\chapter{Consistency on the invariant normal line}\label{s:cons} In this section we prove that, in the balanced root signature $p=k$, the adapted chain of \zcref{s:tube} converges to the continuum semigroup on vectors of the form $\Omega_h \otimes f$, where $\Omega_h$ is the invariant state of the frozen normal transfer and $f$ is a function on the saddle manifold. The precise statement is \zcref{t:cons-limit}. We call a set of consecutive steps of the chain a \emph{leg}. The result will be used in \zcref{s:assembly} for the legs of macroscopic length, after the initial layers in which the endpoints relax to $\Omega_h$. The endpoints themselves are treated in \zcref{s:end}. \paragraph{Fast and slow variables.} We first recall the heuristic of \zcref{s:ov-averaging}, which leads to the consistency relation \eqref{e:ov-consistency}. By \zcref{p:tube-gap}, the frozen normal transfer $\mathbf F_h$, which is the adapted step $A_h$ with the tangent motion and the source removed, contracts closed normal vectors of zero mean by the factor $e^{-c h^2}$ per step, while one step moves the tangent variables, which parametrize the saddle manifold, by a heat flow of duration $\tau_{\rm phys} = \delta_W/2$. The normal variables therefore equilibrate after about $h^{-2}$ steps, long before the tangent variables move over the $h^{-4}$ steps of the chain, and to first order the tangent variables should see them only through their equilibrium state. This is first-order averaging, or adiabatic elimination of fast variables, for operator semigroups, in the sense of Kurtz's limit theorem for perturbed semigroups \cite{kurtz1973limit}; see also \cite[Chapter~1]{ethier1986markov}. One also expects first order to suffice. Let $\Pi$ be the projection onto the line of vectors $\Omega_h \otimes f$, and let $L_H$ be the tangent generator. The fast resolvent on the complement of the line has norm of order $h^{-2}$ in units of steps. The second-order averaging correction to the generator is then of order $\delta_W h^{-2} \|(\Id - \Pi) L_H \Pi\|^2$ per unit of tangent time. We will see in \eqref{e:cons-horizontal-vacuum} that the tangent generator leaves the line only at order $h$, which makes this correction of order $\delta_W = h^4$. \paragraph{The argument.} This picture is only a heuristic, and we do not use an averaging theorem in the proof. Instead, we verify the two hypotheses of the Trotter--Kurtz approximation theorem, which we state in the form we need as \zcref{l:cons-product}. The first is stability of the powers of $A_h$ on all inputs, which is the main result of \zcref{s:stab}, \zcref{t:stab-main}. The second is consistency on the line $\iota_h f = \Omega_h \otimes f$ for $f$ in an operator core, \zcref{p:cons-consistency}. We construct the normal state $\Omega_h$ in \zcref{s:cons-state} by Banach's fixed point theorem, from the contraction of \zcref{p:tube-gap}. The consistency calculation of \zcref{s:cons-consistency} is a Duhamel expansion of one heat link around its frozen normal heat kernel. It needs only a bound for the heat semigroup on the tube over the duration of one link. This bound is of order one and cannot be iterated over the chain; the iterable bound is the stability estimate of \zcref{s:stab}, which is proved from the kernel of the step. The one-link bound rests on two facts that \zcref{s:end} also uses: the horizontal operator is sectorial in group charts, and the actual Gaussian kernel agrees with the Dirichlet heat on the tube, also when the normal coordinates are moved to complex graphs that change in time. We prove both in \zcref{s:cons-heat}. In \zcref{s:cons-tangent}, we combine \zcref{l:tube-casimir}, which identifies the horizontal operator on the saddle manifold as $-(2 s_E^2)^{-1} \mathsf C_k$, with the duration of one link, and we identify the generator that the chain produces on the tangent variables with the generator of the sigma model, which fixes the limit semigroup of \zcref{t:cons-limit} and its time scale. \paragraph{Inputs from Sections~\ref{s:tube} and~\ref{s:stab}.} For the normal fixed state and the normal-line consistency statements, we fix the balanced root signature $p=k$ and work on the real tube of \zcref{d:tube-cycle}, of radius $r$, where $r \le r_*$ is fixed before $h \to 0$. The abstract approximation results of \zcref{s:cons-product} do not depend on a root signature. The analytic heat estimates of \zcref{s:cons-heat} apply in every signature, as specified there. Recall that the points of the tube are \[ R(x, U) = T(x)(R_* + U)T(x)^{-1} , \] where $x$ runs over the saddle manifold $U(k, k|2k)/K$, $T(x)$ is a local section, $R_* = \kappa_E \Id - \iu s_E \Sigma$ is the root matrix, and $U$ is a normal coordinate in the adapted real normal space of \zcref{d:tube-cycle}, with standardized form $U = hZ$. Recall also the normal Wick space $\mathcal H_{\rm nor}$ of \zcref{s:tube-oscillator}, with vacuum $\Omega_0 = 1$, number operator $\mathbf N$, and Gaussian mean $\ell_0$, so that the orthogonal projection onto $\C\Omega_0$ is $\mathbf P_0 f = \ell_0(f)\Omega_0$. We let $\mathcal G^m$ denote the domain of $(\mathbf N + 1)^m$, with the norm $\|f\|_{\mathcal G^m} = \|(\mathbf N + 1)^m f\|$. By \zcref{l:tube-supercharge}, the supercharge $\mathfrak q_{\rm nor}$ is a closed operator. We let $\mathcal K = \ker \mathfrak q_{\rm nor}$ denote the closed subspace of closed normal vectors, and we set $\mathcal K_0 = \mathcal K \cap \ker \ell_0$. Let $\mathbf F_h$ be the frozen normal transfer of \eqref{e:tube-table} and \zcref{s:tube-powers}. Inputs \ref{i:cons-contraction}--\ref{i:cons-localization} below concern the balanced signature; the stability input \ref{i:cons-stability} also holds in every other signature. \begin{enumerate}[label=(N\arabic*)] \item\label{i:cons-contraction} By \zcref{p:tube-gap}, $\mathbf F_h$ maps $\mathcal K$ and $\mathcal K_0$ into themselves, satisfies $\ell_0(\mathbf F_h f) = \ell_0(f)$ for $f \in \mathcal K$, and contracts $\mathcal K_0 \cap \mathcal G^m$ by the factor $e^{-c_m h^2}$ per step in the norm of $\mathcal G^m$, for each of the finitely many prescribed graph orders $m$, as in \eqref{e:tube-graph-contraction}. The exact preservation of the mean is the Ward identity of \zcref{s:ov-averaging}, and \zcref{s:tube-gap} explains why a perturbative argument would not suffice. \item\label{i:cons-graph} By \zcref{l:tube-powers}, $\mathbf F_h$ is bounded on each $\mathcal G^m$, and its vacuum residual satisfies $\|\mathbf F_h \Omega_0 - \Omega_0\|_{\mathcal G^m} \le C_m h^3$, which is \eqref{e:tube-vacuum-residual}. \item\label{i:cons-localization} Localization, \zcref{l:tube-localization}: for a closed normal function $g$ with compactly supported body, the normalized Gaussian Berezin integral of $g$ is the value of its body at the origin. \item\label{i:cons-stability} Stability, \zcref{t:stab-main}: for all $T > 0$ and $q \in \R$, $\sup_{0 \le n \delta_W \le T} \|A_h^n\| \le e^{BT/4}$ on the weighted space $\mathcal H^{\rm tube}_q$ of \zcref{s:stab}, on all inputs, uniformly for the damping in $\mathcal U_2$ and on compact energy sets. \end{enumerate} By \eqref{e:tube-table}, we have $A_h = s_\Gamma T_h$, where $s_\Gamma$ is the multiplication by the physical source factor and $T_h$ is the unsourced step in the symmetric normal gauge. We define the \emph{source potential} $V_h$ on the tube by $s_\Gamma = e^{-\delta_W V_h}$. The frozen normal transfer $\mathbf F_h$ is the normal part of $T_h$; below \eqref{e:cons-steps} we compare these Gaussian steps with their Dirichlet versions. \section{The Trotter--Kurtz approximation theorem}\label{s:cons-product} The following lemma is the discrete-time approximation theorem of Trotter and Kurtz \cite{trotter1958approximation,kurtz1969extensions}, in the core form of \cite[Chapter~1, Section~6]{ethier1986markov}. That theorem assumes contractions. Trotter's Theorem~5.3 \cite{trotter1958approximation} already allows powers that are bounded but not contractive, but it uses hypotheses on the limit of the generators and proves convergence for each fixed time. Since we need the core form of the consistency hypothesis and uniformity in time, we deduce the lemma from the contractive theorem by an equivalent renorming. \begin{lemma}\label{l:cons-product} Let $\mathcal H$ be a Banach space, and let $(U_t)_{t \ge 0}$ be a strongly continuous semigroup on $\mathcal H$ with generator $L$. Let $\mathcal D \subset D(L)$ be a linear subspace that is a core for $L$. For $0 < h \le h_0$, let $\mathcal H_h$ be Banach spaces, let $\iota_h \colon \mathcal H \to \mathcal H_h$ be bounded linear maps with $\sup_h \|\iota_h\| < \infty$, let $A_h$ be bounded operators on $\mathcal H_h$, and let $\delta_h > 0$ with $\delta_h \to 0$ as $h \to 0$. Suppose that for all $T > 0$, \begin{equation}\label{e:cons-power} \sup_{0 < h \le h_0} \ \sup_{0 \le n \delta_h \le T} \|A_h^n\| \le C_T < \infty , \end{equation} and that for all $f \in \mathcal D$, \begin{equation}\label{e:cons-hyp} \lim_{h \to 0} \bigl\| \delta_h^{-1} (A_h \iota_h f - \iota_h f) - \iota_h L f \bigr\| = 0 . \end{equation} Then for all $f \in \mathcal H$ and $T > 0$, \begin{equation}\label{e:cons-conclusion} \lim_{h \to 0} \ \sup_{0 \le n \delta_h \le T} \|A_h^n \iota_h f - \iota_h U_{n \delta_h} f\| = 0 . \end{equation} \end{lemma} We first recall the cited theorem. Let $L_n$ and $L$ be Banach spaces and $\pi_n \colon L \to L_n$ bounded linear maps with $\sup_n \|\pi_n\| < \infty$. Let $T_n$ be linear contractions on $L_n$, let $\eps_n > 0$ with $\eps_n \to 0$, and let $T(t)$ be a strongly continuous contraction semigroup on $L$ whose generator $A$ has a core $D$. Suppose that for each $f \in D$ there exist $f_n \in L_n$ such that $\|f_n - \pi_n f\| \to 0$ and $\|\eps_n^{-1}(T_n - \Id) f_n - \pi_n A f\| \to 0$. Then the theorem of \cite[Chapter~1, Section~6]{ethier1986markov} states that for all $f \in L$ and $T > 0$, \[ \sup_{0 \le t \le T} \|T_n^{\lfloor t/\eps_n \rfloor} \pi_n f - \pi_n T(t) f\| \longrightarrow 0 . \] \begin{proof}[Proof of \zcref{l:cons-product}] It suffices to prove \eqref{e:cons-conclusion} along an arbitrary sequence $h_n \downarrow 0$. We write $A = A_{h_n}$, $\delta = \delta_{h_n}$, and $\iota = \iota_{h_n}$ when the index is clear. We begin by renorming. Let $C_1 \ge 1$ be the constant of \eqref{e:cons-power} with $T = 1$, suppose that $\delta \le 1/2$, and let $n_1 = \lfloor 1/\delta \rfloor$, so that $n_1 \delta \ge 1/2$. Writing $n = j n_1 + i$ with $0 \le i < n_1$, we have $j \le 2 n \delta$, and therefore \[ \|A^n\| \le \|A^{n_1}\|^j \|A^i\| \le C_1^{1 + 2 n \delta} \qquad \text{for all } n \ge 0 . \] Let $M_0 \ge 1$ and $\alpha_0 \in \R$ be such that $\|U_t\| \le M_0 e^{\alpha_0 t}$ for all $t \ge 0$, and fix $\alpha \ge \max(2 \log C_1, \alpha_0)$. We define the norms \[ \|g\|_\sharp = \sup_{n \ge 0} e^{-\alpha n \delta} \|A^n g\| , \qquad \|f\|_* = \sup_{t \ge 0} e^{-\alpha t} \|U_t f\| . \] By the two growth bounds, \[ \|g\| \le \|g\|_\sharp \le C_1 \|g\| , \qquad \|f\| \le \|f\|_* \le M_0 \|f\| , \] uniformly along the sequence. In these norms, $e^{-\alpha \delta} A$ is a contraction, and $e^{-\alpha t} U_t$ is a strongly continuous contraction semigroup. Its generator is $L - \alpha$, with the same domain, and $\mathcal D$ is a core for it. The maps $\iota$ remain uniformly bounded. Next, we verify the hypothesis of the cited theorem with $L_n = \mathcal H_{h_n}$, $\pi_n = \iota_{h_n}$, $\eps_n = \delta_{h_n}$, $T_n = e^{-\alpha \delta} A$, the generator $L - \alpha$, the core $\mathcal D$, and $f_n = \iota f$. For $f \in \mathcal D$, a direct computation gives \[ \begin{aligned} \delta^{-1}(e^{-\alpha \delta} A - \Id)\iota f - \iota (L - \alpha) f &= e^{-\alpha \delta} \bigl[\delta^{-1}(A \iota f - \iota f) - \iota L f\bigr]\\ &\quad + \bigl[\delta^{-1}(e^{-\alpha \delta} - 1) + \alpha\bigr] \iota f + (e^{-\alpha \delta} - 1) \iota L f . \end{aligned} \] The first term tends to zero by \eqref{e:cons-hyp}, and the other two tend to zero since $\iota$ is uniformly bounded. The cited theorem therefore applies. Since $\lfloor t/\delta \rfloor = n$ at $t = n\delta$, its conclusion implies that \[ \sup_{0 \le n \delta \le T} e^{-\alpha n \delta} \|A^n \iota f - \iota U_{n\delta} f\|_\sharp \longrightarrow 0 . \] Multiplying by $e^{\alpha n \delta} \le e^{\alpha T}$ and using $\| \cdot \| \le \| \cdot \|_\sharp$, we obtain \eqref{e:cons-conclusion}. \end{proof} We also need the conclusion uniformly in parameters. Let $\mathcal X$ be a compact metric space of parameters (in our application, energies, sources, and aspect ratios). In the application the spaces depend on the parameters, but they can be identified with one another: we say that a family of Banach spaces $\mathcal H_{h, \mathfrak p}$, $\mathfrak p \in \mathcal X$, is \emph{uniformly identified} with a Banach space $\mathcal H_h$ if there exist linear isomorphisms $J_{h, \mathfrak p} \colon \mathcal H_{h, \mathfrak p} \to \mathcal H_h$ with $\sup_{h, \mathfrak p} (\|J_{h, \mathfrak p}\| + \|J_{h, \mathfrak p}^{-1}\|) < \infty$. \begin{corollary}\label{c:cons-product-param} For $\mathfrak p \in \mathcal X$, let $(U^\mathfrak p_t)$, $L_\mathfrak p$, $\iota_{h, \mathfrak p}$, and $A_{h, \mathfrak p}$ be as in \zcref{l:cons-product}, on a fixed space $\mathcal H$ and on spaces $\mathcal H_{h, \mathfrak p}$ that are uniformly identified with spaces $\mathcal H_h$. Suppose that $\|U^\mathfrak p_t\| \le M_0 e^{\alpha_0 t}$ for all $\mathfrak p$ and $t$, that \eqref{e:cons-power} holds uniformly in $\mathfrak p$, that $\sup_{h, \mathfrak p} \|\iota_{h, \mathfrak p}\| < \infty$, that $\mathcal D$ is a common core on which $\mathfrak p \mapsto L_\mathfrak p f$ is continuous, and that \eqref{e:cons-hyp} holds uniformly in $\mathfrak p$ for each $f \in \mathcal D$. Then \eqref{e:cons-conclusion} holds uniformly in $\mathfrak p \in \mathcal X$. \end{corollary} \begin{proof} Transporting by $J_{h, \mathfrak p}$ changes all the norms by bounded factors, and we may assume that $\mathcal H_{h, \mathfrak p} = \mathcal H_h$. Suppose that the conclusion fails. Then there exist $f \in \mathcal H$, $T > 0$, $\gamma > 0$, $h_n \downarrow 0$, and $\mathfrak p_n \in \mathcal X$ such that the supremum in \eqref{e:cons-conclusion} at $(h_n, \mathfrak p_n)$ is at least $\gamma$. By compactness we may assume that $\mathfrak p_n \to \mathfrak p$. Along this sequence, the consistency \eqref{e:cons-hyp} holds with the fixed generator $L_\mathfrak p$, since \[ \|\iota_{h_n, \mathfrak p_n}(L_{\mathfrak p_n} f - L_\mathfrak p f)\| \le C \|L_{\mathfrak p_n} f - L_\mathfrak p f\| \longrightarrow 0 , \qquad f \in \mathcal D . \] Applying the proof of \zcref{l:cons-product} to the sequence $(A_{h_n, \mathfrak p_n}, \iota_{h_n, \mathfrak p_n})$ and the semigroup $U^\mathfrak p$, we obtain \[ \sup_{0 \le j \delta_{h_n} \le T} \|A_{h_n, \mathfrak p_n}^j \iota_{h_n, \mathfrak p_n} f - \iota_{h_n, \mathfrak p_n} U^\mathfrak p_{j \delta_{h_n}} f\| \longrightarrow 0 . \] The generators $L_{\mathfrak p_n}$ converge to $L_\mathfrak p$ on the common core $\mathcal D$, and the semigroups satisfy a common exponential bound. By the Trotter--Kato theorem for semigroups with a common exponential bound \cite{trotter1958approximation,kato1995perturbation}, whose conclusion is uniform for $t$ in bounded intervals, we have $\sup_{t \le T} \|U^{\mathfrak p_n}_t f - U^\mathfrak p_t f\| \to 0$. Since the maps $\iota_{h_n, \mathfrak p_n}$ are uniformly bounded, the supremum in \eqref{e:cons-conclusion} at $(h_n, \mathfrak p_n)$ tends to zero, which is a contradiction. \end{proof} \section{The normal fixed state}\label{s:cons-state} In this section we construct the invariant state $\Omega_h$ of the frozen normal transfer. It is the vector in the embedding $\iota_h f = \Omega_h \otimes f$, and the initial layer of \zcref{s:assembly} moves both endpoints of the chain to it. The idea is simple. Since $\mathbf F_h$ preserves the mean on closed vectors and contracts closed vectors of zero mean, its restriction to the closed vectors of mean one is a contraction of a complete metric space, and $\Omega_h$ is the fixed point of this restriction. \begin{lemma}\label{l:cons-fixed-state} In the balanced signature $p=k$, there exists a unique $\Omega_h \in \mathcal K$ such that $\ell_0(\Omega_h) = 1$ and $\mathbf F_h \Omega_h = \Omega_h$. It is annihilated by every normal charge. For each prescribed graph order $m$, \begin{equation}\label{e:cons-state-bounds} \mathbf P_0 \Omega_h = \Omega_0 , \qquad \|\Omega_h - \Omega_0\|_{\mathcal G^m} \le C_m h . \end{equation} For all $f \in \mathcal K \cap \mathcal G^m$ and $j \ge 0$, \begin{equation}\label{e:cons-relaxation} \|\mathbf F_h^j f - \ell_0(f) \Omega_h\|_{\mathcal G^m} \le e^{-c_m j h^2} \|f - \ell_0(f) \Omega_h\|_{\mathcal G^m} , \end{equation} where $c_m$ is the rate in \eqref{e:tube-graph-contraction}. The same bound holds for $\mathbf F_h \otimes \Id$ on vectors with values in a fixed auxiliary Hilbert space. For each $h$, the vector $\Omega_h$ is smooth and supported in the support of the output cutoff of $\mathbf F_h$, and its normalized analytic Berezin pairing with itself in the normal variables satisfies \begin{equation}\label{e:cons-pairing} \mathcal B_{\rm nor}(\Omega_h, \Omega_h) = 1 . \end{equation} \end{lemma} \begin{proof} We first construct $\Omega_h$. By \ref{i:cons-contraction}, $\mathbf F_h$ maps $\mathcal K$ into itself and preserves $\ell_0$ there. Since $\mathfrak q_{\rm nor} \Omega_0 = 0$ and $\ell_0(\Omega_0) = 1$, it maps the closed affine subset \[ \Omega_0 + \mathcal K_0 = \{ f \in \mathcal K : \ell_0(f) = 1 \} \] into itself. Fix a prescribed graph order $m$. Note that $(\Omega_0 + \mathcal K_0) \cap \mathcal G^m$ is closed in $\mathcal G^m$, since $\mathfrak q_{\rm nor}$ is closed and $\ell_0$ is bounded, and that $\mathbf F_h$ maps it into itself by \ref{i:cons-graph}. The difference of two of its points lies in $\mathcal K_0 \cap \mathcal G^m$. Using \eqref{e:tube-graph-contraction} with $j = 1$, we obtain \[ \|\mathbf F_h f - \mathbf F_h g\|_{\mathcal G^m} \le e^{-c_m h^2} \|f - g\|_{\mathcal G^m} , \qquad f, g \in (\Omega_0 + \mathcal K_0) \cap \mathcal G^m . \] By Banach's fixed point theorem, $\mathbf F_h$ has a unique fixed point $\Omega_h$ in this set, and \[ \|\Omega_h - \Omega_0\|_{\mathcal G^m} \le \frac{\|\mathbf F_h \Omega_0 - \Omega_0\|_{\mathcal G^m}}{1 - e^{-c_m h^2}} \le \frac{C_m h^3}{c_m h^2/2} , \] where we used \ref{i:cons-graph} and $1 - e^{-x} \ge x/2$ for $0 \le x \le 1$. The fixed point is unique already in $\Omega_0 + \mathcal K_0$, by the case $m = 0$, and it does not depend on $m$. Finally, we have $\mathbf P_0 \Omega_h = \ell_0(\Omega_h) \Omega_0 = \Omega_0$, which completes the proof of \eqref{e:cons-state-bounds}. Next, we