\chapter{The endpoints of the chain}\label{s:end}

In this section we attach the endpoints of the chain to the adapted tube. In the one-sided form of \zcref{s:asm-one-sided}, the chain integral is the pairing of the constant function $1$ with $T_\Gamma^{N - m} f_m$, where $f_m = T_\Gamma^m 1$ and $m = \lceil C_0 W \rceil$. We prove that the right block $f_m$ is, up to an error $C_k W^{A_k} e^{-\gamma_k W}$, a sum over the root signatures of local reference vectors on the tube (\zcref{t:end-entrance}), and that these references, and every family obtained from them by adapted steps, can be compared with the physical chain on the cycle with an exponentially small error (\zcref{c:end-residual}). The comparison itself is carried out in \zcref{s:assembly}. As explained in \zcref{s:ov-ends}, both errors must be of order $e^{-\gamma W}$ for some $\gamma > 0$, since every change of integration cycle at an endpoint is paired in \zcref{s:assembly} with a physical partial chain, whose norm we only know to be $e^{o(W)}$ by \zcref{t:hist-bound}.

\paragraph{The two steps.}
We proceed in two steps. In the first, the block $f_m$ is replaced by local references on the tube with an error $e^{-\gamma_k W}$. Its main ingredients are an Agmon estimate with a time-dependent weight for the physical chain on the cycle, which concentrates $f_m$ near the saddle manifold (\zcref{p:end-escape}), and a Davies--Gaffney estimate on the tube, which compares the physical chain with the references there (\zcref{t:end-entrance}). In the second step, a block of adapted steps with Gaussian weights that increase from $e^{-a|Z|^2}$ to $e^{a|Z|^2}$ turns the negative Gaussian weight of the references into a positive one (\zcref{c:end-ramp}). Both steps are over a block of $m = \lceil C_0 W \rceil$ steps, whose physical heat time $m \tau_{\rm phys} \approx C_0/(2W)$ is short. A term of order $C_k W$ in the generator changes norms by at most a factor $e^{C_k C_0}$ over the block, and plain weighted energy estimates in the raw realization suffice (\zcref{l:end-energy}). The joint normal--tangent analysis of the kernel in \zcref{s:stab} is needed only for the legs of macroscopic length. The point that needs care is the coefficient of $|b|^2$ per step on a complex normal graph of height $b$, which must not depend on the rank; \zcref{r:end-universal} explains why and checks it.

\paragraph{Conventions.}
Three conventions hold throughout this section. First, we assert analytic continuation only for the families generated from the endpoint (\zcref{d:end-generated}) and their finite jets, while the operator bounds hold on arbitrary inputs. Second, real cutoffs are evaluated only at real points. We call the set where a real cutoff takes values strictly between zero and one its \emph{collar}. Every complex deformation and every submean ball lies where the relevant real cutoffs equal one, and the collar of each cutoff is estimated separately on the real cycle. Third, adjoints are positive Hilbert adjoints for the fixed positive coefficient metric. They differ from the analytic Berezin transpose, which defines the left endpoint. Throughout, $h = W^{-1/2}$, $\tau = \tau_{\rm phys} = (2W^2)^{-1}$, and $\eta = \eps \eta_0 \asymp W^{-2}$ is the physical imaginary part. Constants with a subscript $k$ may depend on the rank, on finitely many prescribed jets and polynomial weights, and on every quantity chosen before them in \zcref{tab:end-constants}. The table and the next paragraphs also use three terms, which we define precisely before \zcref{c:end-ramp}. Recall that $Z$ is the standardized normal coordinate, defined after \eqref{e:tube-mu} by $U = hZ$. Roughly, a function $f$ on the tube has Gaussian margin $a$ if $e^{a|Z|^2} f$ has finite norm, a signed ramp is a sequence of such Gaussian weights that increases from $e^{-a|Z|^2}$ to $e^{a|Z|^2}$ in steps of size $O(h^2)$, and a prepared family is one with a positive Gaussian margin.


\paragraph{The order of the constants.}
\zcref{tab:end-constants} lists the constants and radii of this section in the order in which they are chosen. The constants in the first group are universal and are chosen before the rank $k$. At fixed $k$, we then choose the coefficient metric of \zcref{l:cons-sectorial}, the radii in the second group, and finally the lower threshold on $W$. The radii $R_k$, $d_{\rm col}$, $\rho_{\rm pd}$, $r_{\rm phys}$, and $r_1$ are chosen in the order listed, while $r_{\rm esc}$ is subject only to the constraints shown.

\begingroup
\footnotesize
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.13\linewidth}>{\raggedright\arraybackslash}p{0.56\linewidth}>{\raggedright\arraybackslash}p{0.23\linewidth}@{}}
\caption{The constants of \zcref{s:end}, in the order in which they are chosen. Here $s_0 = \inf_{|E| \le 1/2} s_E$.}
\label{tab:end-constants}\\
\toprule
Quantity & Role and constraint & Introduced in\\\midrule
\endfirsthead
\multicolumn{3}{c}{\tablename~\thetable\ (continued)}\\[4pt]
\toprule
Quantity & Role and constraint & Introduced in\\\midrule
\endhead
\midrule
\multicolumn{3}{r}{\emph{Continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\multicolumn{3}{@{}l}{\emph{Universal, chosen before $k$}}\\[2pt]
$a_0$, $v_0$ & largest amplitude of the signed weights $w_a = e^{a|Z|^2}$, and largest change per step in units of $h^2$ & \zcref{l:end-energy}\\
$c_1$, $C_1$ & largest relative graph height $|b|/R_k$, and coefficient of $|\beta_j|^2 + W^2 |\beta_j - \beta_{j - 1}|^2$ per step & \zcref{l:end-energy}\\
$a$ & positive Gaussian margin of prepared families, $0 < a < a_0$ & \zcref{c:end-ramp}\\
$\varrho$ & the shortest signed ramp has $\varrho W$ steps, $\varrho > 2a/v_0$ & \zcref{c:end-ramp}\\
$C_0$ & block length $m = \lceil C_0 W \rceil$, $C_0 \ge \max(1, \varrho)$ & this table\\
$C_n$ & coefficient of $C_0 W |b|^2$ in the exponent of the bound \eqref{e:end-normal-block} for a block of adapted steps on a graph of height $b$, $C_n = 3 C_1$ & \zcref{p:end-coupled}\\
$C_*$ & coefficient of $\|D\|^2 + \|D_F\|^2$ in the shifted onsite bound & \eqref{e:end-strip}\\
$C_{\rm geo}$ & canonical normal height & \zcref{l:end-height}\\
$a_{\rm esc}$ & amplitude of the escape ramp, chosen after $C_0$ & \zcref{p:end-escape}\\
$c_{\rm D}$ & rate of the Davies--Gaffney estimate & \zcref{l:end-davies}\\
$C_{\rm gau}$ & coefficient of $W|b|^2$ from conversion to the physical norm on a normal graph, $C_{\rm gau}=C_Z/4$ & \eqref{e:end-gauge-growth}\\
$c_{\rm sh}$ & rate of the Gaussian decay $e^{-c_{\rm sh} W r_1^2}$ on the support of $\nabla \chi_{\rm phys}$, $c_{\rm sh} = c_{\rm ann}/4$, proportional to $a$ & \zcref{l:end-shell}\\
$E_{\rm geom}$ & energy threshold & \eqref{e:end-Egeom}\\[2pt]
$E_{\rm sep}$, $r_{\rm sep}$ & energy and radius of the Gaussian separation & \zcref{l:cons-separation}\\[2pt]
\multicolumn{3}{@{}l}{\emph{Chosen at fixed rank $k$}}\\[2pt]
$r_k$, $\varkappa_k$ & radius and degree metric of the sectorial bound & \zcref{l:cons-sectorial}\\
$\omega$ & degree-metric parameter of the stability theorem, $\omega \le \min(\varkappa_k, \omega_0)$ with $\omega_0$ of \zcref{p:sigma-regular} & \zcref{t:stab-main}\\
$r$ & cutoff radius of the stability theorem, $r \le r_k$ & \zcref{t:stab-main}\\
graph orders & the finitely many prescribed orders $m$ & \zcref{s:cons-state}\\
$\mathcal U_0 \Subset \mathcal U_1 \Subset \mathcal U_2$ & complex damping balls about $\Lambda$; the closure of $\mathcal U_0$ lies in the open ball of radius $\rho/2$ about $\Lambda$, where $\rho$ is the radius of a ball about $\Lambda$ inside $\mathcal U_2$; the tube is then chosen so that conjugated dampings stay in $\mathcal U_1$ & \eqref{e:end-damping}\\
$\rho$, $\rho_{\rm out}$ & Hardy radii of the escape ramp, with $\rho$ as in the previous row and $\rho_{\rm out} = \rho/2$ & \zcref{p:end-escape}\\
$R_k$ & outer root tube, which carries the normal cutoffs, the normal graphs, and the Taylor remainders; $C_k R_k$ below fixed universal fractions; $R_k \le R_{\rm s}$; $C R_k \le c_0/2$ for the rate $c_0$ of the BB damping and the constant $C$ of the proof of \zcref{l:end-kick}; the identification lemma is applied with $16 R_k \le r_{\rm out}$, $4R_k \le r_{\rm in} \le 3 r_{\rm out}/4$, and $2 r_{\rm out} \le \min(r_k, r_{\rm sep})$ & \zcref{l:end-energy,l:end-kick,p:end-coupled,l:cons-identification,p:end-shifted}\\
$d_{\rm col} = R_k/8$ & distance between the collar of the barrier cutoff and the inner region & \zcref{l:end-barrier}\\
$\rho_{\rm pd} \asymp R_k$ & radius of the polydisk that carries the continued error & \zcref{l:end-continuation}\\
$r_{\rm phys}$ & support radius of the physical cutoffs, $r_{\rm phys}/R_k$ small at fixed $k$ & \zcref{t:end-entrance}\\
$r_1$ & inner radius of the annulus $r_1 \le |U| \le r_{\rm phys}$ on which the physical cutoff varies, $r_1 < r_{\rm phys}$ & \zcref{l:end-shell}\\
$A_{\rm sh}$ & exponent of $W$ in the residual bound \eqref{e:end-residual-bound}, $A_{\rm sh} = A_{\rm ann} + A_1$, depending on $k$, the body weight, the orders of the shell jets, and the prescribed local jets of the family & \eqref{e:end-commutator}\\
$r_{\rm esc}$ & saturation radius of the escape function $\vartheta_B$; $r_{\rm esc} < s_0/2$ and $C_k a_{\rm esc} C_0 r_{\rm esc}$ small & \zcref{s:end-onsite}\\
\end{longtable}
\endgroup

The block length $C_0$ is chosen once and has three roles. In the signed preparation, \zcref{c:end-ramp} and \zcref{p:end-coupled} need $C_0 \ge \max(1, \varrho)$, so that the signed ramp fits into one block. In the escape ramp of \zcref{p:end-escape}, $C_0$ fixes the number of steps over which the escape weight is raised, and the amplitude $a_{\rm esc}$ is chosen afterward with $3 a_{\rm esc}/C_0 < c/4$. In the barrier of \zcref{l:end-barrier}, the physical duration $C_0/(2W)$ of the block gives the decay factor $e^{-c_{\rm D} W d_{\rm col}^2/C_0}$ of the Davies--Gaffney estimate across the collar, which must dominate the growth factor $e^{16 C_{\rm geo}^2 C_* C_0 W (E^2 R_k^2 + R_k^4)}$ that comes from the height of the canonical graph over the collar; this is the second condition in \eqref{e:end-Egeom}. Only the first role constrains $C_0$ itself, and the other two constrain constants chosen after it.

We use the coordinates on the cycle of \zcref{s:hist-step}: the boost $\mathsf g$, the physical normal $\widetilde P = \operatorname{ch}_{\mathsf g} P_0$ of \eqref{e:end-boost-conj}, the escape function $\vartheta$ of \eqref{e:end-escape}, and the parameter balls of \eqref{e:end-damping}, together with the kinetic and onsite bounds \zcref{l:end-kinetic,l:end-onsite}. Recall from \eqref{e:hist-transported} that the transported odd frame consists of the entries of $O_1 = \mathsf g^{-1} \Lambda_B S_{BF}$ and $O_2 = S_{FB} \mathsf g^{-1}$, and that the norms of $\mathcal H_p(\mathcal S)$ are taken in its monomials. We abbreviate $s = s_E$ and $\kappa = \kappa_E$, and in this section all norms of matrices are Hilbert--Schmidt norms unless stated otherwise. The first step of the chain starts from the constant function, which has no finite norm in $\mathcal H_p(\mathcal S)$. Since $K_\tau 1 = 1$, we have $f_1 = w_\Gamma$, and we integrate this first factor directly with a fixed fraction of its BB and FF Gaussians and its damping $e^{-c \eta v}$. Since $v^q e^{-c \eta v} \le C_q \eta^{-q}$ for $q \ge 0$, we obtain $\|w_\Gamma\|_{\mathcal H_p(\mathcal S)} \le C_k \eta^{-A_k}$, also in the parameter Hardy norm used below.

\section{Families generated from the endpoint and their equivariance}\label{s:end-families}

The comparison of \zcref{s:assembly} continues certain functions on the tube from real to complex points. Such a continuation is legitimate only for functions with a known holomorphic structure, and we now describe the class of functions for which we assert it, together with their equivariance under conjugation, which controls their continuation along the complexified orbit.

\paragraph{The realizations on the tube.}
Recall from \zcref{d:tube-raw} that the raw realization is the realization of the chain on functions of the tubular coordinates $(x, U)$ of \eqref{e:tube-map} with the flat measure $\dd U \dd\nu$, in which one link acts by the operator $K^\varpi$ with the kernel \eqref{e:tube-raw-kernel}. The symmetric normal gauge \eqref{e:tube-table} is obtained from it by conjugation with $w_0^{1/2} \gamma_h$. Recall that $w_0$ denotes the source-free onsite weight, that the half-weight $w_0^{1/2}$ factors as in \eqref{e:tube-half-weight} into its exact Gaussian and anharmonic parts, and that the Gaussian gauge $\gamma_h$ is the Gaussian function \eqref{e:tube-gamma} of $U$, which is real on the real tube. By \zcref{l:tube-quadratic}, conjugation by $\gamma_h$ turns the exact quadratic normal transfer, written in the flat realization, into its positive Wick realization. In the symmetric normal gauge we use the positive Wick metric of \zcref{d:tube-wick}, the norm $\| \cdot \|$ in which the Wick basis is orthonormal, together with the orbit measure.

\paragraph{Parameters.}
Recall the linear damping direction $\Gamma$ and the nested complex balls $\mathcal U_0 \Subset \mathcal U_1 \Subset \mathcal U_2$ of \eqref{e:end-damping}. Local inverse-resolvent sources are treated differently. Their directions $\mathbf V = (V_1, \dots, V_d)$ are used only through finite derivatives at zero. We choose a basis of the corresponding finite jet tensor space, and for a family $f_{\Gamma, \mathbf V}$ we write $f_{\Gamma, \ell}$ for its components in this basis. We define the \emph{family norm}
\begin{equation}\label{e:end-family-norm}
 \|f\|_{\mathcal U; p, a}^2 = \int_{\mathcal U} \sum_\ell \bigl\| e^{a|Z|^2} v^p f_{\Gamma, \ell} \bigr\|^2 \dd\Gamma ,
\end{equation}
where $\dd\Gamma$ is the complex volume on the parameter space and all finite odd coefficients are retained in the norm. A family norm on a larger ball controls every fixed parameter derivative on a smaller ball, by Cauchy's formula.

\begin{definition}\label{d:end-generated}
We say that a family $f_{\Gamma, \mathbf V}$ of coefficient functions on the tube is \emph{generated from the endpoint} if it is obtained from the right endpoint $g_R$ of \eqref{e:tube-table}, or from its representative in the raw realization $\varpi$, by applying finitely many of the following operations in chronological order: adapted steps $A_{\Gamma, h}$; the two factors $s_\Gamma$ and $T_h$ of such a step; the half-weights $w_0^{\pm 1/2}$; invariant normal input and output cutoffs; partial heats of duration at most $\tau_{\rm phys}$; and a prescribed finite number of complete local jets.
\end{definition}

The comparison of the physical and adapted chains in \zcref{s:assembly} uses exactly this class. By \eqref{e:end-left} below, the families obtained from $g_L$ by the transposed steps are also generated from the endpoint.

The next lemma states the equivariance of these families under simultaneous conjugation of the matrix point and of the parameters, in its finite and infinitesimal forms, together with its standard consequence: holomorphic continuation along complexified conjugation orbits is the same as continuation in the parameters.

\begin{lemma}\label{l:end-ward}
Let $f_{\Gamma, \mathbf V}$ be a family generated from the endpoint, written in pushed-forward normal and tangent frames, or the physical family $f_{j, \Gamma} = T_\Gamma^j 1$ on the cycle, for which $\mathbf V$ is absent.
\begin{enumerate}[label=(\alph*)]
\item\label{i:end-ward-a} For all real even matrices $S$ near the identity,
\begin{equation}\label{e:end-covariance}
 f_{\Gamma, \mathbf V}(S R S^{-1}) = f_{S^{-1} \Gamma S, \, S^{-1} \mathbf V S}(R) .
\end{equation}
Equivalently, for all constant even matrices $\mathfrak e$, the generator $\mathcal W_{\mathfrak e}$ of the conjugation $R \mapsto e^{t\mathfrak e} R e^{-t\mathfrak e}$ satisfies
\begin{equation}\label{e:end-ward}
 \mathcal W_{\mathfrak e} f_{\Gamma, \mathbf V} = \langle [\Gamma, \mathfrak e], \partial_\Gamma \rangle f_{\Gamma, \mathbf V} + \langle [\mathbf V, \mathfrak e], \partial_{\mathbf V} \rangle f_{\Gamma, \mathbf V} .
\end{equation}
\item\label{i:end-ward-b} For complex $S$, as long as $S^{-1} \Gamma S \in \mathcal U_1$ and the normal point stays in the tube, the right side of \eqref{e:end-covariance} is holomorphic in $S$. At every real output in the interior of the set where the output cutoffs of the family equal one, it is the holomorphic continuation of $R \mapsto f_{\Gamma, \mathbf V}(R)$ along the complexified orbit $\{S R S^{-1}\}$.
\item\label{i:end-ward-c} Let $f$ be generated from the endpoint, let $\Gamma \in \mathcal U_0$, and let $R \mapsto S(R)$ be measurable with $\|S(R)^{\pm 1} - \Id\| \le c_k$ for a sufficiently small $c_k > 0$. Write $f^S_\ell(R)$ for the components of the selected values $f_{S(R)^{-1} \Gamma S(R), \, S(R)^{-1} \mathbf V S(R)}(R)$. Then
\begin{equation}\label{e:end-selected}
 \int \sum_\ell \bigl\| e^{a|Z|^2} v^p f^S_\ell(R) \bigr\|^2 \dd R \le C_k \|f\|_{\mathcal U_2; p, a}^2 .
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We first prove \eqref{e:end-covariance} for a family generated from the endpoint. We change the body integration variables and use the pushed-forward frame $T \mapsto S T$. The ambient Gaussian and the tubular measure are invariant. The normal coordinates, the half-density, the Gaussian gauge, and the invariant cutoffs, including their nilpotent terms, are unchanged, and $g_R$ and $w_0^{1/2}$ are normal class functions by \zcref{l:tube-class}. The identities $\Str(S R S^{-1} \Gamma) = \Str(R S^{-1} \Gamma S)$ and $\det(E - S R S^{-1} - tV) = \det(E - R - t S^{-1} V S)$ imply \eqref{e:end-covariance} for one step and one complete jet, and by induction it holds for their ordered products and for partial heats. The positive source and a fixed part of the weight margin justify the change of variables over the noncompact body.

For the physical family the substitution $R \mapsto S R S^{-1}$ moves the cycle, and we argue infinitesimally, by induction on $j$. The idea is that simultaneous conjugation of the two arguments leaves the heat kernel invariant, and that the conjugation field $\mathcal W_{\mathfrak e}$ can be moved from one argument to the other by an integration by parts over the cycle, since it is holomorphic and has divergence zero. For $j = 0$, the identity \eqref{e:end-ward} is trivial. Let $j \ge 1$, and write $f_{j, \Gamma} = w_\Gamma \, g$ with $g = K_\tau f_{j - 1, \Gamma}$. The factors $w_\Gamma$ and $g$ are holomorphic near $\mathcal S$, the first since $E - R$ is invertible on $\mathcal S$ by \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse}, and the second by the Gaussian bound of the link kernel from complex output points in \zcref{p:end-shifted}\ref{i:end-shifted-cont}. That proposition, and the formula \eqref{e:end-graph-defect} for $d(X, X')$ used below, are proved in \zcref{s:end-canonical} without the present lemma. By the identities used above for one step, we have $w_\Gamma(S R S^{-1}) = w_{S^{-1} \Gamma S}(R)$, and differentiating at $S = e^{t\mathfrak e}$, we obtain \eqref{e:end-ward} for $w_\Gamma$. By the Leibniz rule for the even derivation $\mathcal W_{\mathfrak e}$, it suffices to prove \eqref{e:end-ward} for $g$. The case $j = 1$ is trivial, since $g = K_\tau 1 = 1$, and we let $j \ge 2$.

The kernel $k_\tau(R, R')$ is invariant under the simultaneous conjugation $(R, R') \mapsto (S R S^{-1}, S R' S^{-1})$, since $\Str(S R S^{-1} - S R' S^{-1})^2 = \Str(R - R')^2$. Let $\mathcal W_{\mathfrak e}^{(1)}$ and $\mathcal W_{\mathfrak e}^{(2)}$ denote the field acting on $R$ and on $R'$. Then $\mathcal W_{\mathfrak e}^{(1)} k_\tau = - \mathcal W_{\mathfrak e}^{(2)} k_\tau$, and differentiating under the integral, we find
\begin{equation}\label{e:end-ward-heat}
 \mathcal W_{\mathfrak e} g(R) = - \int_{\mathcal S} \bigl( \mathcal W_{\mathfrak e}^{(2)} k_\tau(R, R') \bigr) f_{j - 1, \Gamma}(R') \dd R' .
\end{equation}

We now integrate by parts on the cycle. We use linear coordinates $(z, \zeta)$ of the ambient space, given by the even and the odd entries, and we write
\[
 \mathcal W_{\mathfrak e} = \sum_\mu a^\mu \partial_{z_\mu} + \sum_b a^b \partial_{\zeta_b} ,
\]
where the coefficients $a^\mu$ and $a^b$ are those of the linear map $Z \mapsto [\mathfrak e, Z]$. This map has divergence zero, since
\[
 [\mathfrak e, Z]_{i l} = (\mathfrak e_{i i} - \mathfrak e_{l l}) Z_{i l} + \sum_{m \ne i} \mathfrak e_{i m} Z_{m l} - \sum_{m \ne l} Z_{i m} \mathfrak e_{m l} ,
\]
and the supertrace of its diagonal part vanishes by the antisymmetry of $\mathfrak e_{i i} - \mathfrak e_{l l}$ in $(i, l)$. It follows that every superfunction $F$ that is holomorphic near $\mathcal S$ satisfies
\[
 \mathcal W_{\mathfrak e} F = \sum_\mu \partial_{z_\mu}(a^\mu F) - \sum_b \partial_{\zeta_b}(a^b F) .
\]
The Berezin integral of the second sum vanishes. Coefficientwise in $\zeta$, the first sum times $\dd z$ is the exterior derivative of the holomorphic form $\sum_\mu \pm a^\mu F \, \iota_{\partial_{z_\mu}} \dd z$. Let $\rho$ be a smooth function on $[0, \infty)$ that equals one on $[0, 1]$ and zero on $[2, \infty)$, and set
\[
 \rho_n(R') = \rho(|y'|/n) ,
\]
where $y' = (P_0', X', M')$ are the body coordinates of $R' \in \mathcal S$. The coefficients $a^\mu$ are linear in $R'$, and the parametrization of $\mathcal S$ has bounded derivatives by \eqref{e:end-divided}. Stokes' theorem on the body of $\mathcal S$, as in the proof of \zcref{l:contour-stokes}, gives
\begin{equation}\label{e:end-ward-ibp}
 \Bigl| \int_{\mathcal S} \rho_n \, \mathcal W_{\mathfrak e} F \dd R' \Bigr| \le C \int_{n \le |y'| \le 2 n} |F|_1 \dd y' ,
\end{equation}
where $|F|_1$ is the sum of the moduli of the coefficients of $F$ in $\zeta$.

