our choice of $K$, and possibly invoking \\autoref{Cor:Bonnet-Myers}, we fix $\\delta> 0$ such that\n\\begin{align*}\n\\big\\vert \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1)) - \\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\big\\vert &\\leq \\varepsilon,\\\\\n\\big\\vert \\tau_{K,N'}^{(t)}(\\uptau(x^0,x^1)) - \\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\big\\vert &\\leq \\varepsilon\n\\end{align*}\nfor every $(x^0,y^0,x^1,y^1)\\in A_\\delta$, where\n\\begin{align}\\label{Eq:A_delta}\nA_\\delta := \\{ z \\in\\mms^4 : \\met(z_1,z_2) + \\met(z_3,z_4) \\leq \\delta\\}.\n\\end{align}\nLastly, somewhat suggestively, for $k\\in\\N$ we define a coupling $\\smash{\\eta_k\\in\\Pi(\\nu_{\\infty,0}^k,\\nu_{\\infty,1}^k)}$, which is notably independent of $\\varepsilon$, by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}\\eta_k(y^0,y^1) &:= \\int_{\\mms^2} \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,0}(x^0)\\,\\varrho_{k,1}(x^1)}\\d\\mathfrak{p}_{x^0}^k(y^0)\\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\varpi_k(x^0,x^1)\\\\\n&\\textcolor{white}{:}= \\int_{\\mms^2} \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0) \\d\\meas_\\infty(y^0)\\\\\n&\\textcolor{white}{:}= \\int_{\\mms^2} \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,0}(x^0)}\\d\\mathfrak{p}_{x^0}^k(y^0)\\d\\mathfrak{r}_{x^1}^1(x^0)\\d\\mathfrak{p}_{y^1}^\\infty(x^1)\\d\\meas_\\infty(y^1).\n\\end{align*}\nThe previous computations then entail\n\\begin{align*}\n&\\int_{\\mms^2} \\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,v_0(x^0)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad \\geq -\\varepsilon+\\int_{A_\\delta} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{A_\\delta^{\\ensuremath{\\mathsf{c}}}} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\\\\n&\\qquad\\qquad = - \\varepsilon + \\int_{\\mms^2} \\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{A_\\delta^{\\ensuremath{\\mathsf{c}}}} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big[\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))- \\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\big]\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\times \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big].\n\\end{align*}\n\n\\textbf{Step 5.5.} Let us estimate the latter error term. We first recall the definition $\\zeta_k := W_2(\\meas_k,\\meas_\\infty)$ from \\eqref{Eq:DEFS}. By definition of $A_\\delta$ and the value $c$ from \\eqref{Eq:c value}, the $\\meas$-a.e.~valid inequality for the density $\\smash{\\rho_{\\infty,1}^{n_k}}$ involved in the definition of $\\smash{\\varrho_{\\infty,1}^k}$ from \\autoref{Le:CM lemma}, as well as H\u00f6lder's inequality,\n\\begin{align*}\n&\\Big\\vert\\!\\int_{A_\\delta^{\\ensuremath{\\mathsf{c}}}} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad \\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big[\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))- \\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\big]\\\\\n&\\qquad\\qquad\\qquad\\qquad \\times \\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\Big\\vert\\\\\n&\\qquad\\qquad \\leq 2\\,c\\,\\delta^{-1}\\!\\int_{\\mms^4} \\d\\mathfrak{p}_{x^1}^k(y^1)\\d\\mathfrak{r}_{x^0}^0(x^1)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\Big[\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big[\\met(x^0,y^0) + \\met(x^1,y^1)\\big]\\,\\frac{\\varrho_{\\infty,0}^k(y^0)\\,\\varrho_{\\infty,1}^k(y^1)}{\\varrho_{k,1}(x^1)}\\Big]\\\\\n&\\qquad\\qquad \\leq_k 2\\,c\\,\\delta^{-1}\\!\\int_{\\mms^2}\\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,\\met(x^0,y^0)\\d\\mathfrak{q}_k(x^0,y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + 2\\,c\\,\\delta^{-1}\\,\\zeta_k^{-1\/N'}\\!\\int_{\\mms} \\rho_{\\infty,1}(y^1)\\,\\met(x^1,y^1)\\d\\mathfrak{q}_k(x^1,y^1)\\\\\n&\\qquad\\qquad \\leq_k 2\\,c\\,\\delta^{-1}\\,\\big\\Vert\\rho_{\\infty,0}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1-1\/N'}\\,\\zeta_k + 2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,1}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\,\\zeta_k^{1-1\/N'}\\!.\\!\\!\\!\\textcolor{white}{\\int}\n\\end{align*}\n\n\\textbf{Step 5.6.} Taking the estimates from Step 5.4 and Step 5.5 together, we get\n\\begin{align*}\n&\\int_{\\mms^2}\\varrho_{\\infty,0}^k(y^0)^{1-1\/N'}\\,v_0(x^0)\\d\\mathfrak{p}_{y^0}^\\infty(x^0)\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad \\geq_k -\\varepsilon + \\int_{\\mms^2}\\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1}\\,\\big\\Vert\\rho_{\\infty,0}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1-1\/N'}\\,\\zeta_k\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1\/N'}\\,\\Vert\\rho_{\\infty,1}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\,\\zeta_k^{1-1\/N'}\\!.\\textcolor{white}{\\int}\n\\end{align*}\nAnalogously, for the second summand in the last part of Step 5.3 we obtain\n\\begin{align*}\n&\\int_{\\mms^2} \\varrho_{\\infty,1}^k(y^1)^{1-1\/N'}\\,v_1(x^1)\\d\\mathfrak{p}_{y^1}^\\infty(x^1)\\d\\meas_\\infty(y^1)\\\\\n&\\qquad\\qquad \\geq_k -\\varepsilon + \\int_{\\mms^2}\\varrho_{\\infty,1}^k(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1}\\,\\big\\Vert\\rho_{\\infty,1}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1-1\/N'}\\,\\zeta_k\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad\\qquad\\qquad - 2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,0}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\,\\zeta_k^{1-1\/N'}\\!.\\textcolor{white}{\\int}\n\\end{align*}\n\n\\textbf{Step 6.} \\textit{Passage to the limit.} Let $(a_k)_{k\\in\\N}$ be a given sequence of normalization constants in $[1,\\infty)$ as provided by \\autoref{Le:CM lemma}, i.e. \n\\begin{align*}\n\\rho_{\\infty,0}^{n_k} &\\leq a_k\\,\\rho_{\\infty,0}\\quad\\meas\\textnormal{-a.e.},\\\\\n\\rho_{\\infty,1}^{n_k} &\\leq a_k\\,\\rho_{\\infty,1}\\quad\\meas\\textnormal{-a.e.}\n\\end{align*}\n\n\\textbf{Step 6.1.} For $k\\in\\N$, we define $\\smash{\\tilde{\\nu}_{\\infty,0}^k,\\tilde{\\nu}_{\\infty,1}^k}\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ by\n\\begin{align*}\n\\tilde{\\nu}_{\\infty,0}^k &:=(a_k^2+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(a_k^2\\,\\rho_{\\infty,0} + \\delta_k+\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\tilde{\\varrho}_{\\infty,0}^k\\,\\meas_\\infty,\\\\\n\\tilde{\\nu}_{\\infty,1}^k &:= (a_k^2+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1}\\,(a_k^2\\,\\rho_{\\infty,1} + \\delta_k+\\varepsilon_k+\\zeta_k)\\,\\meas_\\infty = \\tilde{\\varrho}_{\\infty,1}^k\\,\\meas_\\infty.\n\\end{align*}\nWe turn $\\eta_k$ into a coupling $\\smash{\\alpha_k \\in\\Pi(\\nu_{\\infty,0}^k,\\nu_{\\infty,1}^k)}$ by setting\n\\begin{align*}\n\\alpha_k &:= (a_k^2+\\delta_k+\\varepsilon_k+\\zeta_k)^{-1} \\big[(1+\\delta_k+\\varepsilon_k+\\zeta_k)\\,\\eta_k\\\\\n&\\qquad\\qquad + a_k^2\\,\\mu_{\\infty,0}\\otimes\\mu_{\\infty,1} - \\tilde{\\mu}_{k,0}\\otimes\\tilde{\\mu}_{k,1}\\big].\n\\end{align*}\nThis construction yields\n\\begin{align*}\n&\\int_{\\mms^2}\\varrho_{k,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mms^2}\\varrho_{k,1}^k(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad \\geq_k \\int_{\\mms^2} \\tilde{\\varrho}_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\alpha_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - c\\int_\\mms \\varrho_{\\infty,0}^k(y^0)^{-1\/N'}\\,\\big\\vert \\rho_{\\infty,0}(y^0) - \\tilde{\\rho}_{k,0}(y^0)\\big\\vert\\d\\meas_\\infty(y^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mms^2}\\tilde{\\varrho}_{\\infty,1}^k(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\alpha_k(y^0,y^1)\\\\\n&\\qquad\\qquad\\qquad\\qquad - c\\int_\\mms \\varrho_{\\infty,1}^k(y^1)^{-1\/N'}\\,\\big\\vert \\rho_{\\infty,1}(y^1) - \\tilde{\\rho}_{k,1}(y^1)\\big\\vert\\d\\meas_\\infty(y^1).\n\\end{align*}\n\n\\textbf{Step 6.2.} Taking together the above estimates with those obtained in Step 5 and invoking the $\\Ell^1$-convergence asserted in \\autoref{Le:CM lemma}, we obtain\n\\begin{align*}\n-\\limsup_{k\\to\\infty} {\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) \\geq - \\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\alpha_k) - 2\\varepsilon.\n\\end{align*}\nNow, by Prokhorov's theorem, the sequence $(\\alpha_k)_{k\\in\\N}$ converges weakly to some $\\alpha\\in\\Pi(\\mu_{\\infty,0},\\mu_{\\infty,1})$ up to a nonrelabeled subsequence. Since $\\varepsilon$ was arbitrary and did not influence the construction of $\\alpha_k$, by \\autoref{Le:Const perturb} we thus get\n\\begin{align}\\label{Eq:Eqeqeqeq}\n-\\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k)\\geq -{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\alpha).\n\\end{align}\n\n\\textbf{Step 6.3.} Now we send $k\\to\\infty$ in \\eqref{Eq:TCD COND INVOKING}. For every $k\\in\\N$, there exists some plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{k,0},\\mu_{k,1})}$ representing the curve $(\\mu_{k,t})_{t\\in[0,1]}$. \\autoref{Le:Villani lemma for geodesic} allows us to extract a nonrelabeled subsequence of $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}_k)_{k\\in\\N}}$ converging weakly to some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{\\infty,0},\\mu_{\\infty,1})}$. The assignment $\\mu_{\\infty,t} := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_\\infty$ produces a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{\\infty,t})_{t\\in[0,1]}$ connecting $\\mu_{\\infty,0}$ to $\\mu_{\\infty,1}$. Lower semicontinuity of R\u00e9nyi's entropy on ${\\ensuremath{\\mathscr{P}}}(\\mms)$ by compactness of $\\mms$ and \\eqref{Eq:Eqeqeqeq} thus yield, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:Renyi inequ}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{\\infty,t})\\leq \\limsup_{k\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t}) \\leq \\limsup_{k\\to\\infty}{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\pi_k) \\leq {\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\alpha).\n\\end{align}\n\n\\textbf{Step 7.} \\textit{Proof of the $\\smash{\\ell_p}$-optimality of $\\alpha$.} To conclude the desired property leading towards $\\smash{\\mathrm{TCD}_p(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}_\\infty$, at least under the restrictions of Step 2, we have to prove the causality and the $\\smash{\\ell_p}$-optimality of $\\alpha$. Both is argued similarly to Step 5.4; we concentrate on the proof of the estimate entailing $\\smash{\\ell_p}$-optimality later, and then outline how to modify this argument to prove $\\smash{\\alpha[\\mms_\\ll^2]=1}$.\n\nGiven any $\\varepsilon>0$, fix $\\delta > 0$ with the property\n\\begin{align*}\n\\big\\vert\\uptau(x^0,x^1) - \\uptau(y^0,y^1)\\big\\vert \\leq \\varepsilon\n\\end{align*}\nfor every $(x^0,y^0,x^1,y^1)\\in\\mms^4$ belonging to the set $\\smash{A_\\delta}$ from \\eqref{Eq:A_delta}. As in Step 5.5 and tracing back the definitions of all considered couplings, we obtain\n\\begin{align*}\n&\\int_{\\mms^2}\\uptau^p(y^0,y^1)\\d\\alpha(y^0,y^1) = \\lim_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p(y^0,y^1)\\d\\alpha_k(y^0,y^1)\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty} \\int_{\\mms^2}\\uptau^p(y^0,y^1)\\d\\eta_k(y^0,y^1)\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty} \\int_{\\mms^2} \\uptau^p(x^0,x^1)\\d\\varpi_k(x^0,x^1) - \\varepsilon\\\\\n&\\qquad\\qquad\\qquad\\qquad -2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,0}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\limsup_{k\\to\\infty} W_2(\\meas_k,\\meas_\\infty)\\!\\!\\!\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad\\qquad\\qquad -2\\,c\\,\\delta^{-1}\\,\\Vert\\rho_{\\infty,1}\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}\\limsup_{k\\to\\infty} W_2(\\meas_k,\\meas_\\infty)\\!\\!\\!\\textcolor{white}{\\int}\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p(x^0,x^1)\\d\\pi_k(x^0,x^1)-\\varepsilon\\\\\n&\\qquad\\qquad = \\liminf_{k\\to\\infty} \\int_{\\mms^2}\\uptau^p(x^0,x^1)\\d\\bar{\\pi}_k(x^0,x^1)-\\varepsilon\\\\\n&\\qquad\\qquad \\geq \\liminf_{k\\to\\infty}\\int_{\\mms^2}\\uptau^p(x^0,x^1)\\d\\check{\\pi}_k(x^0,x^1)-\\varepsilon\\geq \\ell_p(\\mu_{\\infty,0},\\mu_{\\infty,1})^p-\\varepsilon.\n\\end{align*}\nIn the equality in the third last step, we have used that both $\\smash{\\pi_k,\\bar{\\pi}_k\\in\\Pi_\\ll(\\mu_{k,0},\\mu_{k,1})}$ are $\\smash{\\ell_p}$-optimal; the last inequality is due to \\eqref{Eq:Lp calc}.\n\nThe relation $\\smash{\\alpha[\\mms_\\ll^2]=1}$ is argued similarly by replacing, given $\\varepsilon > 0$, $\\uptau$ by a nonnegative function $\\phi\\in\\Cont(\\mms^2)$ obeying $\\smash{\\phi(\\mms_\\ll^2)=\\{1\\}}$, $\\sup\\phi(\\mms)\\leq 1$, and\n\\begin{align*}\n\\alpha[\\mms_\\ll^2] \\geq \\int_{\\mms^2}\\phi\\d\\alpha - \\varepsilon.\n\\end{align*}\nBoth results together and the arbitrariness of $\\varepsilon$ yield the claim.\n\n\\textbf{Step 8.} \\textit{Relaxation of the assumptions on $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$.} Now we outline how to get rid of the assumption $\\smash{\\rho_{\\infty,0},\\rho_{\\infty,1}\\in\\Ell^\\infty(\\mms,\\meas_\\infty)}$ from Step 2. \n\nIf $\\smash{\\rho_{\\infty,0}}$ and $\\smash{\\rho_{\\infty,1}}$ are not $\\meas$-essentially bounded, given any $i\\in\\N$ we set\n\\begin{align*}\nE_i := \\{\\rho_{\\infty,0} \\leq i,\\, \\rho_{\\infty,1}\\leq i\\}.\n\\end{align*}\nLet $\\smash{\\pi\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ be an $\\smash{\\ell_p}$-optimal coupling, and set\n\\begin{align*}\n\\pi_i := \\pi[E_i]^{-1}\\,\\pi\\mres E_i\n\\end{align*}\nprovided $\\pi[E_i]>0$. By restriction \\cite[Lem.~2.10]{cavalletti2020}, $\\pi_i$ is an $\\smash{\\ell_p}$-optimal coupling of its marginals $\\smash{\\lambda_{\\infty,0}^i,\\lambda_{\\infty,1}^i\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$; moreover, the pair $\\smash{(\\lambda_{\\infty,0}^i,\\lambda_{\\infty,1}^i)}$ is strongly timelike $p$-dualizable. Let $\\smash{\\dot{\\pi}_i\\in\\Pi_\\ll(\\lambda_{\\infty,0}^i,\\lambda_{\\infty,1}^i)}$ be an $\\smash{\\ell_p}$-optimal coupling as constructed in the previous steps. Define $\\smash{\\beta_i\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ by\n\\begin{align*}\n\\beta_i := \\pi[E_i]\\,\\dot{\\pi}_i + \\pi\\mres E_i^{\\ensuremath{\\mathsf{c}}},\n\\end{align*}\nand note that $\\beta_i$ is an $\\smash{\\ell_p}$-optimal coupling of its marginals. Moreover, we observe that, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{T}}}_{K,N'}^{(t)}(\\dot{\\pi}_i) &\\leq_i -\\int_{\\mms^2} \\rho_{\\infty,0}(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\dot{\\pi}_i(y^0,y^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\rho_{\\infty,1}(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\dot{\\pi}_i(y^0,y^1)\\\\\n&\\leq_i -\\int_{\\mms^2} \\rho_{\\infty,0}(y^0)^{-1\/N'}\\,\\tau_{K,N'}^{(1-t)}(\\uptau(y^0,y^1))\\d\\beta_i(y^0,y^1)\\\\\n&\\qquad\\qquad + c \\int_{E_i^{\\ensuremath{\\mathsf{c}}}} \\rho_{\\infty,0}(y^0)^{-1\/N'}\\d\\pi(y^0,y^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2}\\rho_{\\infty,1}(y^1)^{-1\/N'}\\,\\tau_{K,N'}^{(t)}(\\uptau(y^0,y^1))\\d\\beta_i(y^0,y^1)\\\\\n&\\qquad\\qquad + c\\int_{E_i^{\\ensuremath{\\mathsf{c}}}}\\rho_{\\infty,1}(y^1)^{-1\/N'}\\d\\pi(y^0,y^1).\n\\end{align*}\n\nBy Prokhorov's theorem, $(\\beta_i)_{i\\in\\N}$ converges weakly to some $\\smash{\\beta\\in\\Pi(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ which, thanks to Portmanteau's theorem, is a causal coupling. By stability \\cite[Thm. 2.14]{cavalletti2020}, we thus infer the $\\smash{\\ell_p}$-optimality of $\\beta$. Together with \\autoref{Le:Const perturb} and H\u00f6lder's inequality, this addresses the right-hand side of the desired inequality; the left-hand side is treated with the compactness argument from Step 6.3.\n\n\\textbf{Step 9.} \\textit{Passing from $K$ and $N$ to $K_\\infty$ and $N_\\infty$.} Lastly, the restrictions from Step 4 the previous arguments imply $\\smash{\\mathrm{TCD}_p(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}_\\infty$ for every $KN_\\infty$. Given $\\smash{(\\mu_{\\infty,0},\\mu_{\\infty,1})\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ strongly timelike $p$-dualizable, letting $K$ and $N$ approach $K_\\infty$ and $N_\\infty$, respectively, and using compactness arguments as in Step 6.2 and Step 6.3 gives the existence of a proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{\\infty,t})_{t\\in[0,1]}$ from $\\mu_{\\infty,0}$ to $\\mu_{\\infty,1}$ and a timelike $p$-dualizing coupling $\\smash{\\pi\\in\\Pi_\\ll(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ witnessing the desired inequality for the R\u00e9nyi entropy for every $K < K_\\infty$ and every $N' > N_\\infty$. The semicontinuity properties of both sides in the respective parameters yields $\\smash{\\mathrm{TCD}_p(K_\\infty,N_\\infty)}$. The proof is terminated.\n\\end{proof}\n\n\\begin{remark}\\label{RE:POTEN} Very roughly speaking, the relevant convergences in the above proof are justified by \\emph{uniform continuity}, recall e.g.~Step 5.4. This is considerably weaker than \\emph{Lipschitz continuity} of the quantity inside the distortion coefficients, which has been used in the proof of \\cite[Thm.~3.1]{sturm2006b} in the metric case at a similar point, yet is useless here since $\\uptau$ is no distance. Our proof thus has the potential to admit straightforward adaptations to settings with continuous potentials other than $\\uptau$ which do not come from a distance either.\n\\end{remark}\n\n\\begin{remark}\\label{Re:A remarkable} A remarkable byproduct of the above proof is that both $\\smash{\\mathrm{TCD}_p^*(K,N)}$ and $\\smash{\\mathrm{TCD}_p(K,N)}$ are weakly stable under convergence of \\emph{normalized} Lorentzian pre-length spaces with respect to the following Lorentzian modification of Sturm's transport distance $\\boldsymbol{\\mathsf{D}}$ \\cite[Def. 3.2]{sturm2006a}: it is still defined in terms of $\\met$, but the inherent metric measure isometric embeddings should also respect the given Lorentzian structures, and are required to map into spaces obeying the regularity assumptions from item \\ref{La:Blurr1reg} of \\autoref{Def:Convergence}.\n\\end{remark}\n\n\\subsection{Good geodesics}\\label{Sub:Good TCD} In this section, following \\cite{braun2022}, see also \\cite{cavalletti2017,rajala2012a,rajala2012b}, we show the existence of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics with $\\meas$-densities uniformly in $\\Ell^\\infty$ in time under mild assumptions on their endpoints. This treatise does not require any nonbranching condition on the underlying space. The proofs of the results to follow are mostly analogous to those of \\cite{braun2022} and hence only outlined. \n\nWe regard the corresponding result from \\autoref{Th:Good geos TCD} as a key to develop the notion of \\emph{timelike weak gradients} on Lorentzian geodesic spaces by using so-called \\emph{test plans} --- many of which should exist by \\autoref{Th:Good geos TCD} --- following \\cite{ambrosio2014a}, and to prove a Lorentzian analogue of the \\emph{Sobolev-to-Lipschitz property} \\cite[Subsec.~4.1.3]{gigli2013}.\n\nThe crucial result providing the critical threshold for \\eqref{Eq:Thresh} is the following.\n\n\\begin{lemma}\\label{Le:Lalelu} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for $p\\in(0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ be strongly timelike $p$-dualizable with $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$. Let $D$ be any real number no smaller than $\\sup\\uptau(\\supp\\mu_0\\times\\supp\\mu_1)$. Then there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ such that $\\mu_t=\\rho_t\\,\\meas\\in\\Dom(\\Ent_\\meas)$ for every $t\\in[0,1]$, and\n\\begin{align*}\n\\meas\\big[\\{\\rho_{1\/2}>0\\}\\big] \\geq {\\ensuremath{\\mathrm{e}}}^{-D\\sqrt{K^-N}\/2}\\,\\max\\{ \\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}^{-1}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} Recall that $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies $\\smash{\\mathrm{wTCD}_p^*(K^-,N)}$ by \\autoref{Pr:Consistency TCD}. Let $\\smash{(\\mu_t)_{t\\in[0,1]}}$ be a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ and $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$ be some $\\smash{\\ell_p}$-optimal coupling obeying the inequality defining the condition $\\smash{\\mathrm{wTCD}_p^*(K^-,N)}$ as in \\autoref{Def:TCD*}. Thanks to \\autoref{Le:Stu}, we obtain $\\mu_t\\in\\Dom(\\Ent_\\meas)$ for every $t\\in[0,1]$; in particular, we have $\\smash{\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$. Moreover, by \\autoref{Re:Lower bounds sigma},\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_N(\\mu_{1\/2})&\\leq -\\int_{\\mms^2} \\sigma_{K^-,N}^{(1\/2)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2} \\sigma_{K^-,N}^{(1\/2)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N}\\d\\pi(x^0,x^1)\\\\\n&\\leq -{\\ensuremath{\\mathrm{e}}}^{-D\\sqrt{K^-\/N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}, \\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}^{-1\/N}.\\!\\!\\!\\textcolor{white}{\\int}\n\\end{align*}\nOn the other hand, we have $\\smash{{\\ensuremath{\\mathscr{S}}}_N(\\mu_{1\/2}) \\geq -\\meas[\\{\\rho_{1\/2}>0\\}]^{1\/N}}$ by Jensen's inequality, and rearranging terms provides the claim.\n\\end{proof}\n\n\\begin{theorem}\\label{Th:Good geos TCD} Under the hypotheses of \\autoref{Le:Lalelu}, there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ such that for every $t\\in[0,1]$, $\\smash{\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align}\\label{Eq:Thresh}\n\\Vert\\rho_t\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq {\\ensuremath{\\mathrm{e}}}^{D\\sqrt{K^-N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\},\n\\end{align}\nwhere\n\\begin{align*}\nD := \\sup\\uptau(\\supp\\mu_0\\times\\supp\\mu_1).\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof} We construct the required geodesic at dyadic times in $[0,1]$ by a bisection argument and then define the rest of the geodesic by completion. By induction, given $n\\in\\N_0$ we assume that measures $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}^0, \\dots, {\\ensuremath{\\boldsymbol{\\alpha}}}^n\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ have already been constructed with the following properties. For every $\\smash{k\\in\\{0,\\dots,2^n\\}}$, the pair $\\smash{((\\eval_{(k-1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n,(\\eval_{k2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$ is strongly timelike $p$-dualizable, and\n\\begin{align*}\n\\sup\\uptau(\\supp\\,(\\eval_{(k-1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n\\times\\supp\\,(\\eval_{k2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n) \\leq 2^{-n}\\,D.\n\\end{align*}\n\n\\textbf{Step 1.} \\textit{Minimization of an appropriate functional.