prove that the normal charges annihilate $\Omega_h$. Fix a prescribed order $m \ge 1$. The normal charges are first-order differential operators in the normal variables with linear coefficients, hence bounded from $\mathcal G^1$ to $\mathcal H_{\rm nor}$. The vectors of $(\Omega_0 + \mathcal K_0) \cap \mathcal G^m$ that they all annihilate therefore form a closed subset of $\mathcal G^m$. This subset contains $\Omega_0$, and $\mathbf F_h$ maps it into itself, since the normal Gaussian, the half-weight, and the invariant cutoff commute with the normal charges. By uniqueness, the fixed point of $\mathbf F_h$ in this subset is $\Omega_h$. We now prove the relaxation bound. For $f \in \mathcal K \cap \mathcal G^m$, we have $f - \ell_0(f)\Omega_h \in \mathcal K_0 \cap \mathcal G^m$ and \[ \mathbf F_h^j f - \ell_0(f)\Omega_h = \mathbf F_h^j \bigl(f - \ell_0(f)\Omega_h\bigr) , \] The bound \eqref{e:cons-relaxation} therefore follows from \eqref{e:tube-graph-contraction}, applied to $f - \ell_0(f)\Omega_h$. For vectors with values in an auxiliary Hilbert space, the same argument applies to $(\mathbf F_h|_{\mathcal K_0})^j \otimes \Id$, which has the same norm as $(\mathbf F_h|_{\mathcal K_0})^j$. Finally, we prove \eqref{e:cons-pairing}. For fixed $h > 0$, the Gaussian kernel of $\mathbf F_h$ maps $\mathcal H_{\rm nor}$ into smooth functions, and its output cutoff is smooth. The identity $\Omega_h = \mathbf F_h \Omega_h$ therefore shows that $\Omega_h$ is smooth and supported in the support of that cutoff. Then $\Omega_h^2$ is smooth, compactly supported and closed, since $\mathfrak q_{\rm nor}$ is an odd derivation. Applying \ref{i:cons-localization} to $\Omega_h$ and to $\Omega_h^2$, we find that $\ell_0(\Omega_h)$ and $\mathcal B_{\rm nor}(\Omega_h, \Omega_h)$ are the values at the origin of the bodies of $\Omega_h$ and $\Omega_h^2$. Since the body of $\Omega_h^2$ is the square of the body of $\Omega_h$, we obtain \[ \mathcal B_{\rm nor}(\Omega_h, \Omega_h) = \ell_0(\Omega_h)^2 = 1 . \qedhere \] \end{proof} Only finitely many graph orders are used. They supply the normal moments needed in the consistency calculation, and they make the contributions of the annular parts of cutoffs smaller than every prescribed power of $h$. Indeed, let $a$ be a smooth function of the unscaled normal coordinate with bounded derivatives, supported where $|U| \ge r_1$ for some $r_1 > 0$. On its support we have $|Z| \ge W^{1/2} r_1 = r_1/h$, and every commutator of multiplication by $a$ with $\mathbf N$ is again supported there. For every $L \ge 1$ such that $m + L$ is among the prescribed orders, it follows that \begin{equation}\label{e:cons-annuli} \|a \, \Omega_h\|_{\mathcal G^m} \le C_{m, L}(r_1) \, h^{2L} \|\Omega_h\|_{\mathcal G^{m + L}} \le C_{m, L}(r_1) \, h^{2L} . \end{equation} We include these orders before the tube is chosen. The same bound holds after finitely many derivatives of $a$, and for the nilpotent coefficients of a smooth invariant cutoff. \section{The heat semigroup on the tube}\label{s:cons-heat} The analytic estimates in this section hold in every root signature $p\in\{0,\dots,2k\}$. In this section we prove three facts about one heat link on the adapted tube. First, \zcref{l:cons-sectorial} shows that the coefficients of the horizontal operator are uniform in group charts and that its form is sectorial. Second, \zcref{l:cons-identification} shows that the Gaussian kernel of the link agrees, on vectors supported inside the tube, with the evolution of the Dirichlet problem on a slightly larger tube, also on moving complex normal graphs. Third, \zcref{l:cons-short-heat} bounds this evolution uniformly over the duration of one link. The first two facts are also the analytic inputs of the weighted energy estimate of \zcref{s:end}. The identification uses a lower bound for the real part of the Gaussian exponent between points of the tube, \zcref{l:cons-separation}. \subsection{The generator} Recall from \zcref{s:rep} that the ambient heat semigroup $K_t$ has the kernel $\exp(-\Str(S - S')^2/(4t))$. Its generator is the ambient flat Laplacian $\Delta_{\rm amb}$ of the bilinear form $\Str(\dd S^2)$, in the sense that the kernel satisfies $\partial_t k_t = \Delta_{\rm amb} k_t$, and one link is $K_{\tau_{\rm phys}}$. Recall from \zcref{l:tube-volume} that the tubular map has Berezinian $\varpi^2$, where the half-density $\varpi$ is analytic and normal-harmonic on the tube by \zcref{l:tube-harmonic}. By \eqref{e:tube-decomposition}, the half-density conjugation of $\Delta_{\rm amb}$ is the exact decomposition \begin{equation}\label{e:cons-decomposition} \varpi \Delta_{\rm amb} \varpi^{-1} = \Delta_{\rm nor} + L_H(U) . \end{equation} Here $\Delta_{\rm nor}$ is the flat Laplacian of the normal variables, ordinary and odd, and $L_H(U) = \operatorname{div}_{\nu, \mathbb D}(g_S^{-1} \mathbb D)$ is the horizontal operator of \eqref{e:tube-decomposition}, where $g_S$ denotes the orbit metric \begin{equation}\label{e:cons-orbit-metric} g_S(Y, Y') = \Str\bigl([Y, S][Y', S]\bigr) , \qquad S = R_* + U , \end{equation} on the tangent directions $Y \in \mathfrak m$. By \zcref{l:cons-sectorial}\ref{i:cons-sect-chart} below, the ordinary second-order part of $L_H(U)$ is given by the inverse of $g_S$. All normal derivatives in $L_H(U)$ come from the normal charges of \zcref{s:tube-operator}, which form the infinitesimal action of the stabilizer $K$ on the normal variables. Since $\varpi$ is normal-harmonic, the decomposition has no additional scalar potential. We now fix the chart in which the coefficients are uniform along the noncompact directions of the saddle manifold. Recall that the group $G_0$ of \zcref{s:sigma-target} is the body of the symmetry supergroup $U(k, k|2k)$; it is a real reductive group and hence unimodular. Let $K_0$ be the body of the stabilizer $K$ of \zcref{s:tube-geometry}, and let $\mathfrak m_0$ and $\mathfrak m_1$ be the even and odd parts of the tangent space $\mathfrak m$. We call the coordinates \begin{equation}\label{e:cons-homogeneous-chart} (g_0, \xi, U) \longmapsto R\bigl(g_0 \exp(\xi) K, U\bigr) , \qquad g_0 \in G_0 , \ \xi \in \mathfrak m_1 , \ U \text{ normal}, \end{equation} the \emph{group chart} of the tube; it is the analogue for the tube of the translated charts of \zcref{s:sigma-weighted}. Functions on the tube are functions of $(g_0, \xi, U)$ that are invariant under the right action of $K_0$, which acts on $g_0$ by right multiplication and on $(\xi, U)$ by conjugation. Let $(Y_a)$ be a basis of $\mathfrak m_0$ that is orthonormal for the positive form $4 s_E^2 \Str(Y Y')$ on the real tangent space, and let $X_a$ be the corresponding left-invariant vector fields on $G_0$, so that $X_a f(g_0)$ is the derivative of $t \mapsto f(g_0 e^{t Y_a})$ at $t = 0$. \begin{lemma}\label{l:cons-sectorial} At fixed rank $k$, there exist $r_k > 0$, a degree-metric parameter $\varkappa_k \in (0, 1]$ and a constant $C_k$ such that the following holds for every block-diagonal normal coordinate $U$, real or complex, with $|U| \le r_k$, and for all sufficiently small $h$. Give the coefficient space the positive metric in which the standardized odd normal monomials and the monomials $\varkappa_k^{-|I|}\xi^I$ in the odd tangent coordinates are orthonormal. Then \begin{enumerate}[label=(\alph*)] \item\label{i:cons-sect-chart} in the group chart, the coefficients of $L_H(U)$ do not depend on $g_0 \in G_0$, and its ordinary second-order part is $\sum_{a, b} X_a \, g^{ab}(U, \xi) \, X_b$, where the matrix $g^{ab}$ is the inverse of the orbit metric \eqref{e:cons-orbit-metric} in the basis $(Y_a)$ and its body at $U = 0$ is the identity; \item\label{i:cons-sect-form} for all $u$ in the form domain of the Dirichlet problem on the tube of radius $r_k$, writing $\mathcal E_{\rm tan}(u) = \sum_a \|X_a u\|^2$ and $\mathcal E_{\rm nor}(u) = \|\nabla_U u\|^2$ for the ordinary tangent and normal energies in physical units, \begin{equation}\label{e:cons-sectorial} \begin{aligned} \Re \langle u, L_H(U) u\rangle &\le - \tfrac12 \mathcal E_{\rm tan}(u) + C_k r_k \, \mathcal E_{\rm nor}(u) + C_k h^{-2} \|u\|^2 , \\ |\Im \langle u, L_H(U) u\rangle| &\le C_k (r_k + \varkappa_k + h) \, \mathcal E_{\rm tan}(u) + C_k r_k \, \mathcal E_{\rm nor}(u) + C_k h^{-2} \|u\|^2 . \end{aligned} \end{equation} \end{enumerate} The constants do not depend on $h$ or on the point of the saddle manifold. The same bounds hold in every fixed polynomial body weight $v^p$, with $C_k$ replaced by $C_{k, p}$. They also hold if the squared norm of each standardized odd normal monomial of degree $j$ is multiplied by a prescribed positive number $d_j$, with the constants and the choices of $r_k$ and $\varkappa_k$ allowed to depend on these fixed degree weights. A common choice may be made for any prescribed finite collection of such metrics. \end{lemma} The term $C_k h^{-2}\|u\|^2$ comes from the odd normal variables, which also give the odd part of $\Delta_{\rm nor}$ its norm $C_k h^{-2}$. In every application this lower-order term is integrated over at most $O(W)$ links of duration $\tau_{\rm phys}$, where it contributes a factor $e^{C_k}$. Only the normal block of the generator needs a margin that does not depend on the rank, and we treat that block explicitly where it is used. \begin{proof} We begin with \ref{i:cons-sect-chart}. The operator $\Delta_{\rm amb}$ commutes