We apply \eqref{e:end-ward-ibp} with $F = k_\tau(R, \cdot) \, f_{j - 1, \Gamma}$, for fixed $R \in \mathcal S$. As in the proof of \zcref{l:end-kinetic}, the body of the kernel satisfies
\[
 \bigl| k_\tau(R, R')_\emptyset \bigr| = \exp\Bigl\{ - \frac{W^2}{2} \bigl( \|P_0 - P_0'\|^2 + d(X, X') + \|M - M'\|^2 \bigr) \Bigr\} ,
\]
and its other coefficients carry polynomial factors. The formula \eqref{e:end-graph-defect} implies that $d(X, X') \ge c_X \|X'\| - C_X$, and therefore $k_\tau(R, \cdot)$, times every polynomial in $|y'|$, is square integrable. The function $f_{j - 1, \Gamma}$ lies in $\mathcal H_0(\mathcal S)$, since $j \ge 2$, and the transported odd frame changes coefficient norms only by polynomial factors in $v_{\mathcal S}$. By the Cauchy--Schwarz inequality, $F$ and $\mathcal W_{\mathfrak e} F$ are integrable; for $\mathcal W_{\mathfrak e} f_{j - 1, \Gamma}$ we use the induction hypothesis and Cauchy's formula in $\Gamma$. Letting $n \to \infty$ in \eqref{e:end-ward-ibp}, we find
\[
 \int_{\mathcal S} \bigl( \mathcal W_{\mathfrak e}^{(2)} k_\tau(R, R') \bigr) f_{j - 1, \Gamma}(R') \dd R'
 = - \int_{\mathcal S} k_\tau(R, R') \, \mathcal W_{\mathfrak e} f_{j - 1, \Gamma}(R') \dd R' .
\]
Inserting this into \eqref{e:end-ward-heat}, and using the induction hypothesis and the fact that $k_\tau$ does not depend on $\Gamma$, we obtain $\mathcal W_{\mathfrak e} g = \langle [\Gamma, \mathfrak e], \partial_\Gamma \rangle g$. This is \eqref{e:end-ward} on the cycle. Each side of \eqref{e:end-ward} is holomorphic near $\mathcal S$, which is totally real of maximal dimension, and the identity extends there.

The two forms in \ref{i:end-ward-a} are equivalent. Differentiating \eqref{e:end-covariance} at $S = e^{t\mathfrak e}$, $t = 0$, we obtain \eqref{e:end-ward}. Conversely, \eqref{e:end-ward} at every $(\Gamma, \mathbf V, R)$ implies that the function
\[
 t \mapsto f_{e^{t\mathfrak e} \Gamma e^{-t\mathfrak e}, \, e^{t\mathfrak e} \mathbf V e^{-t\mathfrak e}}(e^{t\mathfrak e} R e^{-t\mathfrak e})
\]
has zero derivative, which is \eqref{e:end-covariance} for $S = e^{t\mathfrak e}$, and products of such $S$ cover a neighborhood of the identity.

For \ref{i:end-ward-b}, the right side of \eqref{e:end-covariance} is holomorphic in complex $S$ by \zcref{l:end-real-family} below, whose proof does not use the present lemma, since the dependence on $\mathbf V$ is a finite multilinear tensor. Fix a real output $R$ in the interior of the set where the output cutoffs equal one. There the output of the last heat is real analytic, and its germ at $R$ extends holomorphically to $S R S^{-1}$ for $S$ in a small complex ball $\mathcal B$ about the identity. The two sides of \eqref{e:end-covariance} are holomorphic on the connected set $\mathcal B$ and agree for real $S$, which form a totally real subspace of maximal dimension. By the identity theorem they agree on $\mathcal B$.

For \ref{i:end-ward-c}, a small ambient $S$ keeps the damping in $\mathcal U_1$ and transforms each jet into a bounded combination of basis jets. We apply \zcref{l:tools-submean} with $\zeta = \Gamma$, with the real body point $R$ as the variable $x$, with the measure $m$ given by the coefficient measure with weight $e^{2a|Z|^2} v^{2p}$, with the selection $\zeta_*(R) = S(R)^{-1} \Gamma S(R)$, and with $\rho$ the distance from $\mathcal U_1$ to the complement of $\mathcal U_2$. The right side of that lemma is at most $|B_\rho|^{-1} \|f\|_{\mathcal U_2; p, a}^2$, which proves \eqref{e:end-selected}. Conjugation by a full supermatrix may create formal BF and FB entries of $\Gamma$ and odd entries of the jet directions. We keep their finite coefficient expansion, using the same finite Duhamel and cofactor identities as for complete source jets; this changes the bound only by fixed powers of $W$ and requires fixed stronger Gaussian and body weights. Only the ordinary BB and FF entries belong to the parameter balls.
\end{proof}

\section{The escape ramp}\label{s:end-escape}

In this section we show that, after one block of $m = \lceil C_0 W \rceil$ steps, the partial chain $f_m = T_\Gamma^m 1$ is concentrated near the saddle manifold, with a Gaussian tail $e^{-c W r^2}$ at distance $r$. The tool is an Agmon estimate with a time-dependent weight \cite{agmon1982lectures}. We conjugate the heat by $e^{A \vartheta}$, where $\vartheta$ is the escape function of \eqref{e:end-escape} and $A$ is a real coefficient of size $O(a_{\rm esc} W)$. The coefficient decreases during each heat interval and is raised over the block using the onsite factors. The time derivative of the weight supplies a negative term, which dominates the positive term that the conjugation creates. Since the heat is not selfadjoint on the cycle, the first-order term of the conjugated generator in the BB block is not skew, and we control it with the equivariance of \zcref{l:end-ward}. In the FF block the cycle is flat and this difficulty does not arise (\zcref{l:end-ff-current}).

\paragraph{The conjugated generator.}
Let $L$ be the generator of the ambient heat pulled to the cycle, as in the proof of \zcref{l:end-kinetic}. The cycle has unit holomorphic Jacobian, so $L$ has divergence form. We first compute its principal part in the BB block. Write $B = DF_X$ for the derivative of $X \mapsto -\Lambda f_0(X)$. Expanding the body of $\Str(\dd S_B^2)$ in the coordinates $(P_0, X)$, we find
\[
 \|\dd P_0\|^2 + \|\dd X\|^2 - \|B \, \dd X\|^2 + 2\iu \langle \dd P_0, B \, \dd X\rangle .
\]
The inverse of this form gives the principal coefficient of the BB body of $L = \nabla \cdot a \nabla$, namely
\begin{equation}\label{e:end-principal}
 a = \begin{pmatrix} \Id - B B^* & -\iu B \\ -\iu B^* & \Id \end{pmatrix} ,
 \qquad
 \Re a = \begin{pmatrix} D_N & 0 \\ 0 & \Id \end{pmatrix} ,
 \qquad
 \Im a = - \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix} ,
\end{equation}
where $D_N = \Id - B B^* = \operatorname{ch}_{\mathsf g}^{-2}$ is the normal real diffusion metric; one checks $a \cdot (\text{form}) = \Id$ by multiplying the block matrices. For a real function $\vartheta$ and a real constant $A$, we have $e^{A\vartheta} \nabla e^{-A\vartheta} = \nabla - A \nabla\vartheta$, and therefore
\begin{equation}\label{e:end-conjugated}
 e^{A\vartheta} L e^{-A\vartheta} = L - A\bigl(\nabla\vartheta \cdot a \nabla + \nabla \cdot (a \nabla\vartheta \, \cdot \,)\bigr) + A^2 \, \nabla\vartheta \cdot a \nabla\vartheta .
\end{equation}
In the first-order part, the term with $\Re a$ is $Y \cdot \nabla + \nabla \cdot (Y \, \cdot \,)$ with the real field $Y = \Re a \nabla\vartheta$, which is exactly skew-adjoint for the flat measure, with no zeroth-order remainder. The term with $\Im a$ is $\iu(2 \mathcal Z \cdot \nabla + \operatorname{div} \mathcal Z)$ with the real field
\[
 \mathcal Z = \Im a \, \nabla\vartheta = -(T_P, V) ,
 \qquad
 T_P = B \nabla_X \vartheta ,
 \quad
 V = B^* \nabla_{P_0} \vartheta .
\]
Taking real parts, we obtain
\begin{equation}\label{e:end-conjugated-form}
 \Re \bigl\langle u, e^{A\vartheta} L e^{-A\vartheta} u\bigr\rangle
 = \Re \langle u, L u\rangle + 2A \Im \langle u, \mathcal Z \cdot \nabla u\rangle + A^2 \int \nabla\vartheta \cdot \Re a \nabla\vartheta \, |u|^2 .
\end{equation}
The last term is the squared gradient of $\vartheta$ in the real diffusion metric. We will see that it is at most $C\vartheta$, and that the current term $\mathcal Z \cdot \nabla$ is the delicate one.

\paragraph{The FF part of the current.}
On the FF block of the cycle, the Hermitian matrices $M$ form a flat Cartesian space, and the FF part of the generator is a real symmetric diffusion. The next lemma computes the FF part of the conjugated form exactly, so that the equivariance is needed only in the BB block.

\begin{lemma}\label{l:end-ff-current}
In the coordinates $(P_0, X, M)$ of the cycle and the transported odd frame, the generator decomposes as $L = L_B + \Delta_M + L_{\rm odd}$, where $\Delta_M$ is the flat Laplacian of the Hilbert--Schmidt metric on Hermitian $2k \times 2k$ matrices, $L_B$ acts only in $(P_0, X)$ and in the odd coordinates through the frame, and $L_{\rm odd}$ acts only in the odd coordinates, with coefficients depending only on $X$. Consequently, for every real constant $A$ and every $u$ in the form domain,
\begin{equation}\label{e:end-ff-current}
 \Re \bigl\langle u, e^{A\vartheta_F} \Delta_M e^{-A\vartheta_F} u\bigr\rangle
 = -\|\nabla_M u\|^2 + A^2 \int |\nabla_M \vartheta_F|^2 |u|^2
 \le -\|\nabla_M u\|^2 + C_F A^2 \int \vartheta_F |u|^2 ,
\end{equation}
where the inner product is that of $\mathcal H_p(\mathcal S)$, of the parameter Hardy space $\mathcal A_\rho(\mathcal H_p(\mathcal S))$ introduced below, or of their weighted versions with any weight that does not depend on $M$, and $C_F$ is the universal constant of \eqref{e:end-escape-gradient}. Moreover, $e^{A \vartheta_F}$ commutes with $L_B$, $L_{\rm odd}$, the body weights $v^p$, the parameter structure, and the BB escape weight.
\end{lemma}

\begin{proof}
We first prove the decomposition. The FF block of a point $S$ of the cycle satisfies $S_{FF} = \kappa \Id + \iu M$, and its holomorphic derivatives are $-\iu$ times the derivatives in $M$. The FF part of the ambient Laplacian of $\Str$ carries the sign $-1$ of the supertrace, which cancels the factor $(-\iu)^2$. The FF part of the generator is therefore $+\Delta_M$, where $\Delta_M$ is the Laplacian of the metric $-\Tr(\iu \, \dd M)^2 = \|\dd M\|^2$, that is, the flat Laplacian for the Hilbert--Schmidt metric in an orthonormal basis of Hermitian matrices. The measure of the cycle is Lebesgue measure in $M$. The ambient Laplacian has no mixed second derivatives between different entries, and the transported odd frame is defined by congruence with the BB boost $\mathsf g = \mathsf g(X)$, which does not depend on $M$. In particular, no derivative in $M$ acts on the frame, and $\Delta_M$ commutes with $L_B$ and $L_{\rm odd}$.

Next, we compute the conjugation. For real $A$ and the real function $\vartheta_F$ of $M$, we have
\[
 e^{A\vartheta_F} \Delta_M e^{-A\vartheta_F} u = \Delta_M u - 2A \nabla_M \vartheta_F \cdot \nabla_M u - A (\Delta_M \vartheta_F) u + A^2 |\nabla_M \vartheta_F|^2 u .
\]
Here $\Delta_M$ acts on coefficient vectors componentwise, since the frame does not depend on $M$, and the coefficient metric is positive. Integrating by parts in $M$, we find that $\Re\langle u, \Delta_M u\rangle = -\|\nabla_M u\|^2$ and
\[
 \Re \bigl\langle u, -2A \nabla_M \vartheta_F \cdot \nabla_M u\bigr\rangle
 = -A \int \nabla_M \vartheta_F \cdot \nabla_M |u|^2
 = A \int (\Delta_M \vartheta_F) |u|^2 ,
\]
which cancels the contribution of $-A(\Delta_M \vartheta_F) u$. This is the identity in \eqref{e:end-ff-current}, and the inequality follows from \eqref{e:end-escape-gradient}. The weights of the norms are independent of $M$ and do not change the integration by parts. The commutation statements hold since $\vartheta_F$ depends only on $M$, while $L_B$, $L_{\rm odd}$, $v$, $\vartheta_B$, and the parameter $\Gamma$ do not involve $M$.
\end{proof}

The identity \eqref{e:end-ff-current} is the Davies--Agmon identity for a real weight and a real symmetric diffusion. The divergence term of the current cancels exactly, which is why the constant is universal. For the rigorous argument below we replace $\vartheta_F$ by $\zeta_n(\vartheta_F)$, where $\zeta_n$ is bounded, smooth, increasing, and concave with $\zeta_n(0) = 0$, $\zeta_n' \le 1$, and $\zeta_n(x) \uparrow x$. Since $\zeta_n' \le 1$ and $x \zeta_n'(x) \le \zeta_n(x)$ by concavity, \eqref{e:end-escape-gradient} implies that
\[
 |\nabla_M \zeta_n(\vartheta_F)|^2 = \zeta_n'(\vartheta_F)^2 |\nabla_M \vartheta_F|^2 \le C_F \zeta_n'(\vartheta_F)^2 \vartheta_F \le C_F \zeta_n(\vartheta_F) .
\] Hence \eqref{e:end-ff-current} holds for the saturated weights with the same constant.

\paragraph{The BB part of the current.}
In the BB block the field $\mathcal Z$ is not small, since $\|V\|$ grows like the boost. We now show that, up to a remainder bounded by the real gradient energy, it is the generator of a conjugation, to which the equivariance \eqref{e:end-ward} applies. Write $G = \nabla_{\widetilde P} \vartheta_B$ and $\Xi = \operatorname{ch}_{\mathsf g} G$. Since $\vartheta_B$ depends on $P_0$ through $\widetilde P = \operatorname{ch}_{\mathsf g} P_0$, we have $\nabla_{P_0} \vartheta_B = \Xi$ and $V = B^* \Xi$, and
\begin{equation}\label{e:end-real-gradient}
 \nabla\vartheta_B \cdot \Re a \nabla\vartheta_B = \langle \Xi, D_N \Xi\rangle + \|\nabla_X \vartheta_B\|^2 = \|G\|^2 + \|\nabla_X \vartheta_B\|^2 .
\end{equation}
Let $\mathcal M = \lambda_f + \rho_f$, where $\lambda_f$ and $\rho_f$ are left and right multiplication by $f_0(X)$, so that $\mathcal M = 2s \cosh(\lambda_{\log \mathsf g} + \rho_{\log \mathsf g}) \operatorname{ch}_{\mathsf g}$. Substituting into \eqref{e:end-divided} in an eigenbasis of $\log \mathsf g$, we obtain
\begin{equation}\label{e:end-M-bounds}
 \|\operatorname{ch}_{\mathsf g} \mathcal M^{-1} V\| \le \|G\|/(2s) ,
 \qquad
 \operatorname{ch}_{\mathsf g} \mathcal M^{-1} \Lambda = \Lambda \operatorname{ch}_{\mathsf g}^{-1}/(2s) .
\end{equation}
Let $\mathfrak e$ be the off-diagonal solution of the dissipative Sylvester equation
\begin{equation}\label{e:end-sylvester}
 -\mathcal M \mathfrak e + \iu \Lambda [\mathfrak e, P_0] = V ,
\end{equation}
whose unweighted inverse has norm at most $(2s)^{-1}$.

We first identify the conjugation field of $\mathfrak e$. On restrictions to $\mathcal S$ of holomorphic functions, the generator $\mathcal W_{\mathfrak e}$ of the conjugation $S \mapsto e^{t\mathfrak e} S e^{-t\mathfrak e}$ acts as $\delta P \cdot \nabla_{P_0} + \delta X \cdot \nabla_X$ plus terms in the odd variables, which we denote by $\mathcal W_{\mathfrak e}^{\rm odd}$, where $(\delta P, \delta X)$ is the tangent vector of $\mathcal S$ whose image under the complexified tangent map $(\delta P, \delta X) \mapsto \delta P - \iu \Lambda (Df_0 \, \delta X + \delta X)$ is $[\mathfrak e, S_B]$. Comparing off-diagonal and block-diagonal parts, we have
\[
 \delta X = \iu \Lambda [\mathfrak e, S_B]_{\rm off} ,
 \qquad
 \delta P = [\mathfrak e, S_B]_{\rm diag} + \iu \Lambda Df_0(\delta X) .
\]
Since $\mathfrak e$ is off-diagonal, $[\mathfrak e, S_B]_{\rm off} = [\mathfrak e, P_0] - \iu [\mathfrak e, \Lambda f_0]$ and $[\mathfrak e, S_B]_{\rm diag} = -\iu [\mathfrak e, \Lambda X] = \iu \Lambda \{\mathfrak e, X\}$, and since $\Lambda \mathfrak e \Lambda = -\mathfrak e$, we have $\Lambda [\mathfrak e, \Lambda f_0] = -\{\mathfrak e, f_0\} = -\mathcal M \mathfrak e$. The Sylvester equation \eqref{e:end-sylvester} implies that $\delta X = V$. Next, differentiating $f_0^2 = s^2 + X^2$ shows that $Df_0(\{\mathfrak e, f_0\}) = \{\mathfrak e, X\}$, so the anticommutator terms cancel, and
\[
 \delta P = -\Lambda Df_0\bigl(\Lambda [\mathfrak e, P_0]\bigr) = \operatorname{th}_{\mathsf g} [\mathfrak e, P_0] .
\]
For the last identity, note that in an eigenbasis of $X$ we have $(Df_0 Y)_{ab} = \tanh((r_a + r_b)/2) Y_{ab}$ by \eqref{e:end-divided}, and that $\Lambda$ maps the eigenvector of $r_a$ to that of $-r_a$, since $\Lambda X \Lambda = -X$; hence $-\Lambda Df_0(\Lambda Y) = \operatorname{th}_{\mathsf g} Y$ for block-off-diagonal $Y$, in particular for $Y = [\mathfrak e, P_0]$. Since $\mathcal Z = -(T_P, V)$, we conclude that
\begin{equation}\label{e:end-current-identity}
 \mathcal Z \cdot \nabla = -\mathcal W_{\mathfrak e} + \mathcal W_{\mathfrak e}^{\rm odd} - (T_P - \delta P) \cdot \nabla_{P_0} .
\end{equation}

We now bound the pieces of \eqref{e:end-current-identity}; all constants are dimension-free. Assume $\|\widetilde P\| \le s/2$, which holds on the support of $\nabla\vartheta_B$ when $r_{\rm esc} < s_0/2$. Multiplying \eqref{e:end-sylvester} by $\operatorname{ch}_{\mathsf g} \mathcal M^{-1}$, and using \eqref{e:end-M-bounds} and $\|[\mathfrak e, P_0]\| \le 2 \|\operatorname{ch}_{\mathsf g} \mathfrak e\| \|\widetilde P\|$, we find
\begin{equation}\label{e:end-ward-current}
 \|\operatorname{ch}_{\mathsf g} \mathfrak e\| \le s^{-1} \|G\| \le C \|\widetilde P\| .
\end{equation}
The inequality $\cosh(a - c) \le 2 \cosh(a - b) \cosh(b - c)$ implies $\|\operatorname{ch}_{\mathsf g}(A_1 A_2)\| \le 2 \|\operatorname{ch}_{\mathsf g} A_1\| \|\operatorname{ch}_{\mathsf g} A_2\|$, and therefore
\[
 \|\operatorname{ch}_{\mathsf g} \delta P\| \le 4 \|\operatorname{ch}_{\mathsf g} \mathfrak e\| \|\widetilde P\| \le 4 s^{-1} \|G\| \|\widetilde P\| .
\]
For $T_P$ we use duality. For a unit block-diagonal $B'$, the direction $V' = DF_X^* \operatorname{ch}_{\mathsf g} B'$ has defect $d_X(\operatorname{ch}_{\mathsf g} B') = \|B'\|^2 = 1$, and $\nabla_X \vartheta_B$ involves the derivative of $\|\widetilde P\|^2 = (\|P_0\|^2 + \|\mathsf g P_0 \mathsf g^{-1}\|^2)/2$. By the second line of \eqref{e:end-connection}, we have
\[
 \|\operatorname{ch}_{\mathsf g} T_P\| = \sup_{\|B'\| = 1} \langle V', \nabla_X \vartheta_B\rangle \le 8 \sup|\chi_0'| \, \|\widetilde P\|^2 \sup_{\|B'\| = 1} \|D\mathsf g(V') \mathsf g^{-1}\|_{\rm op} \le C \|\widetilde P\|^2 .
\]
Combining the last two bounds with $D_N^{-1/2} = \operatorname{ch}_{\mathsf g}$ and $\|G\| \le C \|\widetilde P\|$, we obtain
\begin{equation}\label{e:end-remainder-current}
 \|D_N^{-1/2}(T_P - \delta P)\| \le C \|\widetilde P\|^2 ,
 \qquad
 \langle \Xi, D_N \Xi\rangle = \|G\|^2 .
\end{equation}
Since $\nabla\vartheta_B$ vanishes where $\|\widetilde P\| \ge r_{\rm esc}$, the squared remainder satisfies $\|\widetilde P\|^4 \le C r_{\rm esc}^2 \vartheta_B$, and by \eqref{e:end-real-gradient} the squared gradient of the weight in the real diffusion metric satisfies $\nabla\vartheta_B \cdot \Re a \nabla\vartheta_B \le C(\|G\|^2 + \|\widetilde P\|^4) \le C \vartheta_B$. The same argument gives the bound on the commutator of the conjugation field with $\vartheta_B$,
\[
\begin{aligned}
 |\mathcal W_{\mathfrak e} \vartheta_B|
 &\le |\langle \delta P, \Xi\rangle| + |\langle V, \nabla_X \vartheta_B\rangle| \\
 &\le \|\operatorname{ch}_{\mathsf g} \delta P\| \|G\| + 8 \sup|\chi_0'| \|\widetilde P\|^2 \|D\mathsf g(V) \mathsf g^{-1}\| \le C r_{\rm esc} \vartheta_B ,
\end{aligned}
\]
where for the middle term we used the second line of \eqref{e:end-connection} for $V = DF_X^* \Xi$, whose defect is $\|G\|^2$. Finally, in the transported frame, $\mathcal W_{\mathfrak e}^{\rm odd}$ acts on the odd coordinates by $O_1 \mapsto -\mathsf g^{-1} \mathfrak e \mathsf g \, O_1$ plus the frame term $\mathsf g^{-1} D\mathsf g(V)$. Since $\|\mathsf g^{-1} \mathfrak e \mathsf g\|^2 = 2 \|\operatorname{ch}_{\mathsf g} \mathfrak e\|^2 - \|\mathfrak e\|^2$, this action has norm at most $C \|G\|$ by \eqref{e:end-ward-current} and \eqref{e:end-connection}.