} Set\n\\begin{align*}\nc_{n+1} := {\\ensuremath{\\mathrm{e}}}^{2^{-n-2}D\\sqrt{K^-N}}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}\n\\end{align*}\nand define the functional $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_{n+1}}\\colon {\\ensuremath{\\mathscr{P}}}(\\mms)\\to [0,1]}$ by\n\\begin{align}\\label{Eq:Functional FC}\n{\\ensuremath{\\mathscr{F}}}_{c_{n+1}}(\\mu) := \\big\\Vert (\\rho- c_{n+1})^+\\big\\Vert_{\\Ell^1(\\mms,\\meas)} + \\mu_\\perp[\\mms].\n\\end{align}\nsubject to the decomposition $\\mu = \\rho\\,\\meas + \\mu_\\perp$. It measures how far $\\rho$ deviates from being bounded from above by $c$, and how much the input $\\mu$ fails to be $\\meas$-absolutely continuous. Let $\\smash{k\\in\\{1,\\dots,2^{n+1}-1\\}}$ be odd. As for \\cite[Le.~3.13]{braun2022}, we prove that $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_{n+1}}\\circ \\eval_{1\/2}}$ admits a minimizer $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k^{n+1}\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{(k-1)2^{-n-1}},\\mu_{(k+1)2^{-n-1}}}$). By gluing, we find a measure $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that\n\\begin{align*}\n(\\Restr_{(k-1)2^{-n-1}}^{(k+1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1} = {\\ensuremath{\\boldsymbol{\\pi}}}_k^{n+1}\n\\end{align*}\nfor every $k$ as above. Here, given $s,t\\in [0,1]$ with $sc_{n+1}$, and any odd $k\\in\\smash{\\{1,\\dots,2^{n+1}-1\\}}$, arguing as in \\cite[Prop.~3.14]{braun2022} and using \\autoref{Le:Lalelu} we get\n\\begin{align}\\label{Eq:Fc inequality}\n\\begin{split}\n&\\inf\\{ {\\ensuremath{\\mathscr{F}}}_{c'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}) : {\\ensuremath{\\boldsymbol{\\pi}}}\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau((\\eval_{(k-1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1},\\\\\n&\\qquad\\qquad (\\eval_{(k+1)2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}) \\}=0.\n\\end{split}\n\\end{align}\nThe support of all considered measures belongs to the compact set $J(\\mu_0,\\mu_1)$; thus, using \\cite[Cor.~3.15]{braun2022} we get \\eqref{Eq:Fc inequality} for $c'$ replaced by $c_{n+1}$. Thus, $\\smash{(\\eval_{k2^{-n-1}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^{n+1}} = \\rho_{k2^{-n-1}}\\,\\meas\\in \\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and, computing the geometric sum in the exponent,\n\\begin{align}\\label{Eq:DENS BD}\n\\Vert\\rho_{k2^{-n-1}}\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq {\\ensuremath{\\mathrm{e}}}^{D\\sqrt{K^-N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}, \\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}.\n\\end{align}\n\n\\textbf{Step 3.} \\textit{Completion.} Let $\\smash{({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}}$ be the sequence in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ iteratively constructed according to Step 1 and Step 2. Using \\autoref{Le:Villani lemma for geodesic}, this sequence converges weakly, up to a subsequence, to some $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. In particular, the assignment $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$ defines a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$. Since the functional ${\\ensuremath{\\mathscr{F}}}_c$ given in terms of the threshold\n\\begin{align*}\nc := {\\ensuremath{\\mathrm{e}}}^{D\\sqrt{K^-N}\/2}\\,\\max\\{\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}, \\Vert\\rho_1\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\\}\n\\end{align*}\nanalogously to \\eqref{Eq:Functional FC} is weakly lower semicontinuous in ${\\ensuremath{\\mathscr{P}}}(J(\\mu_0,\\mu_1))$ \\cite[Lem.~3.13]{braun2022}, \\eqref{Eq:DENS BD} yields ${\\ensuremath{\\mathscr{F}}}_c(\\mu_t) =0$ for every $t\\in[0,1]$, which is the claim.\n\\end{proof}\n\n\\subsection{Equivalence with the entropic TCD condition}\\label{Sub:Equiv TCDs} In this section, we prove the equivalence of $\\smash{\\mathrm{TCD}_p^*(K,N)}$ with its entropic counterpart $\\smash{\\mathrm{TCD}_p^e(K,N)}$ introduced in \\cite{cavalletti2020}, provided the underlying space is timelike $p$-essentially nonbranching according to \\autoref{Def:Essentially nonbranching}. We also obtain equivalence of the respective weak with their strong versions; see \\autoref{Th:Equivalence TCD* and TCDe}. Yet another characterization will be obtained in \\autoref{Pr:MDPTS} below.\n\nIn this section, in addition to our standing assumptions on ${\\ensuremath{\\mathscr{X}}}$ we suppose the latter to be timelike $p$-essentially nonbranching for a fixed $p\\in (0,1)$. \n\nFor the convenience of the reader, let us recall the following notions \\cite[Def.~3.2, Prop.~3.3]{cavalletti2020}, for which we define ${\\ensuremath{\\mathscr{U}}}_N \\colon {\\ensuremath{\\mathscr{P}}}(\\mms)\\to [0,\\infty]$, $N\\in (0,\\infty)$, by\n\\begin{align}\\label{Eq:Expo Boltzmann}\n{\\ensuremath{\\mathscr{U}}}_N(\\mu) := {\\ensuremath{\\mathrm{e}}}^{-\\Ent_\\meas(\\mu)\/N}.\n\\end{align}\n\n\\begin{definition}\\label{Def:CD entropic} Let $K\\in\\R$ and $N\\in (0,\\infty)$. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ satis\\-fies the \\emph{entropic timelike curvature-dimension condition} $\\smash{\\mathrm{TCD}_p^e(K,N)}$ if for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1)\\in\\Dom(\\Ent_\\meas)^2$, there exist a timelike proper-time parametrized $\\smash{\\ell_p}$-geo\\-desic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ and $\\mu_1$ as well as a timelike $p$-dualizing coupling $\\pi\\in\\smash{\\Pi_\\ll(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{U}}}_N(\\mu_t) \\geq \\sigma_{K,N}^{(1-t)}\\big[\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)}\\big]\\,{\\ensuremath{\\mathscr{U}}}_N(\\mu_0) + \\sigma_{K,N}^{(t)}\\big[\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)}\\big]\\,{\\ensuremath{\\mathscr{U}}}_N(\\mu_1).\n\\end{align*}\n\\item If the previous claim holds for every strongly timelike $p$-dualizable $(\\mu_0,\\mu_1)\\in({\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas))^2$, we say that ${\\ensuremath{\\mathscr{X}}}$ satisfies the \\emph{weak entropic timelike curvature-dimension condition} $\\smash{\\mathrm{wTCD}_p^e(K,N)}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\\label{Th:Equivalence TCD* and TCDe} The following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{La:1} The condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds.\n\\item\\label{La:1.5} The condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ holds.\n\\item\\label{La:2} For every timelike $p$-dualizable $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists an $\\smash{\\ell_p}$-optimal geodesic plan ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}=\\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}&\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\item\\label{La:3} The condition $\\smash{\\mathrm{TCD}_p^e(K,N)}$ holds.\n\\item\\label{La:3.5} The condition $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ holds.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark} The exceptional set in \\ref{La:2} may depend on $t$ and $N'$.\n\nBy \\autoref{Pr:iii to iv} and the proofs of \\autoref{Pr:ii to iii} and \\autoref{Pr:v to iii}, any of the claims in \\autoref{Th:Equivalence TCD* and TCDe} is equivalent to the following weaker version of \\ref{La:2}: for every pair $\\smash{(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)^2}$ such that $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$ and $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$, there exists $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in[0,1]$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N}&\\geq \\sigma_{K,N}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N} + \\sigma_{K,N}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, where $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$.\n\nAn analogous note applies to \\autoref{Th:Equiv TCD with geo}, \\autoref{Th:Equivalence TMCP* and TMCPe}, and \\autoref{Th:Equivalence TMCP}.\n\\end{remark}\n\nIt is clear that \\ref{La:1} implies \\ref{La:1.5}, and that \\ref{La:3} yields \\ref{La:3.5}. Moreover, \\ref{La:2} implies \\ref{La:1} by integration. The proofs of \\ref{La:1.5} implying \\ref{La:2}, \\ref{La:2} implying \\ref{La:3}, and \\ref{La:3.5} implying \\ref{La:2} are, for the sake of clarity, outsourced to \\autoref{Pr:ii to iii}, \\autoref{Pr:iii to iv}, and \\autoref{Pr:v to iii}, respectively. \n\n\\begin{remark}\\label{Re:Uniq} Below, we occasionally use two results which are only established later in \\autoref{Ch:TMCP}, namely the following.\n\\begin{itemize}\n\\item First, the condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies the timelike measure-contraction property $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ according to \\autoref{Def:TMCP}, cf.~\\autoref{Pr:TMCP to TCD}.\n\\item Second, on timelike $p$-essentially nonbranching spaces satisfying the latter condition --- or $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ according to \\autoref{Def:TMCPe} ---, chronological $\\smash{\\ell_p}$-optimal couplings and $\\smash{\\ell_p}$-optimal geodesic plans with sufficiently well-behaved endpoints are unique, cf.~\\autoref{Th:Uniqueness couplings}, \\autoref{Th:Uniqueness geodesics}, and \\autoref{Re:From TNB to TENB}. \n\\end{itemize}\nAs no result of this section is used in \\autoref{Ch:TMCP}, no circular reasoning occurs.\n\nThe key point of these results here is that the timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics and the $\\smash{\\ell_p}$-optimal couplings from \\autoref{Def:TCD*}, \\autoref{Def:TCD}, as well as \\autoref{Def:TMCP}, which are a priori unrelated, are in fact induced by the same $\\smash{\\ell_p}$-optimal geodesic plan.\n\nThis means that we must have these uniqueness results at our disposal \\emph{a priori} to establish \\autoref{Th:Equivalence TCD* and TCDe}. In particular, these do \\emph{not} follow from \\autoref{Th:Equivalence TCD* and TCDe}, \\autoref{Re:From TNB to TENB}, and \\cite[Thm.~3.19, Thm.~3.20]{cavalletti2020}.\n\\end{remark}\n\n\\begin{proposition}\\label{Pr:ii to iii} Assume $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ for some $K\\in\\R$ and $N\\in [1,\\infty)$. Then for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists some ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}= \\rho_t\\, \\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, where $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$.\n\\end{proposition}\n\n\\begin{proof} \\textbf{Step 1.} \\emph{Reinforcement of the assumptions on $\\mu_0$ and $\\mu_1$.} We first assume that $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$, that $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$, and that $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$. As mentioned above, timelike $p$-dualizable couplings and $\\ell_p$-optimal geodesic plans will then be unique all over Step 1, Step 2, and Step 3 without further notice. \n\n\n\\textbf{Step 2.} \\textit{Restriction.} Observe that the pair $(\\mu_0,\\mu_1)$ is in fact strongly timelike $p$-dualizable thanks to \\autoref{Re:Strong timelike}. Fix a $\\cap$-stable generator $\\{M_n : n\\in\\N\\}$ of the Borel $\\sigma$-field of $\\mms$; to be precise, the $M_n$'s should be chosen to generate the relative Borel $\\sigma$-field of the compact set $\\supp\\mu_0\\cup\\supp\\mu_1$, but we ignore this minor technicality by assuming compactness of $\\mms$ instead until Step 4. Given $n\\in\\N$ we cover $\\mms$ by the mutually disjoint Borel sets \n\\begin{align*}\nL_1 &:= M_1\\cap \\dots \\cap M_n,\\\\\nL_2 &:= M_1\\cap\\dots\\cap M_{n-1}\\cap M_n^{\\ensuremath{\\mathsf{c}}},\\,\\dots\\\\\nL_{2^n} &:= M_1^{\\ensuremath{\\mathsf{c}}} \\cap \\dots \\cap M_n^{\\ensuremath{\\mathsf{c}}}.\n\\end{align*}\nGiven the unique timelike $p$-dualizing coupling $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$ of $\\mu_0$ and $\\mu_1$, let us define $\\smash{\\mu_0^{ij},\\mu_1^{ij}\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas)}$ by \n\\begin{align*}\n\\mu_0^{ij}[A] &:= \\lambda_{ij}^{-1}\\,\\pi[(A\\cap L_i) \\times L_j] = \\varrho_0^{ij}\\,\\meas,\\\\\n\\mu_1^{ij}[A] &:= \\lambda_{ij}^{-1}\\,\\pi[L_i\\times (A\\cap L_j)] = \\varrho_1^{ij}\\,\\meas\n\\end{align*}\nfor $i,j\\in\\{1,\\dots,2^n\\}$ provided $\\lambda_{ij} := \\pi[L_i\\times L_j]> 0$. Then \n\\begin{align}\\label{Eq:MSING}\n\\begin{split}\n\\mu_0^{ij} &\\perp \\mu_0^{i'j'},\\\\\n\\mu_1^{ij} &\\perp \\mu_1^{i'j'}\n\\end{split}\n\\end{align}\nfor every $i,i',j,j'\\in\\{1,\\dots,2^n\\}$ with $i\\neq i'$ or $j\\neq j'$. Also, $\\smash{(\\mu_0^{ij},\\mu_1^{ij})}$ is strongly timelike $p$-dualizable by \\autoref{Re:Strong timelike}. By the consistency part in \\autoref{Pr:Consistency TCD} and the above mentioned uniqueness results, and $\\mathrm{wTCD}_p^*(K,N)$, there exists some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^{ij},\\mu_1^{ij})}$ with \n\\begin{align}\\label{Eq:THa inequ}\n\\begin{split}\n&\\int_{\\mathrm{TGeo}^\\uptau(\\mms)} \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_0^{ij}(\\gamma_0)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}(\\gamma)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{\\mathrm{TGeo}^\\uptau(\\mms)} \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_1^{ij}(\\gamma_1)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}(\\gamma)\\\\\n&\\qquad\\qquad \\leq \\int_{\\mathrm{TGeo}^\\uptau(\\mms)} \\varrho_t^{ij}(\\gamma_t)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}^{ij}(\\gamma)\n\\end{split}\n\\end{align}\nfor every $t\\in [0,1]$ and every $N'\\geq N$; note that since $\\smash{\\mu_0^{ij},\\mu_1^{ij}\\in {\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas)}$ as a consequence of the assumption $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{ij} = \\smash{\\rho_t^{ij}}\\,\\meas \\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ thanks to \\autoref{Le:Stu}.\n\n\\textbf{Step 3.} \\textit{Pasting plans together and conclusion.}\nBy uniqueness, we get\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}} = \\lambda_{11}\\,{\\ensuremath{\\boldsymbol{\\pi}}}^{11} + \\lambda_{12}\\,{\\ensuremath{\\boldsymbol{\\pi}}}^{12} + \\dots + \\lambda_{2^n2^n}\\,{\\ensuremath{\\boldsymbol{\\pi}}}^{2^n2^n}\n\\end{align*}\nfor every $n\\in\\N$, since the right-hand side belongs to $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. In particular, setting $\\smash{\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas}$, $t\\in[0,1]$ --- the $\\meas$-ab\\-solute continuity stemming from \\autoref{Le:Stu} and again uniqueness ---, we have\n\\begin{align*}\n\\rho_t = \\lambda_{11}\\,\\varrho_t^{11} + \\lambda_{12}\\,\\varrho_t^{12} + \\dots + \\lambda_{2^n2^n}\\,\\varrho_t^{2^n2^n}\\quad\\meas\\text{-a.e.}\n\\end{align*}\nMoreover, \\eqref{Eq:MSING} and \\autoref{Le:Mutually singular} imply\n\\begin{align*}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{ij} \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{i'j'}\n\\end{align*}\nfor every $t\\in (0,1)$ and every $i,i',j,j'\\in\\{1,\\dots,2^n\\}$ with $i\\neq i'$ or $j\\neq j'$, i.e.\n\\begin{align*}\n\\meas\\big[\\{\\varrho_t^{ij} > 0\\} \\cap \\{\\varrho_t^{i'j'}>0\\}\\big] = 0.\n\\end{align*}\nSetting $G_{ij} := (\\eval_0,\\eval_1)^{-1}(L_i\\times L_j)$, by construction we obtain from \\eqref{Eq:THa inequ} that\n\\begin{align*}\n&\\int_{G_{ij}} \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\int_{G_{ij}} \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\\\\n&\\qquad\\qquad \\leq \\int_{G_{ij}} \\rho_t(\\gamma_t)^{-1\/N'}\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma).\n\\end{align*}\nSince $i,j\\in\\{1,\\dots,2^n\\}$ and $n\\in\\N$ were arbitrary, the claim follows.\n\n\\textbf{Step 4.} \\textit{Removing the restrictions on $\\mu_0$ and $\\mu_1$.} We initially address the first two restrictions in Step 1, then the last one. \n\n\\textbf{Step 4.1.} If the pair $\\smash{(\\mu_0,\\mu_1) \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2}$ with $\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)$ is merely timelike $p$-dualizable, let $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ be a corresponding timelike $p$-dualizing coupling. Since $\\smash{\\mms_\\ll^2}$ is open and $\\met$ is separable, we cover $\\smash{\\supp\\pi\\cap \\mms_\\ll^2}$ by countably many (not necessarily disjoint) relatively compact rectangles $A_i\\times B_i$ with $\\smash{\\bar{A}_i\\times\\bar{B}_i \\subset \\mms_\\ll^2}$, where $A_i,B_i\\subset\\mms$ are open, $i\\in\\N$. Set $\\pi^0 := 0$, and define the measure $\\pi^i$ on $\\mms$ inductively by\n\\begin{align*}\n\\pi^i := (\\pi - \\pi^{i-1})\\mres (A_i\\times B_i).\n\\end{align*}\nThen $\\smash{\\pi = \\pi^1 + \\pi^2 + \\dots}$ by construction. Define $\\smash{\\hat{\\pi}^i\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms^2)}$ by\n\\begin{align*}\n\\hat{\\pi}^i := \\pi^i[\\mms^2]^{-1}\\,\\pi^i\n\\end{align*}\nprovided $\\smash{\\pi^i[\\mms^2]> 0}$, and consider its marginals\n\\begin{align*}\n\\mu_0^i := (\\pr_1)_\\push\\hat{\\pi}^i = \\varrho_0^i\\,\\meas,\\\\\n\\mu_1^i := (\\pr_2)_\\push\\hat{\\pi}^i = \\varrho_1^i\\,\\meas.\n\\end{align*} \nBy \\autoref{Re:Strong timelike}, the pair $\\smash{(\\mu_0^i,\\mu_1^i)}$ is strongly timelike $p$-dualizable for every $i\\in\\N$. Thus, the previous steps imply the existence of $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^i\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^i,\\mu_1^i)}$ such that for every $t\\in (0,1)$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^i = \\smash{\\varrho_t^i}\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$,\n\\begin{align}\\label{Eq:The Inequality rhoT}\n\\varrho_t^i(\\gamma_t)^{-1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_0^i(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\varrho_1^i(\\gamma_1)^{-1\/N'}\n\\end{align}\nholds for $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^i}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$. By construction,\n\\begin{align*}\n\\mu_0^i &\\perp \\mu_0^j,\\\\\n\\mu_1^i &\\perp \\mu_1^j\n\\end{align*}\nfor every $i,j\\in\\N$ with $i\\neq j$, and from \\autoref{Le:Mutually singular} we derive\n\\begin{align*}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^i \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^j.\n\\end{align*}\nNote that the measure\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}} := \\pi^1[\\mms^2]\\,{\\ensuremath{\\boldsymbol{\\pi}}}^1 + \\pi^2[\\mms^2]\\,{\\ensuremath{\\boldsymbol{\\pi}}}^2 + \\dots\n\\end{align*}\nis an $\\smash{\\ell_p}$-optimal geodesic plan interpolating $\\mu_0$ and $\\mu_1$. It satisfies $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in \\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ for every $t\\in[0,1]$, and \n\\begin{align*}\n\\rho_t = \\pi^1[\\mms^2]\\,\\varrho_t^1 + \\pi^2[\\mms^2]\\,\\varrho_t^2 + \\dots\\quad\\meas\\textnormal{-a.e.}\n\\end{align*}\nTogether with \\eqref{Eq:The Inequality rhoT} this discussion implies, for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$,\n\\begin{align*}\n\\rho_t(\\gamma)^{-1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}.\n\\end{align*}\n\n\\textbf{Step 4.2.} Finally, given any timelike $p$-dualizable pair $(\\mu_0,\\mu_1)= (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2}$, for every $i,j\\in\\N$ we set \n\\begin{align*}\nE_i &:= \\{i-1 \\leq \\rho_0^i < i\\},\\\\\nF_j &:= \\{j-1 \\leq \\rho_1^j < j\\}.\n\\end{align*}\nBy restriction \\cite[Lem. 2.10]{cavalletti2020}, $\\smash{(\\mu_0^i,\\mu_1^j)}$ is timelike $p$-dualizable, where \n\\begin{align*}\n\\mu_0^i &:= \\mu_0[E_i]^{-1}\\,\\mu_0\\mres E_i = \\rho_0^i\\,\\meas,\\\\\n\\mu_1^j &:= \\mu_1[F_j]^{-1}\\,\\mu_1\\mres F_j = \\rho_1^j\\,\\meas\n\\end{align*}\nprovided $\\mu_0[E_i] > 0$ as well as $\\mu_1[F_j]>0$. Clearly $\\smash{\\rho_0^i,\\rho_1^j\\in\\Ell^\\infty(\\mms,\\meas)}$, and a similar argument as above leads to the conclusion. We omit the details.\n\\end{proof}\n\n\\begin{proposition}\\label{Pr:iii to iv} Let $K\\in\\R$ and $N\\in [1,\\infty)$. Assume that for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in\\Dom(\\Ent_\\meas)^2$, there is $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ and\n\\begin{align}\\label{Eq:rhot in}\n\\rho_t(\\gamma_t)^{-1\/N}\\geq \\sigma_{K,N}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N} + \\sigma_{K,N}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N}\n\\end{align}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$. Then ${\\ensuremath{\\mathscr{X}}}$ obeys $\\smash{\\mathrm{TCD}_p^e(K,N)}$.\n\\end{proposition}\n\n\\begin{proof} The assignment $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ creates a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$. Moreover, $\\smash{\\pi := (\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$ forms a timelike $p$-dualizing coupling of $\\mu_0$ and $\\mu_1$. Taking logarithms on both sides of \\eqref{Eq:rhot in} and recalling \\eqref{Eq:Distortion coeff property}, we get\n\\begin{align*}\n-\\frac{1}{N}\\log\\rho_t(\\gamma_t) \\geq {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{1}{N}\\log\\rho_0(\\gamma_0), -\\frac{1}{N}\\log\\rho_1(\\gamma_1), \\frac{K}{N}\\,\\uptau^2(\\gamma_0,\\gamma_1)\\Big]\n\\end{align*}\nfor ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$; recall that the function ${\\ensuremath{\\mathrm{G}}}_t$ is defined in \\eqref{Eq:GtHt}. Integrating this inequality with respect to ${\\ensuremath{\\boldsymbol{\\pi}}}$ and using Jensen's inequality together with joint convexity of ${\\ensuremath{\\mathrm{G}}}_t$ yields\n\\begin{align*}\n-\\frac{1}{N}\\Ent_\\meas(\\mu_t) \\geq {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{1}{N}\\Ent_\\meas(\\mu_0), -\\frac{1}{N}\\Ent_\\meas(\\mu_1), \\frac{K}{N}\\,\\big\\Vert\\uptau\\big\\Vert_{\\Ell^2(\\mms^2,\\pi)}^2\\Big].\n\\end{align*}\nHere, in the case $K>0$ and $\\smash{\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)} = \\infty}$ the right-hand side is consistently interpreted as $\\infty$, which implies $\\Ent_\\meas(\\mu_t) = -\\infty$; for $K<0$ and $\\smash{\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\pi)}}=\\infty$ the right-hand side has an evident interpretation. Exponentiating both sides now yields the desired $\\smash{\\mathrm{TCD}_p^e(K,N)}$ condition.\n\\end{proof}\n\n\\begin{proposition}\\label{Pr:v to iii} Assume $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ for some $K\\in\\R$ and $N\\in [1,\\infty)$. Then for every timelike $p$-dualizable pair $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$, and for every $N'\\geq N$, the inequality \n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}\\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\sigma_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, where $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas$. \n\\end{proposition}\n\n\\begin{proof} The proof is similar to the one of \\autoref{Pr:ii to iii} --- whose notation we retain here ---, whence we only outline the main differences. Again, it suffices to consider the case $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$, $\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$, and $\\smash{\\rho_0,\\rho_1\\in\\Ell^\\infty(\\mms,\\meas)}$. In particular, these restrictions imply $\\mu_0,\\mu_1\\in\\Dom(\\Ent_\\meas)$. The parallel uniqueness results outlined in \\autoref{Re:Uniq} under $\\smash{\\mathrm{TCD}_p^e(K,N)}$ which are implicitly used below are due to \\cite[Thm.