with conjugation by $G$, since the bilinear form $\Str(\dd S^2)$ and the measure $\dd S$ are invariant under it. The tubular map is equivariant, $g R(x, U) g^{-1} = R(g x, U)$, and the half-density $\varpi$ depends only on $U$. Therefore $\varpi \Delta_{\rm amb} \varpi^{-1}$, and with it $L_H(U) = \varpi \Delta_{\rm amb} \varpi^{-1} - \Delta_{\rm nor}$, commutes with the left action of $G_0$, which is $g_0 \mapsto g_1 g_0$ in the group chart. In the frame of left-invariant fields, an operator that commutes with all left translations has coefficients that do not depend on $g_0$. In this frame, the horizontal part of the tubular differential in the direction $Y_a$ is $[Y_a, S]$, and the ordinary principal part is the inverse of the matrix $g_S(Y_a, Y_b)$. The first-order part consists of the fields $X_a$, the odd tangent derivatives, and the infinitesimal stabilizer action on $U$, which is given by the even normal charges $[\mathfrak e, U] \cdot \nabla_U$ with even $\mathfrak e \in \mathfrak k$, and by odd ones that exchange ordinary and odd normal coordinates. All coefficients are polynomial in the odd variables, since there are finitely many of them. They are analytic in $U$, since \eqref{e:cons-orbit-metric} is analytic in $S$ and root separation keeps it invertible on the tube. It remains to compute the principal part at $U = 0$. The scalar part $\kappa_E \Id$ of $R_*$ commutes with $Y$, and for off-diagonal $Y$ we have $\Sigma Y = -Y \Sigma$. Hence \[ [Y, R_*] = 2 \iu s_E \Sigma Y , \qquad [Y, R_*][Y', R_*] = -4 s_E^2 \Sigma Y \Sigma Y' = 4 s_E^2 Y Y' , \] and on the body, the metric \eqref{e:cons-orbit-metric} at $U = 0$ satisfies $g_{R_*}(Y, Y') = 4 s_E^2 \Str(Y Y')$. On the real tangent space of the saddle manifold, the body of $\Str(Y Y')$ is positive definite. Indeed, for $Y_{BB} = \bigl(\begin{smallmatrix} 0 & B \\ B^* & 0 \end{smallmatrix}\bigr)$ in the BB block and $Y_{FF} = \bigl(\begin{smallmatrix} 0 & B \\ -B^* & 0 \end{smallmatrix}\bigr)$ in the FF block, we have \[ \Tr(Y_{BB}^2) = 2 \Tr(B B^*) , \qquad -\Tr(Y_{FF}^2) = 2 \Tr(B B^*) . \] In our basis the body of $g^{ab}(0, 0)$ is therefore the identity matrix, which proves \ref{i:cons-sect-chart}. Next, we bound the coefficients on a fixed analytic tube supplied by root separation, before decreasing its radius. By analyticity, the body of $g^{ab}(U, 0)$ differs from the identity by at most $C_k |U|$; this covers its imaginary part as well, since $U$ may be complex. The nonbody part of $g^{ab}$ is a polynomial without constant term in the odd variables. Multiplication by an odd tangent coordinate has norm $\varkappa_k$, while multiplication by an unscaled odd normal coordinate has norm $C_d h$ in either of the stated normal metrics. Thus the nonbody principal coefficient has norm at most $C_{k,d}(\varkappa_k+h)$. An odd tangent derivative has norm $\varkappa_k^{-1}$, which is fixed before the final tube radius is chosen. We estimate normal charges using their positive Hilbert adjoints. Writing $U$ for the ordinary normal coordinates in the following formulas, an even charge and an odd charge have, respectively, the forms \[ R_{\rm even}=\sum_{i,j} U_i C_{ij}\partial_{U_j}+D, \qquad R_{\rm odd}=h\sum_j A_j\partial_{U_j}+h^{-1}\sum_j U_j B_j, \] with fixed finite coefficient matrices; the matrix $D$ includes the action on the odd normal coordinates. Their formal Hilbert adjoints in the flat positive metric satisfy the same estimates. If $a$ is a coefficient matrix with bounded ordinary derivatives, then, on the tube of radius $r$, \begin{equation}\label{e:cons-charge-adjoint} \|R(au)\|+\|R^*(au)\| \le C\bigl((r+h)\|\nabla_Uu\|+(1+r/h)\|u\|\bigr). \end{equation} The constants are uniform as $r$ decreases inside the initial analytic tube. In particular, on compactly supported smooth vectors, \[ \langle u,aR_1R_2u\rangle=\langle R_1^*(a^*u),R_2u\rangle, \] so a product of two charges contributes at most $C(r+h)^2\mathcal E_{\rm nor}(u)+C h^{-2}\|u\|^2$ for $r,h\le1$. Mixed terms with one $X_a$ and one charge are estimated by the same inequality and Young's inequality. No skew-adjointness of odd charges is used. Each $X_a$ is skew-adjoint on $L^2(G_0)$, since $G_0$ is unimodular. Expand the products of horizontal fields in \eqref{e:tube-horizontal-weak} before integrating by parts in the ordinary variables. We obtain \[ \langle u, L_H(U) u\rangle = - \sum_{a, b} \langle X_a u, g^{ab} X_b u\rangle + \mathcal R(u) , \] where $\mathcal R(u)$ contains the terms involving charges, odd tangent derivatives, and derivatives of coefficients. By the principal coefficient bounds, \[ \begin{gathered} \Re \sum_{a, b} \langle X_a u, g^{ab} X_b u\rangle \ge \bigl(1 - C_k r_k - C_k(\varkappa_k + h)\bigr) \mathcal E_{\rm tan}(u) , \\ \Bigl|\Im \sum_{a, b} \langle X_a u, g^{ab} X_b u\rangle\Bigr| \le C_k(r_k + \varkappa_k + h) \mathcal E_{\rm tan}(u) , \end{gathered} \] and, for every fixed $\varepsilon>0$, \eqref{e:cons-charge-adjoint} and Young's inequality give \[ |\mathcal R(u)| \le \varepsilon\mathcal E_{\rm tan}(u) +C_{\varepsilon,k,d,\varkappa_k}(r+h)^2\mathcal E_{\rm nor}(u) +C_{\varepsilon,k,d,\varkappa_k}h^{-2}\|u\|^2 . \] Fix $\varkappa_k$ sufficiently small for the ordinary tangent principal part. Use $\varepsilon=1/4$ for the real part and $\varepsilon=\varkappa_k$ for the imaginary part, then decrease $r_k$ and take $h\le r_k$ sufficiently small. Since $(r_k+h)^2\le4r_k^2\le4r_k$, this proves \eqref{e:cons-sectorial}, with the constants chosen after the coefficient metric and uniformly as the tube is decreased. In particular, its normal-energy loss can be absorbed by the flat normal Laplacian after a further decrease of $r_k$. A body weight $v^p$ adds first- and zeroth-order terms with bounded coefficients, since $\log v$ has bounded derivatives along the saddle manifold; the same estimates apply. Fixed degree weights $d_j$ change only the constants in odd multiplication, odd differentiation, and their adjoints, which proves the assertion about those metrics. \end{proof} \subsection{Gaussian separation} To identify the actual heat with the Dirichlet heat, we need a lower bound for the real part of the Gaussian exponent between a point near the core of the tube and a point near its boundary. The bound must hold uniformly along the noncompact directions, and it must allow complex normal coordinates of small height, since \zcref{s:end} moves the normal coordinates to complex graphs. Recall from \zcref{d:tube-cycle} that the adapted real normal space of root $\sigma$ is $e^{-\iu \theta_\sigma}$ times the real form, which consists of the Hermitian matrices in the BB block, of $\iu$ times the Hermitian matrices in the FF block, and of free odd entries. Here $\theta_\sigma = \frac12 \arg c_\sigma$ is the phase of \eqref{e:tube-c}, and $|\theta_\sigma| \le C|E|$. For the body of a normal coordinate $U$ we write \[ U = e^{-\iu \theta_\sigma}(u + \iu w) \] on each root block, with $u$ and $w$ in the real form, and we call $u$ the real part and $w$ the height of $U$. \begin{lemma}\label{l:cons-separation} There exist universal constants $r_{\rm sep}, E_{\rm sep}, c > 0$ such that the following holds for $|E| \le E_{\rm sep}$. Let $x, x'$ be points of the real saddle manifold, and let $U, U'$ be normal coordinates with zero odd part, with real parts $u, u'$ and heights $w, w'$, such that $|U|, |U'| \le r_{\rm sep}$. Then \begin{equation}\label{e:cons-separation} \Re \Str\bigl(R(x, U) - R(x', U')\bigr)^2 \ge \tfrac14 \bigl(|u| - |u'|\bigr)^2 - 4 \bigl(|w|^2 + |w'|^2\bigr) + c \, s_E^2 \, \mathfrak d(x, x') , \end{equation} where $\mathfrak d(x, x') \ge 0$ is a function that vanishes only for $x = x'$ and grows exponentially in the distance between the BB parts of $x$ and $x'$. \end{lemma} \begin{proof} The scalar part $\kappa_E \Id$ of $R_*$ cancels in the difference, and we may replace $R_*$ by $-\iu s_E \Sigma$. It suffices to treat the two blocks separately, since the body of $\Str$ is the trace of the BB block minus the trace of the FF block. We begin with the BB block, which carries the noncompact directions. Let $g, g' \in U(k, k)$ be representatives of the BB parts of $x$ and $x'$, and write $S = R_* + U$ and $S' = R_* + U'$ for the BB blocks, so that $R(x, U)^{\mathrm{BB}} = g S g^{-1}$. Conjugation by $g^{-1}$ does not change the trace, and we have \[ \Tr\bigl(g S g^{-1} - g' S' g'^{-1}\bigr)^2 = \Tr\bigl(S - g'' S' g''^{-1}\bigr)^2 , \qquad g'' = g^{-1} g' . \] We use the Cartan decomposition $g'' = k_1 e^A k_2$, with $k_1, k_2$ in the stabilizer $U(k) \times U(k)$ and $A$ Hermitian and off-diagonal. Replacing $S$ by $k_1^{-1} S k_1$ and $S'$ by $k_2 S' k_2^{-1}$ preserves $|u|$, $|u'|$, $|w|$, $|w'|$ and the block structure, and we may assume that $g'' = e^A$. Since $\Tr(S'^2) = \Tr((e^A S' e^{-A})^2)$, and since in $\Tr(S \, e^{\operatorname{ad}_A} S')$ only the even powers of $\operatorname{ad}_A$ contribute (odd powers map block-diagonal matrices to off-diagonal ones), we have \begin{equation}\label{e:cons-sep-expansion} \Tr\bigl(S - e^A S' e^{-A}\bigr)^2 = \Tr(S - S')^2 - 2 \Tr\bigl(S (\cosh(\operatorname{ad}_A) - 1) S'\bigr) . \end{equation} We now compute the main term. On the BB block we have $R_* = -\iu s_E \Lambda$. Since $A \Lambda = -\Lambda A$, we have $\operatorname{ad}_A^2 \Lambda = 4 \Lambda A^2$, and by induction $(\cosh(\operatorname{ad}_A) - 1)\Lambda = \Lambda(\cosh(2A) - \Id)$. Therefore \[ - 2 \Tr\bigl(R_* (\cosh(\operatorname{ad}_A) - 1) R_*\bigr) = 2 s_E^2 \Tr\bigl(\cosh(2A) - \Id\bigr) = 2 s_E^2 \, \mathfrak d_B(A) , \] where we define \[ \mathfrak d_B(A) = \Tr\bigl(\cosh(2A) - \Id\bigr) . \] The quantity $\mathfrak d_B(A)$ is real and nonnegative, since $A$ is Hermitian, and it vanishes only for $A = 0$ and grows exponentially in $\|A\|$. Next, we bound the cross terms in \eqref{e:cons-sep-expansion}. Using that the operator $\operatorname{ad}_A^2$ is symmetric for the bilinear form $\Tr(Y Y')$, we find that the two terms that contain one factor $R_*$ are $2 \iu s_E \Tr(\Lambda(\cosh(2A) - \Id)U')$ and the same expression with $U$. The matrix $\Lambda(\cosh(2A) - \Id)$ is a positive matrix times a unitary one, with trace norm $\mathfrak d_B(A)$. It follows that \[ \begin{aligned} &\bigl|2 \iu s_E \Tr\bigl(\Lambda(\cosh(2A) - \Id)U'\bigr)\bigr| + \bigl|2 \iu s_E \Tr\bigl(\Lambda(\cosh(2A) - \Id)U\bigr)\bigr|\\ &\qquad \le 2 s_E \bigl(\|U\| + \|U'\|\bigr) \mathfrak d_B(A) . \end{aligned} \] For the last term, we use that as an operator on Hilbert--Schmidt matrices, \[ \|\cosh(\operatorname{ad}_A) - 1\| \le \cosh(2\|A\|) - 1 \le \mathfrak d_B(A) , \] which bounds that term by $2 \|U\| \|U'\| \mathfrak d_B(A)$. For $r_{\rm sep} \le s_0/8$, where $s_0 = \inf_{|E| \le 1/2} s_E$, the cross terms are at most $s_E^2 \mathfrak d_B(A)$ in absolute value. It remains to bound $\Re \Tr(S - S')^2 = \Re \Tr(U - U')^2$. On a root block of type $\sigma$, we have \[ \begin{aligned} \Re \Tr\bigl(e^{-\iu \theta_\sigma}((u - u') + \iu (w - w'))\bigr)^2 &= \cos(2\theta_\sigma) \bigl(\|u - u'\|^2 - \|w - w'\|^2\bigr)\\ &\quad + 2 \sin(2\theta_\sigma) \Tr\bigl((u - u')(w - w')\bigr)\\ &\ge \tfrac12 \|u - u'\|^2 - 2 \|w - w'\|^2 , \end{aligned} \] where the inequality holds once $|\theta_\sigma| \le 1/10$, which follows from $|\theta_\sigma| \le C|E|$ for $E_{\rm sep}$ small. Since $\|u - u'\| \ge |\,|u| - |u'|\,|$, the BB block satisfies \eqref{e:cons-separation} with $\mathfrak d = \mathfrak d_B$, with the BB parts of $u$, $u'$, $w$, and $w'$. The FF block is compact, and the argument is the same. The points of the FF block are $\iu$ times Hermitian matrices up to the shift by $\kappa_E$, and on the FF block $-\Tr$ is the positive part of $\Str$. In the Cartan decomposition of $U(2k)$ relative to the FF stabilizer, $A$ is anti-Hermitian, $A = \iu B$ with $B$ Hermitian, and $\cosh(2A) = \cos(2B)$. The main term becomes $2 s_E^2 \Tr(\Id - \cos(2B)) \ge 0$. The Cartan decomposition can be chosen with $\|B\| \le \pi/2$, and the cross terms are then bounded by a small multiple of the main term, as before. Adding the BB and FF blocks, and using $(x + y)^2 \le 2x^2 + 2y^2$ to combine the norms of the two blocks, we conclude that \eqref{e:cons-separation} holds. \end{proof} \subsection{Identification with the Dirichlet heat} We now compare one link of the actual kernel with the Dirichlet evolution of the same generator. We state the comparison on moving normal graphs, which is the form used in \zcref{s:end}; the static case is the case of zero velocity. Let $\psi$ be a smooth nonincreasing function on $[0, \infty)$, equal to one on $[0, 1]$ and to zero on $[4, \infty)$, let $\bar r > 0$, and set $\psi_{\bar r}(U) = \psi(\|U\|_{\rm HS}^2/\bar r^2)$. For a real normal direction $b$ and $\lambda \in [0, 1]$, we define the \emph{compact normal graph} \begin{equation}\label{e:cons-graph} U_\lambda = U + \iu \lambda \psi_{\bar r}(U) b , \qquad \rho_\lambda = \det \frac{\partial U_\lambda}{\partial U} . \end{equation} It moves only the body of the normal coordinate and is the identity outside a compact set; we evaluate functions of the normal variable at the complex points $U_\lambda$ instead of $U$. This is an analytic distortion of the kind used by Hunziker \cite{hunziker1986distortion}. The derivative $D\psi_{\bar r}$ has Hilbert--Schmidt operator norm at most $C/\bar r$, with $C$ independent of the matrix dimension, and $\partial U_\lambda/\partial U - \Id$ has rank one and norm at most $C|b|/\bar r$. Fix a link $[t_0, t_0 + \tau_{\rm phys}]$ and numbers $\lambda_0, \lambda_1 \in [0, 1]$, and let $\lambda(t)$ interpolate linearly between them over the link. Let $\mathcal T_{\rm in} \subset \mathcal T_{\rm out}$ be the tubes of normal radii $r_{\rm in} < r_{\rm out}$, and suppose that $r_{\rm in} \le 3 r_{\rm out}/4$ and $16 \bar r \le r_{\rm out}$, $4\bar r \le r_{\rm in}$, so that the graphs are the identity outside $\mathcal T_{\rm in}$. Let $y = (g_0, \xi, U)$ denote a point of the group chart and $\dd y$ the chart measure, and set \[ R_\lambda(y) = R\bigl(g_0 \exp(\xi) K, U_\lambda\bigr) , \qquad (\varpi \rho^{1/2})_\lambda(y) = \varpi(U_\lambda) \, \rho_\lambda(U)^{1/2} \] for the corresponding point of the complexified tube on the graph of parameter $\lambda$ and for the half-density of that graph. For a coefficient function $f$ supported in $\mathcal T_{\rm in}$, we define the \emph{pulled Gaussian evolution} \begin{equation}\label{e:cons-pulled} u(t, y) = \int k_{t - t_0}\bigl(R_{\lambda(t)}(y), R_{\lambda_0}(y')\bigr) \, (\varpi \rho^{1/2})_{\lambda(t)}(y) \, (\varpi \rho^{1/2})_{\lambda_0}(y') \, f(y') \dd y' , \qquad t > t_0 . \end{equation} To specify the half-density convention, write $\Phi_\lambda(U)=U_\lambda$ and define the graph pull on raw half densities by \[ (\mathcal P_\lambda F)(g_0,\xi,U) =\rho_\lambda(U)^{1/2}F(g_0,\xi,\Phi_\lambda(U)). \] For $\lambda_0>0$, the input in \eqref{e:cons-pulled} has the form $f=\mathcal P_{\lambda_0}F$, where $F$ is holomorphic in the normal variables wherever the graph is deformed. Thus $f/\rho_{\lambda_0}^{1/2}$ is the raw graph value; the second half-Jacobian needed for the graph measure is supplied by $f$. For $\lambda_0=0$ there is no holomorphy requirement on the input. This is the convention used in \zcref{l:end-energy}. The factor $\varpi$ has already been incorporated into the raw half density and into the generator $L_{\rm raw}=\Delta_{\rm nor}+L_H$ of \eqref{e:cons-decomposition}. We let $v(t)$ solve the Dirichlet problem on $\mathcal T_{\rm out}$ for the pulled operator, \begin{equation}\label{e:cons-moving} \begin{aligned} \partial_t v&=(L_t+V_t)v,\qquad v(t_0)=f,\\ L_t&=\mathcal P_{\lambda(t)}L_{\rm raw}\mathcal P_{\lambda(t)}^{-1},\\ V_t&=\mathsf v_t\cdot\nabla_U+\tfrac12\operatorname{div}_U\mathsf v_t, \qquad \mathsf v_t=(D\Phi_{\lambda(t)})^{-1}\partial_t\Phi_{\lambda(t)}. \end{aligned} \end{equation} Here the expression for $L_t$ means the differential operator obtained by the chain rule, including conjugation by $\rho_{\lambda(t)}^{1/2}$, and is extended coefficientwise to the real graph coordinate $U$. \begin{lemma}\label{l:cons-identification} At fixed rank $k$, let $2 r_{\rm out} \le \min(r_k, r_{\rm sep})$ and $|E| \le E_{\rm sep}$. There exist universal constants $c_1, c > 0$ such that the following holds for $|b| \le c_1 \bar r$. The forms of $L_t + V_t$ are sectorial on $\mathcal T_{\rm out}$, uniformly in $t$, and the Dirichlet problem \eqref{e:cons-moving} has a unique weak solution. For all $f$ as above and $t_0 \le t \le t_0 + \tau_{\rm phys}$, \begin{equation}\label{e:cons-identification} \|\1_{\mathcal T_{\rm in}} (u(t) - v(t))\| \le C_k W^{C_k} e^{- c \, r_{\rm out}^2 W^2} \|f\| . \end{equation} The same holds in every fixed polynomial body weight, with $C_k$ depending on the weight. \end{lemma} \begin{proof} We first derive the transport of a half density. Set $q(U)=b\cdot\nabla_U\psi_{\bar r}(U)$, which is real. The rank-one determinant and inverse identities give \begin{equation}\label{e:cons-graph-velocity} \rho_\lambda=1+\iu\lambda q, \qquad \mathsf v_t=\frac{\iu\lambda'(t)\psi_{\bar r}b}{1+\iu\lambda(t)q}, \qquad \|\mathsf v_t\|_\infty\le|\lambda'(t)b|. \end{equation} In particular $\Re\rho_\lambda=1$, so its square root is the branch continuous from $\rho_0^{1/2}=1$. The Jacobian identity $\partial_t\rho_\lambda=\operatorname{div}_U(\rho_\lambda\mathsf v_t)$ implies \[ \dot{\mathcal P}_\lambda\mathcal P_\lambda^{-1} =\mathsf v_t\cdot\nabla_U+ \tfrac12\bigl(\partial_t\log\rho_\lambda -\mathsf v_t\cdot\nabla_U\log\rho_\lambda\bigr) =V_t. \] The second term in parentheses is required because the transport field also differentiates the half-Jacobian. The full ambient half-density pull is $\mathcal P_\lambda M_\varpi$; $M_\varpi$ is independent of $t$, so it adds no term to $\dot{\mathcal P}_\lambda\mathcal P_\lambda^{-1}$. For $s>0$, the integral of $k_s(S,\cdot)$ against the fixed input graph in \eqref{e:cons-pulled} is coefficientwise holomorphic in a sufficiently small complex neighborhood of each output point $S_0=R_{\lambda(t)}(y)$ of the tube. To see this, let $x$ be the saddle coordinate of $S_0$. By \zcref{l:cons-separation} and the smallness of the graph height, the real part of $\Str(S_0-R_{\lambda_0}(y'))^2$ is bounded below by $c\mathfrak d(x,x')-C$. A sufficiently small complex perturbation of $S_0$ preserves this bound with a smaller positive coefficient of $\mathfrak d$: its additional terms are bounded by $C|S-S_0|(1+\mathfrak d(x,x'))$. Every fixed derivative in $S$ or $s$, including the finite odd expansion, introduces only a polynomial in $\mathfrak d$ and is absorbed by this orbit decay. Since the input has bounded normal support and the tangent measure grows at most exponentially in distance, the resulting