\paragraph{The parameter Hardy space.}
Let $\mathcal A_\rho$ be the Hardy space of holomorphic functions on the parameter ball of radius $\rho$ about the normalized base point $\Lambda$, with values in $\mathcal H_p(\mathcal S)$, and let $\mathbf N_\Gamma$ count the homogeneous degree in the parameter. A homogeneous polynomial $p$ of degree $n$ satisfies $\|p\|_{\mathcal A_\rho}^2 = \rho^{2n} \|p\|_{\mathcal A_1}^2$, so for fixed $u$,
\begin{equation}\label{e:end-ovsyannikov}
 \frac{\dd}{\dd t} \|u\|_{\mathcal A_{\rho(t)}}^2 = \frac{2 \rho'(t)}{\rho(t)} \langle u, \mathbf N_\Gamma u\rangle .
\end{equation}
Decreasing the radius $\rho$ during the evolution therefore supplies a negative number form. This is the device of a scale of Banach spaces of Ovsyannikov and Nirenberg--Nishida \cite{ovsjannikov1971nonlinear,nirenberg1972abstract,nishida1977note}. A bounded affine parameter field moves the degree by at most one, and Cauchy--Schwarz between neighboring degrees bounds its quadratic form by $C_k \langle u, (\mathbf N_\Gamma + 1) u\rangle$.

\paragraph{The weighted heat form.}
Let $u = e^{A \vartheta} f_\Gamma$, where $f_\Gamma$ is the physical family and $|A| \le 2 a_{\rm esc} W + C$. We insert \eqref{e:end-current-identity} into the current term of \eqref{e:end-conjugated-form}. The field $\mathfrak e = \mathfrak e(S)$ depends on the point, but $\mathcal W_{\mathfrak e(S)} f(S)$ involves only the value $\mathfrak e(S)$, so the equivariance \eqref{e:end-ward} applies pointwise and gives
\[
 \mathcal W_{\mathfrak e} u = \bigl\langle [\Gamma, \mathfrak e(S)], \partial_\Gamma \bigr\rangle u + A (\mathcal W_{\mathfrak e} \vartheta) u ,
\]
since $\vartheta$ does not depend on $\Gamma$.
By \eqref{e:end-ward-current}, $\|\mathfrak e\| \le \|\operatorname{ch}_{\mathsf g} \mathfrak e\| \le C r_{\rm esc}$, so the parameter field has form at most $C r_{\rm esc} \langle u, (\mathbf N_\Gamma + 1) u\rangle$ on $\mathcal A_\rho$, and the scalar commutator contributes at most $C A^2 r_{\rm esc} \int \vartheta_B |u|^2$. By \eqref{e:end-remainder-current} and Young's inequality in the metric $D_N$, the contribution of the remainder $(T_P - \delta P) \cdot \nabla_{P_0}$ is at most a fraction of the real gradient energy plus $C A^2 r_{\rm esc}^2 \int \vartheta_B |u|^2$, and $\mathcal W_{\mathfrak e}^{\rm odd}$ has norm $C \|G\|$. Collecting these bounds with the real-form calculation of \zcref{l:end-kinetic} and with \zcref{l:end-ff-current} for the FF part, we obtain
\begin{equation}\label{e:end-weighted-heat}
 \Re \bigl\langle u, e^{A\vartheta} L e^{-A\vartheta} u\bigr\rangle_{\mathcal A_\rho}
 \le - c \, \mathcal E_{\rm real}(u) + C_k W \|u\|^2 + C A^2 \int \vartheta |u|^2 + C_k |A| r_{\rm esc} \langle u, (\mathbf N_\Gamma + 1) u\rangle ,
\end{equation}
where $\mathcal E_{\rm real}(u)$ is the real gradient energy of the proof of \zcref{l:end-kinetic} and $C$ is universal. The last term comes from the parameter field. By \eqref{e:end-ovsyannikov}, it is absorbed when $\rho$ decreases at the logarithmic speed $C_k |A| r_{\rm esc}$.

\begin{proposition}\label{p:end-escape}
There exist universal constants $c, a_{\rm esc} > 0$ with $3 a_{\rm esc}/C_0 < c/4$ such that the following holds at fixed rank $k$ after choosing $r_{\rm esc}$ so small that $C_k a_{\rm esc} C_0 r_{\rm esc} \le \log 2$, where $C_k$ is the constant of the last term of \eqref{e:end-weighted-heat}. Let $\rho$ be the radius of a parameter ball about $\Lambda$ inside $\mathcal U_2$, and set $\rho_{\rm out} = \rho/2$. For $m = \lceil C_0 W \rceil$, every body weight $p$ and all large $W$,
\begin{equation}\label{e:end-tail}
 \bigl\| e^{2 a_{\rm esc} W \vartheta} f_{m, \Gamma} \bigr\|_{\mathcal A_{\rho_{\rm out}}(\mathcal H_p(\mathcal S))} \le C_k \eta^{-A_k} .
\end{equation}
\end{proposition}

\begin{proof}
Let $c$ be the onsite gain of \zcref{l:end-onsite}, and choose $a_{\rm esc}$ so small that $C(2a_{\rm esc} W + C)^2 \le \frac{c}{8} W^2$ for the constant $C$ of \eqref{e:end-weighted-heat}, and $3 a_{\rm esc}/C_0 < c/4$. We raise the escape coefficient along the block. We set
\[
 a_1 = -a_{\rm esc} ,
 \qquad
 a_m = 2 a_{\rm esc} ,
 \qquad
 a_j - a_{j - 1} = \frac{3 a_{\rm esc}}{m - 1} \quad (2 \le j \le m) .
\] Since $K_\tau 1 = 1$, we have $f_1 = w_\Gamma$, and since $a_1 < 0$, the bound on $w_\Gamma$ at the start of this section gives $\|e^{W a_1 \vartheta} f_1\|_{\mathcal A_\rho(\mathcal H_p(\mathcal S))} \le \|w_\Gamma\|_{\mathcal A_\rho(\mathcal H_p(\mathcal S))} \le C_k \eta^{-A_k}$; we start the induction at $j = 1$, since the raw constant function has no coefficient Hilbert norm.

For $2 \le j \le m$, during the heat interval of step $j$ we interpolate the coefficient $A(t)$ linearly from $W a_{j - 1}$ to $W a_j - c/2$. Since $m-1=C_0W+O(1)$ and $3a_{\rm esc}/C_0<c/4$, we have $W(a_j-a_{j-1})=(3a_{\rm esc}/C_0)(1+O(W^{-1}))\le c/4$ for all sufficiently large $W$. Its derivative therefore satisfies
\[
 A'(t) = \frac{W(a_j - a_{j - 1}) - c/2}{\tau} \le \Bigl(\frac c4 - \frac c2\Bigr) 2 W^2 = - \frac c2 W^2 .
\]
In the energy identity for $\|e^{A(t)\vartheta} u(t)\|^2_{\mathcal A_{\rho(t)}}$, the term $A'(t) \int \vartheta |u|^2$ is therefore at most $-\frac c2 W^2 \int \vartheta |u|^2$. By our choice of $a_{\rm esc}$, we have $C A^2 \le \frac c8 W^2$, and therefore this term dominates the term $C A^2 \int \vartheta |u|^2$ of \eqref{e:end-weighted-heat}.

The other terms of \eqref{e:end-weighted-heat} are lower order. The term $C_k W \|u\|^2$ integrates over the heat interval to the factor $e^{C_k W \tau} = e^{C_k/W}$, and the number form is absorbed by the decrease of $\rho$. After the heat, the function is bounded in the weight $e^{(W a_j - c/2)\vartheta}$. By \eqref{e:end-onsite}, the whole onsite factor satisfies $|w_\Gamma|_{1; h} \le e^{C_k/W + C_k \eta} e^{-c\vartheta}$, and therefore multiplication by it restores the weight $e^{W a_j \vartheta}$, up to the factor $e^{C_k/W + C_k \eta}$. The order of the factors is the actual onsite-after-heat order of $T_\Gamma$.

Iterating over the $m - 1 \le 2 C_0 W$ steps, these factors multiply to at most $e^{C_k C_0}$, and the total decrease of $\log \rho$ is at most $C_k a_{\rm esc} C_0 r_{\rm esc} \le \log 2$, so the final radius is at least $\rho_{\rm out}$. The universal $a_{\rm esc}$ is not shrunk with the rank, and this completes the proof of \eqref{e:end-tail}.

The calculation above is formal in two respects, which we now remove. First, we saturate $\vartheta_F$ by the bounded concave functions $\zeta_n$ described after \zcref{l:end-ff-current}, whose gradient and current bounds are uniform in $n$. Second, we carry out the calculation for the physical family with compact spatial cutoffs and on a larger parameter ball, retaining the boundary terms. Fixed unscaled derivatives and polynomial moments of the family are finite after the first step, because of the positive damping, and they dominate each cutoff error. We pass to the limit by spatial exhaustion and then by Fatou's lemma in the saturation. The equivariance \eqref{e:end-ward} is used only for the actual family, and the truncations enter only through these boundary terms.
\end{proof}

The zero set of $\vartheta$ is the union of the $2k + 1$ FF signatures of the saddle manifold. Since $\vartheta \ge c r^2$ outside the union of the normal cores of radius $r$, \eqref{e:end-tail} implies the global tail bound $C_k W^{A_k} e^{-c W r^2}$ for $f_m$ outside any fixed sufficiently small union of such cores.

\section{The endpoint factors on the tube}\label{s:end-raw}

We now turn to the adapted tube, with the tubular coordinates $(x, U)$ of \eqref{e:tube-map}, in one root signature. Recall the right endpoint $g_R$ and its analytic dual $g_L$ from \eqref{e:tube-table}, and the rotation phases $e^{-\iu \theta_\sigma}$ of the root supermatrices, with $\theta_\sigma = \frac12 \arg c_\sigma$ as in \eqref{e:tube-c} and $|\theta_\sigma| \le C|E|$. We write $U$ for the coordinate on the adapted real normal space of \zcref{d:tube-cycle} and $Z$ for its standardized version, so that $c_Z W |U|^2 \le |Z|^2 \le C_Z W|U|^2$ with universal constants $c_Z, C_Z$, and we set $w_a = e^{a|Z|^2}$ for $a \in \R$. Let $\widetilde\chi$ be a real invariant normal cutoff with $0 \le \widetilde\chi \le 1$, supported in the tube, such as $\chi_{\rm inv}$.

\begin{lemma}\label{l:end-raw-norms}
In the balanced signature, for every sufficiently small fixed $a > 0$,
\[
 \|e^{-a|Z|^2} \widetilde\chi g_R\| + \|e^{-a|Z|^2} \widetilde\chi g_L\| \le C_{k, a} .
\]
At every other signature the same norms are at most $C_{k, a} W^{A_k}$. The tangent part of the endpoint is the constant function, and for some sufficiently negative $p_0 = p_0(k)$ the same bounds hold in the body weight $v^{p_0}$.
\end{lemma}

\begin{proof}
We compute the quadratic parts first; the full endpoints differ from them by the bounded factors $w_{\rm anh}^{\mp 1/2}$ on the tube (\zcref{r:tube-transport}). In one even coordinate of root $\sigma$, write $U=e^{-\iu\theta_\sigma}x$ with $x$ real. The proof of \zcref{l:tube-quadratic} gives the quadratic half-weight $w_{\rm quad}^{1/2}=e^{-m_\sigma e^{-2\iu\theta_\sigma}x^2/4}$, while the Gaussian gauge maps the flat ground state $e^{-|c_\sigma|x^2/2}$ to the Wick vacuum $\Omega_0=1$. Here $Z=\sqrt{2|c_\sigma|}\,x$, so that $\gamma_h^2=e^{-|Z|^2/2}$. Against this Gaussian density of the positive Wick metric of \zcref{d:tube-wick}, the squared moduli of the quadratic parts of $g_R$ and $g_L$ therefore carry the residual real exponents
\[
 \pm \frac{\Re\bigl(m_\sigma e^{-2\iu\theta_\sigma}\bigr)}{4 |c_\sigma|} |Z|^2 = O(W^{-1}) |Z|^2 ,
\]
since $|c_\sigma| \asymp W$ by \eqref{e:tube-mu}. The odd coefficients contain the factors $(c_\sigma \pm m_\sigma/2) e^{-2\iu\theta_\sigma}/(2|c_\sigma|)$, which are bounded. In the balanced signature each normal root has equal even and odd dimensions, and the standardizing change of variables has canceling Berezinians. The norm is therefore bounded by a constant depending on $k$ times $\int e^{-(2a - C/W)|Z|^2} \dd Z$, uniformly for large $W$. The half-density $\varpi$ and the factors $w_{\rm anh}^{\pm 1/2}$ have bounded coefficients on the tube and preserve this estimate. In an unbalanced signature the same scaling calculation leaves a fixed power of $|c_\sigma|$, hence a fixed power of $W$, and the Gaussian estimate is unchanged; these powers of $W$ are kept until the unbalanced signature is suppressed in \zcref{s:assembly}. The tangent part is the constant function, whose norm is finite in the body weight $v^{p_0}$ for $p_0$ sufficiently negative.
\end{proof}

\paragraph{The left endpoint.}
The analytic left endpoint reduces to the right one. By \eqref{e:tube-table}, we have $g_L = w_0 g_R$ and $A_h = s_\Gamma T_h$, where $T_h$ and $s_\Gamma$ are symmetric for the analytic Berezin pairing. Thus $A_h^{\rm tr} = T_h s_\Gamma$, and
\begin{equation}\label{e:end-left}
 (A_h^{\rm tr})^n g_L = T_h A_h^{n - 1} (s_\Gamma w_0 g_R) ,
 \qquad
 \ell_m = (T_\Gamma^{\rm tr})^m 1 = K_{\tau_{\rm phys}} f_m .
\end{equation}
The second identity is \zcref{p:rep-exact}. Every family obtained from the left endpoint is therefore obtained from the right one, with the input $s_\Gamma w_0 g_R$ in place of $g_R$, followed by one extra unsourced step. Since $w_0^{\pm 1/2}$ are bounded multipliers on the tube and $s_\Gamma$ is the source factor of an adapted step, all the estimates below for families obtained from the right endpoint apply to those obtained from the left endpoint, and we state only the former.

\section{A weighted energy estimate for short blocks}\label{s:end-energy}

A block of $\lceil C_0 W \rceil$ steps has physical heat time of order $C_0/W$, which is why, as explained at the start of this section, we can work with energy estimates in the raw realization. We prove one such estimate, \zcref{l:end-energy}, for signed Gaussian weights and compact normal graphs, and transport it to the symmetric normal gauge by the similarity of \eqref{e:tube-table}, where it gives the signed ramp \zcref{c:end-ramp} and the bound \zcref{p:end-coupled} for a block on a normal graph. Its input on the onsite factor is \zcref{l:end-kick}. In the raw realization, the coefficient of $|b|^2$ per step on a graph of height $b$ comes only from the flat normal metric, the signed weight, the quadratic part of the onsite factor, and the velocity of the graph, and \zcref{r:end-universal} checks that it does not depend on the rank. We also prove the Davies--Gaffney estimate \zcref{l:end-davies}, which \zcref{s:end-attach} uses.

\paragraph{The raw steps.}
We work on the tube of one signature, in the raw realization of \zcref{d:tube-raw}, in which one link acts by the operator $K^\varpi$ of \eqref{e:tube-raw-kernel}, and in the coefficient metric of \zcref{l:cons-sectorial}, in which the odd normal coordinates carry their standardized scaling and the odd tangent coordinates a small degree metric. The adapted step and the frozen normal transfer, with their cutoffs, in the raw realization are
\begin{equation}\label{e:end-raw-steps}
 S_\Gamma = \chi_{\rm inv} \, s_\Gamma w_0 K^\varpi \chi_{\rm inv} ,
 \qquad
 S_N = \chi_{\rm inv} \, w_0 K^\varpi_{\rm nor} \chi_{\rm inv} .
\end{equation}
Here $w_0$ is the physical source-free weight, including its saddle value $\zeta_p=w_0(R_*)$ in signature $p$. Each raw step retains this scalar phase; since $|\zeta_p|=1$, it does not change the norm estimates below.
All the multipliers involved commute, and by \eqref{e:tube-table} we have, for all $n \ge 0$,
\begin{equation}\label{e:end-similarity}
 A_h^n = \gamma_h^{-1} w_0^{-1/2} \, S_\Gamma^n \, w_0^{1/2} \gamma_h ,
 \qquad
 \mathbf F_h^n = \gamma_h^{-1} w_0^{-1/2} \, S_N^n \, w_0^{1/2} \gamma_h ,
\end{equation}
where $A_h$ and $\mathbf F_h$ carry the same cutoffs. Let $\| \cdot \|_{\rm raw}$ be the norm of $L^2$ with respect to $\dd Z$ and the orbit measure, in the coefficient metric above. Recall from \eqref{e:tube-gamma} that the body of $\gamma_h$ is a Gaussian in $U$; multiplication by the body of $\gamma_h^{-1}$ is unitary from the raw metric onto the Wick metric up to a constant factor, and $w_0^{\pm 1/2}$ and the odd part of $\gamma_h^{\pm 1}$ are bounded coefficient multipliers on the tube. It follows that for all $|a| \le 1$,
\begin{equation}\label{e:end-conversion}
 C_k^{-1} \|w_a g\|_{\rm raw} \le \|w_a \gamma_h^{-1} w_0^{-1/2} g\| \le C_k \|w_a g\|_{\rm raw} ,
\end{equation}
where the constant factor of $\gamma_h$ cancels in \eqref{e:end-similarity}. The upper bound also holds for values on a compact normal graph $U \mapsto U_\lambda$ as in \eqref{e:cons-graph}, supported where the cutoffs equal one. Indeed, $w_0^{-1/2}$ is analytic and bounded on the complex tube, and if $Z_\beta$ denotes the standardized version of the height $\beta = \lambda \psi_{R_k}(U) b$, then the Gaussian body of $\gamma_h^{-1}$ satisfies
\[
 |\gamma_h^{-1}(Z + \iu Z_\beta)|^2 e^{-|Z|^2/2} = e^{-|Z_\beta|^2/2} \le 1 .
\]

\paragraph{Graph norms.}
We use the compact normal graphs \eqref{e:cons-graph} with radius $\bar r = R_k$, with $\psi_{R_k}$ supported where $\chi_{\rm inv} = 1$. In the applications the heights are canonical heights at radius at most $r_{\rm phys}$, and \zcref{l:end-height} below gives $|b|/R_k \le C_{\rm geo}(|E| + r_{\rm phys}) r_{\rm phys}/R_k$, which is small once $r_{\rm phys}/R_k$ is small. For $f$ holomorphic in the normal variables where $\psi_{R_k} \ne 0$, we define
\begin{equation}\label{e:end-graph-norm}
 \|f\|_{\lambda; p, a} = \bigl\| w_a(U_\lambda) \, v^p \rho_\lambda^{1/2} \, f(U_\lambda) \bigr\|_{\rm raw} ,
\end{equation}
where $w_a(U_\lambda) = e^{a Z_\lambda^T Z_\lambda}$ is the signed weight evaluated holomorphically, $Z_\lambda$ being the standardized version of $U_\lambda$. For $\lambda = 0$, the norm \eqref{e:end-graph-norm} is $\|w_a v^p f\|_{\rm raw}$ for all $f$.

\paragraph{Onsite factors and jet steps.}
Recall from \zcref{d:grass-norm,l:grass-norm} that the $\ell^1$ coefficient norm $|w|_1$, in the normalized monomials of the coefficient metric, bounds the norm of multiplication by $w$. Besides $S_\Gamma$ and $S_N$, we use the jet steps
\begin{equation}\label{e:end-jet-step}
 S^{(\alpha)} = \chi_{\rm inv} \, s_\Gamma w_0 J_\alpha K^\varpi \chi_{\rm inv} ,
\end{equation}
where $J_\alpha = w_\Gamma^{-1} \partial_Y^\alpha w_{\Gamma, Y} |_{Y = 0}$ is the ratio of a complete local jet of degree $d_\alpha$, a derivative at $Y = 0$ of the marked onsite weight $w_{\Gamma, Y}$ of \eqref{e:rep-weight}, to the onsite weight.