~3.19, Thm.~3.20]{cavalletti2020}, see also \\autoref{Re:From TNB to TENB}.\n\nLet $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ be an $\\smash{\\ell_p}$-optimal geodesic plan interpolating $\\mu_0$ and $\\mu_1$. Given any $i,j\\in\\{1,\\dots,2^n\\}$, we define\n\\begin{align*}\n\\mu_0^{ij} &:= \\lambda_{ij}^{-1}\\,(\\pr_1)_\\push\\big[(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\mres(L_i\\times L_j)\\big] = \\varrho_0^{ij}\\,\\meas\\\\\n\\mu_1^{ij} &:= \\lambda_{ij}^{-1}\\,(\\pr_2)_\\push\\big[(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\mres(L_i\\times L_j)\\big] = \\varrho_1^{ij}\\,\\meas\n\\end{align*}\nprovided $\\smash{\\lambda_{ij} := {\\ensuremath{\\boldsymbol{\\pi}}}[G_{ij}] > 0}$. Recall that $\\smash{\\mathrm{wTCD}_p^e(K,N)}$ implies $\\smash{\\mathrm{wTCD}_p^e(K,N')}$ by \\cite[Lem.~3.10]{cavalletti2020}. Arguing as for \\autoref{Pr:ii to iii}, invoking $\\smash{\\mathrm{wTCD}_p^e(K,N')}$ for the strongly timelike $p$-dualizable pair $\\smash{(\\mu_0^{ij}, \\mu_1^{ij})} \\in \\smash{({\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)\\cap\\Dom(\\Ent_\\meas))^2}$, employing the uniqueness both of ${\\ensuremath{\\boldsymbol{\\pi}}}$ as well as the involved timelike $p$-dualizing coupling $\\pi^{ij}\\in\\smash{\\Pi_\\ll(\\mu_0^{ij},\\mu_1^{ij})}$ by \\autoref{Re:From TNB to TENB}, and taking logarithms on both sides of the resulting ``pasted inequality'' we infer that\n\\begin{align*}\n&-\\frac{\\lambda_{ij}^{-1}}{N'}\\int_{G_{ij}}\\log \\rho_t(\\gamma_t)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\\\\n&\\qquad\\qquad \\geq {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{\\lambda_{ij}^{-1}}{N'}\\int_{G_{ij}} \\log\\rho_0(\\gamma_0)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma),-\\frac{\\lambda_{ij}^{-1}}{N'}\\int_{G_{ij}} \\log\\rho_1(\\gamma_1)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma),\\\\\n&\\qquad\\qquad\\qquad\\qquad \\lambda_{ij}^{-1}\\,\\frac{K}{N'}\\int_{G_{ij}}\\uptau^2(\\gamma_0,\\gamma_1)\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma)\\Big]\\\\\n&\\qquad\\qquad\\geq \\lambda_{ij}^{-1}\\int_{G_{ij}} {\\ensuremath{\\mathrm{G}}}_t\\Big[\\!-\\!\\frac{1}{N'}\\log\\rho_0(\\gamma_0), -\\frac{1}{N'}\\log\\rho_1(\\gamma_1), \\frac{K}{N'}\\,\\uptau^2(\\gamma_0,\\gamma_1)\\Big]\\d{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma).\n\\end{align*}\nHere the deduction that $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ comes directly from the definition of the functional ${\\ensuremath{\\mathscr{U}}}_N$, and the last inequality follows from joint convexity of ${\\ensuremath{\\mathrm{G}}}_t$ and Jensen's inequality. Since $i,j\\in\\{1,\\dots,2^n\\}$ and $n\\in\\N$ were arbitrary, the claim follows by taking exponentials of both integrands, respectively.\n\\end{proof}\n\nBy evident adaptations of the above arguments, we also obtain the following result for the conditions from \\autoref{Def:TCD}.\n\n\\begin{theorem}\\label{Th:Equiv TCD with geo} The following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item The condition $\\smash{\\mathrm{TCD}_p(K,N)}$ holds.\n\\item The condition $\\smash{\\mathrm{wTCD}_p(K,N)}$ holds.\n\\item For every timelike $p$-dualizable $(\\mu_0,\\mu_1) = (\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)^2$ there exists an $\\smash{\\ell_p}$-optimal geodesic plan ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $t\\in [0,1]$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} =\\rho_t\\, \\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'}&\\geq \\tau_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_0(\\gamma_0)^{-1\/N'} + \\tau_{K,N'}^{(t)}(\\uptau(\\gamma_0,\\gamma_1))\\,\\rho_1(\\gamma_1)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{From local to global}\\label{Sub:Local global} The goal of this section is to establish the equivalence of $\\smash{\\mathrm{TCD}_p^*(K,N)}$ to its corresponding local version in \\autoref{Def:TCD loc} below, i.e.~to establish a Lorentzian analogue of the local-to-global property from \\cite[Ch.~5]{bacher2010}.\n\nTo this aim, in addition to our standing assumptions on the base space, suppose again ${\\ensuremath{\\mathscr{X}}}$ to be timelike $p$-essentially nonbranching for $p\\in (0,1)$.\n\n\n\\begin{definition}\\label{Def:TCD loc} Let $K\\in\\R$ and $N\\in[1,\\infty)$. \n\\begin{enumerate}[label=\\textnormal{\\alph*.}]\n\\item We say that ${\\ensuremath{\\mathscr{X}}}$ satisfies $\\smash{\\mathrm{TCD}_p^*(K,N)}$ \\emph{locally}, briefly $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$, if there exists an open cover $(\\mms_i)_{i\\in I}$ of $\\mms$ with the following property. For every $i\\in I$ and every pair $(\\mu_0,\\mu_1)=(\\rho_0\\,\\meas,\\rho_1\\,\\meas)\\in\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms_i,\\meas)^2}$ such that $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset (\\mms_i)_\\ll^2}$, there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$ from $\\mu_0$ to $\\mu_1$ and an $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ such that for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:RED}\n\\begin{split}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2}\\sigma_{K,N'}^{(t)}(\\uptau(x^0,x^1))\\,\\rho_1(x^1)^{-1\/N'}\\d\\pi(x^0,x^1).\n\\end{split}\n\\end{align}\n\\item If the previous statement holds for $\\smash{\\sigma_{K,N'}^{(1-t)}}$ and $\\smash{\\sigma_{K,N'}^{(t)}}$ replaced by $\\smash{\\tau_{K,N'}^{(1-t)}}$ and $\\smash{\\tau_{K,N'}^{(t)}}$, respectively, ${\\ensuremath{\\mathscr{X}}}$ is termed to satisfy $\\smash{\\mathrm{TCD}_p(K,N)}$ \\emph{locally}, briefly $\\smash{\\mathrm{TCD}_{p,\\loc}(K,N)}$.\n\\end{enumerate} \n\\end{definition}\n\n\n\\begin{proposition}\\label{Pr:MDPTS} Given any $K\\in\\R$ and $N\\in[1,\\infty)$, the property $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds if and only if the following does. For every $\\smash{\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\supp\\mu_0\\times\\supp\\mu_1 \\subset\\smash{\\mms_\\ll^2}$ there exists $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0) + \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1),\n\\end{align*}\nwhere\n\\begin{align*}\n\\theta := \\begin{cases} \\sup \\uptau(\\supp\\mu_0\\times\\supp\\mu_1) & \\textnormal{if }K<0,\\\\\n\\inf \\uptau(\\supp\\mu_0\\times\\supp\\mu_1) & \\textnormal{otherwise}.\n\\end{cases}\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof} We only outline the main differences of the proof to its counterpart from \\cite[Prop.~2.10]{bacher2010}. The forward implication is clear. Conversely, similarly to the proof of \\autoref{Th:Good geos TCD}, by successively gluing plans corresponding to ``midpoints'' for which the R\u00e9nyi entropy obeys the given inequality as in the proof of \\cite[Prop.~2.10]{bacher2010}, we inductively construct a sequence $\\smash{({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}}$ in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ such that for every $n\\in\\N$ and every odd $k\\in\\{1,\\dots,2^n-1\\}$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{k2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n) &\\leq \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{(k-1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n)\\\\\n&\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{(k+1)2^{-n}})_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^n)\\\\\n&\\leq \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{(1-(k-1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{((k-1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1)\\\\\n&\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{(1-(k+1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0)\\\\\n&\\qquad\\qquad\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}(2^{-n+1}\\,\\theta)\\,\\sigma_{K,N'}^{((k+1)2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1)\\\\\n&= \\sigma_{K,N'}^{(1-k2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0) + \\sigma_{K,N'}^{(k2^{-n})}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1).\n\\end{align*}\nholds for every $N'\\geq N$. The latter equality is a crucial property of the distortion coefficients $\\smash{\\sigma_{K,N'}^{(r)}}$ \\cite[Lem.~3.1]{rajala2012b} which does not hold for $\\smash{\\tau_{K,N'}^{(r)}}$. Moreover, note that to proceed by induction in this strategy, it is important to know that the respective ``midpoints'' come from $\\smash{\\ell_p}$-optimal geodesic plans, which ensures that all chronology re\\-lations are preserved through this process.\n\nAs in the proof of \\autoref{Th:Good geos TCD}, $\\smash{({\\ensuremath{\\boldsymbol{\\alpha}}}^n)_{n\\in\\N}}$ admits an accumulation point ${\\ensuremath{\\boldsymbol{\\alpha}}} \\in \\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, giving rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ via $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$. Thanks to the weak lower semicontinuity of the R\u00e9nyi entropy for probability measures with uniformly compact support, the latter satisfies, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t)\\leq \\sigma_{K,N'}^{(1-t)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_0) + \\sigma_{K,N'}^{(t)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_1).\n\\end{align*}\n\nAs for \\cite[Prop.~2.8]{bacher2010}, where the replacement of \\cite[Rem.~2.9]{bacher2010} is \\autoref{Le:Stu}, and \\autoref{Pr:ii to iii}, and using \\autoref{Th:Equivalence TCD* and TCDe} we obtain $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{proof}\n\n\\begin{definition}\\label{Def:Chron lp geod} We term $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ \\emph{chronologically $\\smash{\\ell_p}$-geodesic} if every $\\mu_0,\\mu_1\\in\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\supp\\mu_0\\times\\supp\\mu_1\\subset\\smash{\\mms_\\ll^2}$ are joined by a timelike proper-time parame\\-tri\\-zed $\\smash{\\ell_p}$-geodesic $(\\mu_r)_{r\\in[0,1]}$ which consists of $\\meas$-absolutely continuous measures.\n\\end{definition}\n\n\\begin{theorem}\\label{Th:Local to global} Let $K\\in\\R$ and $N\\in[1,\\infty)$. Then $\\smash{\\mathrm{TCD}_p^*(K,N)}$ holds if and only if $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$ is satisfied and $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ is chronologically $\\smash{\\ell_p}$-geodesic.\n\\end{theorem}\n\nWe will prove \\autoref{Th:Local to global} only in the case $K\\geq 0$, the other situation follows by analogous computations. To this aim, we formulate the following property in \\autoref{Def:Cn} for which, given any $m\\in\\N_0$, we define\n\\begin{align*}\nI_m := \\{k\\,2^{-m} : k\\in\\{0,\\dots,2^m\\}\\}.\n\\end{align*}\nMoreover, given a collection $\\mu:= (\\mu_t)_{t\\in[0,1]}$ in ${\\ensuremath{\\mathscr{P}}}(\\mms)$, let $\\smash{G_\\mu^m}$ denote the set of all $\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$ with $\\gamma_t\\in\\supp\\mu_t$ for every $t\\in I_m$.\n\n\\begin{definition}\\label{Def:Cn} Let $C\\subset\\mms$ be a compact set. Given $m\\in\\N_0$, we say that the property ${\\ensuremath{\\mathrm{P}}}_m(C)$ is satisfied if for every timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\mu:=(\\mu_r)_{r\\in[0,1]}$ consisting of $\\meas$-absolutely continuous measures with $\\mu_0,\\mu_1\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2\\cap C^2}$, and every $s,t\\in I_n$ with $t-s =2^{-m}$ there is an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_s,\\mu_t)}$ with\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t),\n\\end{align*}\nwhere\n\\begin{align}\\label{Eq:Thetat}\n\\theta_{s,t}^m := \\inf\\{\\uptau(\\gamma_s,\\gamma_t) : \\gamma\\in G_\\mu^m\\}.\n\\end{align}\n\\end{definition}\n\n\\begin{lemma}\\label{Le:PnC} Retaining the notation of \\autoref{Def:Cn}, ${\\ensuremath{\\mathrm{P}}}_m(C)$ implies ${\\ensuremath{\\mathrm{P}}}_{m-1}(C)$ for every $m\\in\\N$.\n\\end{lemma}\n\n\\begin{proof} Let a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{\\mu:=(\\mu_r)_{r\\in[0,1]}}$ according to ${\\ensuremath{\\mathrm{P}}}_n(C)$ be given. Let $s,t \\in I_{m-1}$ such that $\\smash{t-s = 2^{1-m}}$, and let $\\smash{\\theta:=\\theta_{s,t}^{m-1}}$ be defined with respect to $\\mu$. Inductively, we build a sequence $\\smash{(\\mu^i)_{i\\in\\N_0}}$ of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics $\\smash{\\mu^i := (\\mu_r^i)_{r\\in[0,1]}}$ obeying $\\smash{\\mu^i_r = \\mu_r}$ for every $i\\in\\N_0$ and every $r\\in[0,s]\\cup[t,1]$ as follows. Initially, set $\\smash{\\mu^0 := \\mu}$. \n\n\\textbf{Step 1.} \\textit{Construction for odd $i\\in\\N_0$.} If the element $\\smash{\\mu^{2i}}$ has been constructed for a given $i\\in\\N_0$, as in the proof of \\cite[Thm.~2.10]{ambrosiogigli2013} and using that the respective ``midpoints'' result from an $\\smash{\\ell_p}$-optimal geodesic plan interpolating its endpoints, we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{\\mu^{2i+1}:= (\\mu_r^{2i+1})_{r\\in[0,1]}}$ with the subsequent properties. For every $r\\in[0,s]\\cup[t,1]$, we have $\\smash{\\mu_r^{2i+1}=\\mu_r}$, and moreover, for every $N'\\geq N$, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m-1}}^{2i+1}) &\\leq \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i}),\\\\\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+3\\times 2^{-m-1}}^{2i+1}) &\\leq \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i}) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t).\n\\end{align*}\nHere we have used the property ${\\ensuremath{\\mathrm{P}}}_m(C)$, the inequalities\n\\begin{align*}\n2\\,\\theta_{s,s+2^{-m}}^m &\\geq \\theta,\\\\\n2\\, \\theta_{s+2^{-m},t}^m &\\geq \\theta, \n\\end{align*}\nwhere both quantities on the left-hand sides are defined with respect to $\\smash{\\mu^{2i}}$, and nonincreasingness of $\\smash{\\sigma_{K,N'}^{(1\/2)}(\\vartheta)}$ in $\\vartheta\\geq 0$ by our assumption $K\\geq 0$.\n\n\\textbf{Step 2.} \\textit{Construction for even $i\\in\\N_0$.} Given $\\smash{\\mu^{2i+1}}$ exhibited in Step 1 above, similarly to this step we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{\\mu^{2i+2} := (\\mu_r^{2i+2})_{r\\in[0,1]}}$ with the property that $\\smash{\\mu_r^{2i+2} = \\mu_r^{2i+1}}$ for every $r\\in[0,s+2^{-m-1}]\\cup\\smash{[s+3\\times 2^{-m-1},1]}$ such that for every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i+2}) \\leq \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m-1}}^{2i+1}) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+3\\times 2^{-m-1}}^{2i+1}).\n\\end{align*}\n\n\\textbf{Step 3.} \\textit{Conclusion.} Pasting together the inequalities from Step 1 and Step 2 yields, for every $i\\in\\N_0$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu^{2i+2}_{s+2^{-m}}) &\\leq 2\\,\\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i})\\\\\n&\\qquad\\qquad + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t). \n\\end{align*}\nIterating this inequality gives\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}}^{2i})&\\leq 2^i\\,\\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^2\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{s+2^{-m}})\\\\\n&\\qquad\\qquad + \\frac{1}{2}\\sum_{k=1}^i 2^k\\,\\sigma_{K,N'}^{(1\/2)}\\Big[\\frac{\\theta}{2}\\Big]^{2k}\\,\\big[{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t)\\big].\n\\end{align*}\n\nAs $\\smash{\\mu_0^{2i} = \\mu_0}$ and $\\smash{\\mu_1^{2i}= \\mu_1}$ for every $i\\in\\N_0$ by construction, using \\autoref{Le:Villani lemma for geodesic} we get the existence of a sequence $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}^{2i})_{i\\in\\N_0}}$ in $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ with $\\smash{\\mu_r^{2i} = (\\eval_r)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^{2i}}$ for every $r\\in[0,1]$ which converges weakly, up to a nonrelabeled subsequence, to an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$. By ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, lower semicontinuity of ${\\ensuremath{\\mathscr{S}}}_{N'}$ on ${\\ensuremath{\\mathscr{P}}}(J(\\mu_0,\\mu_1))$, and the same computations as for \\cite[Clm.~5.2]{bacher2010} for the distortion coefficients on the right-hand side of the previous inequality, setting $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t} := (\\Restr_s^t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_s,\\mu_t)}$ we get\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}(\\theta)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t).\n\\end{align*}\nHere $\\smash{\\Restr_s^t}$ is defined in \\eqref{Eq:Restr def}. This accomplishes ${\\ensuremath{\\mathrm{P}}}_{n-1}(C)$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\autoref{Th:Local to global}] The forward implication is a consequence of \\autoref{Le:Stu}. \n\nConcerning the backward implication, by the partition procedure in the proof of \\autoref{Pr:ii to iii} it suffices to prove the desired inequality defining the condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$ for every $\\smash{\\mu_0,\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with $\\supp\\mu_0\\times\\supp\\mu_1\\subset\\smash{\\mms_\\ll^2}$. \n\nThanks to ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, every member of any timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ has support in $J(\\mu_0,\\mu_1)$. By $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$ and by compactness of $J(\\mu_0,\\mu_1)$, there exist $\\delta > 0$, a disjoint cover $L_1,\\dots,L_n\\subset\\mms$ of $J(\\mu_0,\\mu_1)$, where $n\\in\\N$, and closed sets $\\smash{C_k \\subset\\mms}$ containing $\\smash{{\\ensuremath{\\mathsf{B}}}^\\met(L_k,\\delta)}$, $k\\in\\{1,\\dots,n\\}$, such that every two measures in $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(C_k,\\meas)}$ are joined by a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic obeying \\eqref{Eq:RED}. Let $m\\in \\N$ with\n\\begin{align}\\label{Eq:m choice}\n2^{-m} \\leq \\delta.\n\\end{align}\n\nWe claim $\\smash{{\\ensuremath{\\mathrm{P}}}_m(J(\\mu_0,\\mu_1))}$. Let $(\\mu_r)_{r\\in[0,1]}$ be a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ consisting of $\\meas$-absolutely continuous measures. Let ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ represent $(\\mu_r)_{r\\in[0,1]}$, and define $\\smash{\\varpi\\in{\\ensuremath{\\mathscr{P}}}(\\mms^{2^m+1})}$ by\n\\begin{align}\\label{Eq:PROJ}\n\\varpi := (\\eval_0,\\eval_{2^{-m}}, \\eval_{2^{-m+1}},\\dots,\\eval_{(2^m-1)2^{-m}},\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}.\n\\end{align}\nLet $s,t\\in I_m$ with $t-s = 2^{-m}$.\nFor $k\\in\\{1,\\dots,n\\}$, define $\\smash{\\nu_s^k,\\nu_t^k\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ by\n\\begin{align*}\n\\nu_s^k &:= \\alpha_k^{-1}\\,(\\pr_{2^ms})_\\push\\varpi\\mres L_k,\\\\\n\\nu_t^k &:= \\alpha_k^{-1}\\,(\\pr_{2^mt})_\\push\\big[(\\pr_{2^ms},\\pr_{2^mt})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}\\mres (L_k\\times \\mms)\\big],\n\\end{align*}\nprovided $\\smash{\\alpha_k := \\mu_s[L_k]>0}$. Note that $\\smash{\\supp\\nu_s^k\\subset \\bar{L}_k}$ and $\\smash{\\supp\\nu_s^k\\times\\supp\\nu_t^k\\subset\\mms_\\ll^2}$. Since $r:=\\inf\\uptau(\\supp\\mu_0\\times\\supp\\mu_1) > 0$, every curve in $\\mathrm{TGeo}^\\uptau(\\mms)$ starting in $\\supp\\mu_0$ and ending in $\\supp\\mu_1$ belongs to the set $G_r$ defined in \\autoref{Cor:Cptness and equicty}, which is uniformly equi\\-continuous. Hence, using \\eqref{Eq:m choice} and \\eqref{Eq:PROJ} we get $\\smash{\\supp\\nu_s^k, \\supp\\nu_t^k \\subset \\bar{{\\ensuremath{\\mathsf{B}}}}^\\met(L_k,\\delta)\\subset C_k}$. By $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K,N)}$ in the form described above, there is some $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^k\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_s^k,\\nu_t^k)}$ such that\n\\begin{align}\\label{Eq:Mrg}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^k)\\leq \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_s^k) + \\sigma_{K,N}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^k)\n\\end{align}\nfor every $N'\\geq N$, where $\\theta_{s,t}^m$ is defined as in \\eqref{Eq:Thetat} with respect to $\\mu := (\\mu_r)_{r\\in[0,1]}$.\n\nDefine $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_s,\\mu_t)}$ by\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t} := \\alpha_1\\,{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^1 + \\dots + \\alpha_n\\,{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^n.\n\\end{align*}\nBy construction and \\autoref{Le:Mutually singular}, respectively, $\\smash{(\\eval_r)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^1,\\dots,(\\eval_r)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^n}$ are mutually singular for every fixed $r\\in\\{0,1\/2\\}$, whence\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) &= \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_s^1) + \\dots + \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_s^n),\\\\\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) &= \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^1) + \\dots + \\alpha_n^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}^n).\n\\end{align*} \nOn the other hand, since mutual singularity of $\\smash{\\nu_t^1,\\dots,\\nu_t^n}$ may fail, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) \\geq \\alpha_1^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^1) + \\dots + \\alpha_n^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^n).\n\\end{align*}\nMerging these inequalities with \\eqref{Eq:Mrg} yields\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}((\\eval_{1\/2})_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_{s,t}) \\leq \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_s) + \\sigma_{K,N'}^{(1\/2)}(\\theta_{s,t}^m)\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t),\n\\end{align*}\nand the desired property ${\\ensuremath{\\mathrm{P}}}_m(J(\\mu_0,\\mu_1))$ follows.\n\nTo close the proof of \\autoref{Th:Local to global}, let $(\\mu_r)_{r\\in[0,1]}$ be a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0$ to $\\mu_1$ which consists of $\\meas$-absolutely continuous measures, as hypothesized. Iteratively, the property ${\\ensuremath{\\mathrm{P}}}_m(J(\\mu_0,\\mu_1))$ proven above implies ${\\ensuremath{\\mathrm{P}}}_0(J(\\mu_0,\\mu_1))$ by \\autoref{Le:PnC}. By \\autoref{Propos}, this already implies the condition $\\smash{\\mathrm{TCD}_p^*(K,N)}$.