kernels are locally uniformly in $L^2$ in the input variable, also in each fixed polynomial body weight. Cauchy--Schwarz therefore permits differentiation under the integral for the allowed $L^2$ inputs, uniformly for $s$ in compact subsets of $(0,\infty)$. The integral solves $\partial_s=\Delta_S$ on these neighborhoods. Apply the preceding chain rule to this integral, first absorbing $\varpi$ into its raw half density. We obtain $\partial_tu=(L_t+V_t)u$ classically on $\mathcal T_{\rm out}$. We will also use the exact velocity form. In an orthonormal coefficient frame, ordinary integration by parts gives, for compactly supported $f,g$, \begin{equation}\label{e:cons-velocity-form} \int\sum_I\overline{f_I}(V_tg)_I\dd y =\tfrac12\int\mathsf v_t\cdot\sum_I \bigl(\overline{f_I}\nabla_Ug_I -(\nabla_U\overline{f_I})g_I\bigr)\dd y. \end{equation} This identity extends to $H^1_0$ coefficient vectors. It is valid for the complex field $\mathsf v_t$ and any of our fixed positive coefficient metrics. A body weight commutes with $V_t$. By \eqref{e:cons-graph-velocity}, for every $\varepsilon>0$, \begin{equation}\label{e:cons-velocity-bound} |\langle u,V_tu\rangle| \le\varepsilon\mathcal E_{\rm nor}(u) +\frac{|\lambda'b|^2}{4\varepsilon}\|u\|^2. \end{equation} Thus differentiating the Jacobian introduces no separate loss in this energy estimate. We now bound $u$ near the boundary of $\mathcal T_{\rm out}$. We set $d = r_{\rm out} - r_{\rm in}$, and we note that $d \ge r_{\rm out}/4$, since $r_{\rm in} \le 3 r_{\rm out}/4$. Let $\tilde \chi$ be a smooth real function of $|U|$, equal to one where $|U| \le r_{\rm in} + d/3$ and to zero where $|U| \ge r_{\rm in} + 2d/3$, and let $\mathcal A$ be the collar where $\nabla \tilde \chi \ne 0$. For an output point in $\mathcal A$ and an input point in the support of $f$, the real parts of the normal coordinates satisfy $|u| - |u'| \ge d/3$. The output point has height zero, since the graph is the identity there, and the input point has height at most $|b| \le c_1 \bar r \le c_1 r_{\rm out}/16$. By \zcref{l:cons-separation}, the real part of the exponent $\Str(\cdot)^2/(4s)$ of $k_s$ is at least \[ \frac{1}{4s}\Bigl(\frac{d^2}{36} - \frac{c_1^2 r_{\rm out}^2}{64} + c s_E^2 \mathfrak d(x, x')\Bigr) \ge \frac{1}{4s}\Bigl(\frac{r_{\rm out}^2}{1200} + c s_E^2 \mathfrak d(x, x')\Bigr) \] for $c_1 \le 1/10$. The nilpotent part of the exponent is a polynomial in the odd variables whose coefficients are bounded by $C_k(1 + \mathfrak d)$. Its exponential is a finite sum, bounded by $C_k s^{-C_k}$ times a polynomial in $\mathfrak d$, and a fraction of $e^{-c s_E^2 \mathfrak d/(4s)}$ absorbs the polynomial. The same holds for one derivative of the kernel in the output variable. Once $W$ is large, the function $s \mapsto s^{-C_k} e^{-r_{\rm out}^2/(9600 s)}$ is increasing on $(0, \tau_{\rm phys}]$, and $\mathfrak d$ grows exponentially in the distance on the saddle manifold while the invariant measure grows at most exponentially. Combining these bounds with Schur's test, we obtain \begin{equation}\label{e:cons-collar} \sup_{t_0 < t \le t_0 + \tau_{\rm phys}} \bigl(\|\1_{\mathcal A} u(t)\| + \|\1_{\mathcal A} \nabla u(t)\|\bigr) \le C_k W^{C_k} e^{-r_{\rm out}^2 W^2/4800} \|f\| , \end{equation} where we used $\tau_{\rm phys} = (2W^2)^{-1}$. A body weight $v^p$ changes only the constant, since $v$ grows at most exponentially in the distance. Next, we show that the forms are sectorial. The ordinary part of $\Delta_{\rm nor}$ has constant coefficient matrix $D_E=\bigoplus_\sigma e^{2\iu\theta_\sigma}\Id$. If $J_\lambda=D\Phi_\lambda$, its principal coefficient on the graph is $J_\lambda^{-1}D_EJ_\lambda^{-T}$; this formula also allows the graph to mix root blocks. Its distance from $D_E$ is at most $C|b|/\bar r\le Cc_1$, so its form has real part at most $-\frac12\mathcal E_{\rm nor}(u)$ for $c_1$ and $|E|$ small. Differentiating $J_\lambda^{-1}$ and the half-Jacobian also gives first-order and zeroth-order coefficients. All derivatives required here are bounded at fixed $\bar r$, uniformly for $|b|\le c_1\bar r$, and their lower-order form costs are included in $C_k(\bar r)$ below. The horizontal part satisfies \eqref{e:cons-sectorial}, since the normal coordinates of the graph have norm at most $r_{\rm out} + |b| \le 2 r_{\rm out} \le r_k$, and its normal energy term $C_k r_k \mathcal E_{\rm nor}$ is at most $\frac14 \mathcal E_{\rm nor}$ once $r_k$ is small. The odd normal part of $\Delta_{\rm nor}$ is a finite matrix of norm $C_k h^{-2}$ in the standardized metric. Since $|\lambda'|\le\tau_{\rm phys}^{-1}$, \eqref{e:cons-velocity-bound} with $\varepsilon=1/8$ gives \[ |\langle u,V_tu\rangle| \le\tfrac18\mathcal E_{\rm nor}(u) +C\tau_{\rm phys}^{-2}|b|^2\|u\|^2. \] The forms are therefore bounded on the form domain of the Dirichlet problem, which is $H^1_0(\mathcal T_{\rm out})$ with coefficients in the finite exterior space, and satisfy a G\aa rding inequality with constant \[ C_{\rm G} \le C_k(\bar r) h^{-2} + C\tau_{\rm phys}^{-2}|b|^2, \qquad C_{\rm G} \tau_{\rm phys} \le C_k(\bar r) + C W^2 |b|^2 , \] where the static lower-order graph terms are absorbed into $C_k(\bar r)$. The form identity \eqref{e:cons-velocity-form} also bounds the sesquilinear velocity form on $H^1_0$, without a derivative-of-Jacobian term. By the theorem of Lions on evolution equations for time-dependent forms \cite{lions1961equations}, the Dirichlet problem \eqref{e:cons-moving} has a unique weak solution, and its evolution operators $\mathcal D(t, s)$ satisfy the energy inequality $\|\mathcal D(t, s)\| \le e^{C_{\rm G} (t - s)}$. We now compare, first for smooth $f$. The function $w = \tilde \chi u$ lies in the form domain and satisfies \[ \partial_t w = (L_t + V_t) w - [L_t + V_t, \tilde \chi] u , \] where the commutator is a first-order operator with coefficients supported in $\mathcal A$. The graph and its velocity are the identity and zero, respectively, on this outer collar, so $[V_t,\tilde\chi]=0$ there. We next check that $w(t) \to f$ as $t \downarrow t_0$. Let $\mathbf Q_\lambda = D R_\lambda^T \, D R_\lambda$ denote the pullback of the bilinear form $\Str$ by the parametrization $y \mapsto R_\lambda(y)$ of the graph of the tube, and let $s = t - t_0$. In the integral \eqref{e:cons-pulled}, the exponent is $-(y - y')^T \mathbf Q_\lambda(y) (y - y')/(4s)$ up to cubic terms in $y - y'$. The real part of $\mathbf Q_\lambda$ is positive definite near the real tube, and the Laplace method shows that the integral tends to $f(y) \, (\varpi^2 \rho)(y) \Ber(\mathbf Q_\lambda(y))^{-1/2}$, where the branch is fixed by continuity from the real tube and the constants of the Gaussian are the normalized ambient ones, since the flat Gaussian has integral one. By \zcref{l:tube-volume}, the holomorphic Jacobian of the parametrization in the chart satisfies \[ \Ber(\mathbf Q_\lambda)^{1/2} = \Ber(D R_\lambda) = \varpi^2 \rho_\lambda , \] and the limit is $f(y)$, in $L^2$ for smooth $f$. By uniqueness of weak solutions, Duhamel's formula holds, \[ w(t) - v(t) = - \int_{t_0}^t \mathcal D(t, s) [L_s + V_s, \tilde \chi] u(s) \dd s . \] Using \eqref{e:cons-collar} and the energy inequality, we obtain \[ \|w(t) - v(t)\| \le C_k W^{C_k} e^{C W^2 |b|^2 - r_{\rm out}^2 W^2/4800} \|f\| . \] The exponent is at most $-r_{\rm out}^2 W^2/9600$ once $c_1$ is small enough, since $|b| \le c_1 r_{\rm out}/16$. The last display implies \eqref{e:cons-identification} for smooth $f$, since $w = u$ on $\mathcal T_{\rm in}$. For general $f$, we approximate by smooth functions $f_n$. The evolutions $v$ depend continuously on $f$, by the energy inequality. Applied to $f_n$, the bound \eqref{e:cons-identification} implies that \[ \|\1_{\mathcal T_{\rm in}} u_n(t)\| \le \|v_n(t)\| + C_k W^{C_k} e^{-c r_{\rm out}^2 W^2} \|f_n\| , \] and the operators $f \mapsto \1_{\mathcal T_{\rm in}} u(t)$ are therefore uniformly bounded on smooth functions and extend by continuity. This completes the proof of \eqref{e:cons-identification} for all $f$. \end{proof} \subsection{The heat over one link} We now prove the one-link bound for the gauged generator used in the consistency calculation. In the symmetric normal gauge of \zcref{s:tube}, vectors carry the Wick metric of \zcref{s:tube-oscillator}, and the Gaussian gauge $\gamma_h$ relates it to the flat metric. We define the gauged generator \begin{equation}\label{e:cons-gauged} \mathcal L_h = \gamma_h^{-1} \bigl(\Delta_{\rm nor} + L_H(U)\bigr) \gamma_h = L_{N, h} + L_{H, h} , \qquad L_{N, h} = \gamma_h^{-1} \Delta_{\rm nor} \gamma_h , \quad L_{H, h} = L_H(U) . \end{equation} The second identity holds because all normal derivatives of $L_H(U)$ come from the normal charges, which annihilate the normal class function $\gamma_h$ of \zcref{l:tube-class}. We let $L_{H, 0}$ denote the operator $L_H(0)$ acting on tangent functions. On vectors that are invariant under the normal charges, all normal derivatives of $L_H(0)$ vanish, and by \zcref{l:cons-sectorial}\ref{i:cons-sect-chart}, the ordinary second-order part of $L_{H, 0}$ is given by the inverse of the orbit metric $4 s_E^2 \Str(Y Y')$ at the saddle manifold. In the balanced signature, \zcref{l:tube-casimir} shows that $L_{H, 0} = -(2 s_E^2)^{-1} \mathsf C_k$ as operators on superfunctions on $\mathcal M_k$, odd coefficients included. With the half-weight $w_0^{1/2}$ and the invariant cutoff $\chi = \chi_{\rm inv}$ of \zcref{s:tube-cutoff}, define the Dirichlet versions of the unsourced and frozen symmetric steps by \begin{equation}\label{e:cons-steps} T_h^{\rm D} = \chi w_0^{1/2} \, e^{\tau_{\rm phys} \mathcal L_h} \, w_0^{1/2} \chi , \qquad \mathbf F_h^{\rm D} = \chi w_0^{1/2} \, e^{\tau_{\rm phys} L_{N, h}} \, w_0^{1/2} \chi . \end{equation} Here $e^{t \mathcal L_h}$ and $e^{t L_{N, h}}$ denote the Dirichlet evolutions on a fixed tube slightly larger than the support of $\chi$. We retain $T_h$ and $\mathbf F_h$ for the actual Gaussian steps of \eqref{e:tube-table}. Write $B_h(t)$ for the free frozen normal heat on the whole normal space, with the same differential expression $L_{N,h}$. Thus \[ \mathbf F_h=\chi w_0^{1/2} B_h(\tau_{\rm phys})w_0^{1/2}\chi . \] The fixed gap between $\operatorname{supp}\chi$ and the Dirichlet boundary gives \[ \|\mathbf F_h-\mathbf F_h^{\rm D}\|+\|T_h-T_h^{\rm D}\| \le C_k W^{C_k}e^{-cW^2+CW}, \] in every fixed body weight. For the normal step this is the Gaussian boundary estimate; for the full step it is \zcref{l:cons-identification} with the static graph $\lambda_0=\lambda_1=0$, applied between the cutoffs. The factor $e^{CW}$ bounds the Gaussian gauges on the fixed tube. These errors are smaller than every power of $h$. We keep them in the consistency calculation so that the exact fixed-state identity for the original $\mathbf F_h$ remains available. \begin{lemma}\label{l:cons-short-heat} At fixed rank and fixed cutoff radius $r \le r_k$, for every body weight $p$ there exists $C_p$ such that, in the Wick metric with body weight $v^p$, \begin{equation}\label{e:cons-short-heat} \sup_{0 \le t \le \tau_{\rm phys}} \|e^{t \mathcal L_h}\| \le \exp\bigl(C_p \tau_{\rm phys} h^{-2}\bigr) \le 2 \end{equation} for all sufficiently small $h$. The same base-norm bound holds for the Dirichlet frozen normal heat $e^{t L_{N,h}}$ and for the free frozen normal heat $B_h(t)$. The free heat also satisfies \begin{equation}\label{e:cons-free-graph} \|B_h(t)\|_{\mathcal G^m\to\mathcal G^m}\le e^{C_m t h^{-2}}, \qquad 0\le t\le\tau_{\rm phys}, \end{equation} for each fixed prescribed graph order $m$. Consequently, for every vector $g$ in the domain of $\mathcal L_h$ and $0 \le t \le \tau_{\rm phys}$, \begin{equation}\label{e:cons-generator-bound} \|(e^{t \mathcal L_h} - \Id) g\| \le 2 t \, \|\mathcal L_h g\| . \end{equation} \end{lemma} \begin{proof} We transport the positive Wick metric through the full Gaussian gauge. For one standardized odd pair, write $F=a+b\alpha+c\beta+d\alpha\beta$. By \zcref{d:tube-wick}, \[ \|F\|_{\rm W}^2=|a-d|^2+|b|^2+|c|^2+|d|^2. \] The odd Gaussian factor is $\gamma_F=1-\alpha\beta/2$, so \[ \|\gamma_F^{-1}F\|_{\rm W}^2 =\tfrac12|a|^2+|b|^2+|c|^2+2|d|^2. \] For $M$ odd pairs the tensor product of these metrics is the flat positive coefficient metric \begin{equation}\label{e:cons-raw-metric} |F|_*^2=\sum_I2^{|I|-M}|F_I|^2 \end{equation} in standardized odd normal monomials; the tangent degree metric is unchanged. Multiplication by $\gamma_h$ is therefore unitary from the positive Wick space to flat ordinary $L^2$ with metric \eqref{e:cons-raw-metric}, after a common scalar normalization of the ordinary measure. It also preserves the Dirichlet boundary condition and commutes with the body weight $v^p$. Thus $\mathcal L_h$ is unitarily equivalent to $\Delta_{\rm nor}+L_H(U)$ in this flat positive metric. We apply the degree-weighted version of \zcref{l:cons-sectorial}; this uses its coefficient and adjoint estimates directly in \eqref{e:cons-raw-metric}, rather than transferring a numerical-range bound by equivalence of norms. The ordinary part of $\Delta_{\rm nor}$ has form bounded above by $-c_0\mathcal E_{\rm nor}$, since its root coefficients are $e^{2\iu\theta_\sigma}$ with $\cos 2\theta_\sigma$ bounded below. Its odd part is a finite matrix of norm $C_kh^{-2}$. Fix a common coefficient metric and then decrease the fixed tube radius using the unweighted estimates, so that the normal-energy loss in \zcref{l:cons-sectorial} is at most $c_0\mathcal E_{\rm nor}/2$. Conjugating by $v^p$ adds only first-order and zeroth-order terms, since $\log v$ has bounded tangent derivatives and is independent of $U$. By \eqref{e:cons-charge-adjoint} and Young's inequality, these terms are bounded by any fixed small fraction of the two gradient energies plus $C_ph^{-2}\|F\|_*^2$. Thus the tube radius remains independent of $p$, and in every fixed body weight we obtain \[ \Re\langle F,(\Delta_{\rm nor}+L_H)F\rangle_* \le -c\bigl(\mathcal E_{\rm nor}(F)+\mathcal E_{\rm tan}(F)\bigr) +C_ph^{-2}\|F\|_*^2. \] Unitary transport and the Dirichlet energy inequality give $\|e^{t\mathcal L_h}\|\le e^{C_pth^{-2}}$, with no norm-equivalence prefactor. Omitting $L_H$ gives the same base estimate for both frozen normal evolutions, on the tube and on the whole normal space. Since $\tau_{\rm phys}=h^4/2$, this proves \eqref{e:cons-short-heat}. For the free normal heat on $\mathcal G^m$, work first on the whole-space Schwartz core. The generator is $h^{-2}$ times a quadratic oscillator polynomial. Each monomial changes the number $n$ by at most two with amplitude $O(n+1)$. Consequently \[ \bigl\|[(\mathbf N+1)^m,L_{N,h}](\mathbf N+1)^{-m}\bigr\|\le C_mh^{-2}. \] Conjugating by $(\mathbf N+1)^m$ adds this bounded operator to the base generator. The base energy estimate, followed by closure, proves \eqref{e:cons-free-graph}. This graph-space argument concerns the free heat and uses no Dirichlet boundary integration. Finally, \eqref{e:cons-generator-bound} follows from \[ (e^{t \mathcal L_h} - \Id) g = \int_0^t e^{\sigma \mathcal L_h} \mathcal L_h g \dd \sigma \] and \eqref{e:cons-short-heat}. \end{proof} The bound \eqref{e:cons-short-heat} is of order one for a single link and cannot be iterated over the $h^{-4}$ links of the chain. The reason is that the generator of the heat in the finitely many odd normal coordinates is nilpotent; \zcref{r:stab-fock} explains why no metric makes the gauged generator dissipative. The iterable bound \ref{i:cons-stability} comes instead from the kernel of the step. \section{The tangent generator}\label{s:cons-tangent} We return to the balanced signature $p=k$ for the remainder of this chapter. In this section we identify the generator that the adapted chain produces on the tangent variables. The product limit of \zcref{t:cons-limit} needs this identification. The consistency estimate of \zcref{s:cons-consistency} expresses one adapted step on the invariant normal line through the horizontal operator $L_{H, 0}$ and the source potential, combined into the operator $L^{\rm tan}_h$ below. \zcref[S]{p:sigma-regular} constructs the limit semigroup once $L^{\rm tan}_h$ is known to be a multiple of the generator of the sigma model. The identification rests on \zcref{l:tube-casimir}, which shows that $L_{H, 0} = -(2 s_E^2)^{-1} \mathsf C_k$. \zcref[S]{l:asm-time} below combines this identity with the time $\tau_{\rm phys}$ of one link and with the source factor. As a consequence, $j$ steps of the chain run the semigroup of the sigma model for the time $j/N$. Recall the generator of the sigma model from \eqref{e:sigma-generator}, \begin{equation}\label{e:asm-generator} \mathsf G_{X, \Gamma} = X \mathsf C_k + \pi \Str(Q \Gamma) , \end{equation} where $\mathsf C_k$ is the Casimir operator of \zcref{d:sigma-casimir}, $Q$ is the field of \zcref{d:sigma-target}, $\Gamma$ is the source supermatrix, and the second term acts by multiplication. Recall also the operator $L_{H, 0}$, defined after \eqref{e:cons-gauged} as $L_H(0)$ acting on tangent functions, and the source potential $V_h$, defined by $s_\Gamma = e^{-\delta_W V_h}$ on the tube. We let $V_0$ denote the value of $V_h$ at $U = 0$, a function of the tangent variables that acts on tangent functions by multiplication, and we set \[ L^{\rm tan}_h = \frac{\tau_{\rm phys}}{\delta_W} L_{H, 0} - V_0 = \tfrac12 L_{H, 0} - V_0 . \] We will show in \zcref{p:cons-consistency} that, for compactly supported smooth tangent functions $f$, \[ A_h (\Omega_h \otimes f) = \Omega_h \otimes f + \delta_W \, \Omega_h \otimes L^{\rm tan}_h f + O(\delta_W h) \|f\|_{p', m} , \] where $\|f\|_{p', m}$ is the tangent norm of \zcref{s:cons-consistency}. In this expansion, the term $\tau_{\rm phys} L_{H, 0}$ comes from one heat link, which acts on $\Omega_h \otimes f$ as $\Omega_h \otimes e^{\tau_{\rm phys} L_{H, 0}} f$ up to the error, and the term $\delta_W V_0$ comes from the source factor $e^{-\delta_W V_0}$ of one onsite weight on the saddle manifold. \begin{lemma}\label{l:asm-time} In the balanced signature $p=k$, we have \begin{equation}\label{e:asm-time} N \delta_W L^{\rm tan}_h = - \mathsf G_{X_N(E), \Gamma} . \end{equation} Consequently, $j$ steps of the adapted chain correspond to the macroscopic time $j/N$ of the semigroup $e^{-t \mathsf G_{X_N(E), \Gamma}}$. \end{lemma} \begin{proof} We begin with the tangent part. As in \zcref{s:tube-operator}, we identify the tangent functions with the superfunctions on $\mathcal M_k$ through $Q$, so that $R(x, 0) = \kappa_E \Id - \iu s_E Q$ on the balanced saddle manifold. By \zcref{l:tube-casimir}, applied to $L_{H, 0} = L_H(0)$ on these functions, and since $\tau_{\rm phys} = (2 W^2)^{-1}$, we have \begin{equation}\label{e:cons-casimir-multiple} L_{H, 0} = -\frac{1}{2 s_E^2} \, \mathsf C_k , \qquad \tau_{\rm phys} L_{H, 