We first record the damping that the source factor $s_\Gamma$ provides on the parameter ball $\mathcal U_2$. It needs only the coercivity \eqref{e:end-damping} of the parameter balls. Recall from \zcref{s:stab-main} that, at the point $R(x, 0)$ of the saddle manifold, the BB block has the body $\kappa_E \Id - \iu s_E Q_B$, where $Q_B = Q_B(x)$ lies in the BB factor $H_k$ of the underlying manifold, and recall the body weight $v$ of \eqref{e:sigma-weight}.

\begin{lemma}\label{l:end-coercivity}
Let $\Gamma \in \mathcal U_2$. For every $Q_B \in H_k$, with body weight $v = 1 + \Tr(\Lambda_B Q_B)/2$,
\begin{equation}\label{e:end-coercivity}
 \Re \Tr(Q_B \Gamma_{BB}) \ge c_\eps \Tr(\Lambda_B Q_B) \ge c_\eps v ,
\end{equation}
where $c_\eps$ is the constant of \eqref{e:end-damping}. Moreover, there exists $c_0 > 0$, depending on $\mathcal U_2$, such that the source factor satisfies $|s_\Gamma|_1 \le e^{C_k \eta - c_0 \eta v}$ on the real tube.
\end{lemma}

\begin{proof}
We first prove \eqref{e:end-coercivity}. Write $Q_B = g \Lambda_B g^{-1}$ with $g \in U(k, k)$, as in the proof of \zcref{l:sigma-body}. Since $g^{-1} = \Lambda_B g^* \Lambda_B$, the matrix $\Lambda_B Q_B = (\Lambda_B g)(\Lambda_B g)^*$ is positive definite. Using $\Lambda_B^2 = \Id$, we write $\Tr(Q_B \Gamma_{BB}) = \Tr(\Lambda_B Q_B \, \Gamma_{BB} \Lambda_B)$, and the real part of this trace is the trace of $(\Lambda_B Q_B)^{1/2} \Re(\Gamma_{BB} \Lambda_B) (\Lambda_B Q_B)^{1/2}$, where $\Re(\Gamma_{BB} \Lambda_B)$ is the Hermitian part of $\Gamma_{BB} \Lambda_B$. Using \eqref{e:end-damping}, we obtain
\[
 \Re \Tr(Q_B \Gamma_{BB}) \ge c_\eps \Tr(\Lambda_B Q_B) = 2 c_\eps (v - 1) .
\]
By \zcref{l:sigma-body}\ref{i:sigma-body-trace}, we have $v - 1 \ge k \ge 1$, and therefore $2(v - 1) \ge v$. This proves \eqref{e:end-coercivity}.

We now bound the source factor. By \eqref{e:hist-normalized}, the body of the bosonic part of $s_\Gamma$ at $R(x, 0)$ is
\[
 e^{-\iu \eta \Tr((\kappa_E \Id - \iu s_E Q_B) \Gamma_{BB})} = c_\Gamma \, e^{-\eta s_E \Tr(Q_B \Gamma_{BB})} ,
 \qquad
 c_\Gamma = e^{-\iu \eta \kappa_E \Tr \Gamma_{BB}} .
\]
The imaginary part of $\Tr(Q_B \Gamma_{BB})$ gives a factor of modulus one, and $|c_\Gamma| \le e^{C_k \eta}$, since $|\Tr \Gamma_{BB}| \le k/2$ on $\mathcal U_2$. By \eqref{e:end-coercivity}, the modulus of this body is at most $e^{C_k \eta - c_\eps s_0 \eta v}$. The normal remainder of the source has size $\delta_W O_k(|U|) v$, which is bounded by a fixed fraction of the same damping on the tube. The nilpotent source terms carry the factors of the degree metric and damped envelopes of the form $\delta_W v$, and the compact FF terms contribute only a scalar error $O_k(\delta_W)$. These bounds give $|s_\Gamma|_1 \le e^{C_k \eta - c_0 \eta v}$ for some $c_0 > 0$ that depends on $\mathcal U_2$.
\end{proof}

The following lemma bounds the onsite factors $s_\Gamma w_0$, $w_0$ and $s_\Gamma w_0 J_\alpha$ of $S_\Gamma$, $S_N$ and $S^{(\alpha)}$ on the compact normal graphs.

\begin{lemma}\label{l:end-kick}
There exist universal constants $c_{\rm kick}, C_2 > 0$ such that the following holds at fixed rank $k$, after choosing $R_k$ sufficiently small and then $W$ sufficiently large. Let $\Gamma \in \mathcal U_2$, let $b$ be a real normal direction with $|b| \le R_k$, let $\lambda \in [0, 1]$, and let $w$ be one of the onsite factors $s_\Gamma w_0$ and $w_0$. Then at every point $(x, U)$ of the tube with $U$ real and in the support of $\chi_{\rm inv}$,
\begin{equation}\label{e:end-kick}
 |w(U_\lambda)|_1 \le \exp\bigl\{ C_k/W - c_{\rm kick} |U|^2 + C_2 |\beta|^2 + C_k R_k |\beta|^2 \bigr\} ,
 \qquad \beta = \lambda \psi_{R_k}(U) b .
\end{equation}
Moreover, the ratio $J_\alpha$ of \eqref{e:end-jet-step} satisfies $|J_\alpha|_1 \le C_k v^{d_\alpha}$ at the points $T(x)(R_* + U)T(x)^{-1}$ with $U$ complex in the tube and $T(x)$ a complex frame near a real one, and the onsite factor $s_\Gamma w_0 J_\alpha$ satisfies \eqref{e:end-kick} with the additional factor $C_k v^{d_\alpha}$.
\end{lemma}

\begin{proof}
We first bound the body of $w_0$. Recall that the weight $w_0$ is invariant under conjugation, so that on the tube it is a function of $R_* + U$ alone. By the proof of \zcref{l:tube-quadratic}, its normal Hessian at $R_*$ is $m_\sigma \Str(U^2)$ on a root block of type $\sigma$, and on the adapted real normal space of \zcref{d:tube-cycle} this form is $m_\sigma e^{-2\iu\theta_\sigma} |U|^2$. Since $|m_\sigma| \ge 2 s_0^2$ and $|\arg(m_\sigma e^{-2\iu\theta_\sigma})| \le C|E|$, its real part is at least $4 c_{\rm kick} |U|^2$ for a universal $c_{\rm kick} > 0$.

Let $q_2$ be this quadratic form, and write the body of $\log w_0(R_* + U) - \log w_0(R_*)$ as $-q_2(U)/2 + r_{\rm T}(U)$. The remainder $r_{\rm T}$ is analytic on the tube, and
\[
 r_{\rm T}(0) = Dr_{\rm T}(0) = D^2 r_{\rm T}(0) = 0 ,
 \qquad
 \|D^3 r_{\rm T}\| \le C_k .
\]
Taylor's formula gives
\begin{equation}\label{e:end-taylor}
 |r_{\rm T}(U + \iu\beta) - r_{\rm T}(U) - \iu Dr_{\rm T}(U)\beta| \le C_k (R_k + |\beta|) |\beta|^2 ,
 \qquad |Dr_{\rm T}(U)| \le C_k |U|^2 .
\end{equation}
Let $U$ and $\beta$ be real, with $|\beta| \le R_k$ and $C_k R_k \le c_{\rm kick}/4$. Using Young's inequality on the cross term $\Im(m_\sigma e^{-2\iu\theta_\sigma}) \langle U, \beta\rangle$, together with \eqref{e:end-taylor} and $|r_{\rm T}(U)| \le C_k R_k |U|^2$, we obtain
\[
 \begin{gathered}
 \Re\Bigl(-\frac{q_2(U + \iu\beta)}{2}\Bigr) \le -\frac{3 c_{\rm kick}}{2} |U|^2 + C |\beta|^2 , \\
 \Re\Bigl(-\frac{q_2(U + \iu\beta)}{2} + r_{\rm T}(U + \iu\beta)\Bigr) \le -c_{\rm kick} |U|^2 + C_2 |\beta|^2 + 2 C_k R_k |\beta|^2 .
 \end{gathered}
\]
Since $|w_0(R_*)| = 1$ by the saddle normalization, the second bound controls the body of $w_0$ on the graph.

We now bound the remaining factors in \eqref{e:end-kick}. The nonbody coefficients of $w_0$, divided by its body, are analytic and bounded on the tube, since every normal determinant has nonzero body there (\zcref{r:tube-transport}). Each of them contains at least two standardized odd normal coordinates, and hence a factor $h^2$. Next, by \zcref{l:end-coercivity}, the source factor satisfies $|s_\Gamma|_1 \le e^{C_k \eta - c_0 \eta v}$ on the real tube, with $c_0 > 0$ depending on the damping ball $\mathcal U_2$. Moving to complex normal points changes the exponent of $s_\Gamma$ by at most $C \eta |\beta| v \le C R_k \eta v$. We choose $R_k$ so small that $C R_k \le c_0/2$. Then half of the damping absorbs this change, and $|s_\Gamma(U_\lambda)|_1 \le e^{C_k \eta - c_0 \eta v/2}$. Since $\eta \le C W^{-2}$ and $1 + C_k h^2 \le e^{C_k/W}$, these bounds give \eqref{e:end-kick}. Since $|e^{c' \eta v} s_\Gamma(U_\lambda)|_1 \le e^{C_k \eta}$ for $0 \le c' \le c_0/2$, the same argument shows that \eqref{e:end-kick} also holds with $s_\Gamma$ replaced by $e^{c' \eta v} s_\Gamma$, for every $0 \le c' \le c_0/2$.

Finally, we treat a jet step. The ratio $J_\alpha$ is a polynomial of degree $d_\alpha$ in the entries of
\[
 (E - S)^{-1} = T(x) (E - R_* - U)^{-1} T(x)^{-1} .
\]
At the points of the statement, $E - R_* - U$ has a bounded inverse and $\|T(x)\| \|T(x)^{-1}\| \le C v$ by \eqref{e:hist-boost-props}, and therefore these entries are bounded by $C v$. We conclude that $|J_\alpha|_1 \le C_k v^{d_\alpha}$, with no growth in $\langle Z \rangle$, and $s_\Gamma w_0 J_\alpha$ satisfies \eqref{e:end-kick} with the additional factor $C_k v^{d_\alpha}$.
\end{proof}

\begin{lemma}\label{l:end-energy}
There exist universal constants $a_0, v_0, c_1, C_1 > 0$ such that the following holds at fixed rank $k$, after choosing $R_k$ sufficiently small and then $W$ sufficiently large. Let $1 \le n \le C_0 W$, let each of $S_1, \dots, S_n$ be $S_\Gamma$ or $S_N$, with $\Gamma \in \mathcal U_2$, and let $p$ be fixed. Let $a_0', \dots, a_n'$ satisfy $|a_j'| \le a_0$ and $|a_j' - a_{j - 1}'| \le v_0 h^2$, let $b$ be a real normal direction with $|b| \le c_1 R_k$, let $\lambda_0, \dots, \lambda_n \in [0, 1]$, and set $\beta_j = \lambda_j b$. Then
\begin{equation}\label{e:end-energy}
 \|S_n \cdots S_1 f\|_{\lambda_n; p, a_n'}
 \le C_k \exp\Bigl\{ \sum_{j = 1}^n \bigl[ C_1 \bigl(|\beta_j|^2 + W^2 |\beta_j - \beta_{j - 1}|^2\bigr) + C_k R_k |\beta_j|^2 \bigr] \Bigr\} \|f\|_{\lambda_0; p, a_0'}
\end{equation}
for all $f$ if $\lambda_0 = 0$, and for all $f$ that are holomorphic in the normal variables where $\psi_{R_k} \ne 0$ if $\lambda_0 > 0$. The constant $C_k$ may also depend on $C_0$. Moreover, let a fixed number $J$ of the $S_j$ be replaced by jet steps $S^{(\alpha_1)}, \dots, S^{(\alpha_J)}$ of \eqref{e:end-jet-step}, and let $d = d_{\alpha_1} + \dots + d_{\alpha_J}$ be their total degree. Then, for the same $f$,
\begin{equation}\label{e:end-jet-constant}
 \|S_n \cdots S_1 f\|_{\lambda_n; p, a_n'}
 \le C_k^{1 + J} \exp\Bigl\{ \sum_{j = 1}^n \bigl[ C_1 \bigl(|\beta_j|^2 + W^2 |\beta_j - \beta_{j - 1}|^2\bigr) + C_k R_k |\beta_j|^2 \bigr] \Bigr\} \|f\|_{\lambda_0; p + d, a_0'} .
\end{equation}
\end{lemma}

\begin{proof}
We first represent the block by evolutions on moving graphs. The kernel of $S_n \cdots S_1$ is an iterated integral over the integration variables between consecutive steps, which we call the \emph{nodes}. Where the cutoffs equal one, its integrand is holomorphic in the normal variables of each node, since the onsite factors are analytic on the tube and the ambient Gaussian is entire in the matrix point, which is affine in $U$ at each point of the saddle manifold. By Cauchy's theorem, we may move the normal integration at node $j$ to the graph of parameter $\lambda_j$, for $1 \le j < n$, without changing the kernel. The output is holomorphic where its cutoff equals one, and we evaluate it on the graph of parameter $\lambda_n$.

Fix $1 \le j \le n$, let $\lambda(t)$ interpolate linearly from $\lambda_{j - 1}$ to $\lambda_j$ over $0 \le t \le \tau = \tau_{\rm phys}$, and set $\beta(t) = \lambda(t) b$. We use the graph pull $\mathcal P_\lambda$ defined before \eqref{e:cons-moving}: the initial vector for the pulled heat is $\mathcal P_{\lambda_{j-1}}F$, including its half-Jacobian, where $F$ is the raw input of this step. Evaluating the output with $\mathcal P_{\lambda(t)}$ gives \eqref{e:cons-pulled}. By \zcref{l:cons-identification}, it agrees on the support of the cutoffs with the Dirichlet evolution of \eqref{e:cons-moving} on a slightly larger tube, up to an error $C_k W^{C_k} e^{-c R_k^2 W^2}$ times the input norm. The weights below are bounded by $e^{C W}$ on the tube. We may therefore neglect this error, and we work with the Dirichlet evolution $u(t)$ of $\partial_t u = (L_t + V_t) u$, which satisfies the energy identity \cite{lions1961equations}. For $S_N$ the operator $L_t$ is replaced by its normal part; the bounds below hold for both.

We now bound the forms. The ordinary part of $\Delta_{\rm nor}$ has constant coefficients: on a root block of type $\sigma$ it is $e^{2\iu \theta_\sigma}$ times the flat Laplacian of the real coordinate, and on the graph its coefficient matrix differs from this by at most $C|b|/R_k$. The horizontal part satisfies \eqref{e:cons-sectorial} by \zcref{l:cons-sectorial}, and its normal-energy term is absorbed by the normal part for $R_k$ small. The horizontal coefficients at $U + \iu \beta$ differ from those at $U$ by $O_k(|\beta|)$. The odd normal part of $\Delta_{\rm nor}$ has norm $C_k h^{-2}$. The first- and zeroth-order coefficients from the half-densities, the graph Jacobian, and the body weight $v^p$ are bounded by $C_k(R_k)$.

Next, we conjugate by the real norm weights $\psi_t$ specified below, which satisfy $|\nabla_U\psi_t|\le C W|U|$ with universal $C$. Outside the deformed region they are linear combinations of the rootwise Gaussian quadratics $|Z_\sigma|^2$ and $|U|^2$. The holomorphic Gaussian factor in \eqref{e:end-graph-norm} has modulus $e^{a\Re(Z_\lambda^TZ_\lambda)}$. We therefore apply this real multiplier directly to the pulled solution; its evolution contains $\partial_t\psi_t+e^{\psi_t}(L_t+V_t)e^{-\psi_t}$, without any need to conjugate by the complex phase.

The body weight $v^p$ commutes with the normal velocity. Writing $z=e^{\psi_t}v^pu$, we have
\[
 e^{\psi_t}V_te^{-\psi_t}=V_t-\mathsf v_t\cdot\nabla_U\psi_t.
\]
By \eqref{e:cons-velocity-bound}, \eqref{e:cons-graph-velocity}, and Young's inequality, for all $\varepsilon,\mu>0$,
\begin{equation}\label{e:end-moving-velocity}
 \begin{aligned}
 |\langle z,e^{\psi_t}V_te^{-\psi_t}z\rangle|
 &\le\varepsilon\mathcal E_{\rm nor}(z)
      +\mu W^2\||U|z\|^2\\
 &\qquad+C_{\varepsilon,\mu}|\lambda'b|^2\|z\|^2.
 \end{aligned}
\end{equation}
All constants in this velocity estimate are independent of rank, dimension, and the graph radius. In particular, no term proportional to $|\lambda'b|/R_k$ is introduced.

The normal charges require separate weight estimates. Outside $\{|U|\le2R_k\}$ the graph is the identity, its half-Jacobian is one, and every compact even normal charge annihilates the rootwise quadratics in $\psi_t$. Thus the even-charge weight commutators are supported where $|U|\le2R_k$. There their ordinary derivative coefficients are bounded by $C_k(|U|+|\beta|)\le C_kR_k$, giving a commutator bound $C_kR_k|\nabla_U\psi_t|$. An odd charge has ordinary derivative coefficients $O_k(h)$, also after the uniformly bounded graph change, so its weight commutator is $O_k(h|\nabla_U\psi_t|)=O_k(h^{-1}|U|)$ throughout the fixed enlarged tube. Estimate these commutators and their products by the adjoint bounds \eqref{e:cons-charge-adjoint}. Their forms are bounded by a small fraction of the gradient energies, $C_kR_k^2W^2\||U|z\|^2$, and $C_kh^{-2}\|z\|^2$. Fix the universal value of $\mu$ used below before decreasing $R_k$ at fixed rank, and absorb the middle term into a prescribed fraction of $\mu W^2\||U|z\|^2$. The enlarged Dirichlet tube need not have radius comparable to $R_k$. This uses direct odd-charge bounds, rather than invariance of an ordinary radial weight under odd charges.

Combining these estimates with the normal diffusion form, and choosing $c_1$ and $R_k$ sufficiently small, gives
\begin{equation}\label{e:end-energy-form}
\begin{aligned}
 \frac{\dd}{\dd t} \|e^{\psi_t} v^p u\|^2
 &\le 2 \bigl\langle e^{\psi_t} v^p u, \bigl(\partial_t \psi_t + C_5 |\nabla_U \psi_t|^2 + \mu W^2 |U|^2\bigr) e^{\psi_t} v^p u \bigr\rangle\\
 &\qquad + C_\mu \bigl(|\lambda' b|^2 + W^2 |\beta(t)|^2 + C_k h^{-2}\bigr) \|e^{\psi_t} v^p u\|^2 ,
\end{aligned}
\end{equation}
where $C_5$ is universal and $C_\mu$ depends only on $\mu$. The constant $C_5$ is universal because the normal metric is flat. Indeed, for the operator $a \Delta$ with a complex constant $a$ of positive real part and a real weight, the divergence terms in $e^{\psi} a \Delta e^{-\psi}$ cancel exactly, and
\[
 \Re \langle u, e^{\psi} a \Delta e^{-\psi} u\rangle
 \le -(\Re a - |\Im a|) \|\nabla u\|^2 + (\Re a + |\Im a|) \|(\nabla \psi) u\|^2 .
\]
Here $a = e^{2\iu\theta_\sigma}$, and the graph changes the coefficient of $|\nabla_U \psi|^2$ by a factor $1 + C(|E| + |b|/R_k)$. We may thus take $C_5 = 3$, after reserving the small gradient fractions in the preceding estimates. The velocity contribution is \eqref{e:end-moving-velocity}; the rank-dependent weight commutators enter only the allotted $\mu W^2|U|^2$ term and $C_kh^{-2}$ term. Thus the coefficient of $|\lambda'b|^2$ remains universal.

We next choose the weight on the heat interval of step $j$. Let $\alpha(t)$ interpolate linearly from $a_{j - 1}'$ to $a_j'$, and define
\[
 \psi_t = \alpha(t) \, \Re\bigl(Z_{\lambda(t)}^T Z_{\lambda(t)}\bigr) - \frac{c_{\rm kick} t}{2\tau} |U|^2 ,
\]
where $c_{\rm kick} > 0$ is the universal constant of \zcref{l:end-kick}. Since $\Re(Z_\lambda^T Z_\lambda) = |Z|^2 - \lambda^2 \psi_{R_k}^2 |Z_b|^2$, $|b| \le c_1 R_k$ and $|\alpha'| \le 2 v_0 W$, we have $|\nabla_U \psi_t| \le C a_0 W |U| + c_{\rm kick} |U|$ and
\[
 \partial_t \psi_t + C_5 |\nabla_U \psi_t|^2 + \mu W^2 |U|^2
 \le (6 v_0 + C a_0^2 + 2\mu - c_{\rm kick}) W^2 |U|^2 + C \bigl(|\lambda' b|^2 + W^2 |\beta(t)|^2\bigr)
\]
for $W$ large. We choose $\mu = c_{\rm kick}/8$, and $a_0$ and $v_0$ so small that $6 v_0 + C a_0^2 \le c_{\rm kick}/4$; the coefficient of $W^2 |U|^2$ is then negative. Inserting this bound into \eqref{e:end-energy-form} and integrating over $[0, \tau]$, we obtain
\begin{equation}\label{e:end-heat-step}
 \|e^{\psi_\tau} v^p u(\tau)\|
 \le \exp\bigl\{ C \bigl(W^2 |\beta_j - \beta_{j - 1}|^2 + |\beta_{j - 1}|^2 + |\beta_j|^2\bigr) + C_k/W \bigr\} \|e^{\psi_0} v^p u(0)\| ,
\end{equation}
where we used $W^2 \tau = 1/2$, $C_k h^{-2} \tau = C_k/(2W)$ and
\[
 \int_0^\tau |\lambda'|^2 \dd t = \frac{|\lambda_j - \lambda_{j - 1}|^2}{\tau} = 2 W^2 |\lambda_j - \lambda_{j - 1}|^2 .
\] At $t = 0$ the weight is that of $\| \cdot \|_{\lambda_{j - 1}; p, a_{j - 1}'}$, and at $t = \tau$ it is the weight of $\| \cdot \|_{\lambda_j; p, a_j'}$ divided by $e^{c_{\rm kick} |U|^2/2}$. At a shared node the output and input use the same pull $\mathcal P_{\lambda_j}$, so their half-Jacobian conventions agree exactly. No comparison of distinct coefficient metrics is made there. These estimates hold for arbitrary, including nonmonotone, choices of the $\lambda_j$.

The onsite factor $w \in \{s_\Gamma w_0, w_0\}$ of $S_j$ makes up this deficit. By \zcref{l:end-kick}, which applies since $|b| \le c_1 R_k \le R_k$, multiplication by $w$ and by the cutoff of $S_j$ maps the weight $e^{\psi_\tau}$ at $t = \tau$, which is the weight of $\| \cdot \|_{\lambda_j; p, a_j'}$ divided by $e^{c_{\rm kick} |U|^2/2}$, to the weight of $\| \cdot \|_{\lambda_j; p, a_j'}$, and it changes the norm by at most $\exp\{C_k/W + C_2 |\beta_j|^2 + C_k R_k |\beta_j|^2\}$. Multiplying the resulting bounds over $j$, and using $|\beta_{j - 1}|^2 \le 2 |\beta_j|^2 + 2 |\beta_j - \beta_{j - 1}|^2$ and $\sum_j C_k/W \le C_k C_0$, we obtain \eqref{e:end-energy}.