\n\\end{proof}\n\nThe following \\autoref{Cor:K-} easily follows from \\autoref{Th:Local to global} and the stability result from \\autoref{Th:Stability TCD}. \\autoref{Pr:Pr} is proven along the same lines as \\cite[Prop.~5.5]{bacher2010} by using uniform continuity of $\\uptau$ on compact subsets of $\\smash{\\mms^2}$; the point is that for small $\\vartheta\\geq 0$, the quantities $\\smash{\\sigma_{K,N}^{(r)}(\\vartheta)}$ and $\\smash{\\tau_{K,N}^{(r)}(\\vartheta)}$ ``almost coincide''. See also the computations in \\cite{deng}.\n\n\\begin{corollary}\\label{Cor:K-} Let $K\\in\\R$ and $N\\in[1,\\infty)$. Then chronological $\\smash{\\ell_p}$-geodesy of $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and the condition $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K',N)}$ hold for every $K' < K$ if and only if $\\smash{\\mathrm{TCD}_p^*(K,N)}$ is satisfied.\n\\end{corollary}\n\n\\begin{proposition}\\label{Pr:Pr} Assume $\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ to be chronologically $\\smash{\\ell_p}$-geodesic. Given any $K\\in\\R$ and $N\\in[1,\\infty)$, $\\smash{\\mathrm{TCD}_{p,\\loc}^*(K',N)}$ holds for every $K'0$, the rescaled measured Lorentzian pre-length space $(\\mms, a\\met, b\\meas,\\ll,\\leq, \\theta\\uptau)$ satisfies $\\smash{\\mathrm{TMCP}_p^*(K\/\\theta^2,N)}$.\n\\end{enumerate}\n\nAnalogous statements hold for the $\\smash{\\mathrm{TMCP}_p(K,N)}$ condition.\n\\end{proposition}\n\n\\begin{remark}\\label{Re:Unlikely} An analogue of \\autoref{Le:Stu} is unlikely to hold under $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ or $\\smash{\\mathrm{TMCP}_p(K,N)}$. In particular, we do not know a priori whether the measures $\\mu_t$, $t\\in[0,1)$, of a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic as in \\autoref{Def:TMCP} are always --- or can be chosen to be --- $\\meas$-absolutely continuous. Instead, we establish the existence of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics consisting of $\\meas$-absolutely continuous measures whose densities are uniformly $\\Ell^\\infty$ in time under $\\smash{\\mathrm{TMCP}_p^*(K,N)}$, at least if $\\smash{\\mu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ has $\\meas$-essentially bounded density, by a variational method following \\cite{braun2022}, see also \\cite{cavalletti2017,rajala2012a,rajala2012b} in \\autoref{Sub:Good}; cf.~\\autoref{Th:Good TMCP}. By uniqueness of timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics under timelike essential nonbranching conditions, see \\autoref{Th:Uniqueness geodesics}, this suffices to show an equivalence result analogous to \\autoref{Th:Equivalence TCD* and TCDe} for our timelike measure-contraction property, see \\autoref{Th:Equivalence TMCP* and TMCPe} and \\autoref{Th:Equivalence TMCP}.\n\\end{remark}\n\nThe link of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ and $\\smash{\\mathrm{TMCP}_p(K,N)}$ to their respective $\\mathrm{TCD}$ counterparts is more subtle. The proof of the corresponding \\autoref{Pr:TMCP to TCD} follows \\cite[Prop.~3.11]{cavalletti2020}. We need the following result whose proof is a straigtforward adaptation of the argument for \\cite[Lem.~3.3]{sturm2006b}, recall also \\autoref{Le:Const perturb}. The difference to \\cite{sturm2006b} is that we only require the marginal whose density is contained in the respective integrand to be constant.\n\n\\begin{lemma}\\label{Le:USC lemma} Let $K\\in\\R$ and $N\\in[1,\\infty)$. Let $\\rho\\colon\\mms\\to [0,\\infty)$ be a Borel function with $\\mu:= \\rho\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$. Moreover, let $(\\pi_n)_{n\\in\\N}$ be a sequence in ${\\ensuremath{\\mathscr{P}}}(\\mms^2)$ converging weakly to $\\pi\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ such that for some $i\\in\\{1,2\\}$, we have \n\\begin{align*}\n(\\pr_i)_\\push\\pi_n = \\mu\n\\end{align*}\nfor every $n\\in\\N$. Then for every every $r\\in[0,1]$,\n\\begin{align*}\n&\\int_{\\mms^2} \\tau_{K,N}^{(r)}(\\uptau(x^0,x^1))\\,(\\rho\\circ\\pr_i)(x^0,x^1)^{-1\/N} \\d\\pi(x^0,x^1)\\\\\n&\\qquad\\qquad\\leq \\liminf_{n\\to\\infty}\\int_{\\mms^2} \\tau_{K,N}^{(r)}(\\uptau(x^0,x^1))\\,(\\rho\\circ\\pr_i)(x^0,x^1)^{-1\/N}\\d\\pi_n(x^0,x^1).\n\\end{align*}\n\nAn analogous assertion holds with $\\smash{\\tau_{K,N}^{(r)}}$ replaced by $\\smash{\\sigma_{K,N}^{(r)}}$.\n\\end{lemma}\n\n\\begin{proposition}\\label{Pr:TMCP to TCD} The following hold for every $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{LART} The condition $\\smash{\\mathrm{wTCD}_p(K,N)}$ implies $\\smash{\\mathrm{TMCP}_p(K,N)}$.\n\\item\\label{LARTT} The condition $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ implies $\\smash{\\mathrm{TMCP}_p^*(K,N)}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} We only prove \\ref{LART}, the proof of \\ref{LARTT} is similar.\n\n\\textbf{Step 1.} \\textit{Approximation of $\\mu_0$ and $\\mu_1$.} \nGiven any $\\varepsilon\\in (0,1)$, let $C_\\varepsilon \\subset\\mms$ be a compact set with $\\mu_0[C_\\varepsilon] \\geq 1-\\varepsilon$ and $\\smash{C_\\varepsilon \\times \\{x_1\\}\\subset\\mms_\\ll^2}$. Define $\\smash{\\mu_0^\\varepsilon\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ by\n\\begin{align*}\n\\mu_0^\\varepsilon := \\mu_0[C_\\varepsilon]^{-1}\\,\\mu_0\\mres C_\\varepsilon = \\rho_0^\\varepsilon\\,\\meas\n\\end{align*}\nMoreover, since $\\smash{\\mms_\\ll^2}$ is open while $C_\\varepsilon\\times\\{x_1\\}$ is compact, there exists $\\eta > 0$ such that $\\smash{C_\\varepsilon\\times {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\eta)\\subset\\mms_\\ll^2}$. For $\\delta\\in (0,\\eta)$, we set \n\\begin{align*}\n\\mu_1^\\delta := \\meas\\big[{\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta)\\big]^{-1}\\,\\meas\\mres {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\delta) = \\rho_1^\\delta\\,\\meas.\n\\end{align*}\n\nBy \\autoref{Re:Strong timelike}, the pair $\\smash{(\\mu_0^\\varepsilon,\\mu_1^\\delta)}$ is strongly timelike $p$-dualizable, and using $\\smash{\\mathrm{wTCD}_p(K,N)}$ we find a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_t^{\\varepsilon,\\delta})_{t\\in[0,1]}}$ connecting $\\smash{\\mu_0^\\varepsilon}$ to $\\smash{\\mu_1^\\delta}$ as well as a timelike $p$-dualizing $\\smash{\\pi^{\\varepsilon,\\delta}\\in\\Pi_\\ll(\\mu_0^\\varepsilon,\\mu_1^\\delta)}$, with support in $\\smash{C_\\varepsilon\\times {\\ensuremath{\\mathsf{B}}}^\\met(x_1,\\eta)}$, such that for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^{\\varepsilon,\\delta}) &\\leq -\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta}(x^0,x^1)\\\\\n&\\qquad\\qquad -\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_1^\\delta(x^1)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta}(x^0,x^1)\\\\\n&\\leq -\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta}(x^0,x^1).\n\\end{align*}\n\n\\textbf{Step 2.} \\textit{Sending $\\delta \\to 0$.} Given any $\\varepsilon > 0$ and a fixed sequence $(\\delta_n)_{n\\in\\N}$ decreasing to $0$, from $\\smash{(\\mu_t^{\\varepsilon,\\delta_n})_{t\\in[0,1]}}$, $n\\in\\N$, we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic from $\\mu_0^\\varepsilon$ to $\\mu_1$ as follows. Let $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^{\\varepsilon,\\delta_n}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^\\varepsilon,\\mu_1^{\\delta_n})}$ represent $\\smash{(\\mu_t^{\\varepsilon,\\delta_n})_{t\\in[0,1]}}$. As $\\smash{(\\mu_1^{\\delta_n})_{n\\in\\N}}$ converges weakly to $\\mu_1$, this sequence is tight, and so is $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}^{\\varepsilon,\\delta_n})_{n\\in\\N}}$ by \\autoref{Le:Villani lemma for geodesic}. Let $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}^\\varepsilon\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0^\\varepsilon,\\mu_1)}$ be a weak limit of a nonrelabeled subsequence. Then the assignment $\\smash{\\mu_t^\\varepsilon := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}^\\varepsilon}$, $t\\in[0,1]$, gives rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic connecting $\\smash{\\mu_0^\\varepsilon}$ to $\\mu_1$.\n\n\nEvery weak limit point of $\\smash{(\\pi^{\\varepsilon,\\delta_n})_{n\\in\\N}}$ equals $\\smash{\\mu_0^\\varepsilon\\otimes\\mu_1 = \\mu_0^\\varepsilon\\otimes\\delta_{x_1}}$.\nTherefore, ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, weak lower semicontinuity of $\\smash{{\\ensuremath{\\mathscr{S}}}_{N'}}$ on measures with uniformly bounded support, and \\autoref{Le:USC lemma} yield, for every such $t$ and every $N'\\geq N$, \n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^\\varepsilon) &\\leq \\limsup_{n\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^{\\varepsilon,\\delta_n})\\\\\n&\\leq -\\liminf_{n\\to\\infty}\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\pi^{\\varepsilon,\\delta_n}(x^0,x^1)\\\\\n&\\leq -\\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_1))\\,\\rho_0^\\varepsilon(x^0)^{-1\/N'}\\d\\mu_0^\\varepsilon(x^0)\n\\end{align*}\n\n\\textbf{Step 3.} \\textit{Sending $\\varepsilon \\to 0$.} Given a sequence $(\\varepsilon_n)_{n\\in\\N}$ decreasing to $0$, note that $\\smash{(\\mu_0^{\\varepsilon_n})_{n\\in\\N}}$ converges weakly to $\\mu_0$. Similarly to Step 2, from $\\smash{(\\mu_t^{\\varepsilon_n})_{n\\in\\N}}$ we construct a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ connecting $\\mu_0$ to $\\mu_1$. As in Step 2 and using Fatou's lemma we get, for every $t\\in[0,1]$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq \\limsup_{n\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^{\\varepsilon_n})\\\\\n&\\leq -\\liminf_{n\\to\\infty}\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_1))\\,\\rho_0^{\\varepsilon_n}(x^0)^{-1\/N'}\\d\\mu_0^{\\varepsilon_n}(x^0)\\\\\n&\\leq -\\int_{\\mms^2}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\mu_0(x^0).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{remark}\\label{Re:Geom inequ TMCP} With essentially identical proofs, the respective versions of the timelike Bonnet--Myers inequality, \\autoref{Cor:Bonnet-Myers} and \\autoref{Cor:Reduced BM}, and the timelike Bishop--Gromov inequality, \\autoref{Th:BG} and \\autoref{Th:Reduced BG}, hold as well under $\\smash{\\mathrm{TMCP}_p(K,N)}$ and $\\smash{\\mathrm{TMCP}_p^*(K,N)}$. We only need to assume ${\\ensuremath{\\mathscr{X}}}$ to be a causally closed and globally hyperpolic Lorentzian geodesic space.\n\\end{remark}\n\n\\subsection{Stability} Next, we discuss the stability of the notions from \\autoref{Def:TMCP}. In contrast to the weak stability from \\autoref{Th:Stability TCD}, both timelike measure-contraction properties are stable under the convergence introduced in \\autoref{Def:Convergence}.\n\n\\begin{theorem}\\label{Th:Stability TMCP} Assume the convergence of $({\\ensuremath{\\mathscr{X}}}_k)_{k\\in\\N}$ to ${\\ensuremath{\\mathscr{X}}}_\\infty$ as in \\autoref{Def:Convergence}. Moreover, let $(K_k,N_k)_{k\\in\\N}$ be a sequence in $\\R\\times [1,\\infty)$ converging to $(K_\\infty,N_\\infty)\\in\\R\\times[1,\\infty)$. Suppose the existence of $p\\in (0,1)$ such that ${\\ensuremath{\\mathscr{X}}}_k$ obeys $\\mathrm{TMCP}_p(K_k,N_k)$ for every $k\\in\\N$. \nThen $\\smash{{\\ensuremath{\\mathscr{X}}}_\\infty}$ satisfies $\\smash{\\mathrm{TMCP}_p(K_\\infty,N_\\infty)}$. \n\nThe analogous statement in which $\\smash{\\mathrm{TMCP}_p(K_k,N_k)}$ is respectively replaced by $\\smash{\\mathrm{TMCP}_p^*(K_k,N_k)}$, $k\\in\\N_\\infty$, holds as well.\n\\end{theorem}\n\n\\begin{proof} It suffices to prove the first statement, the second is argued analogously. \n\nIn this proof, we often adopt notations from the proof of \\autoref{Th:Stability TCD} without explicit notice. In particular, we again identify $\\mms_k$ with its image $\\iota_k(\\mms_k)$ in $\\mms$ and $\\meas_k$ with its push-forward $(\\iota_k)_\\push\\meas_k$ for every $k\\in\\N_\\infty$. \n\n\\textbf{Step 1.} \\textit{Reduction to compact $\\mms$.} Owing to \\autoref{Re:mae x1}, given any $\\mu_{\\infty,0} = \\rho_{\\infty,0}\\,\\meas_\\infty\\in {\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas_\\infty)$ and $x_{\\infty,1}\\in I^+(\\mu_{\\infty,0}) \\cap \\supp\\meas_\\infty$, as for \\autoref{Th:Stability TCD} we use the compactness of $\\supp\\mu_{\\infty,0}$ and $\\supp\\mu_{\\infty,1}$, where $\\smash{\\mu_{\\infty,1} := \\delta_{x_{\\infty,1}}}$, to assume without restriction that $\\mms$ is compact, and that $\\meas_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms)$ for every $k\\in\\N_\\infty$. All measures considered below will thus be compactly supported. Moreover, we may and will suppose that $W_2(\\meas_k,\\meas_\\infty)\\to 0$ as $k\\to\\infty$.\n\n\\textbf{Step 2.} \\textit{Restriction of the assumptions on $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$.} We will first assume that $\\tau(\\cdot,x_{\\infty,1})$ is bounded away from zero on $\\supp\\mu_{\\infty,0}$, that $\\rho_{\\infty,0}\\in\\Ell^\\infty(\\mms,\\meas_\\infty)$, and that $x_{\\infty,1}$ can be approximated with respect to $\\met$ by a sequence $(x_{k,1})_{k\\in\\N}$ of points $x_{k,1}\\in \\supp\\meas_k$ such that $x_{\\infty,1} \\in I^-(x_{k,1})$ for every $k\\in\\N$. The general case is discussed in Step 7 below; we note for now that this conclusion will not conflict with our reductions from Step 1.\n\n\\textbf{Step 3.} \\textit{Construction of a chronological recovery sequence.}\nIn this step, given the sequence $(x_{k,1})_{k\\in\\N}$ from Step 2 we construct a sequence $(\\mu_{k,0})_{k\\in\\N}$ of measures $\\smash{\\mu_{k,0} = \\rho_{k,0}\\,\\meas_k\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$, $k\\in\\N$, such that $\\mu_{k,0}\\to\\mu_{\\infty,0}$ weakly as $k\\to \\infty$ possibly up to extracting a subsequence, and $x_{k,1}\\in I^+(\\mu_{k,0})$ for every $k\\in\\N$. The constructed sequence will allow for the correct behavior of all functionals under consideration, cf.~Step 5 below.\n\nGiven any $k\\in\\N$, let $\\mathfrak{q}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ be a $W_2$-optimal coupling of $\\meas_k$ and $\\meas_\\infty$. We disintegrate $\\mathfrak{q}_k$ with respect to $\\pr_1$, writing\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}} \\mathfrak{q}_k(x,y) = {\\ensuremath{\\mathrm{d}}}\\mathfrak{p}_x^k(y)\\d\\meas_k(x).\n\\end{align*}\nLet $\\smash{\\mathfrak{p}^k\\colon {\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)\\to{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ denote the canonically induced map.\n\nGiven any $k\\in\\N$, define $\\tilde{\\mu}_{k,0}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)$ by\n\\begin{align*}\n\\tilde{\\mu}_{k,0} := \\mathfrak{p}^k(\\mu_{\\infty,0}) = \\tilde{\\rho}_{k,0}\\,\\meas_k.\n\\end{align*}\nNote that the measure $\\mathfrak{r}_k := (\\rho_{\\infty,0}\\circ\\pr_2)\\,\\mathfrak{q}_k\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)$ constitutes a coupling of $\\tilde{\\mu}_{k,0}$ and $\\mu_{\\infty,0}$. With the $W_2$-optimality of $\\mathfrak{q}_k$, this implies\n\\begin{align*}\nW_2(\\tilde{\\mu}_{k,0},\\mu_{\\infty,0}) \\leq \\big\\Vert\\rho_{\\infty,0}\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}^{1\/2}\\,W_2(\\meas_k,\\meas_\\infty),\n\\end{align*}\nand consequently $\\smash{\\tilde{\\mu}_{k,0}\\to \\mu_{\\infty,0}}$ weakly as $k\\to\\infty$. Hence, by our assumption on the se\\-quence $(x_{k,1})_{k\\in\\N}$ from Step 2 and Portmanteau's theorem,\n\\begin{align*}\n\\liminf_{k\\to\\infty} \\tilde{\\mu}_{k,0}[I^-(x_{k,1})] \\geq \\liminf_{k\\to\\infty} \\tilde{\\mu}_{k,0}[I^-(x_{\\infty,1})] \\geq \\mu_{\\infty,0}[I^-(x_{\\infty,1})] =1.\n\\end{align*}\nUp to passing to a subsequence we may and will thus assume $\\smash{\\tilde{\\mu}_{k,0}[I^-(x_{k,1})] >0}$ for every $k\\in\\N$. We then define $\\smash{\\mu_{k,0}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_k)}$ by\n\\begin{align*}\n\\mu_{k,0} := \\tilde{\\mu}_{k,0}[I^-(x_{k,1})]^{-1} \\tilde{\\mu}_{k,0} \\mres I^-(x_{k,1}).\n\\end{align*}\nBy construction, the sequence $(\\mu_{k,0})_{k\\in\\N}$ converges to $\\mu_{\\infty,0}$ weakly as $k\\to\\infty$, and we have $x_{k,1}\\in I^+(\\mu_{k,0})$ for every $k\\in\\N$, as desired.\n\n\\textbf{Step 4.} \\textit{Invoking the $\\mathrm{TMCP}$ condition.} Fix $K\\in\\R$ and $N\\in (1,\\infty)$ such that $K< K_\\infty$ and $N>N_\\infty$. Up to passing to a subsequence if necessary, we may and will thus assume that $KN_k$ for every $k\\in\\N$.\n\nBy \\autoref{Pr:Consistency TMCP}, for every $k\\in\\N$ there exists a timelike proper-time para\\-metrized $\\smash{\\ell_p}$-geodesic $(\\mu_{k,t})_{t\\in[0,1]}$ connecting $\\mu_{k,0}$ and $\\smash{\\mu_{k,1} = \\delta_{x_{k,1}}}$ such that for every $t\\in [0,1)$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:TMCP cond}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t}) \\leq -\\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{k,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'} \\d\\mu_{k,0}(x^0).\n\\end{align}\n\n\\textbf{Step 5.} \\textit{Estimating the previous right-hand side.} We estimate the negative of the right-hand side of \\eqref{Eq:TMCP cond} from below, up to errors which become arbitrarily small. By construction of $\\mu_{k,0}$, we find a sequence $(a_k)_{k\\in\\N}$ of normalization constants converging to $1$ such that $\\smash{\\rho_{k,0} \\leq a_k\\,\\tilde{\\rho}_{k,0}}$ $\\meas_k$-a.e.~for every $k\\in\\N$.\n\n\\textbf{Step 5.1.} First, we argue that $x_{k,1}$ can be replaced $x_{\\infty,1}$ up to a small error. By our choice of $K$ and $N$ and the timelike Bonnet--Myers inequality from \\autoref{Re:Geom inequ TMCP},\n\\begin{align*}\nc := \\sup\\tau_{K,N'}^{(1-t)}\\circ\\uptau(\\mms^2)\n\\end{align*}\nis finite. Furthermore, the involved function is jointly uniformly continuous. Thus, given any $\\varepsilon> 0$, by uniform continuity we obtain, for sufficiently large $k\\in\\N$,\n\\begin{align*}\n&\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{k,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)\\\\\n&\\qquad\\qquad \\geq \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)-\\varepsilon.\n\\end{align*}\n\n\\textbf{Step 5.2.} We modify $\\smash{\\tilde{\\mu}_{k,0}}$ into the measure\n\\begin{align*}\n\\nu_{k,0} := (1+\\delta_k)^{-1}\\,(\\tilde{\\rho}_{k,0}+\\delta_k)\\,\\meas_k =\\varrho_{k,0}\\,\\meas_k,\n\\end{align*} \nwhere $\\delta_k\\in[0,1]$ is defined by\n\\begin{align*}\n\\delta_k = \\tilde{\\mu}_{k,0}[I^-(x_{k,1})^{\\ensuremath{\\mathsf{c}}}].\n\\end{align*}\nNote that $(\\delta_k)_{k\\in\\N}$ converges to $0$. Employing the inequality $\\rho_{k,0} \\leq a_k\\,\\varrho_{k,0}$ $\\meas_k$-a.e., the definition of $\\mu_{k,0}$, and a similar notation as in the proof of \\autoref{Th:Stability TCD}, \n\\begin{align}\\label{Eq:Integral rechnung}\n&\\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'} \\d\\mu_{k,0}(x^0)\\nonumber\\\\\n&\\qquad\\qquad \\geq_k\\int_{\\mms}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)\\nonumber\\\\\n&\\qquad\\qquad \\geq_k\\int_{\\mms}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad\\qquad\\qquad - \\int_{I^-(x_{k,1})^{\\ensuremath{\\mathsf{c}}}} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\tilde{\\mu}_{k,0}(x^0)\\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad - \\delta_k\\int_{\\mms}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0).\\nonumber\n\\end{align}\n\n\\textbf{Step 5.3.} Possibly invoking the timelike Bishop--Gromov inequality outlined in \\autoref{Re:Geom inequ TMCP}, we obtain the estimates\n\\begin{align*}\nc\\,\\delta_k^{1-1\/N'} &\\geq \\int_{I^-(x_{k,1})^{\\ensuremath{\\mathsf{c}}}}\\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\tilde{\\mu}_{k,0}(x^0)\\\\\nc\\,\\delta_k^{1-1\/N'} &\\geq \\delta_k\\int_{\\mms} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\meas_k(x^0).\n\\end{align*}\nHence, our main task is to estimate the integral in \\eqref{Eq:Integral rechnung} from below. To this aim, by definition of $\\nu_{k,0}$ and Jensen's inequality, we obtain\n\\begin{align*}\n&\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad = \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\Big[\\!\\int_\\mms (\\rho_{\\infty,0}(y^0)+\\delta_k)\\d\\mathfrak{p}_{x^0}^k(y^0)\\Big]^{1-1\/N'}\\!\\d\\meas_k(x^0)\\\\\n&\\qquad\\qquad \\geq \\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{\\infty,0}(y^0)^{1-1\/N'}\\d\\mathfrak{q}_k(x^0,y^0).\n\\end{align*}\nLet $(\\phi_i)_{i\\in\\N}$ be a sequence in $\\Cont_\\bounded(\\mms)$ such that\n\\begin{align*}\n\\big\\Vert \\phi_i- \\rho_{\\infty,0}^{1-1\/N'}\\big\\Vert_{\\Ell^1(\\mms,\\meas_\\infty)} \\leq 2^{-i}\n\\end{align*}\nas well as $\\sup\\phi_i(\\mms) \\leq \\Vert\\rho_{\\infty,0} \\Vert_{\\Ell^\\infty(\\mms,\\meas_\\infty)}$ for every $i\\in\\N$. Then we get\n\\begin{align*}\n&\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad \\geq \\int_{\\mms^2} \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\phi_i(y^0)\\d\\mathfrak{q}_k(x^0,y^0) - 2^{-i}\\,c.\n\\end{align*}\n\n\\textbf{Step 5.4.} By tightness and stability of $W_2$-optimal couplings \\cite[Lem.~4.3, Lem. 4.4]{villani2009}, $(\\mathfrak{q}_k)_{k\\in\\N}$ converges weakly to the dia\\-gonal coupling of $\\meas_\\infty$ and $\\meas_\\infty$ along a nonrelabeled subsequence. In particular, by Lebesgue's theorem,\n\\begin{align*}\n&\\liminf_{k\\to\\infty} \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\varrho_{k,0}(x^0)^{-1\/N'}\\d\\nu_{k,0}(x^0)\\\\\n&\\qquad\\qquad \\geq \\liminf_{i\\to\\infty} \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\phi_i(x^0)\\d\\meas_\\infty(x^0) - c\\,\\limsup_{i\\to\\infty}2^{-i}\\\\\n&\\qquad\\qquad =\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{\\infty,0}(x^0)^{-1\/N'}\\d\\mu_{\\infty,0}(x^0).\n\\end{align*}\n\n\\textbf{Step 6.} \\textit{Conclusion.} Let $({\\ensuremath{\\boldsymbol{\\pi}}}_k)_{k\\in\\N}$ be a sequence of $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_k\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{k,0},\\mu_{k,1})}$ representing the $\\smash{\\ell_p}$-geodesic $(\\mu_{k,t})_{t\\in[0,1]}$ in Step 4, $k\\in\\N$. By compactness of $\\mms$ and \\autoref{Le:Villani lemma for geodesic}, this sequence converges to an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_\\infty\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_{\\infty,0},\\mu_{\\infty,1})}$ along a nonrelabeled subsequence. In particular, the assignment $\\smash{\\mu_{\\infty,t} := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_\\infty}$ gives rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_{\\infty,t})_{t\\in[0,1]}$ from $\\mu_{\\infty,0}$ to $\\mu_{\\infty,1}$; note that every optimal coupling of $\\mu_{\\infty,0}$ and $\\mu_{\\infty,1}$ is concentrated on $\\smash{\\mms_\\ll^2}$. Using weak lower semicontinuity of the R\u00e9nyi entropy and \\eqref{Eq:TMCP cond}, given any $\\varepsilon > 0$ this yields, for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{\\infty,t}) &\\leq \\limsup_{k\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{k,t})\\\\\n&\\leq -\\liminf_{k\\to\\infty} \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{k,1}))\\,\\rho_{k,0}(x^0)^{-1\/N'}\\d\\mu_{k,0}(x^0)\\\\\n&\\leq \\varepsilon - \\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}))\\,\\rho_{\\infty,0}(x^0)^{-1\/N'}\\d\\mu_{\\infty,0}(x^0).