0} = -\frac{1}{4 s_E^2 W^2} \, \mathsf C_k . \end{equation} Using $4 s_E^2 = 4 - E^2$, we obtain \[ N \tau_{\rm phys} L_{H, 0} = - \frac{N}{(4 - E^2) W^2} \, \mathsf C_k = - X_N(E) \mathsf C_k , \] where in the last step we used the definition of $X_N(E)$ in \eqref{e:scaling}. We next compute the source term. Recall from \eqref{e:rep-weight} that one onsite weight carries the source factor $e^{-\iu \eta_0 \Str(S \Gamma)}$, where $\eta_0 = a_N^{-1}$ by \zcref{d:rep-sources}. On the saddle manifold we have $S = \kappa_E \Id - \iu s_E Q$, and $\Str \Gamma = 0$, since $\sum_j u_j = 0$ for the source supermatrix $\Gamma = \Gamma(\mathbf u, \mathbf b)$. Therefore \[ \begin{gathered} \delta_W V_0 = \iu \eta_0 \Str\bigl( (\kappa_E \Id - \iu s_E Q) \Gamma \bigr) = \frac{s_E}{a_N} \Str(Q \Gamma) , \\ N \delta_W V_0 = \frac{N s_E}{a_N} \Str(Q \Gamma) = \pi \Str(Q \Gamma) , \end{gathered} \] where we used $a_N = N \rho(E)$ and $\pi \rho(E) = s_E$. Combining the two computations with $N \delta_W L^{\rm tan}_h = N \tau_{\rm phys} L_{H, 0} - N \delta_W V_0$ and \eqref{e:asm-generator}, we obtain \eqref{e:asm-time}. \end{proof} \begin{remark}\label{r:cons-link-variance} The constant in \eqref{e:cons-casimir-multiple} can also be read off, heuristically, from the variance of a single link, which explains the time $(4 s_E^2 W^2)^{-1}$ of \zcref{s:sigma-heuristic}. In one bosonic retarded--advanced plane, write $Q = e^{2Y} \Lambda$ with $Y = \bigl(\begin{smallmatrix} 0 & z \\ \bar z & 0 \end{smallmatrix}\bigr)$ and $z \in \C$. By \eqref{e:sigma-body-metric}, the real and imaginary parts of $2z$ are orthonormal coordinates at $\Lambda$. To first order, the saddle point $\kappa_E \Id - \iu s_E Q$ moves by $-2 \iu s_E Y \Lambda$, and since $\Lambda Y \Lambda = -Y$, the link factor of \zcref{d:rep-chain} restricted to this displacement is \[ \exp\Bigl( -\frac{W^2}{2} \Str\bigl( (2 \iu s_E Y \Lambda)^2 \bigr) \Bigr) = \exp\bigl( -4 s_E^2 W^2 |z|^2 \bigr) . \] Each real component of $2z$ therefore has variance $(2 s_E^2 W^2)^{-1}$ under one link. This is the variance $2t$ of the heat semigroup of $-\mathsf C_k$, whose body is $\Delta_{\mathcal M_k^0}$, at the time $t = (4 s_E^2 W^2)^{-1}$ of \eqref{e:cons-casimir-multiple}. \end{remark} \section{Consistency}\label{s:cons-consistency} We now prove consistency on the line of vectors $\iota_h f = \Omega_h \otimes f$. Since $\Omega_h$ is invariant under the normal charges by \zcref{l:cons-fixed-state}, the product $\Omega_h(U) f(g_0, \xi)$ is invariant under the body stabilizer in the group chart, and $\Omega_h \otimes f$ is a well-defined function on the tube. For $p' \in \R$ and an integer $m \ge 0$, we define the tangent norm \[ \|f\|_{p', m} = \sum_{j \le m} \ \sum_{a_1, \dots, a_j} \|v^{p'} \mathbf D_{a_1} \cdots \mathbf D_{a_j} f\| , \] where the $\mathbf D_a$ run over the left-invariant fields $X_a$ and the odd tangent derivatives. It is finite for compactly supported smooth tangent functions. We will need the following bounds for the normal heat applied to the fixed state. Choose a second invariant normal cutoff $\rho$ equal to one on a neighborhood of $\operatorname{supp}\chi$, with support strictly inside the enlarged Dirichlet tube. It includes its nilpotent terms, as do the cutoffs of \zcref{s:tube-cutoff}. We localize the free heat with $\rho$ so that normal-charge invariance and oscillator graph norms are retained, while every vector used below has compact Dirichlet support. \begin{lemma}\label{l:cons-inner-normal} For $0 \le s \le \tau_{\rm phys}$, set $g_h=w_0^{1/2}\chi\Omega_h$ and $\Omega_{h,s}=\rho B_h(s)g_h$. Then $\Omega_{h,s}$ is invariant under the normal charges, has compact support in the enlarged tube, and for each fixed prescribed graph order $m$, \begin{equation}\label{e:cons-inner-normal} \|U \Omega_{h, s}\|_{\mathcal G^m} \le C h , \qquad \|\Omega_{h, s} - \Omega_h\|_{\mathcal G^m} \le C h^2 . \end{equation} Here $U$ denotes an ordinary or odd unscaled normal coordinate. Moreover, with $L_{N,h}$ acting by its differential expression on these compactly supported vectors, \begin{equation}\label{e:cons-normal-defect} (\partial_s-L_{N,h})\Omega_{h,s}=R_{h,s}, \qquad R_{h,s}=-[L_{N,h},\rho]B_h(s)g_h, \qquad \sup_{0\le s\le\tau_{\rm phys}}\|R_{h,s}\|_{\mathcal G^m}=O(h^L) \end{equation} for every prescribed $m$ and $L$. \end{lemma} \begin{proof} The free normal heat, the half-weight $w_0^{1/2}$, and the invariant cutoffs commute with the normal charges. Since $\Omega_h$ is invariant, so is $\Omega_{h,s}$. For the second bound, recall that $w_0^{1/2}(0) = 1$ and that $w_0^{1/2}$ has a critical point at the origin, since the onsite weight $w_0$ has one at the saddle. Hence $w_0^{1/2} - 1 = O(|U|^2)$ on the tube, and since $U = O(h |Z|)$, \[ \|(w_0^{1/2} \chi - 1) \Omega_h\|_{\mathcal G^m} \le C \bigl\| |U|^2 \Omega_h\bigr\|_{\mathcal G^m} + \|(\chi - 1) \Omega_h\|_{\mathcal G^m} \le C h^2 \|\Omega_h\|_{\mathcal G^{m + 1}} + C h^{2} , \] where in the last step we used \eqref{e:cons-annuli} for the cutoff. The whole-space generator $L_{N,h}$ is bounded from $\mathcal G^{m+1}$ to $\mathcal G^m$ by $C_mh^{-2}$. By \eqref{e:cons-free-graph}, for sufficiently small $h$, \[ \|(B_h(s)-\Id)g\|_{\mathcal G^m} \le 2s\|L_{N,h}g\|_{\mathcal G^m} \le C_m s h^{-2}\|g\|_{\mathcal G^{m+1}}. \] The fixed physical cutoff $\rho$ is uniformly bounded on each prescribed $\mathcal G^m$: its standardized derivatives gain powers of $h$, and its nilpotent coefficients are finite cutoff derivatives. Since $\rho g_h=g_h$, we have $\Omega_{h,s}-\Omega_h=\rho(B_h(s)-\Id)g_h+(g_h-\Omega_h)$. The preceding bounds and \eqref{e:cons-state-bounds} therefore give the second estimate. For the first, each unscaled normal coordinate is $h$ times a standardized one, which is bounded from $\mathcal G^{m + 1}$ to $\mathcal G^m$. Using the second estimate with $m + 1$ in place of $m$, we obtain \[ \|U \Omega_{h, s}\|_{\mathcal G^m} \le C h \|\Omega_{h, s}\|_{\mathcal G^{m + 1}} \le C h . \] Differentiating $\rho B_h(s)g_h$ gives the defect identity. Its commutator is supported where $\rho$ varies, at a fixed positive normal distance from $\operatorname{supp}g_h\subset\operatorname{supp}\chi$. The differentiated free Gaussian kernel there is bounded by a fixed power of $h^{-1}$ times $e^{-c/h^4}$, uniformly for $0 0$, \[ \lim_{h \to 0} \ \sup_{0 \le n \delta_W \le T} \|A_h^n \iota_h f - \iota_h U_{n \delta_W} f\| = 0 , \] uniformly on compact sets of energies, sources, and aspect ratios. \end{theorem} \begin{proof} We apply \zcref{l:cons-product} with $\mathcal H = \mathcal H^{\rm tan}_p$, the weighted space $\mathcal H^{\rm tube}_p$ of \zcref{t:stab-main} as $\mathcal H_h$, $\delta_h = \delta_W$, the maps $\iota_h f = \Omega_h \otimes f$, and the core $\mathcal D$ of compactly supported smooth superfunctions. The maps $\iota_h$ are uniformly bounded, since $\|\Omega_h\| \le 1 + C h$ by \eqref{e:cons-state-bounds} and the coefficient norms of the odd tangent coordinates in the two frames are equivalent, with constants that do not depend on $h$. The power bound \eqref{e:cons-power} is the stability estimate \ref{i:cons-stability}. For $f \in \mathcal D$, the norm $\|f\|_{p', m}$ is finite, and \eqref{e:cons-target} implies \eqref{e:cons-hyp}, since its error term divided by $\delta_W$ is $O(h) \|f\|_{p', m}$. The conclusion of \zcref{l:cons-product} is the limit at fixed parameters. For the uniformity, we apply \zcref{c:cons-product-param}. The adapted spaces for different parameters are identified with one space through the chart $(g_0, \xi, Z)$, and their norms are uniformly equivalent on compact parameter sets, since $c_\sigma$, $\theta_\sigma$, and $\gamma_h$ depend continuously on $E$. The generators depend continuously on the parameters on the common core, and the semigroups obey a common exponential bound on compact parameter sets, by \zcref{p:sigma-regular} with constants that are uniform for $X$, $\eps$, and $\Gamma$ in compact sets. Finally, the stability and consistency estimates are uniform there. \end{proof} \section{The initial layer}\label{s:cons-layer} \zcref[S]{t:cons-limit} applies to vectors on the invariant normal line, but the endpoints of the chain are not of this form. In \zcref{s:assembly}, \zcref{l:asm-initial-layer} moves both endpoints onto the line in an initial layer of order $h^{-2} \log(1/h)$ steps, whose tangent time tends to zero. Its proof relaxes the frozen path with the bound \eqref{e:cons-relaxation} of \zcref{l:cons-fixed-state}. We record here a second property of $\mathbf F_h$, which concerns the left endpoint. The original frozen normal transfer $\mathbf F_h$ of \eqref{e:tube-table}, with its cutoffs, is its own analytic transpose for the analytic Berezin pairing in the normal variables (a transpose for this indefinite pairing, not a Hilbert adjoint for the positive Wick metric). Indeed, the Wick density $\gamma_h^2$ of the pairing cancels the two Gaussian gauges in its free-heat kernel. This leaves the kernel $w_0^{1/2}(U) K^\varpi_{\rm nor}(U, U') w_0^{1/2}(U')$ between identical cutoffs, which is symmetric under the interchange of its endpoints, with the Berezin orientation retained. The transposed chain, which carries the left endpoint $g_L$, therefore has the same frozen normal transfer as the chain itself, and $g_L$ relaxes under the same contraction \eqref{e:cons-relaxation} as $g_R$.