For a jet step $S^{(\alpha_i)}$, \zcref{l:end-kick} gives the additional factor $C_k v^{d_{\alpha_i}}$. We absorb it by lowering the body weight by $d_{\alpha_i}$ from that step on, and the rest of the argument is unchanged. The input then carries the body weight $p + d$, and the $J$ jet steps contribute the factor $C_k^J$, which proves \eqref{e:end-jet-constant}.
\end{proof}

The same energy method with an exponential weight in the normal coordinate gives a Davies--Gaffney estimate \cite{davies1989heat}, which \zcref{s:end-attach} uses to show that a disturbance created at normal distance of order one from the saddle manifold is exponentially small near it after one block. We state it on the tube of one signature $\alpha$, as in \zcref{s:end-attach}.

\begin{lemma}\label{l:end-davies}
There exists a universal constant $C > 0$ such that the following holds at fixed rank $k$, after choosing $R_k$ sufficiently small and then $W$ sufficiently large. Let $\alpha$ be a signature and $p \in \R$. Let $L_\alpha^{\rm raw}$ be the generator of the raw local heat, that is, of the ambient heat in the raw realization on a slightly larger Dirichlet tube of the signature $\alpha$, with all odd coordinates scaled by $h$, and let $\mathcal E_\alpha^{\rm real}(u) = \mathcal E_{\rm tan}(u) + \mathcal E_{\rm nor}(u)$ be the real gradient energy of \zcref{l:cons-sectorial}. Let $d_{\rm n}$ be a real function of the normal coordinate with Lipschitz constant one whose gradient $\nabla_U d_{\rm n}$ vanishes outside $\{|U| \le 2 R_k\}$, and let $a_{\rm D} \in \R$. Then there exists $c > 0$ such that, in the raw norm with the body weight $v^p$, for all $u$ in the form domain and all $t \ge 0$,
\begin{equation}\label{e:end-davies}
 \Re \langle u, L_\alpha^{\rm raw} u\rangle \le -c \, \mathcal E_\alpha^{\rm real}(u) + C_k W \|u\|^2 ,
 \qquad
 \|e^{a_{\rm D} d_{\rm n}} e^{t L_\alpha^{\rm raw}} e^{-a_{\rm D} d_{\rm n}}\| \le e^{(C_k W + C a_{\rm D}^2) t} .
\end{equation}
Moreover, for the onsite factors $w \in \{s_\Gamma w_0, w_0\}$ with $\Gamma \in \mathcal U_2$, the local step satisfies
\[
 \|e^{a_{\rm D} d_{\rm n}} w \, e^{\tau_{\rm phys} L_\alpha^{\rm raw}} e^{-a_{\rm D} d_{\rm n}}\| \le e^{C_k/W + (C_k W + C a_{\rm D}^2) \tau_{\rm phys}} .
\]
\end{lemma}

\begin{proof}
We first prove the form bound, using the orthogonal normal and horizontal decomposition \eqref{e:cons-decomposition}. By \zcref{l:cons-sectorial}, the ordinary principal part is a uniformly accretive perturbation of the flat normal metric and of the orbit metric of the group chart, with a universal margin in the normal block. A nilpotent ordinary principal coefficient contains two odd coordinates, so that it is $O(h^2)$, and it is absorbed in that margin. In the coordinates scaled by $h$, one odd normal derivative has norm $h^{-1}$, and the odd second-order part has norm at most $C_k h^{-2} = C_k W$. By Young's inequality, the remaining finite matrix terms and the terms of the transported frame are bounded by a small multiple of $\mathcal E_\alpha^{\rm real}(u)$ plus $C_k W \|u\|^2$, and so are the terms that the body weight $v^p$ adds, with constants depending on $p$. This proves the first bound in \eqref{e:end-davies}.

For the second bound we use Davies' method of exponential weights \cite{davies1989heat}. The weight $e^{a_{\rm D} d_{\rm n}}$ depends only on the normal coordinate. On the flat normal part, which is $e^{2\iu\theta_\sigma}$ times the flat Laplacian on a root block of type $\sigma$, the divergence terms cancel exactly, as in the proof of \zcref{l:end-energy}, and since $|\nabla_U d_{\rm n}| \le 1$ we have
\[
 \Re \langle u, e^{a_{\rm D} d_{\rm n}} e^{2\iu\theta_\sigma} \Delta e^{-a_{\rm D} d_{\rm n}} u\rangle
 \le -(\cos 2\theta_\sigma - |\sin 2\theta_\sigma|) \|\nabla u\|^2 + 2 a_{\rm D}^2 \|u\|^2 .
\]
In $L_H(U)$, the normal derivatives come only from the normal charges, which are first-order operators in $U$ with coefficients of size $O(|U|)$. In the form of the conjugated operator, their action on $e^{\pm a_{\rm D} d_{\rm n}}$ produces terms that carry the factor $a_{\rm D} \nabla_U d_{\rm n}$. By hypothesis, this factor vanishes outside $\{|U| \le 2 R_k\}$, where the coefficients of the normal charges are at most $C_k R_k$. By Young's inequality, these terms are therefore bounded by a small multiple of $\mathcal E_{\rm tan}(u)$ plus $C_k (2 R_k)^2 a_{\rm D}^2 \|u\|^2$, and $4 C_k R_k^2 \le 1$ once $R_k$ is small. Therefore
\[
 \Re \langle u, e^{a_{\rm D} d_{\rm n}} L_\alpha^{\rm raw} e^{-a_{\rm D} d_{\rm n}} u\rangle \le C_k W \|u\|^2 + C a_{\rm D}^2 \|u\|^2
\]
with a universal $C$, and the energy inequality for $e^{a_{\rm D} d_{\rm n}} e^{t L_\alpha^{\rm raw}} e^{-a_{\rm D} d_{\rm n}} u$ gives the second bound in \eqref{e:end-davies}. Finally, multiplication by $e^{a_{\rm D} d_{\rm n}}$ commutes with every onsite multiplier, and by \zcref{l:end-kick} with $b = 0$, multiplication by $w$ has norm at most $e^{C_k/W}$. This implies the bound for the local step.
\end{proof}

We set $c_{\rm D} = 1/(4C)$, with the universal constant $C$ of \zcref{l:end-davies}. It is the rate of the Gaussian decay in the barrier \eqref{e:end-barrier} below.

Since the energy interval of \zcref{t:interface} depends on the coefficient $C_1$ of \zcref{l:end-energy} through \eqref{e:end-Egeom} below, we now list its sources and check that it does not depend on the rank.

\begin{remark}\label{r:end-universal}
Summing \eqref{e:end-heat-step} and \eqref{e:end-kick} over the steps, with $|\beta_{j - 1}|^2 \le 2|\beta_j|^2 + 2|\beta_j - \beta_{j - 1}|^2$, shows that we may take $C_1 = 3C + C_2$, where $C$ is the constant of \eqref{e:end-heat-step} and $C_2$ that of \eqref{e:end-kick}. The constant $C$ comes from the time derivative of the weight, with the universal parameters $a_0$, $v_0$, and $c_{\rm kick}$, from the velocity term, through Young's inequality against the flat normal energy, and from the flat normal metric through $C_5$. The constant $C_2$ comes from the quadratic part $q_2$ of $\log w_0$, which is the scalar $m_\sigma$ times $\Str(U^2)$, and from Young's inequality with the universal $c_{\rm kick}$.

None of these involves the dimension of the normal space: the flat normal metric and the scalar Hessian act on the Hilbert--Schmidt norm, and the half-divergence in the moving-graph velocity cancels exactly in \eqref{e:cons-velocity-form}. The rank-one inverse bound \eqref{e:cons-graph-velocity} makes the velocity cost in \eqref{e:end-moving-velocity} universal, with no loss proportional to the total variation of the graph heights. The rank appears only in three ways: through the cubic Taylor remainder $r_{\rm T}$ and the horizontal coefficients at complex points, which come with a factor $R_k$ and appear as $C_k R_k |\beta_j|^2$; through the smallness of $|b|/R_k$ and $R_k$ needed to absorb the graph terms of the horizontal operator; and through prefactors $e^{C_k/W}$ per step, which include the odd-charge weight commutators and give $C_k$ over a block. The constants $c_1$, $a_0$, and $v_0$ are universal for the same reasons.

Consequently the constant $C_n = 3 C_1$ of \eqref{e:end-normal-block} below is universal. The constants $C_{\rm geo}$, $C_*$, and $c_{\rm D}$ are dimension-free as well, by \zcref{l:end-height}, by \zcref{p:end-shifted}, whose FF part rests on \zcref{l:end-ff-strip}, and by \zcref{l:end-davies}, whose constant comes from the flat normal metric and from the normal charges. The normal charges act on the Davies weight only where its gradient is nonzero, which by the hypothesis of that lemma is inside $\{|U| \le 2 R_k\}$, and there they contribute at most $C_k R_k^2 a_{\rm D}^2 \le a_{\rm D}^2$, for the parameter $a_{\rm D}$ of that lemma, once $R_k$ is small. Therefore the energy threshold $E_{\rm geom}$ of \eqref{e:end-Egeom} is universal.
\end{remark}

We now introduce the terms in which the rest of Part IV states its Gaussian bounds. Let $a \in \R$. We say that a function $f$ on the tube has \emph{Gaussian margin} $a$ if $\|w_a f\| = \|e^{a|Z|^2} f\|$ is finite, in the body weights at hand. By \zcref{l:end-raw-norms}, the endpoint factors $g_R$ and $g_L$ have every sufficiently small negative margin $-a$, but no positive margin. We call a sequence $a_0', \dots, a_n'$ with $a_0' = -a$, $a_n' = a$, and $|a_j' - a_{j - 1}'| \le v_0 h^2$, as in \zcref{l:end-energy}, a \emph{signed ramp} of length $n$, and we call a family \emph{prepared} if it has a positive Gaussian margin $a$ with $0 < a < a_0$. In the symmetric normal gauge, \zcref{l:end-energy} gives the following bound along a signed ramp, which turns a negative margin into a positive one.

\begin{corollary}\label{c:end-ramp}
Choose $0 < a < a_0$ and $\varrho > 0$ so that $2a/\varrho < v_0$. For $\varrho W \le n \le C_0 W$,
\[
 \|w_a \mathbf F_h^n w_a\| + \|w_a A_h^n w_a\| \le C_{k, a, \varrho, C_0} ,
\]
uniformly for $\Gamma \in \mathcal U_2$ and in every fixed body weight. Constant Gaussian weights $w_{a'}$ with $|a'| \le a_0$ are likewise stable over $0 \le n \le C_0 W$. If a fixed local source multiplier $m_{\rm src}$ has coefficient growth $C_k v^d \langle Z \rangle^{d'}$, then for $0 < a' < a$ it maps the input norm with Gaussian weight $w_a$ and body weight $v^{p + d}$ to the norm with $w_{a'}$ and body weight $v^p$, with a norm that does not depend on $h$. If $d' = 0$, we may take $a' = a$.
\end{corollary}

\begin{proof}
We take $a_j' = -a + 2aj/n$, so that
\[
 |a_j' - a_{j - 1}'| = \frac{2a}{n} \le \frac{2a}{\varrho W} \le v_0 h^2 .
\] Applying \zcref{l:end-energy} with $b = 0$, we obtain $\|w_a S_\Gamma^n w_a\|_{\rm raw} \le C_k$ and the same bound for $S_N$, in every body weight $v^p$. By \eqref{e:end-similarity} and \eqref{e:end-conversion}, the same bounds hold for $A_h^n$ and $\mathbf F_h^n$. The assertion for constant weights uses $a_j' = a'$ for all $j$. Finally, $\sup_Z \langle Z\rangle^{d'} e^{-(a - a')|Z|^2} < \infty$, which proves the multiplier bound, and for $d' = 0$ the multiplier commutes with $w_a$.
\end{proof}

We now separate the positive margin produced by the signed ramp from the negative Gaussian norms of the endpoint factors $g_R$ and $g_L$.

\begin{corollary}\label{c:end-margin}
Let $a$ and $\varrho$ be as in \zcref{c:end-ramp}, and let $\varrho W \le n \le C_0 W$. In the balanced signature,
\[
 \|w_a \mathbf F_h^n g_R\| + \|w_a \mathbf F_h^n g_L\| \le C_{k, a, \varrho, C_0} ,
\]
and at every other signature the same norms are at most $C_{k, a, \varrho, C_0} W^{A_k}$. The same bounds hold in the body weight $v^{p_0}$ of \zcref{l:end-raw-norms}.
\end{corollary}

\begin{proof}
Let $\widetilde\chi$ be a real cutoff as in \zcref{l:end-raw-norms} that equals one on the support of $\chi_{\rm inv}$, so that $\mathbf F_h = \mathbf F_h \widetilde\chi$. Then
\[
 \|w_a \mathbf F_h^n g\| \le \|w_a \mathbf F_h^n w_a\| \, \|e^{-a|Z|^2} \widetilde\chi g\| ,
\]
and \zcref{c:end-ramp} and \zcref{l:end-raw-norms} give the bounds. The weight $v^{p_0}$ commutes with $\mathbf F_h$.
\end{proof}

We now state the bound for a block of adapted steps on a normal graph in the symmetric normal gauge. Let $b$ be a real normal direction with $|b| \le c_1 R_k$, and let $U_1 = U + \iu \psi_{R_k}(U) b$ be the compact normal graph \eqref{e:cons-graph} with $\bar r = R_k$ and $\lambda = 1$. It has the constant height $b$ on the inner tube $\{|U| \le R_k\}$, and it is the identity wherever $\chi_{\rm inv} \ne 1$. For $a \in \R$, a body weight $p$, and a function $f$ on the tube that is holomorphic in the normal variables where $\psi_{R_k} \ne 0$, we define the graph norm in the symmetric normal gauge
\begin{equation}\label{e:end-gauge-graph-norm}
 \|f\|_{b; p, a} = \bigl\| w_a(U_1) \, v^p \rho_1^{1/2} \, f(U_1) \bigr\| ,
\end{equation}
where the norm on the right is the positive Wick metric of \zcref{d:tube-wick}. For $b = 0$, the norm \eqref{e:end-gauge-graph-norm} is $\|w_a v^p f\|$. By the version of \eqref{e:end-conversion} on graphs, $\|f\|_{b; p, a} \le C_k \|w_0^{1/2} \gamma_h f\|_{1; p, a}$, with the raw graph norm \eqref{e:end-graph-norm}.

\begin{proposition}\label{p:end-coupled}
There exist universal constants $a, C_n > 0$ and a universal $C_0$ such that, after choosing the coefficient metric and $R_k$ at fixed rank, the following holds for the adapted kernel. Let $m = \lceil C_0 W \rceil$, let $\Gamma \in \mathcal U_2$, let $p \in \R$, and let $b$ be a real normal direction with $|b| \le c_1 R_k$. Then for all $g$ on the real tube,
\begin{equation}\label{e:end-normal-block}
 \|A_h^m g\|_{b; p, a} \le C_k(R_k) \exp\bigl\{ C_n C_0 W |b|^2 + C_k W R_k |b|^2 \bigr\} \|w_{-a} v^p g\| ,
\end{equation}
with $C_n = 3 C_1$. The input has the Gaussian margin $-a$, and the output has the Gaussian margin $a$, on the real tube ($b = 0$) and on the graph. The estimates are uniform on the nested damping balls. For an input $f$ in the raw realization that is holomorphic in the normal variables where $\psi_{R_k} \ne 0$ and has $\|f\|_{1; p, a} < \infty$, using the same normal graph at every node gives, for every $0 \le j \le C_0 W$, including $j = 0$,
\[
 \|S_\Gamma^j f\|_{1; p, a} \le C_k \exp\bigl\{ (C_1 + C_k R_k) j |b|^2 \bigr\} \|f\|_{1; p, a} .
\]
If a prescribed finite number of complete local jets are inserted as jet steps \eqref{e:end-jet-step}, these bounds hold with the changes of \eqref{e:end-jet-constant}.
\end{proposition}

\begin{proof}
We keep every body variable on its real cycle and deform only the normal coordinates of the intermediate nodes, as in the proof of \zcref{l:end-energy}, with $\lambda_j = j/m$ for $0 \le j \le m$. The graph vanishes wherever an original real cutoff varies, so the input is real. We use the signed weights $a_j' = -a + 2aj/m$, running from $-a$ to $a$; this is possible since $m \ge \varrho W$. The heights $\beta_j = \lambda_j b$ satisfy
\[
 \sum_{j = 1}^m \bigl(|\beta_j|^2 + W^2 |\beta_j - \beta_{j - 1}|^2\bigr) \le \Bigl(m + \frac{W^2}{m}\Bigr) |b|^2 \le 3 C_0 W |b|^2 ,
\]
where we used $m \le 2 C_0 W$ and $W^2/m \le W/C_0 \le C_0 W$, since $C_0 \ge 1$. Similarly,
\[
 \sum_{j = 1}^m R_k |\beta_j|^2 \le 2 C_0 R_k W |b|^2 .
\]
Inserting these bounds into \eqref{e:end-energy}, we obtain the raw form of \eqref{e:end-normal-block}, for $S_\Gamma^m$ applied to $w_0^{1/2} \gamma_h g$. By \eqref{e:end-similarity}, we have $A_h^m g = \gamma_h^{-1} w_0^{-1/2} S_\Gamma^m (w_0^{1/2} \gamma_h g)$. The lower bound in \eqref{e:end-conversion} bounds the raw input norm by $C_k \|w_{-a} v^p g\|$, and its version on graphs bounds $\|A_h^m g\|_{b; p, a}$ by $C_k$ times the raw output norm on the graph. This proves \eqref{e:end-normal-block} with $C_n = 3 C_1$. The source appears only in the onsite bound \zcref{l:end-kick}, which holds uniformly for $\Gamma \in \mathcal U_2$, and it adds no rank-dependent term proportional to $W |b|^2$. By \zcref{r:end-universal}, the constants $C_1$ and $C_n = 3 C_1$ are universal.

For the assertion with the same graph at all nodes, we use \zcref{l:end-energy} with $\lambda_j = 1$ for all $j$, including the input, and with $a_j' = a$ for all $j$. The velocity terms vanish, and the exponent in \eqref{e:end-energy} reduces to $(C_1 + C_k R_k) j |b|^2$, for every $0 \le j \le C_0 W$. A complete local jet enters as a jet step, and in both arguments \eqref{e:end-jet-constant} replaces \eqref{e:end-energy} for blocks with jet steps. Finally, normal derivatives of order $\ell$ of the output, obtained by Cauchy's estimates on polydisks of radius $c/W$ in the normal variables, are bounded with an additional factor $C W^\ell$, and body derivatives require fixed stronger polynomial weights. This completes the proof.
\end{proof}

The estimates above use the source factor only through \eqref{e:end-kick}. The following lemma shows that all of them are uniform on the damping balls.

\begin{lemma}\label{l:end-real-family}
The power bounds of \zcref{t:stab-main}, the body-weight gain of \zcref{l:asm-weight-gain} and the signed normal preparation of \zcref{c:end-ramp} and \zcref{p:end-coupled} hold uniformly for $\Gamma \in \mathcal U_2$, in every fixed body weight, with the fixed changes of body weight that each finite jet requires. In particular, a leg of macroscopic length with positive damping, followed by a final block of $C_0 W$ steps, gives a family with a positive Gaussian margin in the family norm \eqref{e:end-family-norm}. For the even damping family and the finite BB and FF jet components of the original sources, this bound is $O(1)$ in the balanced signature. For the odd components of these jets, the bound holds with additional fixed powers of $W$ and fixed losses of body weight, and the polynomial factors of the unbalanced signatures are kept until they are suppressed in \zcref{s:assembly}. Moreover, multiplication by $s_\Gamma$ is holomorphic in $\Gamma \in \mathcal U_2$ as a map into the bounded operators on each weighted space, and every finite product of the operations of \zcref{d:end-generated} is holomorphic in $\Gamma \in \mathcal U_2$ as a Banach-space-valued function.
\end{lemma}

\begin{proof}
For the power bounds there is nothing to prove, since \zcref{t:stab-main} holds uniformly for complex damping in $\mathcal U_2$, with the complex frozen kick of \eqref{e:stab-kick} in its reference contraction; the constant $c_\Gamma$ of \eqref{e:stab-scalar} is kept out of the kick and gives a scalar term of the four-term bound. The other estimates use the source factor only through the onsite bound \eqref{e:end-kick} and through the damping, and we now check that these are uniform on $\mathcal U_2$.

By \zcref{l:end-coercivity}, the source factor has the damping $|s_\Gamma|_1 \le e^{C_k \eta - c_0 \eta v}$ uniformly on $\mathcal U_2$. The fixed derivatives of the exponent $\Tr(Q_B \Gamma_{BB})$ of \eqref{e:end-coercivity} along the orbit are bounded by $C_{k, j} v$, since conjugated unit tangent generators insert bounded commutators and $\|\Gamma\|$ is bounded. The onsite bound \eqref{e:end-kick} is thus uniform on the parameter ball, as \zcref{l:end-kick} states, and so are the estimates of \zcref{c:end-ramp} and \zcref{p:end-coupled}, which use the source only through it. The same positive damping proves the body-weight gain of \zcref{l:asm-weight-gain}, for the step and for its adjoint. A finite jet uses its fixed changes of body weight and Gaussian margin once, and the raw inputs have the bounds of \zcref{l:end-raw-norms}.

We next prove that $s_\Gamma$ is holomorphic. By \eqref{e:rep-weight}, $s_\Gamma = e^{-\iu \eta_0 \Str(S \Gamma)}$, and the exponent is linear in $\Gamma$. For a small even matrix $\delta$ the quotient $s_{\Gamma + \delta}/s_\Gamma$ is the exponential of a function that is linear in $\delta$. In the normalization \eqref{e:hist-normalized}, this function is at most $C \eta |\delta| v$ on the tube, as is the damping exponent, since $\|S_{BB}\| \le C v$ by \eqref{e:hist-boost-props}. By \zcref{l:end-coercivity}, we have $|s_\Gamma|_1 \le e^{C_k \eta - c_0 \eta v}$, and therefore
\[
 |s_{\Gamma + \delta} - s_\Gamma|_1
 \le |s_\Gamma|_1 \, C \eta |\delta| v \, e^{C \eta |\delta| v}
 \le C |\delta| \, e^{C_k \eta} \sup_{v \ge 1} \eta v \, e^{-(c_0 - C|\delta|) \eta v} ,
\]
which is at most $C' |\delta|$, uniformly in $v$, for $C|\delta| \le c_0/2$. The same expansion to second order in $\delta$ shows that the multiplication by $s_\Gamma$ is complex differentiable in the operator norm. The heat factors, the half-weights, and the cutoffs do not depend on $\Gamma$. It follows that every finite product of the operations of \zcref{d:end-generated} is holomorphic in the ordinary damping entries, with values in the weighted spaces. Integrating the uniform bounds over the ball proves the bound for \eqref{e:end-family-norm}. All local BB sources remain formal finite derivatives at zero.
\end{proof}

\section{Canonical coordinates near the saddle manifold}\label{s:end-canonical}

The physical chain lives on the cycle $\mathcal S$ and the adapted chain on the tube. To compare them near the saddle manifold, we write each point of the cycle in tubular coordinates. In the FF block, the cycle and the real tube share the compact orbit and differ only by the rotation $e^{-\iu\theta_\sigma}$ of the normal axes. In the BB block, the point of the cycle does not lie on the real tube, and a small complex conjugation, which we call the canonical gauge, brings it to a complex point of the tube. The normal height of that complex point must be bounded by $C_{\rm geo}(|E| R + R^2)$ at normal radius $R$, with a constant $C_{\rm geo}$ that does not depend on the rank, the dimension, or the boost.