\n\\end{align*}\n\n\n\n\n\\textbf{Step 7.} \\textit{Relaxation of the assumptions on $\\mu_\\infty,0$ and $\\mu_{\\infty,1}$.} First, we argue how to construct the hypothesized sequence $(x_{k,1})_{k\\in\\N}$ from Step 2. The idea is to approximate points which lie ``in between'' $\\supp\\mu_{\\infty,0}$ and $x_{\\infty,1}$ from the future. For these points, the above discussion applies, and we will be able to conclude the desired $\\mathrm{TMCP}$ property by a tightness argument.\n\n\\textbf{Step 7.1.} As $\\inf\\tau(\\supp\\mu_{\\infty,0}, x_{\\infty,1})>0$, given any $i\\in \\N$ we may and will fix a sequence $\\smash{(x_{\\infty,1}^i)_{i\\in\\N}}$ of points $\\smash{x_{\\infty,1}^i}\\in I(\\mu_{\\infty,0},\\mu_{\\infty,1})\\cap\\supp\\meas_\\infty$ converging to $\\smash{x_{\\infty,1}}$. Since $I(x_{\\infty,1}^i,x_{\\infty,1})$ is open and nonempty, for every $i\\in\\N$ we construct a sequence $\\smash{(x_{\\infty,1}^{i,j})_{j\\in\\N}}$ such that $\\smash{x_{\\infty,1}^{i,j} \\in I(x_{\\infty,1}^i, x_{\\infty,1})\\cap\\supp\\meas_\\infty}$ and\n\\begin{align*}\nx_{\\infty,1}^{i,j} \\ll x_{\\infty,1}^{i,j-1}\n\\end{align*}\nfor every $j\\in\\N$. Since $\\smash{x_{\\infty,1}^{i,j}\\in\\supp\\meas_\\infty}$, the weak convergence of $(\\meas_k)_{k\\in\\N}$ implies the existence of a sequence $\\smash{(x_{k,1}^{i,j})_{k\\in\\N}}$ of points $\\smash{x_{k,1}^{i,j}\\in\\supp\\meas_k}$ which converges to $\\smash{x_{\\infty,1}^{i,j}}$. In particular, for a sufficiently large integer $k_j\\in \\N$, we have \n\\begin{align*}\nx_{\\infty,1}^i\\ll x_{\\infty,1}^{i,j} \\ll x_{k_j,1}^{i,j}\n\\end{align*}\nfor every $i,j\\in\\N$, as well as $\\smash{\\met(x_{k_j}^{i,j}, x_{\\infty,1}^i) \\to 0}$ as $j\\to\\infty$ for every $i\\in\\N$. \n\nThe above arguments applied for a fixed $i\\in\\N$ (along a suitable subsequence in $k$ which depends on $i$) yield the existence of a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_{\\infty,t}^i)_{t\\in[0,1]}}$ from $\\mu_{\\infty,0}$ to $\\smash{\\mu_{\\infty,1}^i:= \\delta_{x_{\\infty,1}^i}}$ such that for every $t\\in[0,1)$ and every $N'\\geq N_\\infty$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_{\\infty,t}^i) \\leq -\\int_\\mms \\tau_{K,N'}^{(1-t)}(\\uptau(x^0,x_{\\infty,1}^i))\\,\\rho_{\\infty,0}(x^0)^{-1\/N'}\\d\\mu_{\\infty,0}(x^0).\n\\end{align*}\nSimilarly to Step 6 and using lower semicontinuity of R\u00e9nyi's entropy and Fatou's lemma, we find a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_{\\infty,t})_{t\\in[0,1]}}$ connecting $\\smash{\\mu_{\\infty,0}}$ to $\\smash{\\mu_{\\infty,1}}$ such that the previous inequality holds for $\\smash{\\mu_{\\infty,t}^i}$ and $\\smash{x_{\\infty,1}^i}$ replaced by $\\smash{\\mu_{\\infty,t}}$ and $\\smash{x_{\\infty,1}}$, respectively, for every $t\\in[0,1)$ and every $N'\\geq N_\\infty$.\n\n\\textbf{Step 7.2.} Finally, we remove the two assumptions on $\\mu_{\\infty,0}$ from Step 2. Let $\\mu_{\\infty,0}\\in\\smash{{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ and $\\smash{x_{\\infty,1}\\in I^+(\\mu_{\\infty,0})\\cap\\supp\\meas_\\infty}$. Given sufficiently large $n,m\\in\\N$, define $\\smash{\\mu_{\\infty,0}^n,\\mu_{\\infty,0}^{n,m}\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas_\\infty)}$ by\n\\begin{align*}\n\\mu_{\\infty,0}^n &:= \\mu_{\\infty,0}\\big[\\{\\uptau(\\cdot, x_{\\infty,1}) \\geq 2^{-n}\\}\\big]^{-1}\\,\\mu_{\\infty,0} \\mres \\{\\uptau(\\cdot, x_{\\infty,1}) \\geq 2^{-n}\\} = \\rho_{\\infty,0}^n\\,\\meas_\\infty,\\textcolor{white}{\\big\\vert^{0}}\\\\\n\\mu_{\\infty,0}^{n,m} &:= \\big\\Vert \\!\\min\\{\\rho_{\\infty,0}^n,m\\}\\big\\Vert_{\\Ell^1(\\mms,\\meas_\\infty)}^{-1}\\,\\min\\{\\rho_{\\infty,0}^n,m\\}\\,\\meas_\\infty.\n\\end{align*}\nBy continuity of $\\uptau$, $\\smash{\\mu_{\\infty,0}^{n,m}}$ obeys the hypotheses from Step 2 for every $n,m\\in\\N$. A diagonal procedure gives rise to a map $m\\colon \\N\\to\\N$ such that $\\smash{(\\tilde{\\mu}_{\\infty,0}^n)_{n\\in\\N}}$, where \n\\begin{align*}\n\\tilde{\\mu}_{\\infty,0}^n := \\mu_{\\infty,0}^{n,m_n},\n\\end{align*}\nconverges weakly to $\\smash{\\mu_{\\infty,0}}$ as $n\\to\\infty$.\n\nThe above discussion thus yields the desired inequality along some timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic connecting $\\smash{\\tilde{\\mu}_{\\infty,0}^n}$ to $\\smash{\\mu_{\\infty,1} := \\delta_{x_{\\infty,1}}}$. The usual tightness and lower semicontinuity argument directly gives the claim; note that the convergence of the corresponding right-hand sides is granted by Levi's theorem.\n\n\\textbf{Step 8.} \\textit{Passage from $K$ and $N$ to $K_\\infty$ and $N_\\infty$.} According to Step 4, all in all we thus infer $\\smash{\\mathrm{TMCP}_p(K,N)}$ for ${\\ensuremath{\\mathscr{X}}}_\\infty$ for every $KN_\\infty$, and we deduce $\\smash{\\mathrm{TMCP}_p(K_\\infty,N_\\infty)}$ as in Step 9 in the proof of \\autoref{Th:Stability TCD}.\n\\end{proof}\n\n\\subsection{Good geodesics}\\label{Sub:Good} Now we prove an analogue of \\autoref{Th:Good geos TCD}, assuming the $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ condition. As indicated in \\autoref{Re:Unlikely}, this result will be used in the next \\autoref{Sec:Equiv TMCP's} to prove the equivalence of the latter condition with the entropic timelike measure-contraction property from \\cite{cavalletti2020}.\n\nAs in \\autoref{Sub:Good TCD}, we only outline the proof and refer to \\cite[Sec.~4.3]{braun2022}, see also \\cite{cavalletti2017}, for a similar discussion in the entropic case.\n\nThe following result is proven similarly to \\autoref{Le:Lalelu}.\n\n\\begin{lemma}\\label{Le:Lululu} Assume the $\\smash{\\mathrm{wTCD}_p^*(K,N)}$ condition for some $p\\in (0,1)$, $K\\in\\R$, and $N\\in[1,\\infty)$. Let $\\smash{\\mu_0 = \\rho_0\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$, and let $x_1\\in I^+(\\mu_0)$. Finally, let $D$ be any real number no smaller than $\\sup\\uptau(\\supp\\mu_0\\times\\{x_1\\})$. Then there is a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\smash{\\mu_1 := \\delta_{x_1}}$ such that for every $t\\in [0,1)$, denoting by $\\rho_t$ the $\\meas$-ab\\-solutely continuous part of $\\mu_t$,\n\\begin{align*}\n\\meas\\big[\\{\\rho_t>0\\}\\big] \\geq (1-t)^N\\,{\\ensuremath{\\mathrm{e}}}^{-tD\\sqrt{K^-N}}\\,\\big\\Vert\\rho_0\\big\\Vert_{\\Ell^\\infty(\\mms,\\meas)}^{-1}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{theorem}\\label{Th:Good TMCP} Under the hypotheses of \\autoref{Le:Lululu}, there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\smash{\\mu_1:= \\delta_{x_1}}$ such that for every $t\\in[0,1]$, $\\smash{\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align*}\n\\Vert\\rho_t\\Vert_{\\Ell^\\infty(\\mms,\\meas)} \\leq \\frac{1}{(1-t)^N}\\,{\\ensuremath{\\mathrm{e}}}^{Dt\\sqrt{K^-N}}\\,\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)},\n\\end{align*}\nwhere\n\\begin{align*}\nD:= \\sup\\uptau(\\supp\\mu_0\\times\\{x_1\\}).\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof} As for \\cite[Thm.~4.18]{braun2022}, we first observe that the bisection argument from the proof of \\autoref{Th:Good geos TCD} does not apply here since $\\smash{\\mu_1\\notin{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, while we aim to establish that $\\smash{\\mu_t \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$ for every $t\\in[0,1)$.\n\nGiven $n\\in\\N$ and $k\\in\\N_0$, we set $\\smash{s_n^k := (1-2^{-n})^k}$. Fix $n\\in\\N$, and assume that a measure $\\smash{{\\ensuremath{\\boldsymbol{\\beta}}}_n^k\\in \\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,\\mu_0)}$ [sic] has already been defined in such a way that for every $i\\in\\{1,\\dots,k\\}$, we have $\\smash{(\\eval_{s_n^i})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k \\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and\n\\begin{align*}\n\\sup\\uptau(\\supp\\,(\\eval_{s_n^i})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k\\times\\{x_1\\})\\leq 2^{-n}\\,s_n^{i-1}\\,D.\n\\end{align*}\n\n\\textbf{Step 1.} \\textit{Minimization of an appropriate functional.} Set\n\\begin{align*}\nc_n^{k+1} := \\frac{1}{(1-2^{-n})^N}\\,{\\ensuremath{\\mathrm{e}}}^{2^{-n}s_n^kD\\sqrt{K^-N}}\\,\\Vert\\rho_0\\Vert_{\\Ell^\\infty(\\mms,\\meas)}\n\\end{align*}\nand let the functional $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_n^{k+1}}}$ be defined as in \\eqref{Eq:Functional FC} above. As in \\cite[Le.~3.13]{braun2022}, the functional $\\smash{{\\ensuremath{\\mathscr{F}}}_{c_n^{k+1}}\\circ \\eval_{2^{-n}}}$ admits some minimizer $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_n^{k+1}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau((\\eval_{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k,\\mu_1)}$. Let ${\\ensuremath{\\boldsymbol{\\sigma}}}_{k+1}^n\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,(\\eval_{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k)}$ be the $\\smash{\\ell_p}$-optimal geodesic plan obtained by ``time-reversal'' of $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_n^{k+1}}$. By gluing, we build $\\smash{{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_1,\\mu_0)}$ with\n\\begin{align*}\n(\\Restr_0^{s_n^k})_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1} &= {\\ensuremath{\\boldsymbol{\\pi}}}_n^{k+1},\\\\\n(\\Restr_{s_n^k}^1)_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^{k+1} &= (\\Restr_{s_n^k}^1)_\\push{\\ensuremath{\\boldsymbol{\\beta}}}_n^k.\n\\end{align*}\nHere, given any $s,t\\in[0,1]$ with $s0$, recall \\eqref{Eq:BLUBBB} and \\eqref{Eq:Blubbbb}, we get $\\mu_t = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)$, and $\\Vert\\rho_t\\Vert_{\\Ell^\\infty(\\mms,\\meas)}$ obeys the desired upper bound for every $t\\in[0,1)$.\n\\end{proof}\n\n\\subsection{Uniqueness of $\\ell_p$-optimal couplings and $\\ell_p$-geodesics}\\label{Sub:Uniqueneess} In this section, we prove uniqueness of \\emph{chronological} $\\smash{\\ell_p}$-optimal couplings (if existent), \\autoref{Th:Uniqueness couplings}, as well as of $\\smash{\\ell_p}$-optimal geodesic plans, \\autoref{Th:Uniqueness geodesics}. We follow \\cite[Sec.~3.4]{cavalletti2020}, see also \\cite{cavalletti2017}. A byproduct of our discussion is an extension of the results from \\cite{cavalletti2020} from the timelike nonbranching to the timelike \\emph{essential} nonbranching case, cf.~\\autoref{Re:From TNB to TENB}.\n\nIn this section, in addition to our standing assumptions, let ${\\ensuremath{\\mathscr{X}}}$ be timelike $p$-essentially nonbranching for some fixed $p\\in (0,1)$. \n\n\\begin{lemma}\\label{Le:Uniqueness Diracs} Assume $\\smash{\\mathrm{TMCP}^*_p(K,N)}$ for some $K\\in\\R$ and $N\\in [1,\\infty)$. Let $\\mu_0 = \\rho_0\\,\\meas\\in\\smash{{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$, and define $\\mu_1 := \\lambda_1\\,\\delta_{x_1} + \\dots + \\lambda_n\\,\\delta_{x_n}$ for $\\lambda_1,\\dots,\\lambda_n\\in (0,1]$ with $\\lambda_1+\\dots+\\lambda_n=1$ and pairwise distinct $x_1,\\dots,x_n\\in\\mms$. Let $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ be an $\\smash{\\ell_p}$-optimal coupling. Then there is a $\\mu_0$-measurable map $T\\colon \\supp\\mu_0\\to\\mms$ with \n\\begin{align*}\n\\pi = (\\Id,T)_\\push\\mu_0,\n\\end{align*}\nand consequently,\n\\begin{align*}\n\\ell_p(\\mu_0,\\mu_1)^p = \\int_\\mms \\uptau(x,T(x))^p\\d\\mu_0(x).\n\\end{align*}\n\nIn particular, $\\pi$ is the unique chronological $\\smash{\\ell_p}$-optimal coupling of $\\mu_0$ and $\\mu_1$.\n\\end{lemma}\n\n\\begin{proof} By a standard argument, cf.~the proof of \\cite[Lem.~3.17]{cavalletti2020}, it suffices to prove the existence of a $\\mu_0$-measurable map $T\\colon \\supp\\mu_0\\to\\mms$ such that $\\pi=(\\Id,T)_\\push\\mu_0$. To this aim, define the compact set $E\\subset\\supp\\mu_0$ by\n\\begin{align*}\nE := \\{x\\in \\mms : \\#[(\\{x\\}\\times\\mms)\\cap \\supp\\pi] \\geq 2\\}.\n\\end{align*}\nWe claim that $\\mu_0[E] = 0$, which directly gives the desired $T$.\n\nAssume to the contrapositive that $\\mu_0[E]>0$. We first reduce the discussion to the uniform distribution on some subset of $\\mms$ as follows. Up to shrinking $E$, we assume without restriction that $\\varepsilon\\leq \\rho_0\\leq 1\/\\varepsilon$ $\\meas$-a.e.~on $E$ for some $\\varepsilon > 0$. A further possible shrinking of $E$ entails the existence of distinct points $z_1,z_2\\in\\{x_1,\\dots,x_n\\}$ and well-defined maps $T_1,T_2\\colon E\\to \\mms$ with $T_1(x) = z_1$ and $T_2(x)=z_2$ for every $x\\in E$. We may and will additionally assume that $\\smash{E\\times\\{z_1\\}, E\\times\\{z_2\\}\\subset \\mms_\\ll^2}$. By restric\\-tion \\cite[Lem.~2.10]{cavalletti2020}, the couplings\n\\begin{align*}\n\\pi_1 &:= \\meas[E]^{-1}\\,\\One_{E\\times\\{z_1\\}}\\,(\\rho_0\\circ\\pr_1)^{-1}\\,\\pi,\\\\\n\\pi_2 &:= \\meas[E]^{-1}\\,\\One_{E\\times\\{z_2\\}}\\,(\\rho_0\\circ\\pr_1)^{-1}\\,\\pi\n\\end{align*}\nare $\\smash{\\ell_p}$-optimal with $\\smash{\\pi_1\\in \\Pi_\\ll(\\nu_0,\\delta_{z_1})}$ and $\\smash{\\pi_2\\in\\Pi_\\ll(\\nu_0,\\delta_{z_2})}$, respectively, where\n\\begin{align*}\n\\nu_0 := \\meas[E]^{-1}\\,\\meas\\mres E = \\varrho_0\\,\\meas.\n\\end{align*}\n\nFrom now on, we will work with $\\nu_0$ instead of $\\mu_0$. Let $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_1\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_0,\\delta_{z_1})}$ and $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_2\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_0,\\delta_{z_2})}$ be $\\smash{\\ell_p}$-optimal geodesic plans which represent timelike proper-time parametrized $\\smash{\\ell_p}$-geodesics witnessing the defining inequality of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$. Note that $(\\eval_0)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_1$ and $(\\eval_0)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_2$ are mutually singular since\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}}_i[\\{\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms) : \\gamma_1 = z_i\\}] = 1\n\\end{align*}\nfor every $i\\in\\{1,2\\}$, and $z_1\\neq z_2$. By \\autoref{Le:Mutually singular}, \n\\begin{align}\\label{Eq:Mutual singularity}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_1 \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_2\n\\end{align}\nfor every $t\\in(0,1]$. Given $i\\in\\{1,2\\}$, let $\\smash{\\varrho_t^i}$ denote the density of the $\\meas$-absolutely continuous part of $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_1$, which is nontrivial by $\\smash{\\mathrm{TMCP}_p^*(K,N)}$. By \\autoref{Re:Lower bounds sigma}, \n\\begin{align*}\n\\int_\\mms (\\varrho_t^i)^{1-1\/N'}\\d\\meas &\\geq (1-t)\\,{\\ensuremath{\\mathrm{e}}}^{-tD\\sqrt{K^-\/N'}}\\int_\\mms \\varrho_0^{1-1\/N'}\\d\\meas\\\\\n&= (1-t)\\,{\\ensuremath{\\mathrm{e}}}^{-tD\\sqrt{K^-\/N'}}\\,\\meas[E]^{1\/N'}\\textcolor{white}{\\int}\n\\end{align*}\nfor some $N'>1$. On the other hand, by Jensen's inequality,\n\\begin{align*}\n\\int_\\mms (\\varrho_t^i)^{1-1\/N'}\\d\\meas &\\leq \\meas\\big[\\{\\varrho_t^i > 0\\}\\big]\\,\\meas\\big[\\{\\varrho_t^i > 0\\}\\big]^{-1}\\int_{\\{\\varrho_t^i > 0\\}} (\\varrho_t^i)^{1-1\/N'}\\d\\meas\\\\\n&\\leq \\meas\\big[\\{\\varrho_t^i > 0\\}\\big]\\,\\Big[\\meas\\big[\\{\\varrho_t^i > 0\\}\\big]^{-1}\\int_{\\{\\varrho_t^i>0\\}}\\varrho_t^i\\d\\meas\\Big]^{1-1\/N'}\\\\\n&\\leq \\meas\\big[\\{\\varrho_t^i > 0\\}\\big]^{1\/N'}.\\textcolor{white}{\\int}\n\\end{align*}\nThese inequalities imply\n\\begin{align}\\label{Eq:mm}\n\\liminf_{t\\to 0} \\meas\\big[\\{\\varrho_t^i > 0\\}\\big] \\geq \\meas[E]= \\meas\\big[\\{\\varrho_0^i > 0\\}\\big].\n\\end{align}\n\nBy ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, the sets\n\\begin{align*}\nF &:= \\{x\\in \\mms : x\\in\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_i\\textnormal{ for some }t\\in[0,1],\\, i\\in\\{1,2\\}\\},\\\\\nG_\\delta &:= \\{x\\in F : \\uptau(y,x) \\leq \\delta\\textnormal{ for some }y\\in E\\}\n\\end{align*}\nare compact for every $\\delta>0$. Since $\\meas[G_\\delta]\\to \\meas[E]$ as $\\delta\\to 0$ by Lebesgue's theorem, there exists some $\\delta_0 \\in (0,1)$ such that\n\\begin{align}\\label{Eq:mmm}\n\\meas[G_{\\delta_0}] \\leq \\frac{3}{2}\\,\\meas[E].\n\\end{align}\nBy construction, for every $i\\in\\{1,2\\}$, every $t\\in (0,1)$, and $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_i$-a.e.~$x\\in \\mms$ there exists $\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$ such that $x=\\gamma_t$, $\\gamma_0\\in E$, and $\\smash{\\gamma_1 = z_i}$. By definition of $\\smash{G_{\\delta_0}}$, we thus obtain $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_i[G_{\\delta_0}] = 1$ for every $t\\in[0,\\delta_0]$. Hence, \\eqref{Eq:mm} and \\eqref{Eq:mm} imply\n\\begin{align*}\n\\meas\\big[\\{\\rho_s^1 > 0\\} \\cap \\{\\rho_s^2 > 0\\}\\big] > 0\n\\end{align*}\nfor some $s\\in (0,\\delta_0)$. However, this contradicts \\eqref{Eq:Mutual singularity}.\n\\end{proof}\n\n\\begin{lemma}\\label{Pr:More than TMCP} Let $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ hold for some $K\\in\\R$ and $N\\in[1,\\infty)$. Let $\\smash{\\mu_0 = \\rho_0\\,\\meas \\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ and $\\mu_1\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ with $\\smash{\\supp\\mu_0\\times\\supp\\mu_1\\subset\\mms_\\ll^2}$. Then there exist a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ connecting $\\mu_0$ to $\\mu_1$ as well as an $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ such that for every $t\\in [0,1)$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:Claim conv TMCP}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) \\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1).\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \\textbf{Step 1.} \\textit{Approximation of $\\mu_1$.} Let $B_1,\\dots,B_n\\subset\\mms$, $n\\in\\N$, be a given Borel partition of $\\supp\\mu_1$ with $\\mu_1[B_i]>0$ for every $i\\in\\{1,\\dots,n\\}$. Given such an $i$, fix $\\smash{x_1^i\\in\\supp\\mu_1}$ and set $\\lambda_i := \\mu_1[B_i]$ as well as\n\\begin{align*}\n\\mu_1^n := \\lambda_1\\,\\delta_{x_1^1} + \\dots + \\lambda_n\\,\\delta_{x_1^n}.\n\\end{align*}\nSince every coupling of $\\mu_0$ and $\\smash{\\mu_1^n}$ is chronological, an $\\smash{\\ell_p}$-optimal coupling of these exists uniquely by \\autoref{Le:Uniqueness Diracs}. Let $\\smash{\\pi_n\\in \\Pi_\\ll(\\mu_0,\\mu_1^n)}$ be this coupling, given by a $\\mu_0$-measurable map $T_n\\colon \\supp\\mu_0\\to\\mms$. For $i\\in\\{1,\\dots,n\\}$, we define \n\\begin{align*}\nA_i := T_n^{-1}(\\{x_1^i\\})\\times\\{x_1^i\\},\n\\end{align*}\nand observe that $A_1,\\dots,A_n\\subset \\mms^2$ constitutes a Borel partition of $\\supp\\pi_n$. \n\nDefine $\\smash{\\pi_n^i\\in{\\ensuremath{\\mathscr{P}}}(\\mms^2)}$, $\\smash{\\nu_0^i\\in {\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$, and $\\smash{\\nu_1^i\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)}$ by\n\\begin{align*}\n\\pi_n^i &:= \\lambda_i^{-1}\\,\\pi_n\\mres A_i,\\\\\n\\nu_0^i &:= (\\pr_1)_\\push\\pi_n^i = \\varrho_0^i\\,\\meas,\\\\\n\\nu_1^i &:= (\\pr_2)_\\push\\pi_n^i = \\delta_{x_1^i}.\n\\end{align*}\nBy construction, we have $\\mu_0 = \\lambda_1\\,\\nu_0^i + \\dots + \\lambda_n\\,\\nu_0^n$ and thus\n\\begin{align*}\n\\rho_0 = \\lambda_1\\,\\varrho_0^i + \\dots + \\lambda_n\\,\\varrho_0^n\\quad\\meas\\textnormal{-a.e.}\n\\end{align*}\nMoreover $\\smash{\\supp\\nu_0^i \\cap \\supp\\nu_0^j = \\emptyset}$ for every $i,j\\in\\{1,\\dots,n\\}$ with $i\\neq j$, whence\n\\begin{align}\\label{Eq:MUT SING}\n\\nu_0^i\\perp\\nu_0^j.\n\\end{align}\n\n\\textbf{Step 2.} \\textit{Invoking the $\\mathrm{TMCP}$ condition.} As $\\smash{x_1^i\\in I^+(\\nu_0^i)}$ for every $i\\in\\{1,\\dots,n\\}$, using the $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ condition there exists a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\nu_t^i)_{t\\in[0,1]}}$ from $\\smash{\\nu_0^i}$ to $\\smash{\\nu_1^i}$ such that for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align}\\label{Eq:Interm TMCP}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\nu_t^i) \\leq -\\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0, x_1^i))\\,\\rho_0^i(x^0)^{1-1\/N'}\\d\\meas(x^0).\n\\end{align}\n\nWith this information, we now build a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $\\smash{(\\mu_t^n)_{t\\in [0,1]}}$ from $\\mu_0$ to $\\smash{\\mu_1^n}$ for which \\eqref{Eq:Claim conv TMCP} holds with $\\pi$ replaced by $\\pi_n$. By definition, $\\smash{(\\nu_t^i)_{t\\in[0,1]}}$ is represented by some $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}^i\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\nu_0^i,\\nu_1^i)}$. Thanks to \\eqref{Eq:MUT SING} and \\autoref{Le:Mutually singular}, we obtain\n\\begin{align}\\label{Eq:MUT SING II}\n(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^i \\perp (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^j\n\\end{align}\nfor every $t\\in(0,1]$ and every $i,j\\in\\{1,\\dots,n\\}$ with $i\\neq j$. Then\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}}_n := \\lambda_1\\,{\\ensuremath{\\boldsymbol{\\alpha}}}^1 + \\dots + \\lambda_n\\,{\\ensuremath{\\boldsymbol{\\alpha}}}^n\n\\end{align*}\nis an $\\smash{\\ell_p}$-optimal geodesic plan from $\\mu_0$ to $\\smash{\\mu_1^n}$, and the assignment $\\smash{\\mu_t^n := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_n}$ gives rise to a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic connecting these probability measures. Let $\\smash{\\varrho_t^i}$ denote the nontrivial density of the $\\meas$-absolutely continuous part of $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}^i}$. Then we obtain that\n\\begin{align*}\n\\rho_t^n\\,\\meas := \\big[\\lambda_1\\,\\varrho_t^1 + \\dots + \\lambda_n\\,\\varrho_t^n\\big]\\,\\meas\n\\end{align*}\nis the $\\meas$-absolutely continuous part of $\\mu_t^n$, $t\\in[0,1]$. Now \\eqref{Eq:MUT SING II} ensures \n\\begin{align*}\n\\meas\\big[\\{\\varrho_t^i > 0\\} \\cap \\{\\varrho_t^j > 0\\}\\big] = 0.\n\\end{align*}\nfor every $t\\in [0,1]$ and every $i,j\\in\\{1,\\dots,n\\}$ with $i\\neq j$. Hence, using \\eqref{Eq:Interm TMCP} and \\eqref{Eq:MUT SING} we get, for every $t\\in[0,1)$ and every $N'\\geq N$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^n) &=\n\\sum_{i=1}^n \\lambda_i^{1-1\/N'}\\,{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^i)\\\\\n&\\leq -\\sum_{i=1}^n\\lambda_i^{1-1\/N'}\\!\\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x_1^i))\\,\\varrho_0^i(x^0)^{-1\/N'}(x^0)\\d\\nu_0^i(x^0)\\\\\n&= -\\sum_{i,i'=1}^n \\int_\\mms \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x_1^i))\\,\\big[\\lambda_{i'}\\,\\varrho_0^{i'}(x^0)\\big]^{-1\/N'}\\d\\big[\\lambda_i\\,\\nu_0^i\\big](x^0)\\\\\n&= -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi_n(x^0,x^1).