\paragraph{The canonical equation.}
We use the coordinates of \zcref{s:end-cycle}. By \eqref{e:end-boost-conj}, the BB block of a point of the cycle satisfies $S_B = \mathsf g^{-1} R_{\mathsf g}(\widetilde P) \mathsf g$, where for a block-diagonal $n_B$ we write
\[
 R_{\mathsf g}(n_B) = R_{*, B} + n_B + \operatorname{th}_{\mathsf g} n_B .
\]
For an off-diagonal matrix $T$ we set $Y_{\mathsf g}(T) = \operatorname{ch}_{\mathsf g}^{-1} T$ and $T^\natural = \operatorname{Ad}_{\mathsf g} Y_{\mathsf g}(T) = T + \operatorname{th}_{\mathsf g} T$. Since $\operatorname{ad}_{\log \mathsf g}$ is selfadjoint for the Hilbert--Schmidt inner product, $\|\operatorname{ch}_{\mathsf g}^{-1}\| \le 1$ and $\|\operatorname{th}_{\mathsf g}\| \le 1$. Hence $\|Y_{\mathsf g}(T)\| \le \|T\|$, and $\|T^\natural\| \le \sqrt 2 \|T\|$ since its two block parities are orthogonal. Let $\Pi_{\rm off}$ and $\Pi_{\mathfrak k}$ be the projections onto off-diagonal and block-diagonal matrices. We look for an off-diagonal $T = T_{\mathsf g}(n_B)$ such that
\begin{equation}\label{e:end-canonical-equation}
 \Pi_{\rm off}\bigl(e^{T^\natural} R_{\mathsf g}(n_B) e^{-T^\natural}\bigr) = 0 ,
 \qquad
 U_{\mathsf g}(n_B) = \Pi_{\mathfrak k}\bigl(e^{T^\natural} R_{\mathsf g}(n_B) e^{-T^\natural}\bigr) - R_{*, B} .
\end{equation}
We call $U_{\mathsf g}(n_B)$ the canonical normal coordinate of the BB block. Since $e^{T^\natural} = \mathsf g e^{Y_{\mathsf g}(T)} \mathsf g^{-1}$, the BB block of the point of the cycle is then
\begin{equation}\label{e:end-canonical-frame}
 S_B = T_{\rm can}(R_{*, B} + U_{\mathsf g}(n_B)) T_{\rm can}^{-1} ,
 \qquad
 T_{\rm can} = S_{\mathsf g}^{-1} \mathsf g^{-1} ,
 \quad S_{\mathsf g} = e^{Y_{\mathsf g}(T_{\mathsf g}(n_B))} ,
\end{equation}
so that the canonical frame differs from the real frame $\mathsf g^{-1}$ by the small complex gauge $S_{\mathsf g}^{-1}$.

\begin{lemma}\label{l:end-height}
Let $E_0 \le 1/2$. There exist universal constants $r_{\rm c}, C > 0$ and $C_{\rm geo}$ such that the following holds for all $|E| \le E_0$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:end-height-bb} \textup{(BB block.)} For every Hermitian off-diagonal $X$ and every block-diagonal $n_B$ with $\|n_B\| \le r_{\rm c}$, there exists a unique off-diagonal $T_{\mathsf g}(n_B)$ with $\|T_{\mathsf g}(n_B)\| \le s^{-1}\|n_B\|$ that solves \eqref{e:end-canonical-equation}. It is holomorphic in $n_B$, and
\[
 \|U_{\mathsf g}(n_B) - n_B\| \le C \|n_B\|^2 , \qquad \|S_{\mathsf g}^{\pm 1} - \Id\| \le C \|n_B\| .
\]
If moreover $n_B$ is Hermitian with $\|n_B\| \le R \le r_{\rm c}$, then the height of $U_{\mathsf g}(n_B)$ above the real tube satisfies $\|w_B\| \le C_{\rm geo}(|E| R + R^2)$.
\item\label{i:end-height-ff} \textup{(FF block.)} Let $M$ be Hermitian and $p \in \{0, \dots, 2k\}$, and suppose that $M = V(-s\Sigma_F + n_F)V^*$ with $V$ unitary, $\Sigma_F = \diag(\Id_p, -\Id_{2k - p})$ and $n_F$ Hermitian and block diagonal with $\|n_F\| \le R \le r_{\rm c}$. Then the FF block of the point of the cycle is the point of the complexified tube with the real orbit coordinate $V$ and the normal coordinate $U_F = \iu n_F$, and its height above the real tube satisfies $\|w_F\| = |\sin \theta_\sigma| \, \|n_F\| \le C_{\rm geo} |E| R$, blockwise for the roots $\sigma$.
\end{enumerate}
Consequently, the body of the canonical normal coordinate of a point of the cycle at normal radius $R \le r_{\rm c}$ has height at most $C_{\rm geo}(|E| R + R^2)$. The constants do not depend on $k$, on $X$ or on the matrix dimension.
\end{lemma}

Part \ref{i:end-height-ff} computes the rotation of the FF normal axes, which is of order $|E| R$. In the FF block no complex gauge is needed. Indeed, a Hermitian matrix $M = V(-s\Sigma_F + n_F)V^*$ as in \ref{i:end-height-ff}, whose eigenvalues are close to $\pm s$, is block diagonalized exactly by the unitary matrix $V$. The cycle point is therefore a point of the tube with a real orbit coordinate, and its normal coordinate $\iu n_F$ lies on the unrotated real form. Its height above the adapted real form $e^{-\iu\theta_\sigma} \iu \operatorname{Herm}$ comes only from the rotation, which is of order $|E|$ since the FF normal Hessian $m_\sigma$ has argument of order $|E|$.

\begin{proof}
We begin with \ref{i:end-height-bb}, which is a quantitative implicit function theorem in the Hilbert--Schmidt norm; we check that every constant is independent of the dimension and of the boost. Write $T^\natural$ as above and $\tilde n_B = n_B + \operatorname{th}_{\mathsf g} n_B$, so that $R_{\mathsf g}(n_B) = R_{*, B} + \tilde n_B$ and $\|\tilde n_B\| \le \sqrt 2 \|n_B\|$. The matrix $\operatorname{th}_{\mathsf g} T$ is block diagonal and commutes with $\Lambda$, which gives
\[
 [T^\natural, R_{*, B}] = [T, R_{*, B}] = 2\iu s \Lambda T .
\] The linear map $\mathcal L T = 2 \iu s \Lambda T$ preserves the off-diagonal matrices, and $\|\mathcal L^{-1}\| = (2s)^{-1}$.

Expanding $e^{T^\natural} R_{\mathsf g} e^{-T^\natural} = e^{\operatorname{ad}_{T^\natural}} R_{\mathsf g}$, we have
\[
 \Pi_{\rm off}\bigl(e^{\operatorname{ad}_{T^\natural}} R_{\mathsf g}(n_B)\bigr) = \mathcal L T + \operatorname{th}_{\mathsf g} n_B + \mathcal R(T, n_B) ,
 \]
 where
 \[
 \mathcal R(T, n_B) = \Pi_{\rm off}\Bigl([T^\natural, \tilde n_B] + \sum_{n \ge 2} \frac{\operatorname{ad}_{T^\natural}^n R_{\mathsf g}(n_B)}{n!}\Bigr) .
\]
Since $\|\operatorname{ad}_{T^\natural}^n R_{*, B}\| \le (2\|T^\natural\|)^{n - 1} 2s \|T\|$ and $\|T^\natural\| \le \sqrt 2 \|T\|$, there exists a universal $C_{\mathcal R}$ such that for $\|T\|, \|n_B\| \le 1$,
\[
 \begin{gathered}
 \|\mathcal R(T, n_B)\| \le C_{\mathcal R}(\|T\| \|n_B\| + \|T\|^2) , \\
 \|\mathcal R(T, n_B) - \mathcal R(T', n_B)\| \le C_{\mathcal R}(\|T\| + \|T'\| + \|n_B\|)\|T - T'\| .
 \end{gathered}
\]

The equation \eqref{e:end-canonical-equation} is therefore the fixed point equation
\[
 T = -\mathcal L^{-1}\bigl(\operatorname{th}_{\mathsf g} n_B + \mathcal R(T, n_B)\bigr) .
\]
For $\|n_B\| \le r_{\rm c}$ with $r_{\rm c}$ small, depending only on $C_{\mathcal R}$ and $s_0$, the right side maps the ball $\{\|T\| \le s^{-1}\|n_B\|\}$ into itself and is a contraction there. This gives $T_{\mathsf g}(n_B)$, which is holomorphic in $n_B$ as a locally uniform limit of holomorphic iterates.

Next, we compare $U_{\mathsf g}(n_B)$ with $n_B$. We have $\Pi_{\mathfrak k} \tilde n_B = n_B$, and $\Pi_{\mathfrak k}[T^\natural, R_{*, B}] = 0$ since $\mathcal L T$ is off-diagonal. Hence
\[
 U_{\mathsf g}(n_B) - n_B = \Pi_{\mathfrak k}\Bigl([T^\natural, \tilde n_B] + \sum_{n \ge 2} \frac{\operatorname{ad}_{T^\natural}^n R_{\mathsf g}(n_B)}{n!}\Bigr) ,
\]
whose norm is at most $C(\|T\| \|n_B\| + \|T\|^2) \le C\|n_B\|^2$. The bound on $S_{\mathsf g}^{\pm 1} - \Id$ follows from $\|Y_{\mathsf g}(T)\| \le \|T\| \le s^{-1}\|n_B\|$.

For the height, let $n_B$ be Hermitian. Since $s^2 = 1 - E^2/4$, we have $m_\sigma = 1 - r_\sigma^2 = 2s^2 - \iu \sigma E s$ and $c_\sigma^2 = m_\sigma(W^2 + m_\sigma/4)$ by \eqref{e:tube-mu}, so that
\[
 |\arg c_\sigma| \le \tfrac12 \arctan\frac{|E|}{2s} + \frac{C|E|}{W^2} \le C|E| , \qquad |\theta_\sigma| \le C|E| .
\]
On the root block of type $\sigma$, $n_B = e^{-\iu\theta_\sigma}(e^{\iu\theta_\sigma} n_B)$, and $e^{\iu\theta_\sigma} n_B = \cos\theta_\sigma n_B + \iu \sin\theta_\sigma n_B$, and the height of $n_B$ is $\sin\theta_\sigma$ times its block of type $\sigma$. Therefore the height $w_B$ of $U_{\mathsf g}(n_B)$ satisfies
\[
 \|w_B\| \le |\sin\theta_\sigma| \|n_B\| + \|U_{\mathsf g}(n_B) - n_B\| \le C|E| \|n_B\| + C \|n_B\|^2 ,
\]
which proves \ref{i:end-height-bb}. Every constant depends only on $s_0$ and on the inequality $\|[A, Y]\| \le 2\|A\|_{\rm op}\|Y\|$, which does not depend on the dimension.

We now prove \ref{i:end-height-ff}. The FF block of the point of the cycle is $\kappa \Id + \iu M = V(\kappa \Id - \iu s \Sigma_F + \iu n_F)V^*$, and $\kappa \Id - \iu s\Sigma_F$ is the FF block $R_{*, F}$ of the root matrix of signature $p$. Thus the point is $V(R_{*, F} + U_F)V^{-1}$ with $V$ in the real compact orbit group and the block-diagonal normal coordinate $U_F = \iu n_F$, which is the tubular form \eqref{e:tube-map} of the point; no complex conjugation is needed. The real form of the FF normal coordinates is $\iu$ times the Hermitian matrices, and on the root block of type $\sigma$ we have $\iu n_F = e^{-\iu\theta_\sigma}\bigl(\cos\theta_\sigma \, \iu n_F + \iu \sin\theta_\sigma \, \iu n_F\bigr)$. The real part of $U_F$ is $\cos\theta_\sigma \, \iu n_F$, and its height is $w_F = \sin\theta_\sigma \, \iu n_F$. Thus $\|w_F\| = |\sin\theta_\sigma| \|n_F\| \le C|E| R$, with the universal constant of the bound on $\theta_\sigma$ above. This proves \ref{i:end-height-ff}.

Finally, the body of the full supermatrix splits into its BB and FF blocks, and the body of the canonical map is the product of the two constructions above. Taking $C_{\rm geo}$ larger than the constants in \ref{i:end-height-bb} and \ref{i:end-height-ff}, and using $\|w\|^2 = \|w_B\|^2 + \|w_F\|^2$, we obtain the last assertion with $\sqrt 2 C_{\rm geo}$ in place of $C_{\rm geo}$, and we rename the constant.
\end{proof}

The construction extends to the full supermatrix by induction in the total odd degree. First apply the body frames of \zcref{l:end-height}, so that the transformed supermatrix is $R_*+U_{\rm b}+N$, where $U_{\rm b}$ is the body normal coordinate, $[\Sigma,U_{\rm b}]=0$, $\|U_{\rm b}\|\le C r_{\rm c}$, and $N$ has zero body. We seek an additional even nilpotent conjugation $e^Y$, with $\Sigma Y=-Y\Sigma$, that removes all entries between the two root spaces. At each positive degree, its unknown component satisfies a linear equation with the full body Jacobian
\[
 \mathcal J_{\rm b}Y=[Y,R_*+U_{\rm b}]
 =\mathcal L_\Sigma Y+[Y,U_{\rm b}],
 \qquad \mathcal L_\Sigma Y=2\iu s\Sigma Y .
\]
This formula applies to the BB, FF, BF and FB entries between the root spaces. In the Hilbert--Schmidt coefficient norm, $\|\mathcal L_\Sigma^{-1}\|=(2s)^{-1}$ and $\|\mathcal J_{\rm b}-\mathcal L_\Sigma\|\le2\|U_{\rm b}\|_{\rm op}\le C r_{\rm c}$. After a universal decrease of $r_{\rm c}$, a Neumann series therefore gives $\|\mathcal J_{\rm b}^{-1}\|\le(2s-Cr_{\rm c})^{-1}$. The right side at each degree is a polynomial in coefficients already determined. There are finitely many degrees at fixed rank, and this constructs every nilpotent normal shift and odd frame entry, with fixed-rank polynomial bounds in the body weight on each smaller tube. The body map is unchanged, so only its heights enter the energy threshold; the nilpotent parts are finite coefficient jets with rank-dependent bounds.

To fix the full frame, let $T_{\rm b}$ be the product of the BB canonical body frame and the FF unitary frame, and let $T_{\rm r}$ be the corresponding real body frame. We include the correction just constructed by setting $\widehat T_{\rm can}=T_{\rm b}e^{-Y}$ and $\widehat S=T_{\rm r}\widehat T_{\rm can}^{-1}$. Then the original supermatrix is $\widehat T_{\rm can}(R_*+U)\widehat T_{\rm can}^{-1}$ with $[\Sigma,U]=0$, and $\widehat T_{\rm can}=\widehat S^{-1}T_{\rm r}$. The body of $\widehat S$ is $S_{\mathsf g}$ in the BB block and the identity in the FF block.

We let $\Can$ denote the resulting canonical map, which takes a function on the tube, with its half-density, its Gaussian gauge, and its full nilpotent structure, to a function near the saddle manifold on the cycle. By \zcref{l:end-ward}\ref{i:end-ward-b} and the full frame identity above, for a family generated from the endpoint the composition with $\Can$ is the family at the conjugated damping $\widehat S\Gamma\widehat S^{-1}$ and directions $\widehat S\mathbf V\widehat S^{-1}$, evaluated at the real frame; the pushed frame keeps the normal coordinates unchanged. The body selection uses $S_{\mathsf g}$ as in \eqref{e:end-canonical-frame}, and the nilpotent corrections are retained through their finite coefficient expansion, as in \zcref{l:end-ward}\ref{i:end-ward-c}.

\paragraph{The physical cutoff.}
We now fix the real physical cutoff $\chi_{\rm phys}$ on the cycle, which equals one near the saddle manifold and vanishes at normal distance $r_{\rm phys}$ from it. Its body is a radial function of the real part of the canonical normal coordinate. Its nilpotent part is obtained by transporting a cutoff on the real tube along a homotopy of body cycles from the cycle to the real tube, which \zcref{s:assembly} also uses to return the cycle of integration to the tube.

Let $\chi$ be a smooth function on $[0, \infty)$ with values in $[0, 1]$, equal to one on $[0, r_1]$ and to zero on $[r_{\rm phys}, \infty)$. For a complex normal coordinate $U$ we write $U = U^{\rm re} + \iu U^{\rm im}$, where $U^{\rm re}$ and $U^{\rm im}$ lie in the real normal space of the adapted tube, so that $U^{\rm im}$ is the height of \zcref{l:end-height}. On the real tube we write $\chi_{\rm tube} = \chi(|U|)$.

We parametrize $\mathcal S$ near the saddle manifold by $y = (X, \widetilde P, M)$ as in \zcref{s:end-cycle}, together with the odd entries of the supermatrix, which we call the ambient odd coordinates, and we let $\mathcal Y$ be the parameter domain of $y$. We write $n_B = \widetilde P$, we let $T^\natural$ and $U_{\mathsf g}(n_B)$ be as in \zcref{l:end-height}\ref{i:end-height-bb}, and we write $M = V(-s \Sigma_F + n_F) V^*$ as in \zcref{l:end-height}\ref{i:end-height-ff}. For $0 \le t \le 1$, let $\Phi_t(y)$ be the body point with the blocks
\begin{equation}\label{e:asm-homotopy}
\begin{aligned}
 \Phi_t(y)_{\rm BB} &= \mathsf g^{-1} e^{-t T^\natural} \bigl( R_{*, B} + U_{\mathsf g}(n_B)^{\rm re} + \iu t \, U_{\mathsf g}(n_B)^{\rm im} \bigr) e^{t T^\natural} \mathsf g , \\
 \Phi_t(y)_{\rm FF} &= V \bigl( R_{*, F} + (\iu n_F)^{\rm re} + \iu t \, (\iu n_F)^{\rm im} \bigr) V^* ,
\end{aligned}
\end{equation}
and write $\Phi(t, y) = \Phi_t(y)$. We write $U_{\rm can}$ for the canonical normal coordinate of the point $y$ of the cycle, which is $U_{\mathsf g}(n_B)$ in the BB block and $\iu n_F$ in the FF block, and $U^{\rm re}_{\rm can}$ for its real part.

\begin{lemma}\label{l:asm-homotopy}
Let $0 \le t \le 1$ and $y \in \mathcal Y$, and write $U = U_{\mathsf g}(n_B)$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:asm-homotopy-frame} The BB block of $\Phi_t(y)$ has the frame $S_{\mathsf g}^{-t} \mathsf g^{-1}$ and the normal coordinate $U^{\rm re} + \iu t \, U^{\rm im}$, and its FF block has the frame $V$ and the normal coordinate $(\iu n_F)^{\rm re} + \iu t \, (\iu n_F)^{\rm im}$.
\item\label{i:asm-homotopy-ends} The point $\Phi_1(y)$ is the body of the point $y$ of $\mathcal S$, and $\Phi_0$ is a diffeomorphism from a neighborhood of the saddle manifold in $\mathcal S$ onto a neighborhood of the saddle manifold in the real tube.
\item\label{i:asm-homotopy-bounds} There exists $A_\Phi$ such that $|\partial_t \Phi_t(y)|$ and $\|D \Phi_t(y)\|^{\dim - 1}$ are bounded by $C_k v_{\mathcal S}^{A_\Phi}$, where $D \Phi_t$ is the derivative in $y$ and $\dim$ is the real dimension of the body of $\mathcal S$.
\end{enumerate}
\end{lemma}

\begin{proof}
Recall that $S_{\mathsf g} = e^{Y_{\mathsf g}}$, with $Y_{\mathsf g} = Y_{\mathsf g}(T_{\mathsf g}(n_B))$, and that $e^{T^\natural} = \mathsf g e^{Y_{\mathsf g}} \mathsf g^{-1}$. Hence $\mathsf g^{-1} e^{-t T^\natural} = S_{\mathsf g}^{-t} \mathsf g^{-1}$, and \ref{i:asm-homotopy-frame} follows from \eqref{e:asm-homotopy}. At $t = 1$, the BB block has the canonical frame
\[
 T_{\rm can} = S_{\mathsf g}^{-1} \mathsf g^{-1} = \mathsf g^{-1} e^{-T^\natural}
\]
of \eqref{e:end-canonical-frame} and the normal coordinate $U^{\rm re} + \iu U^{\rm im} = U$. Therefore \eqref{e:end-canonical-frame} and \zcref{l:end-height}\ref{i:end-height-ff} show that $\Phi_1(y)$ is the body of the point $y$ of $\mathcal S$. At $t = 0$, the frames $\mathsf g^{-1}$ and $V$ and the normal coordinates are real, and \zcref{l:end-height} implies that $\Phi_0$ is a diffeomorphism onto a neighborhood of the saddle manifold in the real tube. This proves \ref{i:asm-homotopy-ends}.

For \ref{i:asm-homotopy-bounds}, recall that $\Phi_t$ is built from $\mathsf g^{\pm 1}$, whose norms are at most $(3 v_{\mathcal S})^{1/2}$ by \eqref{e:hist-boost-props} and whose logarithmic derivatives are bounded by \eqref{e:end-connection}, and from $T^\natural$ and $U_{\mathsf g}(n_B)$, which are holomorphic in $n_B$ with bounds uniform in the boost by \zcref{l:end-height}. For instance, we have
\[
 \partial_t \Phi_t(y)_{\rm BB} = \mathsf g^{-1} e^{-t T^\natural} \bigl( -[T^\natural, R_{*, B} + U^{\rm re} + \iu t U^{\rm im}] + \iu U^{\rm im} \bigr) e^{t T^\natural} \mathsf g ,
 \qquad
 \|T^\natural\| \le \sqrt 2 \, s^{-1} \|n_B\| .
\]
The other derivatives are products of the same factors, and each factor is bounded by a power of $v_{\mathcal S}$.
\end{proof}

We fix the nilpotent part of the physical cutoff by transporting $\chi_{\rm tube}$ along the homotopy. In the ambient odd coordinates, the nilpotent coefficients of $\chi_{\rm tube} = \chi(|U|)$ are polynomials in the nilpotent part of $U$ with coefficients $\chi^{(j)}(|U|)$, $j \ge 1$. On $[0, 1] \times \mathcal Y$ we set
\begin{equation}\label{e:asm-cutoff}
 \chi_\Phi(t, y) = \chi_{\rm tube}(\Phi_0(y)) ,
 \qquad
 \chi_{\rm phys}(y) = \chi_\Phi(1, y) ,
\end{equation}
coefficientwise in the ambient odd coordinates. By \zcref{l:asm-homotopy}\ref{i:asm-homotopy-frame} at $t = 0$, the cutoff $\chi_{\rm phys}$ has the body $\chi(|U^{\rm re}_{\rm can}|)$. Therefore $\chi_{\rm phys} = 1$ where $|U_{\rm can}| \le r_1$, and $\chi_{\rm phys}$, together with all its nilpotent coefficients, vanishes where $|U^{\rm re}_{\rm can}| \ge r_{\rm phys}$. The nilpotent coefficients of $\chi_\Phi$ and the derivatives of all its coefficients are supported in the annulus $\{y : r_1 \le |U^{\rm re}_{\rm can}(y)| \le r_{\rm phys}\}$, which we call the annulus where $\chi_{\rm phys}$ varies.

\paragraph{The strip of the actual cycle.}
The physical chain on the cycle is compared with the references on the tube through normal graphs of the cycle itself. For a constant small Hermitian block-diagonal matrix $D$, we shift the physical normal $\widetilde P$ to $\widetilde P + \iu D$, that is, $P_0$ to $P_0 + \iu P_D$ at fixed $X$, where we write $P_D = \operatorname{ch}_{\mathsf g}^{-1} D$. This shift has unit Jacobian, since only the block-diagonal coordinate is translated. For a Hermitian $D_F$, we also shift the FF block from $\kappa \Id + \iu M$ to $\kappa \Id + \iu M + D_F$. We write $\mathcal S_D$ for the shifted cycle, and we define $w_{\Gamma, D, D_F}$ to be the onsite weight at the shifted point, as a function of the coordinates $(P_0, X, M)$ and the odd coordinates. Recall the defect $d_X(\Xi)$ of \eqref{e:end-defect}, and let
\[
 d(X, X') = \|X - X'\|^2 - \|f_0(X) - f_0(X')\|^2
\]
denote the defect of the BB graph between $X$ and $X'$, which is nonnegative by \eqref{e:contour-edge-identity}. The following proposition shows that the partial chains continue to the shifted cycles, with bounds that grow like $e^{C_* j (\|D\|^2 + \|D_F\|^2)}$, and it gives the resulting bound near the saddle manifold in the coordinates of the tube.

\begin{proposition}\label{p:end-shifted}
There exist universal constants $r_{\rm s} \in (0, s_0/4]$, $R_{\rm s}, C, C_*, c' > 0$ such that the following holds at fixed rank $k$ for all sufficiently large $W$. Let $\Gamma \in \mathcal U_2$, let $p \in \R$, and let $D$ be a block-diagonal Hermitian matrix and $D_F$ a Hermitian matrix with $\|D\| + \|D_F\| \le r_{\rm s}$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:end-shifted-defect} For every block-diagonal Hermitian matrix $\Xi$,
\begin{equation}\label{e:end-shifted-defect}
 \|D P_D^* \Xi\| \le C \|D\| d_X(\Xi)^{1/2} ,
 \qquad
 |\langle DF_X^* \Xi, D P_D^* \Xi\rangle| \le C \|D\| d_X(\Xi) ,
\end{equation}
and the ambient heat pulled to $\mathcal S_D$ satisfies the bound $\exp[(C_k W + C_{k, p}) t]$ of \zcref{l:end-kinetic} on $\mathcal H_p(\mathcal S)$, for $0 \le t \le \tau_{\rm phys}$.
\item\label{i:end-shifted-onsite} The shifted onsite weight satisfies
\begin{equation}\label{e:end-strip}
 |w_{\Gamma, D, D_F}|_{1; h} \le e^{C_*(\|D\|^2 + \|D_F\|^2) + C_k/W + C_k \eta} \, e^{-c' \vartheta - c' \eta v_{\mathcal S}} .
\end{equation}
\item\label{i:end-shifted-cont} For every $j \ge 1$, the partial chain $f_j = T_\Gamma^j 1$ extends holomorphically from $\mathcal S$ to a neighborhood of $\mathcal S$ that contains $\mathcal S_D$, and $f_{j + 1}|_{\mathcal S_D}$ is the image of $f_j|_{\mathcal S_D}$ under the shifted transfer operator, the heat on $\mathcal S_D$ followed by multiplication by $w_{\Gamma, D, D_F}$. For $1 \le j \le C_0 W$,
\begin{equation}\label{e:end-rough-strip}
 \|f_j|_{\mathcal S_D}\|_{\mathcal H_p(\mathcal S)} \le C_k W^{A_k} \exp\bigl[C_k C_0 + C_* j (\|D\|^2 + \|D_F\|^2)\bigr] .
\end{equation}
\item\label{i:end-shifted-collar} Let $0 < R \le R_{\rm s}$ and $1 \le j \le C_0 W$. The partial chain $f_j$, written in the coordinates of the tube through the canonical map, has norm on the set $\{|U| \le R\}$ of the tube at most
\begin{equation}\label{e:end-actual-collar}
 C_k W^{A_k} \exp\bigl[C_k C_0 + 16 C_{\rm geo}^2 C_* C_0 W (E^2 R^2 + R^4)\bigr] ,
\end{equation}
uniformly for $\Gamma$ in a compact parameter ball strictly inside the parameter ball of \zcref{p:end-escape}.
\end{enumerate}
\end{proposition}

\begin{proof}
We first prove \ref{i:end-shifted-defect}. Let $\Xi$ be the Fourier variable of $P_0$, as in \zcref{l:end-kinetic}. Since $\operatorname{ch}_{\mathsf g} = \frac12(\operatorname{Ad}_{\mathsf g} + \operatorname{Ad}_{\mathsf g^{-1}})$, its derivative in a direction $V$ is
\[
 (D_V \operatorname{ch}_{\mathsf g}) Y = \tfrac12\bigl([D_V\mathsf g \, \mathsf g^{-1}, \mathsf g Y \mathsf g^{-1}] + [\mathsf g^{-1} Y \mathsf g, \mathsf g^{-1} D_V \mathsf g]\bigr) .
\]
For block-diagonal $Y$ we have $\|\operatorname{Ad}_{\mathsf g^{\pm 1}} Y\|^2 = 2 \|\operatorname{ch}_{\mathsf g} Y\|^2 - \|Y\|^2$, and therefore
\[
 \|(D_V \operatorname{ch}_{\mathsf g}) P_D\| \le \sqrt 2 \bigl(\|D_V\mathsf g \, \mathsf g^{-1}\| + \|\mathsf g^{-1} D_V \mathsf g\|\bigr) \|D\| .
\]
Differentiating $\operatorname{ch}_{\mathsf g} P_D = D$, we obtain $\langle \Xi, D_V P_D\rangle = -\langle \operatorname{ch}_{\mathsf g}^{-1} \Xi, (D_V \operatorname{ch}_{\mathsf g}) P_D\rangle$. Using the first line of \eqref{e:end-connection} for unit directions $V$, and the second line for $V = DF_X^* \Xi$, we obtain \eqref{e:end-shifted-defect}. The second, stronger bound is needed for the directions $\Xi$ whose defect $d_X(\Xi)$ is small compared with $\|\Xi\|^2$. In the Fourier variable the shift replaces $\phi_\Xi$ by $\phi_\Xi - \langle \Xi, P_D(X)\rangle$, which is still real, and \eqref{e:end-shifted-defect} implies that the shifted defect satisfies
\[
 \|\Xi\|^2 - \|D(F_X + P_D)^* \Xi\|^2 \ge \bigl(1 - 2C\|D\| - C^2\|D\|^2\bigr) d_X(\Xi) .
\]
For $r_{\rm s}$ small, the right side is at least $d_X(\Xi)/2 \ge 0$, and the proof of \zcref{l:end-kinetic} gives the shifted kinetic estimate in \ref{i:end-shifted-defect}.

We next prove \ref{i:end-shifted-onsite}, beginning with the shifted BB onsite factor. Let $\kappa = E/2$ and set
\[
 L = \mathsf g^{-1}\Lambda P_0 \mathsf g^{-1} ,
 \qquad
 L_D = \mathsf g^{-1}\Lambda P_D \mathsf g^{-1} = \Lambda \operatorname{Ad}_{\mathsf g}(P_D) ,
 \qquad
 A_\kappa = s\Id + \iu(L - \kappa\Lambda) .
\]
The holomorphic BB phase is, up to its saddle constant, $\widetilde V_\kappa(P_0) = \frac12 \Tr P_0^2 + \kappa \Tr P_0 - \iu s \Tr L + \Tr \log A_\kappa$, with real part $V_\kappa$. Its derivative in the direction $\iu P_D$ has real part $\Tr[(s\Id - \Re A_\kappa^{-1}) L_D]$. Let $A = \Id + L^2$ and $\mathcal M_L = A^{-1/2}\{\Lambda, L\}A^{-1/2}$. The inequality $2|\langle \Lambda u, L u\rangle| \le \|u\|^2 + \|L u\|^2$ implies that $\|\mathcal M_L\|_{\rm op} \le 1$. With $H_L = L - \kappa\Lambda$, we have $\Re A_\kappa^{-1} = s(s^2 + H_L^2)^{-1}$ and $s^2 + H_L^2 = A^{1/2}(\Id - \kappa \mathcal M_L)A^{1/2}$, by $\kappa^2 + s^2 = 1$. Therefore
\[
 \Re A_\kappa^{-1} = s A^{-1/2}(\Id - \kappa \mathcal M_L)^{-1} A^{-1/2} .
\]

We also use $\|\Id - A^{-1}\|^2 \le \Tr \log A$ and $\|L A^{-1/2}\|^2 \le \Tr \log A$, which follow from $x/(1 + x) \le \log(1 + x)$, so that $\|\mathcal M_L\| \le 2 (\Tr \log A)^{1/2}$. It follows that
\[
\begin{aligned}
 \|s\Id - \Re A_\kappa^{-1}\|
 &\le s\|\Id - A^{-1}\| + s\bigl\|A^{-1/2}\bigl((\Id - \kappa \mathcal M_L)^{-1} - \Id\bigr)A^{-1/2}\bigr\| \\
 &\le \Bigl(1 + \frac{2|\kappa|}{1 - |\kappa|}\Bigr) (\Tr \log A)^{1/2} .
\end{aligned}
\]
By the concavity bound \eqref{e:hist-coercive} for the first relation, and since $L_D = \Lambda(D + \operatorname{th}_{\mathsf g} D)$ is an orthogonal sum with $\|\operatorname{th}_{\mathsf g} D\|^2 = \|D\|^2 - \|P_D\|^2$ for the second, we have
\[
 \Tr \log A = 2\Bigl(V_0 - \frac12 \Tr P_0^2\Bigr) \le 4 V_\kappa ,
 \qquad
 \|L_D\|^2 = 2\|D\|^2 - \|P_D\|^2 \le 2\|D\|^2 ,
\]
and Young's inequality bounds the first variation by $\mu V_\kappa + C_\mu \|D\|^2$. Let $0 \le t \le 1$. Since $\sqrt 2 \|D\| \le s_0/2$ and $\|L_D\|_{\rm op} \le \|L_D\| \le \sqrt 2 \|D\|$, the Hermitian part $s\Id - t L_D$ of $A_\kappa - t L_D$ is at least $s\Id/2$; in particular the BB block stays invertible on $\mathcal S_D$, so the shifted factor has no poles. The second variation of the logarithm is bounded by $C \|L_D\|^2 \le 2C\|D\|^2$, and the Gaussian second variation is $-\|P_D\|^2$. Taylor's formula yields, with dimension-free constants,
\[
 \Re \widetilde V_\kappa(P_0 + \iu P_D) \ge (1 - \mu) V_\kappa(P_0) - C_\mu \|D\|^2 .
\]

For the FF block, \zcref{l:end-ff-strip} with the shift $D_F$ bounds the body. For the nonbody coefficients we use, as in the proof of \zcref{l:end-ff-strip}, Hadamard's bound by the column norms $y_i^{1/2}$ in the eigenbasis of $M$, since the joint diagonalization used in \zcref{l:end-onsite} is not available for the shifted factor; the damping is treated as in \zcref{l:end-onsite}. Combining these bounds, we obtain \eqref{e:end-strip}.

We now prove \ref{i:end-shifted-cont}, that is, we identify the partial chains on the shifted cycle with the holomorphic continuation of the partial chains on $\mathcal S$. Let $f$ be one of the partial chains $f_j = T_\Gamma^j 1$. For a point $S$ of $\mathcal S_D$, we define $f(S)$ by the same iterated integral over $\mathcal S$, with the complex output point $S$. The link Gaussian from the complex output $S$ to a point $S' \in \mathcal S$ has, up to the FF and odd parts, the real exponent
\[
 \|P_0 - P_0'\|^2 - \|P_D\|^2 + d(X, X') + 2 \Tr\bigl(P_D \Lambda (f_0(X) - f_0(X'))\bigr) .
\]
The defect and the cross term grow at the same exponential rate as $X'$ moves away from $X$, and the integral converges because $\|D\|$ is small.

To see this, we write $\mathsf g = \mathsf g_X$ and $\mathsf g' = \mathsf g_{X'}$, and we let $\mathsf g \mathsf g'^{-1} = k_1 e^Y k_2$ be the Cartan decomposition of this element of $U(k, k)$, with $k_1$ and $k_2$ block diagonal and unitary, and with $Y$ Hermitian and off-diagonal, with eigenvalues $\pm y_j$ for $1 \le j \le k$. By \eqref{e:contour-edge-identity}, we have
\begin{equation}\label{e:end-graph-defect}
 d(X, X') = s^2 \Tr\bigl(\mathsf g^2 \mathsf g'^{-2} + \mathsf g^{-2} \mathsf g'^2 - 2 \Id\bigr) = 4 s^2 \sum_{j = 1}^k (\cosh 2 y_j - 1) .
\end{equation}
The matrix $f_0(X')$ is the block-diagonal part of $s \mathsf g'^2$, by \eqref{e:end-boost}. Since $P_D$ is block diagonal, $\mathsf g \Lambda = \Lambda \mathsf g^{-1}$ and $\operatorname{Ad}_{\mathsf g} P_D = D + \operatorname{th}_{\mathsf g} D$, we find
\[
 2 \Tr\bigl(P_D \Lambda (f_0(X) - f_0(X'))\bigr) = 2 s \Tr\bigl((D + \operatorname{th}_{\mathsf g} D)(\Lambda - k_1 \Lambda e^{-2 Y} k_1^{-1})\bigr) .
\]
We have $\|D + \operatorname{th}_{\mathsf g} D\| \le \sqrt 2 \|D\|$, the matrix $k_1$ commutes with $\Lambda$, and
\[
 \|\Lambda - \Lambda e^{-2 Y}\|^2 = 4 \sum_{j = 1}^k \cosh 2 y_j \, (\cosh 2 y_j - 1) \le \frac{d(X, X')}{s^2} + \frac{d(X, X')^2}{4 s^4} .
\]
By the Cauchy--Schwarz inequality for the Hilbert--Schmidt inner product, it follows that
\[
 \bigl|2 \Tr\bigl(P_D \Lambda (f_0(X) - f_0(X'))\bigr)\bigr| \le \sqrt 2 \|D\| \bigl(s^{-1} d(X, X') + 2 d(X, X')^{1/2}\bigr) ,
\]
uniformly in the boost. Since $\sqrt 2 \|D\| \le s_0/2$, Young's inequality and $\|P_D\| \le \|D\|$ show that the real exponent is at least $\|P_0 - P_0'\|^2 + d(X, X')/4 - 9 \|D\|^2$. In particular, the integral converges locally uniformly, and $f$ is holomorphic near $\mathcal S$.

Next, the heat on $\mathcal S_D$ applied to the restriction of $f$ is the restriction of $K_\tau f$. Indeed, at fixed $(X', M')$ and fixed odd coordinates, we may shift the $P_0'$-integral back by Cauchy's theorem, since the integrand is holomorphic in $P_0'$ on the strip $\|\Im \widetilde P'\| \le \|D\|$, has no poles there by the invertibility above, and has the Gaussian decay $e^{-\|P_0'\|^2/2 + \|P_D\|^2/2}$ uniformly on the strip. The FF shift is treated in the same way in $M'$. By induction on $j$, $f_{j + 1}|_{\mathcal S_D}$ is the shifted onsite weight times $(K_\tau f_j)|_{\mathcal S_D}$, which is the image of $f_j|_{\mathcal S_D}$ under the shifted transfer operator.

We start from $\|w_{\Gamma, D, D_F}\|_{\mathcal H_p(\mathcal S)} \le C_k \eta^{-A_k} e^{C_* \|D\|^2}$, and we apply the shifted kinetic estimate of \ref{i:end-shifted-defect} and the bound \eqref{e:end-strip} at each step. Since $e^{-c' \vartheta} \le 1$ and the factors $e^{C_k/W + C_k \eta}$ multiply to at most $e^{C_k C_0}$ over $j \le C_0 W$ steps, we obtain \eqref{e:end-rough-strip}.

Finally, we prove \ref{i:end-shifted-collar}. The canonical gauge and the normal height selected by the canonical map both depend on the output point. We control both selections by \zcref{l:tools-submean}, applied jointly in $\zeta = (\Gamma, D, D_F)$ before the real body is integrated, on a compact parameter ball strictly inside the Hardy ball of \zcref{s:end-escape}, on which the Bergman integral is bounded by the Hardy norm. If the selected heights have norm at most $B$, we use a ball of radius $B$ in $(D, D_F)$ about the selected heights and a fixed radius in $\Gamma$, after a linear rescaling as in \zcref{r:tools-ellipsoid}. For complex $D$, the imaginary part of $D$ is a real translation of $P_0$ with unit Jacobian at fixed $X$. It acts as a real magnetic potential, which does not change the real part of the form and preserves the norm of $\mathcal H_p(\mathcal S)$, and the genuine imaginary strip is $\Re D$, of norm at most $2B$. For $R_{\rm s}$ small, $2B \le r_{\rm s}/2$, and therefore \eqref{e:end-rough-strip} holds on that ball with $4B^2$ in place of $\|D\|^2 + \|D_F\|^2$. The volume ratio of the submean lemma is a constant that depends on $k$ but not on $B$ or $W$. Fubini's theorem bounds the norm of the selected values by that constant times $e^{4 C_* m B^2}$ times the bound \eqref{e:end-rough-strip}, where $m \le 2 C_0 W$. The nilpotent canonical corrections are kept coefficientwise, and they change the bound by fixed powers of $W$, once, at the output. By \zcref{l:end-height}, we have $B^2 \le 2C_{\rm geo}^2(E^2 R^2 + R^4)$ on the set $\{|U| \le R\}$, and therefore
\[
 4 C_* m B^2 \le 16 C_{\rm geo}^2 C_* C_0 W (E^2 R^2 + R^4) .
\]
This proves \eqref{e:end-actual-collar}.
\end{proof}

\paragraph{Curved graphs.}
Bounds at constant height imply bounds on curved graphs, with a prefactor that depends on $k$. Let $F$ be Hilbert-space valued and holomorphic in the complex normal coordinate $w \in \C^d$, with all half-densities included, and let $b(x)$ be measurable with $|b(x)| \le B$. Suppose that $\int \|F(x + \iu v)\|^2 \dd x \le M^2$ for all $|v| \le 2B$. We apply \zcref{l:tools-submean} with $\zeta = w$, the function $(w, x) \mapsto F(x + w)$, the selection $\zeta_*(x) = \iu b(x)$, and $\rho = B$. The ball $B(\iu b(x), B)$ lies in $\{|\Re w| < B, \, |\Im w| < 2B\}$, and Fubini's theorem in $y = x + \Re w$ and $v = \Im w$ gives
\begin{equation}\label{e:end-curved}
 \int \|F(x + \iu b(x))\|^2 \dd x \le \frac{\operatorname{vol}_{\R^d}(B) \operatorname{vol}_{\R^d}(2B)}{\operatorname{vol}_{\R^{2d}}(B)} M^2 .
\end{equation}
The ratio depends on $d$ but not on $B$ or $W$, and it enters once, at the output, and not at the nodes. On a chart where the cutoffs equal one the submean balls lie where the real cutoff equals one.

\section{The energy threshold and the physical residual}\label{s:end-residual}

We now fix the energy threshold and show that a family of finite prepared norm, continued to the cycle by the canonical map, has an exponentially small residual on the annulus $r_1 \le |U| \le r_{\rm phys}$, where the physical cutoff varies. We begin with the prepared norm.

\paragraph{The prepared norm.}
We fix the constant $C_k$ in the exponent of \eqref{e:end-normal-block}, and we set
\[
 C_{\rm prep} = C_n C_0 + C_k R_k ,
\]
so that this exponent is $C_{\rm prep} W |b|^2$. For a family $f_{\Gamma, \mathbf V}$ generated from the endpoint that is holomorphic in the normal variables where $\psi_{R_k} \ne 0$, for a parameter ball $\mathcal U$ and a body weight $p$, we define the \emph{prepared norm}
\begin{equation}\label{e:end-prep-norm}
 \|f\|_{{\rm prep}, \mathcal U; p}^2 = \sup_{|b| \le c_1 R_k} e^{-2 C_{\rm prep} W |b|^2} \int_{\mathcal U} \sum_\ell \|f_{\Gamma, \ell}\|_{b; p, a}^2 \dd\Gamma ,
\end{equation}
with the components $f_{\Gamma, \ell}$ of \eqref{e:end-family-norm} and the Gaussian margin $a$ of \zcref{p:end-coupled}. The term with $b = 0$ is the family norm \eqref{e:end-family-norm}, so that $\|f\|_{\mathcal U; p, a} \le \|f\|_{{\rm prep}, \mathcal U; p}$. Integrating \eqref{e:end-normal-block} over $\Gamma \in \mathcal U_2$ and summing over the components, we find that the output of a block of $m = \lceil C_0 W \rceil$ adapted steps satisfies
\begin{equation}\label{e:end-prep-block}
 \|A_h^m g\|_{{\rm prep}, \mathcal U_2; p} \le C_k(R_k) \|g\|_{\mathcal U_2; p, -a}
\end{equation}
for every family $g$ generated from the endpoint, where on the right is the family norm \eqref{e:end-family-norm} with the Gaussian margin $-a$. By the last assertion of \zcref{p:end-coupled}, \eqref{e:end-prep-block} also holds for a block that contains finitely many jet steps, with the changes of \eqref{e:end-jet-constant}.

The prepared norm controls the values of a family on the complex normal graphs of height $|b| \le c_1 R_k$. A positive Gaussian margin on the real tube, which makes a family prepared, does not bound these values. The residual bounds of this section use them, and they are therefore stated for families of finite prepared norm, such as the outputs of blocks in \eqref{e:end-prep-block}.