\n\\end{align*}\n\n\\textbf{Step 3.} \\textit{Conclusion.} If $\\supp\\mu_1$ consists of finitely many points, the claim simply follows by choosing $n\\in\\N$ such that $\\smash{\\mu_1 = \\mu_1^n}$. \n\nTherefore, only the case $\\#\\supp\\mu_1=\\infty$ remains to be studied. By \\cite[Thm.~2.16]{cavalletti2020}, there exists a sequence $\\smash{(\\bar{\\mu}_1^n)_{n\\in\\N}}$ in ${\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ which converges weakly to $\\mu_1$ such that $\\smash{\\bar{\\mu}_1^n\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\mu_1^n)}$ for every $n\\in\\N$, and for every sequence $(\\pi_n)_{n\\in\\N}$ of $\\smash{\\ell_p}$-optimal couplings $\\smash{\\pi_n \\in\\Pi_\\leq(\\mu_0,\\bar{\\mu}_1^n)}$, every weak limit of any subsequence of the latter is $\\smash{\\ell_p}$-optimal for $\\mu_0$ and $\\mu_1$. In particular, the previous discussion applies to $\\smash{\\bar{\\mu}_1^n}$ in place of $\\smash{\\mu_1^n}$, $n\\in\\N$, and yields sequences $\\smash{({\\ensuremath{\\boldsymbol{\\pi}}}_n)_{n\\in\\N}}$ and $(\\pi_n)_{n\\in\\N}$ of $\\smash{\\ell_p}$-optimal geodesic plans $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}_n\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\bar{\\mu}_1^n)}$ and $\\smash{\\ell_p}$-optimal couplings $\\smash{\\pi_n\\in\\Pi_\\ll(\\mu_0,\\mu_1^n)}$ as in Step 2, respectively. By ${\\ensuremath{\\mathscr{K}}}$-global hyperbolicity, Prokhorov's theorem, and \\autoref{Le:Villani lemma for geodesic} these sequences converge weakly to some $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ and an $\\smash{\\ell_p}$-optimal coupling $\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$, respectively, up to a nonrelabeled subsequence. Define a timelike proper-time parametrized $\\smash{\\ell_p}$-geodesic $(\\mu_t)_{t\\in[0,1]}$ from $\\mu_0$ to $\\mu_1$ by $\\mu_t := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$. Employing weak lower semicontinuity of ${\\ensuremath{\\mathscr{S}}}_{N'}$ on measures with uniformly bounded support and \\autoref{Le:USC lemma}, we then obtain, for every $t\\in[0,1)$, every $N'\\geq N$, and with $\\smash{\\mu_t^n := (\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_n}$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t) &\\leq \\limsup_{n\\to\\infty} {\\ensuremath{\\mathscr{S}}}_{N'}(\\mu_t^n)\\\\\n&\\leq -\\liminf_{n\\to\\infty} \\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi_n(x^0,x^1)\\\\\n&\\leq -\\int_{\\mms^2} \\sigma_{K,N'}^{(1-t)}(\\uptau(x^0,x^1))\\,\\rho_0(x^0)^{-1\/N'}\\d\\pi(x^0,x^1).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{remark} Arguing as in Step 4 in the proof of \\cite[Prop.~3.18]{cavalletti2020} as well as Step 2 above, the conclusion of \\autoref{Pr:More than TMCP} holds under the following weaker assumption: the measures $\\mu_0$ and $\\mu_1$ admit an $\\smash{\\ell_p}$-optimal coupling $\\varpi\\in\\Pi_\\ll(\\mu_0,\\mu_1)$ such that $\\smash{\\supp\\pi\\subset\\mms_\\ll^2}$. We will not need this extension in the sequel.\n\\end{remark}\n\nThe proof of the following \\autoref{Th:Uniqueness couplings} follows the lines of \\autoref{Le:Uniqueness Diracs} modulo some modifications we briefly discuss.\n\n\\begin{theorem}\\label{Th:Uniqueness couplings} Assume $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ for $K\\in\\R$ and $N\\in [1,\\infty)$. Suppose the pair $(\\mu_0,\\mu_1)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)\\times{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ to be timelike $p$-dualizable by $\\smash{\\pi\\in\\Pi_\\ll(\\mu_0,\\mu_1)}$. Then there exists a $\\mu_0$-measurable map $T\\colon \\mms \\to \\mms$ such that\n\\begin{align*}\n\\pi = (\\Id,T)_\\push\\mu_0,\n\\end{align*}\nand consequently,\n\\begin{align*}\n\\ell_p(\\mu_0,\\mu_1)^p = \\int_\\mms \\uptau(x,T(x))^p\\d\\mu_0(x).\n\\end{align*}\n\nIn particular, $\\pi$ is the unique chronological $\\ell_p$-optimal coupling of $\\mu_0$ and $\\mu_1$.\n\\end{theorem}\n\n\\begin{proof} As in the proof of \\autoref{Le:Uniqueness Diracs}, it suffices to prove the existence of $T$. Let $\\Gamma\\subset\\mms_\\ll^2$ be an $\\smash{\\ell_p}$-cyclically monotone set with $\\pi[\\Gamma]=1$ \\cite[Prop.~2.8]{cavalletti2020}, and set\n\\begin{align*}\nE := \\{x\\in\\mms : \\pr_2[\\Gamma \\cap (\\{x\\}\\times \\mms)] \\geq 2\\},\n\\end{align*}\nwhich is a Suslin set. We claim that $\\mu_0[S]=0$, which directly gives the desired $T$.\n\nAssume to the contrapositive that $\\mu_0[S]>0$. By the von Neumann selection theorem \\cite[Thm.~9.1.3]{bogachev2007b}, there exist $\\mu_0$-measurable maps $T_1,T_2\\colon E\\to \\mms$ whose graphs are both contained in $\\Gamma$ and such that $T_1(x) \\neq T_2(x)$ for every $x\\in E$. By Lusin's theorem, there exists a compact set $C\\subset E$ with $\\mu_0[C]>0$ such that the restrictions $\\smash{T_1\\big\\vert_C}$ and $\\smash{T_2\\big\\vert_C}$ are continuous. This yields\n\\begin{align*}\n\\min\\{\\met(T_1(x),T_2(x)) : x\\in C\\} > 0.\n\\end{align*}\nIn particular, there exist $z_1,z_2\\in\\mms$ and $r>0$ with $\\met(z_1,z_2)>r$, and a compact set $C'\\subset C$ with $\\mu_0[C]>0$ as well as $\\smash{T_1(C')\\subset{\\ensuremath{\\mathsf{B}}}^\\met(z_1,r\/2)}$ and $\\smash{T_2(C')\\subset{\\ensuremath{\\mathsf{B}}}^\\met(z_2,r\/2)}$. Up to possibly shrinking the radius $r$, we may and will assume without restriction that $\\smash{C'\\times [\\bar{{\\ensuremath{\\mathsf{B}}}}^\\met(z_1,r\/2)\\cup \\bar{{\\ensuremath{\\mathsf{B}}}}^\\met(z_2,r\/2)] \\subset\\mms_\\ll^2}$.\n\nAs in the proof of \\autoref{Le:Uniqueness Diracs}, we may further shrink $C'$, hence assume that $\\rho_0$ is bounded and bounded away from zero on $C'$. Consider $\\smash{\\nu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$ with\n\\begin{align*}\n\\nu_0 := \\meas[C']^{-1}\\,\\meas\\mres C'.\n\\end{align*}\nFurthermore, define $\\smash{\\mu_1^1,\\mu_1^2\\in{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)}$ by\n\\begin{align*}\n\\mu_1^1 &:= (T_1)_\\push\\mu_0,\\\\\n\\mu_1^2 &:= (T_2)_\\push\\mu_0.\n\\end{align*}\nBy construction $\\smash{\\supp\\mu_1^1 \\cap \\supp\\mu_1^2 = \\emptyset}$, and the pairs $\\smash{(\\nu_0,\\mu_1^1)}$ and $\\smash{(\\nu_0,\\mu_1^2)}$ are both strongly $p$-timelike dualizable by \\autoref{Re:Strong timelike}. \\autoref{Pr:More than TMCP} applies to both pairs; following the proof of \\autoref{Le:Uniqueness Diracs} from now on gives the claim.\n\\end{proof}\n\n\\begin{theorem}\\label{Th:Uniqueness geodesics} Assume $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ for some $K\\in\\R$ and $N\\in[1,\\infty)$. Suppose the pair $(\\mu_0,\\mu_1)\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)\\times{\\ensuremath{\\mathscr{P}}}_\\comp(\\mms)$ to be timelike $p$-dualizable. Then for every ${\\ensuremath{\\boldsymbol{\\pi}}}\\in\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ there is a $\\mu_0$-measurable map $\\smash{\\mathfrak{T} \\colon \\supp\\mu_0\\to \\mathrm{TGeo}^\\uptau(\\mms)}$ with\n\\begin{align*}\n{\\ensuremath{\\boldsymbol{\\pi}}} = \\mathfrak{T}_\\push\\mu_0.\n\\end{align*}\n\nIn particular, the set $\\smash{\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ is a singleton.\n\\end{theorem}\n\n\\begin{proof} By the usual argument outlined in the proof of \\autoref{Le:Uniqueness Diracs}, it suffices to prove that every $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$ is induced by a map $\\mathfrak{T}$ as above.\n\nAssume to the contrapositive that one such ${\\ensuremath{\\boldsymbol{\\pi}}}$, henceforth fixed, is not given by a map. We disintegrate ${\\ensuremath{\\boldsymbol{\\pi}}}$ with respect to $\\eval_0$, writing, with some abuse of notation,\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}{\\ensuremath{\\boldsymbol{\\pi}}}(\\gamma) = {\\ensuremath{\\mathrm{d}}}{\\ensuremath{\\boldsymbol{\\pi}}}_x(\\gamma)\\d\\mu_0(x)\n\\end{align*}\nfor a $\\mu_0$-measurable map ${\\ensuremath{\\boldsymbol{\\pi}}}\\colon \\supp\\mu_0\\to {\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}^\\uptau(\\mms))$. By assumption on ${\\ensuremath{\\boldsymbol{\\pi}}}$, there is a compact set $C\\subset\\supp\\mu_0$ such that $\\mu_0[C]>0$ and $\\#\\supp {\\ensuremath{\\boldsymbol{\\pi}}}_x \\geq 2$ for every $x\\in C$.\nHence, for every $x\\in C$ there exists $t_x\\in (0,1)$ such that $\\#\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_x \\geq 2$; since $\\smash{\\mathrm{TGeo}^\\uptau(\\mms)\\subset\\Cont([0,1];\\mms)}$ and $\\smash{\\supp{\\ensuremath{\\boldsymbol{\\pi}}}_x \\subset \\mathrm{TGeo}^\\uptau(\\mms)}$, there exists an open interval $I_x\\subset (0,1)$ with $t_x\\in I_x$ such that $\\#\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_x \\geq 2$ for every $t\\in I_x$.\n\nGiven any $t\\in \\Q\\cap (0,1)$, we define\n\\begin{align*}\nC_t &:= \\{x \\in C : \\#\\supp\\,(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}_x\\geq 2\\}.\n\\end{align*}\nSince $C_t$ unites to $C$ when ranging over $t\\in \\Q\\cup (0,1)$, we find $s\\in \\Q\\cap (0,1)$ with $\\smash{\\mu_0[C_s] >0}$. Finally, we define $\\smash{{\\ensuremath{\\boldsymbol{\\alpha}}}\\in{\\ensuremath{\\mathscr{P}}}(\\mathrm{TGeo}^\\uptau(\\mms))}$ by\n\\begin{align*}\n{\\ensuremath{\\mathrm{d}}}{\\ensuremath{\\boldsymbol{\\alpha}}}(\\gamma) = \\mu_0[C_s]^{-1}\\,\\One_{C_s}(x)\\d{\\ensuremath{\\boldsymbol{\\pi}}}_x(\\gamma)\\d\\mu_0(x).\n\\end{align*}\nBy restriction \\cite[Lem.~2.10]{cavalletti2020}, $(\\eval_0,\\eval_1)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$ is an $\\smash{\\ell_p}$-optimal coupling of its marginals, the first one of which is $\\smash{\\mu_0[C_s]^{-1}\\,\\mu_0\\mres C_s\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)}$. Since ${\\ensuremath{\\boldsymbol{\\alpha}}}$ is concentrated on $\\mathrm{TGeo}^\\uptau(\\mms)$, a standard argument using the reverse triangle inequality \\eqref{Eq:Reverse tau} for $\\uptau$ implies that $(\\eval_0,\\eval_s)_\\push{\\ensuremath{\\boldsymbol{\\alpha}}}$ is a chronological $\\smash{\\ell_p}$-optimal coupling of its marginals. By definition of $C_s$, it is not given by a map, which contradicts \\autoref{Th:Uniqueness couplings}.\n\\end{proof}\n\n\\begin{remark}\\label{Re:From TNB to TENB} The above arguments --- which, as mentioned, are leaned on \\cite{cavalletti2020} --- show that the uniqueness results of \\cite[Thm.~3.19, Thm.~3.20]{cavalletti2020} remain true if the timelike nonbranching assumption therein is replaced by the hypothesis of timelike $p$-essential nonbranching for the considered exponent $p\\in (0,1)$.\n\\end{remark}\n\n\\subsection{Equivalence with the entropic TMCP condition}\\label{Sec:Equiv TMCP's} In this section, additionally to our standing assumptions we suppose ${\\ensuremath{\\mathscr{X}}}$ to be timelike $p$-essentially nonbranching for some fixed $p\\in (0,1)$. Building upon the results in \\autoref{Sub:Good} and \\autoref{Sub:Uniqueneess}, we establish the equivalence of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ with the entropic timelike measure-contraction property from \\cite[Def.~3.7]{cavalletti2020}, restated in \\autoref{Def:TMCPe}. A byproduct of our argumentation is a pathwise characterization of $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ and $\\smash{\\mathrm{TMCP}_p^e(K,N)}$, cf.~\\autoref{Th:Equivalence TMCP* and TMCPe}, which is also available for $\\smash{\\mathrm{TMCP}_p(K,N)}$ according to \\autoref{Th:Equivalence TMCP}.\n\nRecall the definition \\eqref{Eq:Expo Boltzmann} of the exponentiated Boltzmann entropy ${\\ensuremath{\\mathscr{U}}}_N$.\n\n\\begin{definition}\\label{Def:TMCPe} Let $K\\in\\R$ and $N\\in (0,\\infty)$. We say that ${\\ensuremath{\\mathscr{X}}}$ satis\\-fies the \\emph{entropic timelike measure-contraction property} $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ if for every $\\mu_0\\in{\\ensuremath{\\mathscr{P}}}_\\comp^{\\mathrm{ac}}(\\mms,\\meas)$ and every $x_1\\in I^+(\\mu_0)$, there exists an $\\smash{\\ell_p}$-geo\\-desic $(\\mu_t)_{t\\in [0,1]}$ connecting $\\mu_0$ and $\\smash{\\mu_1:= \\delta_{x_1}}$ such that for every $t\\in [0,1]$,\n\\begin{align*}\n{\\ensuremath{\\mathscr{U}}}_N(\\mu_t) \\geq \\sigma_{K,N}^{(1-t)}\\big[\\Vert\\uptau\\Vert_{\\Ell^2(\\mms^2,\\mu_0\\otimes\\mu_1)}\\big]\\,{\\ensuremath{\\mathscr{U}}}_N(\\mu_0).\n\\end{align*}\n\\end{definition}\n\n\\begin{theorem}\\label{Th:Equivalence TMCP* and TMCPe} The following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item\\label{LAAA} The condition $\\smash{\\mathrm{TMCP}_p^*(K,N)}$ holds.\n\\item\\label{LAAAA} For every $\\mu_0 = \\rho_0\\,\\meas \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ and every $x_1\\in I^+(\\mu_0)$ there is an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, where $\\smash{\\mu_1 := \\delta_{x_1}}$, such that for every $t\\in[0,1)$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas \\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'} \\geq \\sigma_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,x_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\item\\label{LAAAAA} The condition $\\smash{\\mathrm{TMCP}_p^e(K,N)}$ holds.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} We outline the necessary adaptations of the arguments in \\autoref{Sub:Equiv TCDs} and leave the details to the reader. \n\nBy integration, \\ref{LAAAA} implies \\ref{LAAA}. The implication of \\ref{LAAA} to \\ref{LAAAA} is argued analogously to \\autoref{Pr:ii to iii}. Note that to obtain the asserted absolute continuity of $(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}}$ with respect to $\\meas$ for every $t\\in[0,1)$, we have combined \\autoref{Th:Good TMCP} with the uniqueness \\autoref{Th:Uniqueness geodesics}. The step from \\ref{LAAAA} to \\ref{LAAAAA} follows since\n\\begin{align*}\n-\\frac{1}{N}\\log\\rho_t(\\gamma_t) \\geq {\\ensuremath{\\mathrm{H}}}_t\\Big[\\!-\\!\\frac{1}{N}\\log\\rho_0(\\gamma_0),\\frac{K}{N}\\,\\uptau^2(\\gamma_0,x_1)\\Big]\n\\end{align*}\nfor ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$, using that the function ${\\ensuremath{\\mathrm{H}}}_t$ defined in \\eqref{Eq:GtHt} has similar convexity properties as the function ${\\ensuremath{\\mathrm{G}}}_t$ used to derive \\autoref{Pr:iii to iv}, and then arguing as in the proof of the latter result. Similarly, the implication from \\ref{LAAAAA} to \\ref{LAAAA} is shown analogously to the proof of \\autoref{Pr:v to iii}.\n\\end{proof}\n\nEvident adaptations of the previous arguments give the following.\n\n\\begin{theorem}\\label{Th:Equivalence TMCP} The following statements are equivalent for every given $K\\in\\R$ and $N\\in[1,\\infty)$.\n\\begin{enumerate}[label=\\textnormal{\\textcolor{black}{(}\\roman*\\textcolor{black}{)}}]\n\\item The condition $\\smash{\\mathrm{TMCP}_p(K,N)}$ holds.\n\\item For every $\\mu_0 =\\rho_0\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)$ and every $x_1\\in I^+(\\mu_0)$ there is an $\\smash{\\ell_p}$-optimal geodesic plan $\\smash{{\\ensuremath{\\boldsymbol{\\pi}}}\\in\\mathrm{OptTGeo}_{\\ell_p}^\\uptau(\\mu_0,\\mu_1)}$, where $\\smash{\\mu_1 := \\delta_{x_1}}$, such that for every $t\\in[0,1)$, we have $\\smash{(\\eval_t)_\\push{\\ensuremath{\\boldsymbol{\\pi}}} = \\rho_t\\,\\meas\\in{\\ensuremath{\\mathscr{P}}}^{\\mathrm{ac}}(\\mms,\\meas)}$, and for every $N'\\geq N$, the inequality\n\\begin{align*}\n\\rho_t(\\gamma_t)^{-1\/N'} \\geq \\tau_{K,N'}^{(1-t)}(\\uptau(\\gamma_0,x_1))\\,\\rho_0(\\gamma_0)^{-1\/N'}\n\\end{align*}\nholds for ${\\ensuremath{\\boldsymbol{\\pi}}}$-a.e.~$\\gamma\\in\\mathrm{TGeo}^\\uptau(\\mms)$.\n\\end{enumerate}\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOff-resonant Raman scattering is a robust approach to light-atom interfaces.\nOne of the methods is to induce spontaneous Stokes scattering in which\npairs of photons and collective atomic excitations - a two-mode squeezed\nstate - are created. These excitations can be stored and later retrieved\nin the anti-Stokes process \\cite{VanderWal2003,Bashkansky2012}. This approach is\ncommonly known to be a basic building block of the DLCZ protocol \\cite{Duan2001}.\n\nTypically rubidium and cesium have been used as atomic systems in\nboth warm and cold atomic ensembles \\cite{Chrapkiewicz2012,Michelberger2015,Dabrowski2014}.\nThese systems are coupled to light at near-IR wavelengths,\nsuch as 795 and 780 nm for rubidium D1 and D2 lines. Coupling to new wavelengths holds a promise to greatly extend capabilities of quantum memories. This\ncan be accomplished by non-linear frequency conversion in the four-wave\nmixing (4WM) process using strong resonant non-linearities in atoms\nthanks to transitions between excited states. Such processes\nhave been used to demonstrate frequency conversion in rubidium \\cite{Donvalkar2014,Becerra2008,Parniak2015,Akulshin2009a,Khadka2012} or to generate photon pairs in the\ncascaded spontaneous 4WM process \\cite{Willis2010,Srivathsan2013a,Chaneliere2006}. Multiphoton\nprocesses are also a developing method to interface light and Rydberg\natoms \\cite{Kondo2015,Huber2014a,Kolle2012}.\n\n\nChaneli\\`ere \\textit{et al.} \\cite{Chaneliere2006} proposed to combine the\nprocesses of Raman scattering and 4WM by first creating photon pairs\nin a cascaded spontaneous 4WM in one atomic ensemble and then storing\nphotons as collective atomic excitations in another cold ensemble,\nand an experiment was recently realized \\cite{Zhang2016}.\nAs a result they obtained a two-mode squeezed state of atomic excitations\nand telecom photons. Another approach was to frequency-convert light\ngenerated in quantum memory with 4WM \\cite{Radnaev2010a}, in order to create a frequency-converted\nquantum memory. In all cases one atomic ensemble was used for storage,\nand another for frequency conversion or photon generation. \n\n\\begin{figure}\n\\includegraphics[scale=1.02]{fig1}\\protect\\caption{(a) Configuration of atomic levels and fields we use to realize the\nfour-photon interface, (b) pulse sequence used in the experiment,\n(c) the central part of experimental setup demonstrating phase-matching\ngeometry and (d) trace of the 1 GHz FSR Fabry-P\\'erot interferometer signal showing the frequency of 4Ph field being different from what one would expect from the closed-loop process. Rows in (b) correspond to different beam paths presented\nin (c).}\n\\label{fig:schemat}\n\\end{figure}\n\n\nIn this paper we realize a Raman-like interface based\non 4WM in warm rubidium vapors driven by ground-state atomic coherence. The process we present may be in principle\nused to generate correlated pairs of collective atomic excitations\nand photons coupled to transition between two excited states in a\nsingle, four-photon process in a single atomic ensemble. Transition between two excited states corresponds to 776-nm light\nas illustrated in Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(a)}. As the two intermediate states we use $5\\mathrm{P}_{3\/2}$\nand $5\\mathrm{D}_{5\/2}$.\n\nThe paper is organized as follows. In Sec. \\ref{sec:theory} we discuss the principles behind our idea. In Sec. \\ref{sec:experimental} we describe the experimental setup and methods we use to verify our findings. Finally, we give the results of our studies of the four-wave mixing interface, namely correlations and statistical properties in Sec. \\ref{sec:corr} and detuning dependencies in Sec. \\ref{sec:spectral}. We conclude the paper and give prospects for future developments in Sec. \\ref{sec:concl}.\n\n\\section{General idea}\\label{sec:theory}\nIn our experiment we generate ground-state atomic coherence $\\rho_{gh}$\nand light denoted by 2Ph in a two-photon stimulated Raman Stokes process, seeded by vacuum fluctuations. The advantage of this approach is the fact that it is a well-established and effective way to prepare atomic-ground state coherence. In particular, it may be used in different regimes, starting from the single-photon and single-excitation spontaneous regime as in the DLCZ protocol \\cite{Duan2001,Chrapkiewicz2012}, through the linear gain regime \\cite{Raymer1981,Duncan1990} and even in the nonlinear gain-saturation regime \\cite{Walmsley1985,Lewenstein1984,Trippenbach1985}. In the two latter cases, macroscopic ground-state coherence is generated \\cite{Zhang2014c}. The generated coherence and the number of scattered photons will be highly random but correlated. The atomic coherence is not averaged out to zero due to atomic motion, since the buffer gas makes the motion diffusive \\cite{Raymer2004}. Moreover, the generated Raman field remains coherent with the driving field, so phase fluctuations of the driving field do not disturb the process \\cite{Wu2010}. In particular, the generated macroscopic ground-state coherence may be probed \\cite{Chen2010}, read-out \\cite{VanderWal2003}, or may enhance further stimulated Raman process \\cite{Yuan2010}.\n\nIn this experiment, we observe concurrent generation of 776-nm light denoted by 4Ph in\na four-photon process analogous to stimulated Raman scattering driven by\nground-state coherence. It does not occur spontaneously in the macroscopic\nregime due to small gain. However, with macroscopic $\\rho_{gh}$ generated in the two-photon stimulated Raman process, the\ndriving fields $a$, $b$ and $c$ couple ground-state atomic coherence\nto the weak optical 4Ph field. In other words, the 4Ph process is stimulated by pre-existing atomic coherence. In the leading order in drive beam\nfields Rabi frequencies $\\Omega_{i}$ the atomic polarization resulting\nin emission of 4Ph signal field is:\n\n\\begin{equation}\n\\mathbf{P}_\\mathrm{4Ph}=-n\\mathbf{d}_{e_{3}e_{2}}\\rho_{gh}\\frac{{\\Omega_{a}\\Omega_{b}\\Omega_{c}^{*}}}{4\\Delta_{g}\\delta\\Delta_{h}},\n\\label{eq:P2ndrho}\n\\end{equation}\n\n\nwhere $n$ is the atom number density and $\\mathbf{d}_{ij}$ is the\ndipole moments of respective transition. From this formula it follows\nwhich detunings play role.\n\nFor the experimental observation \\textit{a priori} knowledge of polarization\nproperties is crucial. To find it, we add contributions from\nall possible paths through intermediate hyperfine states. Since the\ndetunings from intermediate $5\\mathrm{P}_{3\/2}$ state are much larger\nthan respective hyperfine splittings we ignored the latter. Same approximation\nis adopted for the highest excited state $5\\mathrm{D}_{5\/2}$.\nEven though respective detuning $\\delta$ it is of the order of several\nMHz, similar to the hyperfine splitting of the highest excited state\n$5\\mathrm{D}_{5\/2}$, the hyperfine structure is completely unresolved\ndue to significant pressure broadening \\cite{Zameroski2014} in 0.5~torr krypton as buffer gas we use. Consequently, we may omit any detuning\ndependance and calculate the unnormalized polarization vector $\\boldsymbol{\\epsilon}$\nof the signal light using path-averaging formalism we developed in\n\\cite{Parniak2015}, by decalling the definitions of Rabi frequencies\nin Eq.~\\hyperref[eq:P2ndrho]{(\\ref*{eq:P2ndrho})}:\n\n\\begin{equation}\n\\boldsymbol{\\epsilon}_\\mathrm{4Ph}\\propto\\sum_{F,\\:m_{F}}\\mathbf{d}_{e_{3}e_{2}}(\\mathbf{E}_{a}\\cdot\\mathbf{d}_{ge_{1}}\\mathbf{E}_{b}\\cdot\\mathbf{d}_{e_{1}e_{2}}\\mathbf{E}_{c}^{*}\\cdot\\mathbf{d}_{e_{3}h}^{*}),\n\\label{eq:polaryzacje}\n\\end{equation}\n\n\nwhere $\\mathbf{E}_{i}$ are electric fields of respective beams $i$.\nSummation is carried over all possible magnetic sublevels ($F_{i}$,\n$m_{F_{i}}$) of all intermediate states $|e_{1}\\rangle$, $|e_{2}\\rangle$\nand $|e_{3}\\rangle$. \n\\section{Experimental Methods}\\label{sec:experimental}\nThe experimental setup is built around the rubidium-87 vapor cell\nheated to 100$^{\\circ}$C, corresponding to atom number density $n=7.5\\times10^{12}\\ \\mathrm{cm^{-3}}$ and optical density of 1600 at D2 line for optically pumped ensemble. For the two lowest, long-lived states,\nwe use two hyperfine components of rubidium ground-state manifold,\nnamely $|5\\mathrm{S}_{1\/2},F=1\\rangle=|g\\rangle$ and $|5\\mathrm{S}_{1\/2},F=2\\rangle=|h\\rangle$.\nFor the $|e_{1}\\rangle$ and $|e_{3}\\rangle$ states we take hyperfine\ncomponents of the $5\\mathrm{P}_{3\/2}$ manifold, and for the highest\nexcited state $|e_{2}\\rangle$ we take the $5\\mathrm{D}_{5\/2}$ manifold.\nAtoms are initially pumped into $|g\\rangle$ state using 400 $\\mu$s\noptical pumping pulse (at 795 nm). Next, three square-shaped driving pulses of 4 $\\mu$s duration each\nare applied simultaneously, as shown in Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(b)}. Inside\nthe cell, beams are collimated and have diameters of 400~$\\mu$m,\nbeing well-overlapped over a 2-cm-long cylindrical region. They intersect\nat a small angle of 14 mrad, with 780-nm beams nearly counter-propagating\nwith respect to the 776-nm beams, as presented in Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(c)}.\nPowers of driving beams $a$, $b$ and $c$ are 10 mW, 45 mW and 8\nmW, respectively. \n\nThe 780-nm two-photon Raman signal 2Ph co-propagates with the driving\nfield $a$. It is separated using a Wollaston polarizer (Wol) and\ndetected on a fast photodiode (PD), with $10^4$ signal to driving field leakage ratio. The four-photon signal 4Ph, is\nemitted in the direction given by the phase matching condition $\\mathbf{k}_{a}+\\mathbf{k}_{b}=\\mathbf{k}_{\\mathrm{{4Ph}}}+\\mathbf{k}_{c}+\\mathbf{K}$. The wavevector $\\mathbf{K}$ of spin-wave generated in the 2Ph process Raman scattering equals $\\mathbf{K}=\\mathbf{k}_a-\\mathbf{k}_{\\mathrm{2Ph}}$. We note that both longitidual and transverse components of the spin-wave $\\mathbf{K}$ are much smaller than that of light wavevectors, and consequently it has only an insignificant effect on the 4Ph signal emission direction. This particular, nearly co-propagating geometry allows us to couple the same spin-wave of wavevector $\\mathbf{K}$ to both the 2Ph and 4Ph processes.\nIn addition to the desired 4Ph signal, 776-nm light coming from the\nclosed-loop process in which field $c$ couples level $|e_{3}\\rangle$\ndirectly to $|g\\rangle$ is emitted in the same direction \\cite{Parniak2015}\nand at a frequency differing by only 6.8~GHz. However,\nwhen driving fields $a$, $b$ and $c$ are $x$-, $y$- and $y$-\npolarized, respectively, the 4Ph signal light is $x$-polarized, while\nthe closed-loop 4WM light is $y$-polarized, where $x\\perp y$. This arrangement enables\nfiltering out the 4Ph signal from both stray 776-nm driving light\nand the closed-loop 4WM light with a polarizing beam-splitter (PBS, $10^2$ extinction).\nTo suppress residual drive laser background at 780 nm 4Ph signal goes through\nan interference filter (transmission of 80$\\%$ for 776 nm and $10^{-3}$ for 780 nm) and is detected by an avalanche photodiode\n(APD). We were able to obtain signal to background ratio of up to $10^2$. \n\nBy rotating the half-wave\nplate ($\\lambda\/2$) we can easily switch between observing 4Ph\nand closed-loop signal. For detunings optimal for the 4Ph process, we register less than 10 nW of the closed-loop 4WM light. The two signals display different temporal characteristics,\nalso only 4Ph is correlated with the 2Ph light. \n\nWe verify the frequencies of 4Ph and closed-loop light using\na scanning Fabry-P\\'erot confocal interferometer of 1 GHz free spactral range (FSR) inserted in the 4Ph beam path. Fig.~\\hyperref[fig:schemat]{\\ref*{fig:schemat}(d)} show the trace obtained by scanning the interferometer for detuning difference $\\Delta_g+\\Delta_h=2\\pi\\times3.43$ GHz. Leakage of driving field $b$, induced by slight misalignement, is used as frequency reference. The middle peak corresponds to the 4Ph signal, with a solid line indicating expected frequency. Dashed line corresponds the frequency one would expect from the closed-loop signal. \n\n\n\\section{Results}\n\\subsection{Statistics and Correlations}\\label{sec:corr}\n\\begin{figure}[b]\n\\begin{centering}\n\\includegraphics[scale=1.2]{fig2}\n\\par\\end{centering}\n\n\\protect\\caption{(a - 2Ph, b - 4Ph) Averaged signal intensities (solid lines) with several\nsingle realizations (dashed lines), denoted by same colors in (a)\nand (b), demonstrating visible strong correlations and (c) calculated\ncross-correlation $C(t)$ between signals of four-photon (4Ph) and\nthe two-photon processes (2Ph). }\n\\label{fig:corr}\n\\end{figure}\n\n\n\\begin{figure}[b]\n\\begin{centering}\n\\includegraphics[scale=0.7]{fig3}\n\\par\\end{centering}\n\n\\protect\\caption{Joint statistics $P(\\mathcal{E}_\\mathrm{2Ph},\\mathcal{E}_\\mathrm{4Ph})$ together with marginal distributions of registered pulse energies for (a) short driving time of 200 ns, yield nearly single-mode thermal statistics (note the logarythimc scale in this plot), and (b) longer driving time of 4 $\\mu$s with observable pulse energy stabilization well described by multimode thermal statistics with mode number $\\mathcal{M}$ of 4.1 for 2Ph pulses and 1.6 for 4Ph pulses. Solid curves correspond to fitted thermal distributions.}\n\\label{fig:corrmap}\n\\end{figure}\n\nIn our experiment we remain in the macroscopic scattered light intensity\nregime. When strong Raman driving field is present, atoms are transferred\nto $|h\\rangle$ simultaneously with scattering of\n2Ph photons and buildup of large atomic coherence $\\rho_{gh}$.\nThe temporal shape of the 2Ph pulse is an exponential only at the\nvery beginning, which we observe in Fig.~\\hyperref[fig:corr]{\\ref*{fig:corr}(a)}. Not only pulse energies, but also the shapes fluctuate significantly from shot to shot, as the process is\nseeded by vacuum fluctuations \\cite{Raymer1989}. However, the\n4Ph pulse, presented in Fig.~\\hyperref[fig:corr]{\\ref*{fig:corr}(b)}, nearly exactly follows\nthe 2Ph pulse. We calculate temporal correlations between the 2Ph\nsignal, which is known to be proportional to the ground-state atomic\ncoherence $\\rho_{gh}$, and the 4Ph signal. Normalized intensity\ncorrelation at time $t$ between the 2Ph signal $I_{2\\mathrm{Ph}}(t)$\nand the 4Ph signal $I_{4\\mathrm{{Ph}}}(t)$ is calculated according\nto the formula $C(t)=\\langle\\Delta I_{\\mathrm{{2Ph}}}(t)\\Delta I_{\\mathrm{{4Ph}}}(t)\\rangle\/\\sqrt{\\langle\\Delta I_{\\mathrm{{2Ph}}}^{2}(t)\\rangle\\langle\\Delta I_{\\mathrm{{4Ph}}}^{2}(t)\\rangle}$\nby averaging over 500 realizations, where the standard deviations\n$\\langle\\Delta I_{\\mathrm{{2Ph}}}^{2}(t)\\rangle$ and $\\langle\\Delta I_{\\mathrm{{4Ph}}}^{2}(t)\\rangle$\nare corrected for electronic noise. Figure \\hyperref[fig:corr]{\\ref*{fig:corr}(c)} presents\nthe cross-correlation $C(t)$. We observe that correlations are high\nduring the entire process, which proves that at any time both processes\ninteract with the same atomic coherence $\\rho_{gh}$, similarly as in some previous works in $\\Lambda$-level configurations \\cite{Chen2010,Yang2012b}.\nIn particular, we are able to measure high correlation >0.9 at the very\nbeginning of the pulses, were light intensities are low. This regime\nis quite promising for further quantum applications. To estimate the\nuncertainty of calculated correlations, we divided data into 10 equal\nsets of 50 repetitions and calculated the correlation inside each\nset to finally obtain the uncertainty by calculating standard\ndeviation of results from all sets. \n\nNext, we study statistics and correlations of pulse energies in detail. We consider short and long pulse duration regimes. Figure \\hyperref[fig:corrmap]{\\ref*{fig:corrmap}(a)} corresponds to short driving time of 200 ns. In this regime light generated in both 2Ph and 4Ph processes is well-characterized by single-mode thermal energy distribution $P(\\mathcal{E})=\\langle\\mathcal{E}\\rangle^{-1} \\exp(-\\mathcal{E}\/\\langle\\mathcal{E}\\rangle)$ \\cite{Raymer1982}, where $\\mathcal{E}$ is the total scattered light energy in a single realization. This observation shows that we excite only a single transverse-spatial mode, as intended by using an adequately small size of Raman driving beam $a$ \\cite{Raymer1985a,Duncan1990}. Thermal distribution yields very high pulse energy fluctuations (namely, mean energy is equal to standard deviation), that are due to vacuum fluctuations of electromagnetic field and quantum fluctuations of atomic state \\cite{Walmsley1985,Walmsley1983}. Still, we observe that energies of 2Ph and 4Ph pulses are highly correlated, which is demonstrated by joint statistics $P(\\mathcal{E}_\\mathrm{2Ph},\\mathcal{E}_\\mathrm{4Ph})$. The residual spread is mainly due to detection noise of both signals.\n\nIn the second regime [Fig. \\hyperref[fig:corrmap]{\\ref*{fig:corrmap}(b)}] the driving pulses of 4~$\\mu$s are longer than in the previous scheme. The relative fluctuations become smaller due to gain saturation. We found the marginal statistics to be well described by multimode thermal distributions (with number of modes $\\mathcal{M}$) given by \n\\begin{equation}\nP(\\mathcal{E})=\\frac{\\mathcal{M}}{\\langle\\mathcal{E}\\rangle\\Gamma(\\mathcal{M})}\\left(\\frac{\\mathcal{E}\\mathcal{M}}{\\langle\\mathcal{E}\\rangle}\\right)^{\\mathcal{M}-1}\\exp(-\\mathcal{E}\\mathcal{M}\/\\langle\\mathcal{E}\\rangle).\n\\end{equation}\nThe joint statistics $P(\\mathcal{E}_\\mathrm{2Ph},\\mathcal{E}_\\mathrm{4Ph})$ demonstrate clear correlations, which are here less influenced by detection noise than previously, as the pulse energies are higher.\n\n\n\\begin{figure}[b]\n\\begin{centering}\n\\includegraphics[scale=0.75]{fig4}\n\\par\\end{centering}\n\\protect\\caption{(a) Scheme of experiment to demonstrate storage of ground-state atomic coherence for $\\tau=450$ ns. (b) Measured two-point correlation $C(t_1,t_2)$ between 2Ph and 4Ph signals with average registered signal powers. Off-diagonal elements of the correlation map demonstrate that light fields interact with the same atomic coherence before and after the time delay. }\n\\label{fig:corrmaptemporal}\n\\end{figure}\n\nFinally, we check that correlations are indeed mediated by ground-state atomic coherence by interrupting the scattering process for a dark period of $\\tau=450$ ns. This is proved by strong correlations between the intensities of light scattered before and after the dark period, observed in both 2Ph and 4Ph processes. \n\nAfter the atoms are optically pumped as in the original scheme, we drive the processes with 150 ns pulses of the three driving fields $a$, $b$ and $c$. After a dark period of $\\tau=450$ ns, we drive the process for another 200 ns. The coherence $\\rho_{gh}$ generated in the first stage induces optical polarization at both the 4Ph field frequency (as in Eq. \\ref{eq:P2ndrho}) and 2Ph field frequency:\n\\begin{equation}\n\\mathbf{P}_\\mathrm{2Ph}=-n\\mathbf{d}_{e_h e_1}\\rho_{gh}\\frac{\\Omega_a}{\\Delta_g},\n\\label{eq:2ph}\n\\end{equation}\nresulting in stimulated Raman emission. The full two-point correlation map presented in Fig.~\\hyperref[fig:corrmaptemporal]{\\ref*{fig:corrmaptemporal}(b)} is calculated as $C(t_1,t_2)=\\langle\\Delta I_{\\mathrm{{4Ph}}}(t_1)\\Delta I_{\\mathrm{{2Ph}}}(t_2)\\rangle\/\\sqrt{\\langle\\Delta I_{\\mathrm{{4Ph}}}^{2}(t_1)\\rangle\\langle\\Delta I_{\\mathrm{{2Ph}}}^{2}(t_2)\\rangle}$. Apart from the diagonal correlated areas at $t_1\\approx t_2 \\approx 100$ ns and 800 ns, we observe the anti-diagonal terms corresponding to correlations between the two pulses. Due to spontaneous emission and collisional dephasing all the excited atomic states decay quickly, with their lifetime limited to 20 ns. This proves that we store information in the ground-state atomic coherence. Storage time $\\tau$ is limited mainly by atomic motion. In fact, the stored ground-state coherence and in turn correlations decay in multiple ways, e.g. by diffusive spatial spread of atomic coherence \\cite{Chrapkiewicz2014b,Parniak2014} and by influx of optically pumped atoms into the interaction region.\nFinally, we mention that very similar results are obtained if the system is driven by field $a$ only in the first stage of the sequence and in turn only the 2Ph field and atomic coherence are generated. In the second stage of the sequence, where all driving field are present, we observe that both the 2Ph and 4Ph signals are correlated with the 2Ph signal emitted in the first stage.\n\nAgreement of the above measured statistics with theoretical predictions and multiple previous experiments proves that the correlations do arise from vacuum fluctuations. With driving power stable to within less than 1\\% and laser frequency stable to within 1 MHz, these external contributions to fluctuations can be neglected. Large magnitude of fluctuations, with nearly perfect correlations between 2Ph and 4Ph signals together with demonstrated storage of correlated signals allow to reject phase-noise to amplitude-noise conversion as a source of correlations \\cite{Cruz2006}.\n\nAll of the above results were measured for $\\Delta_g\/2\\pi=1000$ MHz, $\\Delta_h\/2\\pi=1200$ MHz and $\\delta\/2\\pi=-50$ MHz.\n\n\n\\subsection{Detuning dependance}\\label{sec:spectral}\n\n\\begin{figure}\n\\includegraphics[scale=1.03]{fig5}\\protect\\caption{(a) Averaged (over 500 realizations) pulse shapes\nfor the intensities of the 2Ph and the 4Ph for a set of single-photon\ndetunings $\\Delta_{g}$ for constant $\\delta\/2\\pi=-50$\nMHz and $\\Delta_h\/2\\pi=1500$ MHz, (b) pulses full-width at half-maxima (FWHM), (c) their energies and (d) pulse energy correlation between the two processes as\na function of field $a$ single photon detuning $\\Delta_{g}$. Subsequent\nplots of 2Ph (4Ph) signals in (a) are shifted by 120 $\\mu$W (40 nW).}\n\\label{fig:shapes}\n\\end{figure}\n\n\nNow we switch to verifying properties of 4Ph signal for various drive\nfield detunings. The influence of field $a$ detuning $\\Delta_{g}$\nis seen in Fig. \\ref{fig:shapes}. A number of pronounced effects\ncomes about as this laser drives the Raman scattering and produces\nground-state atomic coherence $\\rho_{gh}$. Initially the 2Ph signal grows exponentially. The corresponding\nRaman gain coefficient is strongly dependent on drive field detuning\n$\\Delta_{g}$. \nThe final effect is shortening of 2Ph pulses closer to resonance, as shown in Fig.~\\hyperref[fig:shapes]{\\ref*{fig:shapes}(b)}. The 4Ph pulse follows\nthe ground-state coherence $\\rho_{gh}$ and 2Ph pulse as shown\nin the previous section, however its maximum is somewhat delayed.\nWe attribute this effect to internal atom dynamics at high drive intensity\nlevels which is not captured by Eq.~\\hyperref[eq:P2ndrho]{(\\ref*{eq:P2ndrho})}. Energies\nof pulses are also higher closer to resonance, although the two-photon\nRaman pulse energy saturates due to absorption losses [see Fig.~\\hyperref[fig:shapes]{\\ref*{fig:shapes}(c)}].\n\nImportant insight is provided by calculating energy correlation between\nthe 2Ph and 4Ph light pulses, which fluctuate significantly from shot\nto shot. Correlation is calculated as $C=\\langle\\Delta \\mathcal{E}_{\\mathrm{{2Ph}}}\\Delta \\mathcal{E}_{\\mathrm{{4Ph}}}\\rangle\/\\sqrt{\\langle\\Delta \\mathcal{E}_{\\mathrm{{2Ph}}}^{2}\\rangle\\langle\\Delta \\mathcal{E}_{\\mathrm{{4Ph}}}^{2}\\rangle}$,\nwhere $\\mathcal{E}_{\\mathrm{{2Ph}}}$ and $\\mathcal{E}_{\\mathrm{{4Ph}}}$ are total energies\nof 2Ph and 4Ph light pulses, respectively, in a single realization.\nThe averaging $\\langle.\\rangle$ is done over 500 realizations of\nthe process and results are plotted in Fig.~\\hyperref[fig:shapes]{\\ref*{fig:shapes}(d)}.\nStrong correlations at various detunings reinforce the observation that indeed we are able\nto couple 4Ph optical field to the ground-state coherence, since number\nof photons in the 2Ph pulse is proportional to generated atomic coherence $|\\rho_{gh}|^2$.\nWe attribute the drop in correlations close to resonance line to absorption\nlosses. Finally, we estimate the efficiency of conversion from the ground-state atomic coherence $\\eta=\\mathcal{E}_\\mathrm{4Ph}\/\\mathcal{E}_\\mathrm{2Ph}$ to the 4Ph field to be $5\\times10^{-4}$. By comparing Eq.~(\\ref{eq:P2ndrho}) with analogous expression for the 2Ph process given by Eq. \\ref{eq:2ph}, we obtain $\\eta\\approx10^{-4}$ as well. This figure of merit could be improved by choosing different experimental geometries, or laser-cooling of the atomic ensemble.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=1.15]{fig6} \n\\par\\end{centering}\n\n\\protect\\caption{Experimental average pulse energies of 2Ph and 4Ph signals as a function\nof field $a$ detuning $\\Delta_{g}$ (measured from $F=1\\rightarrow F'=1$\nresonance) and field $b$ detuning $\\delta-\\Delta_{g}$ around the\ntwo-photon resonance. As we change the field $a$ detuning both single\nphoton detuning $\\Delta_{g}$ and the two-photon detuning $\\delta$\nchange accordingly. Dots (squares) represent maxima (minima) of the 4Ph (2Ph)\nsignal, while $\\delta=0$ line indicates the two-photon absorption resonance.}\n\\label{fig:detuning}\n\\end{figure}\n\n\nTo capture the physics in the vicinity of the two-photon resonance\nwe study 4Ph and 2Ph pulse energies as a function of detunings of\nfields $a$ and $b$. First we scan the field $b$ detuning $\\delta-\\Delta_{g}$\nacross the two-photon resonance line (see Fig. \\ref{fig:detuning}).\nWe observed strong suppression of 2Ph signal (minima are marked by squares in Fig. \\ref{fig:detuning}) due to two-photon absorption\n(TPA) in ladder configuration \\cite{Moon2013}. As a consequence,\nless atomic coherence $\\rho_{gh}$ is generated and the 4Ph signal\nis also reduced. In turn, the 4Ph shifted maxima position (marked by dots) are due to a trade-off\nbetween TPA losses and 4WM efficiency, as the latter is highest at\nthe two-photon resonance (marked by $\\delta=0$ line) according to Eq.~\\hyperref[eq:P2ndrho]{(\\ref*{eq:P2ndrho})}. The peak appears only on one side of the resonance due to the phase-matching condition being influenced by atomic dispersion \\cite{Zibrov2002}. \nAdditionally, we observe expected broadening of 60 MHz of two-photon resonance due to buffer gas collisions.\nBy changing the field $a$ detuning $\\Delta_{g}$ we see expected\nshifting of the two-photon resonance. We checked that even at sub-optimal two-photon detuning $\\delta$ the 2Ph and 4Ph signals are correlated, but the correlations become harder to measure as the 4Ph signals becomes very weak. \n\n\\begin{figure}\n\\centering{}\\includegraphics[scale=1.15]{fig7}\\protect\\caption{Four-photon signal pulse energy as a function of the detuning $\\Delta_{h}$\n(measured from $F=2\\rightarrow F'=2$ resonance) of field $c$ for\noptimal conditions of other lasers ($\\Delta_{g}\/2\\pi=900$~MHz and\n$\\delta\/2\\pi=-50$ MHz). Absorption line corresponds to the $F=2$\nhyperfine component of the ground state, where the right side of the\nplot is the red-detuned side. Drop at around $\\Delta_{h}\/2\\pi=$-900\nMHz corresponds exactly to 6.8 GHz detuning between the\ntwo 780-nm lasers, or $\\Delta_{g}+\\Delta_{h}=0$.}\n\\label{fig:deltah}\n\\end{figure}\n\n\nContrary to the above, changing the detuning $\\Delta_{h}$ of the\n780-nm driving field $c$ has only mild effect on the 4Ph signal.\nFig.~\\ref{fig:deltah} presents 4Ph signal pulse energy as a function\nof $\\Delta_{h}$ while other lasers were tuned for maximal signal\n($\\Delta_{g}\/2\\pi=900$ MHz and $\\delta\/2\\pi=-50$ MHz). Since the\n4Ph field frequency adapts to match the energy conservation for the\n$|h\\rangle\\rightarrow|e_{3}\\rangle\\rightarrow|e_{2}\\rangle$ two-photon\ntransition, the frequency of the driving field $c$ is not critical.\nThe laser must only be off-resonant, so the driving field is not absorbed\nand does not disturb the ground-state coherence. We also observe\t\na marked narrow drop in 4Ph process efficiency when the detuning between\nfields $a$ and $c$ is exactly the ground-state hyperfine splitting, or equivalently $\\Delta_{g}+\\Delta_{h}=0$,\nwhich is due to $\\Lambda$ configuration two-photon resonance yielding strong interaction with the ground-state coherence. \n\n\\section{Conclusions}\\label{sec:concl}\nThe experiment we performed is a proof-of-principle\nof a light atom interface that enables coupling of long-lived\nground-state atomic coherence and light resonant with transition between\nexcited states. The non-linear process we discussed is a novel type\nof process with typical characteristics of both Raman scattering and\n4WM. \nThe observation of inverted-Y type nonlinear four-photon process involving ground-state coherence, performed in a very different regime in cold atoms, has been so far reported only in \\cite{Ding2012}.\nHere, we generated ground-state atomic coherence via the well known\ntwo-photon process. We demonstrated ability to couple the\nvery same atomic coherence to optical field resonant with transition\nbetween two excited states via a four-photon process. This was verified\nby measuring high correlations between 2Ph and 4Ph fields, as well as frequency and\npolarization characteristics of the four-photon process. \n\nWe studied the behavior of pulse shapes as a function\nof driving laser detunings. Among many results, we found that maximum\nsignal is achieved when lasers are detuned from the two-photon resonance\nby approximately $\\delta\/2\\pi=-50$ MHz, which is a trade-off\nbetween TPA spoiling the generation of atomic coherence, and the efficiency\nof the four-photon process. This results demonstrate that we are able\nto control the 4WM process with ground-state coherence, which constitutes\na novel type of Raman scattering driven by three non-degenerate fields,\nin analogy to hyper-Raman scattering where the scattering process is driven\nby two degenerate fields.\n\nHere we used light at 776 nm coupled to $5\\mathrm{P}_{3\/2}\\rightarrow5\\mathrm{D}_{5\/2}$\ntransition. Using different states, such as $4\\mathrm{D}_{3\/2}$ as\nthe highest excited state, and $5\\mathrm{P}_{1\/2}$ and $5\\mathrm{P}_{3\/2}$\nas two intermediate states, would enable coupling telecom light (at 1475.7 nm or 1529.3 nm). Such process could be used as a building\nblock for a telecom quantum repeater or memory \\cite{Michelberger2015,Chaneliere2006,Zhang2016,Radnaev2010a}. By applying\nexternal weak quantum field as the 4Ph field, the system may serve\nas an atomic quantum memory, based on a highly non-linear process\nas in \\cite{DeOliveira2015}, but still linear in the input field\namplitude. It may also solve a variety of filtering problems \\cite{Michelberger2015,Dabrowski2014},\nsince many similar configurations exists (e.g. with $5\\mathrm{P}_{3\/2}$,\n$5\\mathrm{D}_{3\/2}$ and $5\\mathrm{P}_{1\/2}$ as intermediate states) in which all driving lasers operate at different wavelengths\nthan the signal. Even thought the 2Ph field was measured to be much stronger than the 4Ph field, we note that using a cold atomic ensemble would\noffer selectivity in intermediate states of the process and small\ndetuning. Thanks to selection rules, exclusively the 4Ph process could be driven. This would be the requirement for generating pairs of photons and collective atomic excitations in the 4Ph process only. Additionally, 4WM character of the\nprocess enables engineering of phase-matching condition, namely changing\nangles between incident driving beams to address different spin-wave excitations \\cite{Chrapkiewicz2012},\nto explore spatially-multimode capabilities of the system. In future\nstudies of the process we propose to address patterns \nunachievable\nby typical Raman light-atom interface based on $\\Lambda$ level configuration.\n\n\n\\begin{acknowledgments}\nWe acknowledge R. Chrapkiewicz, M. D\\k{a}browski and J. Nunn for insightful discussions and K. Banaszek and T. Stacewicz for their generous support.\nThis work was supported by Polish Ministry of Science and Higher Education ``Diamentowy Grant''\nProject No. DI2013 011943 and National Science Center Grant\nNo. 2011\/03\/D\/ST2\/01941.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{\\fontsize{10}{15}\\selectfont A. Details of STS spectra and DFT calculations}\n\n\\textbf{We provide additional information about the STS spectra, Mott states of the theoretical models and interlayer coupling effect.}\n\n\\renewcommand{\\figurename}{SFIG.}\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=14.5cm]{Sfig1} }\n\\caption{ \\label{fig1}\nWell filtered STM image of the NC phase.\nDashed lines separate two different domains and domain wall.\nThe solid lines represent the same area as Fig. 2(c).\n}\n\\end{figure}\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=16cm]{Sfig2} }\n\\caption{ \\label{fig1}\nSTS spectra of the NC phase and theoretical DOS of DW-1 structure. (a) STS spectra, (b) Total DOS as a function of the electronic temperature parameter $\\sigma$ (eV). (c) Local DOS of domain and domain wall regions ($\\sigma$ = 0.15 and 0.2 eV). \n}\n\\end{figure}\n\n\\begin{figure} [ht]\n\\centering{ \\includegraphics[width=16cm]{Sfig3} }\n\\caption{ \\label{fig2}\nLocal density of states at center Ta atom of David stars (a) DW-1 and (b) Hexagonal domain model.\n}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=16cm]{Sfig4} }\n\\caption{ \\label{fig3}\nOne possible tri-layer stacking order in the C phase. (a) atomic structure, (b) and (c) are the band structure and total DOS without and with electron-electron correlations, respectively. The interlayer distance of 5.9 {\\AA} is fixed.\n}\n\\end{figure}\n\n\\begin{figure} [h!]\n\\centering{ \\includegraphics[width=16cm]{Sfig5} }\n\\caption{ \\label{fig4}\nTwo different bilayer stacking order in the DW-1 structure. (a) on-top stacking and (b) shifted stacking. Top and bottom panels are atomic structure and LDOS, respectively. All atoms are fixed at their atomic position in the single-layer DW-1 structure. The interlayer distance of 5.9 {\\AA} is fixed.\n}\n\\end{figure}\n\n\\clearpage\n\n\\section{\\fontsize{10}{15}\\selectfont B. Details of Honeycomb Network Model}\nIn this supplementary material, we present the details of the honeycomb network model. Here we apply the strategy of Ref. \\cite{Efimkin} to the honeycomb lattice.\n\n\\subsection{Construction of Network Model}\n\nWe theoretically model the system by a regular array of one-dimensional metals living on the links of a honeycomb lattice. The construction of this model is motivated from the following experimental observations present in the main text. \n\\begin{enumerate}\n\\item \\textbf{Emergent honeycomb lattice}: The domains of the NC-CDW state form a regular honeycomb lattice, and the domain walls are the links of this honeycomb lattice.\n\\item \\textbf{Metallic domain walls}: The domain walls trap finite local density of states near the Fermi level (Note that this is generically expected for any domain wall of a charge-density wave since the domain wall carries in-gap states whose origin are topological \\cite{su79}). \n\\end{enumerate} \n\nMotivated from these, we consider a regular array of one-dimensional metals living on the links of a honeycomb lattice. Similar network models of one-dimensional metals have been studied to some degree in the context of quantum Hall plateau transitions, known as ``Chalker-Coddington model\" \\cite{chal88}, and recently have been revived to explain the physics of the twisted bilayer graphene at a small twisting angle \\cite{Efimkin}. We apply this latest theoretical progress to model the network on the honeycomb lattice. \n\nSome details of the model are following: \n\\begin{enumerate}\n\\item \\textbf{Degrees of freedom}: On each link $a=x,y,z$ of the honeycomb lattice, we assign the two wavefunctions $\\psi_a$ and $\\psi_{\\bar{a}}$. Here $\\psi_a$ represent the chiral mode propagating from an A-sublattice to its neighboring B-sublattice and $\\psi_{\\bar{a}}$ for the mode propagating from a B-sublattice to its neighboring A-sublattice (See Fig. \\ref{fig1}).\nMicroscopically they correspond to the low-energy modes near the Fermi momentum of one-dimensional metals propagating along the links. (Here we suppress the spin index for the modes because we are mainly interested in the spectral properties of the network model.)\n\\item \\textbf{Scattering between wavefunctions}: We assume that the modes propagate coherently within each link and scatter only at the nodes of the honeycomb lattice. We further assume that there are three-fold rotation and two-fold mirror symmetries at each nodes, and the scattering between the modes respects the crystal symmetries. \n\\end{enumerate}\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[width=0.5\\textwidth]{Sfig6.pdf}\n\\caption{Pictorial Representation of Network Model: (A) labeling of the links by $x,y,z$ on real space. Each link has the two degrees of freedom. One is from the neighboring A-sublattice [See (B)] and another is from the neighboring B-sublattice [See (C)]. They are written as $\\psi_a$ and $\\psi_{\\bar{a}}$, and the arrows represent the propagating directions of the modes. Here $\\hat{e}_a, a= x,y,z$ is the vector connecting the A-sublattice to the B-sublattice.}\\label{fig1}\n\\end{figure}\n\nHaving these in mind, we have the following scattering problem at the A-sublattice. \n\\begin{align}\n\\left[\n\\begin{array}{c}\n\\psi_x (\\bm{R}) \\\\\n\\psi_y (\\bm{R}) \\\\ \n\\psi_z (\\bm{R}) \n\\end{array} \\right] = e^{-i\\frac{E}{v_F \\hbar} L} \\cdot \\hat{T}_A \\cdot \\left[\n\\begin{array}{c}\n\\psi_{\\bar{x}} (\\bm{R} +\\hat{e}_x) \\\\\n\\psi_{\\bar{y}} (\\bm{R} + \\hat{e}_y) \\\\ \n\\psi_{\\bar{z}} (\\bm{R} +\\hat{e}_z) \n\\end{array} \\right]\n\\end{align}\nHere, the left-hand side $\\psi_a (\\bm{R}), a= x,y,z$ represents the out-going modes from the A-sublattice, which is related by a scattering matrix $\\hat{T}_A$ to the in-coming modes $\\psi_{\\bar{a}} (\\bm{R}), a = x,y,z$ appearing on the right-hand side (See Fig. \\ref{fig1}). The additional phase factor $\\sim \\exp (-i\\frac{E}{v_F \\hbar} L)$ is the phase accumulated by the incoming modes while it propagates coherently from the neighboring B-sublattices to the A-sublattice at $\\bm{R}$. Here $v_F$ is the Fermi velocity within the one-dimensional metal, which is expected to be similar to that of the bulk electron, and $L$ is the length of the link. The scattering matrix $\\hat{T}_A$ can be fixed by the three-fold rotation as well as the two-fold mirrors at the node. With the unitarity of the scattering matrix, we find \n\\begin{align}\n\\hat{T}_A = e^{i\\chi_A} \\left[\n\\begin{array}{ccc}\nT_A & t_A & t_A \\\\\nt_A & T_A & t_A \\\\ \nt_A & t_A & T_A \n\\end{array} \\right], ~~ |T_A| \\in \\Big[\\frac{1}{3}, ~1\\Big], ~ t_A = e^{i\\phi_A} \\sqrt{\\frac{1-|T_A|^2}{2}}, \n\\end{align}\nwith $\\phi_A = \\cos^{-1}(\\frac{|t_A|}{2|T_A|})$. Similarly we have the following scattering problem at the B-sublattice.\n\\begin{align}\n\\left[\n\\begin{array}{c}\n\\psi_{\\bar{x}} (\\bm{R}) \\\\\n\\psi_{\\bar{y}} (\\bm{R}) \\\\ \n\\psi_{\\bar{z}} (\\bm{R}) \n\\end{array} \\right] = e^{-i\\frac{E}{v_F \\hbar} L} \\cdot \\hat{T}_B \\cdot \\left[\n\\begin{array}{c}\n\\psi_{x} (\\bm{R} -\\hat{e}_x) \\\\\n\\psi_{y} (\\bm{R} - \\hat{e}_y) \\\\ \n\\psi_{z} (\\bm{R} -\\hat{e}_z) \n\\end{array} \\right],\n\\end{align}\nwhere $\\hat{T}_B$ has the same structure as the $\\hat{T}_A$. \n\nNow we can perform the Fourier transformation on $\\bm{R}$ and solve these scattering problems. \n\\begin{align}\n\\bm{\\Psi}_{\\bm{q}} = e^{-i\\frac{E_{\\bm{q}}}{v_F \\hbar} L} \\hat{T}_{\\bm{q}} \\cdot \\bm{\\Psi}_{\\bm{q}}, ~~ \\bm{\\Psi}_{\\bm{q}} = \\left[\n\\begin{array}{c}\n\\psi_{x} (\\bm{q}) \\\\\n\\psi_{y} (\\bm{q}) \\\\ \n\\psi_{z} (\\bm{q}) \\\\\n\\psi_{\\bar{x}} (\\bm{q}) \\\\\n\\psi_{\\bar{y}} (\\bm{q}) \\\\ \n\\psi_{\\bar{z}} (\\bm{q}) \n\\end{array} \\right], ~~ \n\\hat{T}_{\\bm{q}} = \\left[\n\\begin{array}{cc}\n0 & \\hat{T}_A \\cdot \\hat{V}_{\\bm{q}} \\\\ \n\\hat{T}_B \\cdot \\hat{V}_{\\bm{q}}^* & 0 \n\\end{array} \\right],\n\\label{Energy}\n\\end{align}\nwhere $\\hat{V}_{\\bm{q}} = $ diag $[\\exp (i\\bm{q}\\cdot \\hat{e}_x), ~\\exp (i\\bm{q}\\cdot \\hat{e}_y), ~\\exp (i\\bm{q}\\cdot \\hat{e}_z)]$. Hence, the energy spectrum can be obtained by diagonalizing $\\hat{T}_{\\bm{q}}$, which is again an unitary matrix. In terms of the eigenvalues $e^{i \\epsilon_{j} (\\bm{q})}, j= 1, 2, \\cdots 6$ of $\\hat{T}_{\\bm{q}}$, we have the energy spectrum: \n\\begin{align}\nE_{j, \\bm{q}}^n = 2\\pi \\frac{v_F \\hbar}{L} n + \\frac{v_F \\hbar}{L} \\epsilon_{j} (\\bm{q}), ~~j = 1,2, \\cdots 6.\n\\end{align}\nHere $n \\in \\mathbb{Z}$ and thus the minibands are repeating in the energy in period of $2\\pi \\frac{v_F \\hbar}{L}$. Mathematically this repetition in $n$ originates from the ambiguity of $\\epsilon_j (\\bm{q})$ by $2\\pi$ appearing in the eigenvalues $e^{i \\epsilon_{j} (\\bm{q})}, j= 1, 2, \\cdots 6$. Physically this repetition can be traced back to the excitation energy of the microscopic one-dimensional modes with the same momentum $\\bm{q}$, i.e., for a given $\\bm{q}$, there are multiple different one-dimensional modes with energy $2\\pi \\frac{v_F \\hbar}{L} n, ~ n\\in \\mathbb{Z}$. Thus we expect that the energy spectrum given by $\\frac{v_F \\hbar}{L}\\epsilon_j (\\bm{q})$ will repeat in energy with a period $2\\pi \\frac{v_F \\hbar}{L}$ and entirely fills up the bulk CDW gap. Below we will analyze only one period of the band spectrum. \n\n\\subsection{Band Spectrum}\nWe first consider the case where we have a full symmetry of the honeycomb lattice, i.e., $\\hat{T}_A = \\hat{T}_B$. As apparent from the Fig \\ref{fig2}, the spectrum features (i) Dirac cones at the $K$ and $K'$ points, (ii) flat bands, and (iii) quadratic band touchings at the $\\Gamma$ point. Now we discuss the stabilities of these features. \n\\begin{enumerate}\n\\item \\textbf{Dirac Cones at the $K$ and $K'$ points}: The Dirac cones are protected by the sublattice symmetry as in the graphene. It is easily removed by breaking the symmetry, i.e., $T_A \\neq T_B$. See the spectrum in Fig \\ref{fig2}.\n\\item \\textbf{Quadratic Band Touching at the $\\Gamma$ point}: The quadratic band touchings can be protected by the six-fold rotation symmetry \\cite{sun09}. However, even when the symmetry is broken (while keeping the three-fold rotation and mirror symmetries are kept), the band touchings are robust within our network model. See the Fig \\ref{fig2}. \n\\item \\textbf{``Flat-ness\" of Flat bands}: The flat-ness of the bands cannot be protected. However, within our network model (with the three-fold rotation $C_3$ and mirror symmetries), we find that it is robust. See the Fig \\ref{fig2}. \n\\end{enumerate}\n\n\n\\begin{figure}[htbp]\n\\centering\\includegraphics[width=0.5\\textwidth]{Sfig7.pdf}\n\\caption{Energy Spectrum of Honeycomb Network Model. Here we plot the spectrum of $\\epsilon_j (\\bm{q})$ of the scattering matrix $\\hat{T}_{\\bm{q}}$ along $M \\to K \\to \\Gamma \\to M$. (A) $C_6$-symmetric spectrum $\\hat{T}_A = \\hat{T}_B$. It is straightforward to note the Dirac fermion at the $K$ point, quadratic band touching at $\\Gamma$ point, and also the flat bands. (B) $C_3$-symmetric spectrum $\\hat{T}_A \\neq \\hat{T}_B$. Here the Dirac band touching is removed.}\\label{fig2}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\\subsection{Symmetry Analysis of Quadratic Band Touching} \nHere we perform the symmetry analysis of the quadratic band touchings at the $\\Gamma$ point. By explicitly diagonalizing Eq.\\eqref{Energy} at the $\\Gamma$ point (with $\\hat{T}_A = \\hat{T}_B$), we obtain the wavefunctions of the degenerate states. They are labeled as $|\\Psi_a \\rangle, a =1,2$ (with $\\langle \\Psi_a | \\Psi_b \\rangle = \\delta_{ab}$), from which we reconstruct the representations of the three-fold rotation $C_3$, the x-mirror $R_x: x \\to -x$, and the six-fold rotation $C_6$, i.e., $\\{[C_3]_p, [R_x]_p, [C_6]_p\\}$ within these bands. For example, we obtain the three-fold rotation $[C_3]_p$ within the two states by computing\n\\begin{align}\\label{Ex}\n[C_3]_p = \\left[\n\\begin{array}{cc}\n\\langle \\Psi_1 | \\hat{C}_3 |\\Psi_1 \\rangle & \\langle \\Psi_2 | \\hat{C}_3 |\\Psi_1 \\rangle \\\\ \n\\langle \\Psi_1 | \\hat{C}_3 |\\Psi_2 \\rangle & \\langle \\Psi_2 | \\hat{C}_3 |\\Psi_2 \\rangle\n\\end{array} \\right],\n\\end{align} \nin which $\\hat{C}_3$ is the representation of the three-fold rotation in the six-component $\\Psi_{\\bm{q}}$ in Eq.\\eqref{Energy}. \n\n\n\n\nBy computing these explicitly, we find\n\\begin{align}\n[C_3]_p = e^{\\frac{2\\pi i}{3} \\sigma^z}, ~ [R_x]_p = e^{-\\frac{\\pi i}{3} \\sigma^z} \\sigma^x, ~[C_6]_p = e^{-\\frac{2\\pi i}{3} \\sigma^z}, \n\\end{align}\nwhere $\\sigma^a, a= x,y,z$ is the Pauli matrix acting on the space spanned by $\\{|\\Psi_1 \\rangle, |\\Psi_2 \\rangle\\}$.\n\nWith these, we can write down symmetry-allowed Hamiltonian near the $\\Gamma$ point. First, by imposing $[C_3]_p$ and $[R_x]_p$, we find that no perturbation is allowed to split $|\\Psi_1 \\rangle$ and $|\\Psi_2 \\rangle$ at the $\\Gamma$ point. \n\\begin{align}\n[C_3]_p^{\\dagger} H_0 [C_3]_p = H_0, ~~[R_x]_p^{\\dagger} H_0 [R_x]_p = H_0, ~ \\to H_0 \\propto \\mu \\sigma^0\n\\end{align}\nHence the degeneracy cannot be removed when $[C_3]_p$ and $[R_x]_p$ are imposed. On the other hand, near the $\\Gamma$ point, we find that the linear band touching is allowed. \n\\begin{align}\n[C_3]_p^{\\dagger} H (\\bm{k}) [C_3]_p = H (C_3^{-1}[\\bm{k}]), ~~[R_x]_p^{\\dagger} H(k_x, k_y) [R_x]_p = H(-k_x, k_y),\n\\end{align}\nallows $H(\\bm{k}) \\propto k_x \\hat{s}^x + k_y \\hat{s}^y$ (where $(\\hat{s}_x, \\hat{s}_y)$ are the Pauli matrices obtained by properly rotating $\\sigma^x$ and $\\sigma^y$.). Hence, the quadratic band touching cannot be protected by $[C_3]_p$ and $[R_x]_p$. Nevertheless, within our network model, the touching is found to be robust though the touching is not protected by the symmetries. \n\nWe can show that we need the six-fold rotation symmetry $[C_6]_p$ to protect the quadratic band touching and this is consistent with Ref. \\cite{sun09}. Thus, on imposing $[C_6]_p$, we can fix the Hamiltonian as \n\\begin{align}\nH = \\epsilon_0 (|\\bm{k}|) + \\Big( \\frac{k_x^2 - k_y^2}{2m} \\sigma^x + \\frac{2k_x k_y}{2m} \\sigma^y \\Big).\n\\end{align}\nTo match the band spectrum seen in the model, we have $\\epsilon_0 (|\\bm{k}|) = \\frac{\\bm{k}^2}{2m}$ and thus the lower band is completely flat and the density of state at the zero energy is divergent. \n\n\\subsection{Comparison with Twisted Graphene Bilayer}\nHere we extend our discussion in the main text on the similarity between our network system and the theoretical models \\cite{Efimkin, bist11} for the twisted graphene bilayers. In particular, we compare ours with the network model in Ref. \\cite{Efimkin} and a continuum Dirac fermion model in Ref. \\cite{bist11}.\n\nTo start with, we find that our network system is close to the network model appeared in Ref \\cite{Efimkin}. In Ref. \\cite{Efimkin}, the twisted graphene bilayer at a small twisting angle has been considered. When the twisting angle is small, there is a periodic array of domain walls separating the locally AA-stacked regions and the locally AB-stacked regions. Ref. \\cite{Efimkin} argued that these domain walls trap localized one-dimensional metallic channels. These one-dimensional modes scatter at the nodes, which form a triangular lattice (in our case, the nodes form a honeycomb lattice). The structure of their model is quite similar to ours and indeed Ref. \\cite{Efimkin} obtained a similar spectrum as ours: Dirac fermions, nearly flat bands, as well as van-Hove singularities. \n\nWe can also make a comparison of our network system with the continuum Dirac theory of magic-angle twisted bilayer graphene in Ref. \\cite{bist11}. In this approach, the Dirac fermions coming from the top and bottom layers interfere each other, and as a result, the bands become flat. Theoretically, this flat spectrum is speculated to be the source of surprising correlation-driven phenomena seen in the experiments \\cite{cao18, lian18, po18}. One may note that our network model appears to be different than the continuum Dirac theory of Ref. \\cite{bist11}. Despite of the difference in the theoretical treatments, we emphasize that our network model and the result of Ref. \\cite{bist11} share the strikingly-similar features in spectrum: Dirac fermions, flat bands and associated singularities in density of states, which are believed to play an essential role in the correlation physics.\n\nBoth the twisted bilayer graphene and our honeycomb network have weak disorders \\cite{brih12, raza16}. For example, there are some imperfect hexagons in our network and imperfect triangles in twisted bilayer graphene. Naively one expects that such weak disorders would immediately localize the flat bands and completely destroy associated many-body physics. However, the previous study \\cite{chal10} surprisingly found that the flat bands do not get immediately localized but become critical. This implies that the flat bands are stronger against disorders than we naively expect. Though a more thorough investigation is desirable, we expect from the reference \\cite{chal10} that the flat bands retain relatively flat spectrum even with the weak disorders and hence is expected to remain very susceptible to many-body physics.\n\nIn summary, we have shown that the two systems, twisted graphene bilayer and our network system, share the surprising similarities including the flat bands and a large density of states, which are the key to the exotic correlation-driven phenomena. \n\n\\subsection{Interlayer coupling}\n\nIn this subsection, we consider the effect of interlayer coupling to the electronic structures in the conducting network, and we will argue that, in general, the interlayer couplings between the layers will little affect the emergent electronic structures.\n\nFor the concrete-ness of our theoretical discussion, we first assume that the charge-density wave domains in the nearly commensurate phase remain insulating even after the inclusions of interlayer couplings [see SFig.5]. With this in hand, all the lowest-energy electronic states are in the domain walls in the conducting networks, and the interlayer couplings will introduce the coupling between these metallic modes inside the conducting networks living in different layers.\n\nAmong various possible couplings, the most important coupling, which can largely modify the band structure, is the electron hopping process between the layers. Note that this is proportional to the wavefunction overlap between the states of domain walls in different layers, and the states are highly localized within each domain walls. Hence, the effect of coupling will be strongly suppressed if not the networks are almost exactly overlapping to each other when seen from c-axis. From the available literature \\cite{cho16}, we note that the networks in different layers are not correlated to each other and thus we expect that the emergent band structure of the low-energy theory will not be affected much by the interlayer coupling. \n \n\n\\newcommand{\\AP}[3]{Adv.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\CMS}[3]{Comput.\\ Mater.\\ Sci. \\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PR}[3]{Phys.\\ Rev.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PRL}[3]{Phys.\\ Rev.\\ Lett.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PRB}[3]{Phys.\\ Rev.\\ B\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NA}[3]{Nature\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAP}[3]{Nat.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAM}[3]{Nat.\\ Mater.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAC}[3]{Nat.\\ Commun.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NAN}[3]{Nat.\\ Nanotechnol.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NARM}[3]{Nat.\\ Rev.\\ Mater.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\NT}[3]{Nanotechnology {\\bf #1}, #2 (#3)}\n\\newcommand{\\JP}[3]{J.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\JAP}[3]{J.\\ Appl.\\ Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\JPSJ}[3]{J.\\ Phys.\\ Soc.\\ Jpn.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\JPC}[3]{J.\\ Phys.\\ C: \\ Solid State Phys.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PNAS}[3]{Proc.\\ Natl.\\ Acad.\\ Sci. {\\bf #1}, #2 (#3)}\n\\newcommand{\\PRSL}[3]{Proc.\\ R.\\ Soc.\\ Lond. A {\\bf #1}, #2 (#3)}\n\\newcommand{\\PBC}[3]{Physica\\ B+C\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\PAC}[3]{Pure Appl.\\ Chem. \\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\SCI}[3]{Science\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\SCA}[3]{Sci.\\ Adv.\\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\RPP}[3]{Rep.\\ Prog.\\ Phys. \\ {\\bf #1}, #2 (#3)}\n\\newcommand{\\SCR}[3]{Sci.\\ Ref.\\ {\\bf #1}, #2 (#3)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}