\paragraph{The energy threshold.}
The Gaussian weight $e^{a|Z|^2}$ of the prepared norm, evaluated at the complex normal point $U + \iu b$, has the real exponent
\[
 a(|Z|^2 - |Z_b|^2) \ge c_{\rm ann} W |U|^2 - C_{\rm ann} W |b|^2 ,
\]
with the universal constants $c_{\rm ann} = a c_Z$ and $C_{\rm ann} = a C_Z$. Conversion from the symmetric normal gauge to the physical image under $\Can$ also multiplies by $\gamma_h$, besides the bounded half-weight and half-density factors. The ordinary body of this Gaussian satisfies
\begin{equation}\label{e:end-gauge-growth}
 |\gamma_h(U+\iu b)_\emptyset|^2
 = e^{-(|Z|^2-|Z_b|^2)/2},
 \qquad
 e^{|Z_b|^2/4}\le e^{C_{\rm gau}W|b|^2},
 \qquad C_{\rm gau}=C_Z/4.
\end{equation}
Since the ordinary Wick density is $e^{-|Z|^2/2}$, the conversion of a norm on this graph therefore costs $e^{|Z_b|^2/4}$. The odd Gaussian factors are bounded coefficient multipliers; the other changes of half-density and frame require only fixed constants and polynomial body weights. Thus $C_{\rm gau}$ is a separate, universal height cost, while $C_{\rm ann}=aC_Z$ records only the signed Gaussian weight. Recall the universal rate $c_{\rm D}$ of the Davies--Gaffney estimate \zcref{l:end-davies}, which gives the decay term $-c_{\rm D} W d_{\rm col}^2/C_0$ in the exponent of the barrier \eqref{e:end-barrier} below, and recall that the collar distance of the barrier is $d_{\rm col} = R_k/8$. We choose $E_{\rm geom} > 0$ so small that
\begin{equation}\label{e:end-Egeom}
 4 (C_n C_0 + C_{\rm ann}+C_{\rm gau}) C_{\rm geo}^2 E_{\rm geom}^2 \le \frac{c_{\rm ann}}{4} ,
 \qquad
 8192 \, C_{\rm geo}^2 C_* C_0^2 E_{\rm geom}^2 \le c_{\rm D} .
\end{equation}
The first condition is used in the proof of \zcref{l:end-shell}. The second makes the term $16 C_{\rm geo}^2 C_* C_0 W E_{\rm geom}^2 R_k^2$ of the barrier \eqref{e:end-barrier} at most an eighth of the decay term $c_{\rm D} W d_{\rm col}^2/C_0 = c_{\rm D} W R_k^2/(64 C_0)$ of its exponent. Since $c_{\rm ann}$ is proportional to $a$, the two conditions amount to
\[
 E_{\rm geom} \lesssim C_{\rm geo}^{-1} \Bigl(\frac{a}{C_n C_0+C_{\rm ann}+C_{\rm gau}}\Bigr)^{1/2} ,
 \qquad
 E_{\rm geom} \lesssim \frac{1}{C_{\rm geo} C_0} \Bigl(\frac{c_{\rm D}}{C_*}\Bigr)^{1/2} ,
\]
where $0 < a < a_0$ and $C_0 \ge \varrho > 2a/v_0$. Every constant in this chain is universal and depends only on constants chosen before it in \zcref{tab:end-constants}: $C_n$ by \zcref{r:end-universal}, $C_{\rm gau}=C_Z/4$ by the standardized normal-coordinate bounds, $C_{\rm geo}$ by \zcref{l:end-height}, $C_*$ by \eqref{e:end-strip} and $c_{\rm D}$ by \zcref{l:end-davies}. Therefore $E_{\rm geom}$ is universal. We do not give it explicitly, since these universal constants are not computed. We set $c_{\rm sh} = c_{\rm ann}/4$.

\paragraph{The shell jets.}
Let $K_{{\rm ad}, s}$ denote the adapted heat over time $s$, and $\mathbf 1_{\rm ann}$ the indicator of the annulus $r_1 \le |U^{\rm re}_{\rm can}| \le r_{\rm phys}$ where the physical cutoff varies, defined after \eqref{e:asm-cutoff}. Let $\| \cdot \|_{\rm jets}$ be the sum of the $\mathcal H_p(\mathcal S)$ norms of the finitely many normal and tangent derivatives used in \zcref{s:assembly}, together with the derivatives of order at most one; we call these derivatives, restricted to the annulus, the \emph{shell jets}. Recall the canonical map $\Can$ of \zcref{s:end-canonical} and the physical cutoff $\chi_{\rm phys}$ of \eqref{e:asm-cutoff}, and set $\Can_{\rm phys} = \chi_{\rm phys} \Can$. The following proposition, with its two lemmas, shows that a family of finite prepared norm \eqref{e:end-prep-norm}, continued to the cycle by the canonical map, has an exponentially small heat residual: the physical step applied to its canonical image differs from the canonical image of the adapted step only through the commutator of the physical heat with $\chi_{\rm phys}$, which lives on the annulus, and there the family is exponentially small.

\begin{lemma}\label{l:end-shell}
Let $|E| \le E_{\rm geom}$, choose the nested radii of \zcref{tab:end-constants} at fixed rank, and let $p \in \R$. There exist $A_{\rm ann} \ge 0$ and $p_{\rm ann} \ge p$, depending only on $k$, $p$, the orders of the shell jets, and the prescribed local jets of the family, such that the following holds for all sufficiently large $W$. Let $f = f_{\Gamma, \mathbf V}$ be a family generated from the endpoint that is holomorphic in the normal variables where $\psi_{R_k} \ne 0$, and let $\Gamma \in \mathcal U_0$. Then
\begin{equation}\label{e:end-shell}
 \sup_{0 \le s \le \tau_{\rm phys}} \bigl\| \mathbf 1_{\rm ann} \, \Can K_{{\rm ad}, s}(\chi_{\rm inv} f_{\Gamma, \mathbf V}) \bigr\|_{\rm jets}
 \le C_k W^{A_{\rm ann}} e^{-c_{\rm sh} W r_1^2} \|f\|_{{\rm prep}, \mathcal U_2; p_{\rm ann}} .
\end{equation}
\end{lemma}

\begin{proof}
We first continue the family. By \zcref{l:end-height} and \eqref{e:end-canonical-frame}, $S_{\mathsf g} - \Id$ and its logarithm $Y_{\mathsf g}$ are $O(|U|)$ uniformly in the boost. The path $t \mapsto e^{t Y_{\mathsf g}}$, $t \in [0, 1]$, therefore stays in a neighborhood of the identity of size $O(|U|)$. By \zcref{l:end-ward}\ref{i:end-ward-b}, the continuation of the body along this path is the family on the real tube at damping $S_{\mathsf g} \Gamma S_{\mathsf g}^{-1}$ and directions $S_{\mathsf g} V_a S_{\mathsf g}^{-1}$. We choose the tube so small that these damping values stay in $\mathcal U_1$ for $\Gamma \in \mathcal U_0$.

We now evaluate at the canonical point. The parameter and the normal height that the canonical map selects both depend on the output, and we control the two selections together. We cover the annulus by the shells $\Omega_s = \{s \le |U| \le (1 + \theta)s\}$, where $(1 + \theta)^6 \le 2$. By \zcref{l:end-height}, the canonical height on $\Omega_s$ is at most
\[
 B_s = C_{\rm geo}\bigl(|E|(1 + \theta)s + (1 + \theta)^2 s^2\bigr) .
\]
The first condition in \eqref{e:end-Egeom} and $c_Z \le C_Z$ give $C_{\rm geo} |E| \le 1/4$, and therefore $B_s \le s$ for $s \le r_{\rm phys}$ once $r_{\rm phys}$ is small. On each shell we apply \zcref{l:tools-submean} jointly in $\zeta = (\Gamma, w)$, where $w$ is the complex normal coordinate, with the selection $\zeta_*(U) = (S_{\mathsf g}\Gamma S_{\mathsf g}^{-1}, \iu b(U))$, after rescaling $w$ so that the ball has radius $\theta B_s$ in $w$ and, in $\Gamma$, the radius equal to the distance from $\mathcal U_1$ to the complement of $\mathcal U_2$, as in \zcref{r:tools-ellipsoid}. The submean balls lie inside the region where $\chi_{\rm inv} = 1$ and $\psi_{R_k} = 1$. We apply the submean estimate to the physical half-densities, including the Gaussian gauge, so that their ordinary normal measure is Lebesgue measure. As in \eqref{e:end-curved}, Fubini's theorem reduces the estimate, up to a volume ratio depending only on $k$, to physical norms at constant heights $|b'| \le (1 + \theta) B_s$, restricted to the real points at distance at most $\theta B_s$ from $\Omega_s$. We compare these norms with the graph norms $\|f_{\Gamma', \ell}\|_{b'; p_{\rm ann}, a}$ of \eqref{e:end-gauge-graph-norm}, accounting for \eqref{e:end-gauge-growth} and the signed Gaussian weight below. By the definition \eqref{e:end-prep-norm}, the integrals of the squared graph norms over $\mathcal U_2$ are at most $e^{2 C_{\rm prep} W |b'|^2} \|f\|_{{\rm prep}, \mathcal U_2; p_{\rm ann}}^2$.

The derivatives in the shell jets change these bounds by fixed factors. By Cauchy's estimates on polydisks of radius $c/W$, normal derivatives of order $\ell$ are bounded with an additional factor $C W^\ell$, and the finite odd canonical corrections, which we retain as finite jets, give further fixed powers of $W$. Derivatives of the Gaussian gauge also cost only fixed powers of $W$ on the fixed tube. By the equivariance \eqref{e:end-ward}, tangent derivatives become parameter derivatives, and fixed body derivatives require fixed polynomial weights, which the body weight $p_{\rm ann} \ge p$ provides. These factors enter once, at the output, and not at the nodes. We take $A_{\rm ann}$ to be the sum of the resulting exponents of $W$ and of the exponent of the final partial heat below.

We now collect the exponents on $\Omega_s$. The real points $U'$ of the submean balls satisfy $|U'| \ge (1 - \theta) s$, since $B_s \le s$. At the point $U' + \iu b'$ of the inner tube, the Gaussian weight of the prepared norm has modulus $e^{a(|Z'|^2 - |Z_{b'}|^2)}$, where $Z'$ and $Z_{b'}$ are the standardized versions of $U'$ and $b'$, and $|Z'|^2 \ge c_Z W |U'|^2$, $|Z_{b'}|^2 \le C_Z W |b'|^2$. Removing this weight, converting to the physical norm by \eqref{e:end-gauge-growth}, and using the factor $e^{C_{\rm prep} W |b'|^2}$ contribute the factors
\[
\begin{gathered}
 \exp\bigl\{-c_{\rm ann} (1 - \theta)^2 W s^2 + (C_{\rm ann}+C_{\rm gau}) W (1 + \theta)^2 B_s^2\bigr\} ,
 \\
 \exp\bigl\{(C_n C_0 + C_k R_k) W (1 + \theta)^2 B_s^2\bigr\} ,
\end{gathered}
\]
where $c_{\rm ann} = a c_Z$, $C_{\rm ann} = a C_Z$, and $C_{\rm gau}=C_Z/4$. Since $(1 + \theta)^6 \le 2$, we have
\[
 (1 + \theta)^2 B_s^2 \le 2 C_{\rm geo}^2 (1 + \theta)^6 (E^2 s^2 + s^4) \le 4 C_{\rm geo}^2 (E^2 s^2 + s^4) ,
\]
and the first condition in \eqref{e:end-Egeom} implies that the part of the exponent proportional to $(C_n C_0 + C_{\rm ann}+C_{\rm gau}) E^2$ is at most $c_{\rm ann} W s^2/4$. The quartic part and the terms with $C_k R_k$ are at most $c_{\rm ann} W s^2/8$ once $r_{\rm phys}$ and $R_k$ are small at fixed rank. Since $(1 - \theta)^2 \ge 1 - 2\theta$ and $2\theta \le 3/8$, the exponent is at most
\[
 -c_{\rm ann} \Bigl(1 - 2\theta - \frac14 - \frac18\Bigr) W s^2 \le -\frac{c_{\rm ann}}{4} W s^2 \le -c_{\rm sh} W r_1^2
\]
on $\Omega_s$, and the finite number of shells changes only the constant. This bound is also uniform along the homotopy \eqref{e:asm-homotopy}: its selected heights are $t b(U)$ with $0\le t\le1$, hence no larger than $B_s$, and its conjugated dampings remain in $\mathcal U_1$.

A final partial heat satisfies the same bound. Indeed, its normalized Gaussian either stays in the region where $\chi_{\rm inv} = 1$, or makes a jump across a fixed gap, which contributes at most $C_k W^{A_{\rm ann}} e^{-c W^2 r_1^2}$. The bound is uniform down to zero heat time, since the input is holomorphic in the normal variables on polydisks of a fixed radius about the real points of the annulus. This proves \eqref{e:end-shell}. Since $W \ge 1$ and the prepared norm increases with the body weight, the bound remains true when $A_{\rm ann}$ and $p_{\rm ann}$ are replaced by larger numbers.
\end{proof}

\paragraph{The exponent and the body weight of the residual.}
Let $L$ denote the generator of the physical heat on the cycle, as in \zcref{s:end-escape}. By the construction of $\chi_{\rm phys}$ in \eqref{e:asm-cutoff}, the commutator $[L, \chi_{\rm phys}]$ is a first-order operator whose coefficients are supported on the annulus where $\chi_{\rm phys}$ varies. At fixed rank, these coefficients are bounded by fixed powers of $W$, from the odd coordinates, and by fixed powers of the body weight, from the boost, and the shell jets contain the derivatives of order at most one. Hence there exist $A_1 \ge 0$ and $q \ge 0$, depending only on $k$, such that
\begin{equation}\label{e:end-commutator}
 \|[L, \chi_{\rm phys}] u\|_{\mathcal H_p(\mathcal S)} \le C_k W^{A_1} \|\mathbf 1_{\rm ann} u\|_{{\rm jets}, p + q} ,
\end{equation}
where the norm on the right is that of the shell jets in the body weight $p + q$. Let $A_{\rm ann}$ and $p_{\rm ann}$ be the exponent and the body weight of \zcref{l:end-shell} for the body weight $p + q$. We set
\[
 A_{\rm sh} = A_{\rm ann} + A_1 ,
 \qquad
 p' = p_{\rm ann} .
\]
They depend only on $k$, $p$, the orders of the shell jets, and the prescribed local jets of the family, and $p' \ge p + q \ge p$. Since $v \ge 1$ and $q \ge 0$, the norm of the shell jets in the body weight $p$ is at most their norm in the body weight $p + q$. Therefore \eqref{e:end-shell} for $p + q$ implies \eqref{e:end-shell} for $p$ with $A_{\rm sh}$ and $p'$ in place of $A_{\rm ann}$ and $p_{\rm ann}$. We use these $A_{\rm sh}$ and $p'$ in the rest of this section.

\begin{lemma}\label{l:end-heat-residual}
Let $E$, the radii, $p$, $f$ and $\Gamma$ be as in \zcref{l:end-shell}, let $A_{\rm sh}$ and $p'$ be as defined before this lemma, and suppose that $\|f\|_{{\rm prep}, \mathcal U_2; p'} < \infty$. For $0 \le s \le \tau_{\rm phys}$, let $u_s = \Can K_{{\rm ad}, s}(\chi_{\rm inv} f)$. Then the physical step $T_\Gamma$ and the adapted step $A_h$ satisfy the coefficientwise identity
\begin{equation}\label{e:end-heat-residual}
 (T_\Gamma \Can_{\rm phys} - \Can_{\rm phys} A_h) f = M_{w_\Gamma} \int_0^{\tau_{\rm phys}} K_{\tau_{\rm phys} - s} \, [L, \chi_{\rm phys}] \, u_s \dd s ,
\end{equation}
where $L$ is the generator of the physical heat and $M_{w_\Gamma}$ is the multiplication by $w_\Gamma$.
\end{lemma}

\begin{proof}
We first say what $u_s$ means at the points of the cycle, which are complex points of the tube. Let
\[
 g = \varpi \, w_0^{1/2} \gamma_h \, \chi_{\rm inv} f
\]
be the raw input on the real tube, let $k_s$ be the kernel of the ambient heat over time $s$, and let $s > 0$. For a real frame $x$ and a complex normal coordinate $U$ with $|U| \le r_{\rm sep}$, we set
\begin{equation}\label{e:end-us}
 u_s(x, U) = \int k_s\bigl(R(x, U), R(y')\bigr) \, g(y') \dd y' ,
\end{equation}
where $y' = (x', U')$ runs over the real tube with the flat measure $\dd U' \dd\nu(x')$. By \eqref{e:cons-separation}, the real part of the exponent of $k_s$ is bounded above by $-c \, s_E^2 \, \mathfrak d(x, x')/(4s)$ plus $|\Im U|^2/s$, and the odd coordinates contribute only polynomial factors. Therefore the integral converges locally uniformly, and $u_s$ is holomorphic in $U$. To evaluate $u_s$ at the canonical point of a point of the cycle, we use \eqref{e:end-us} for the family at the conjugated damping $\widehat S \Gamma \widehat S^{-1}$ and directions $\widehat S \mathbf V \widehat S^{-1}$, including the finite nilpotent expansion, as in the definition of $\Can$. The partial heat $K_{{\rm ad}, s}(\chi_{\rm inv} f)$ is a family generated from the endpoint, and by \zcref{l:end-ward}\ref{i:end-ward-b} this is its canonical image $\Can K_{{\rm ad}, s}(\chi_{\rm inv} f)$. By the same lemma, $u_s$ is holomorphic along the complexified orbits. Together with its holomorphy in $U$, this makes $u_s$ a holomorphic function of the ambient point near the canonical points.

We now show that $u_s$ satisfies the heat equation $\partial_s u_s = \Delta_{\rm amb} u_s$ near the canonical points, where $\Delta_{\rm amb}$ acts on the holomorphic extension just described; the points $R(x, U)$ with $x$ real and $U$ complex do not form an open set. The real orbit directions and the complex normal directions at such a point span the ambient tangent space over $\C$. The holomorphic derivatives of a holomorphic function at the point are therefore determined by its derivatives in these directions, and $\Delta_{\rm amb}$ acts as the operator $\varpi^{-1}(\Delta_{\rm nor} + L_H(U)) \varpi$ of \eqref{e:tube-decomposition} in the coordinates $(x, U)$. Recall from \zcref{s:cons-heat} that $k_s$ is entire in its first argument and satisfies $\partial_s k_s = \Delta_{\rm amb} k_s$. We differentiate \eqref{e:end-us} under the integral sign in $s$, in $x$ and in $U$, which is justified by \eqref{e:cons-separation} locally uniformly for $s > 0$, and we obtain $\partial_s u_s = \Delta_{\rm amb} u_s$ at the points $R(x, U)$ with $x$ real and $|U| \le r_{\rm sep}$. The function $\partial_s u_s - \Delta_{\rm amb} u_s$ is holomorphic on the connected open set of points $S^{-1} R(x, U) S$ with $x$ real, $U$ complex and $S$ near the identity, and it vanishes on the real tube, which is a totally real submanifold of maximal dimension. By the identity theorem, it vanishes on this set, which contains the canonical points. The generator $L$ of the physical heat is the restriction of $\Delta_{\rm amb}$ to the cycle, so the chain rule gives $\partial_s u_s = L u_s$ on the cycle.

The identity at time $\tau = \tau_{\rm phys}$ is exact. By \eqref{e:tube-table} and \eqref{e:tube-raw-kernel}, with the cutoffs of \eqref{e:end-raw-steps},
\[
 A_h f = \chi_{\rm inv} \, s_\Gamma \gamma_h^{-1} w_0^{1/2} \varpi \cdot (u_\tau \circ R)
\]
on the real tube. The canonical map multiplies by the inverse half-density and the inverse gauge $\varpi^{-1} w_0^{1/2} \gamma_h$, every factor other than $\chi_{\rm inv}$ is holomorphic in the normal variables, and $\chi_{\rm inv} = 1$ on the support of $\chi_{\rm phys}$. Since $s_\Gamma w_0 = w_\Gamma$, we obtain
\begin{equation}\label{e:end-exact-tau}
 \Can_{\rm phys} A_h f = \chi_{\rm phys} \, w_\Gamma \, u_\tau .
\end{equation}
Both sides are continued to complex orbit points by \zcref{l:end-ward}\ref{i:end-ward-b}, so they agree there by the identity theorem.

We next apply Duhamel's formula on $[\eps, \tau]$, for $0 < \eps < \tau$. The explicit Gaussian form of $k_s$ bounds $u_s$ and its derivatives on the support of $\chi_{\rm phys}$, at fixed $W$ and uniformly for $\eps \le s \le \tau$. Hence $\chi_{\rm phys} u_s$ lies in the domain of $L$ in $\mathcal H_p(\mathcal S)$, and $s \mapsto \chi_{\rm phys} u_s$ is continuously differentiable there, with derivative $\chi_{\rm phys} L u_s$. Since $\partial_s (K_{\tau - s} \chi_{\rm phys} u_s) = -K_{\tau - s} [L, \chi_{\rm phys}] u_s$, we obtain
\begin{equation}\label{e:end-duhamel-eps}
 K_{\tau - \eps} (\chi_{\rm phys} u_\eps) - \chi_{\rm phys} u_\tau = \int_\eps^\tau K_{\tau - s} \, [L, \chi_{\rm phys}] \, u_s \dd s .
\end{equation}

It remains to let $\eps \downarrow 0$. We show that
\begin{equation}\label{e:end-initial-limit}
 \lim_{\eps \downarrow 0} \bigl\| \chi_{\rm phys} (u_\eps - \Can f) \bigr\|_{\mathcal H_p(\mathcal S)} = 0 .
\end{equation}
Fix a point of the cycle in the support of $\chi_{\rm phys}$, with canonical frame $x$ and canonical normal coordinate $U + \iu b$. In \eqref{e:end-us}, the orbit coordinate of the input is already in the real frame, at the conjugated damping. By Cauchy's theorem in the normal input variables, we move the normal input onto the constant-height graph through the output height $b$. This deformation takes place where $\chi_{\rm inv} = 1$ and the graph cutoff equals one, except for inputs at a normal distance at least $d > 0$ from the output, where $d$ depends only on the nested radii. By \eqref{e:cons-separation}, these inputs contribute $O(e^{-c d^2/\eps})$, which tends to zero.

After this deformation, $u_\eps$ is the pulled Gaussian evolution \eqref{e:cons-pulled} on the constant-height graph, with constant parameter and with the values of $f$ on the same graph as initial data. As $\eps \downarrow 0$ it tends to $f$ in $L^2$ of the graph. For smooth $f$ this is the Laplace method in the proof of \zcref{l:cons-identification}, and for general $f$ it follows by density, since that proof bounds the maps $f \mapsto \1_{\mathcal T_{\rm in}} u(t)$ uniformly in $t$. The damping and the height of the graph depend on the output. As in the proof of \zcref{l:end-shell}, we control this selection by \zcref{l:tools-submean} in $\zeta = (\Gamma, w)$, which bounds the selected values by an integral of graph norms over a ball in $(\Gamma, w)$. Each integrand tends to zero as $\eps \downarrow 0$. It is bounded by a constant times the squared graph norm of $f$, which is integrable over the ball since $\|f\|_{{\rm prep}, \mathcal U_2; p'} < \infty$. By dominated convergence, we obtain \eqref{e:end-initial-limit}.

We now let $\eps \downarrow 0$ in \eqref{e:end-duhamel-eps}. The left side converges by \eqref{e:end-initial-limit}, together with the uniform bound and the strong continuity of $K_t$ from \zcref{l:end-kinetic}. By \eqref{e:end-commutator} and \zcref{l:end-shell} in the body weight $p + q$, which applies since $\|f\|_{{\rm prep}, \mathcal U_2; p'} < \infty$ for the body weight $p' = p_{\rm ann}$ defined before this lemma, we have
\[
 \int_0^\tau \|[L, \chi_{\rm phys}] u_s\| \dd s < \infty ,
\]
and the right side converges. We obtain
\[
 K_\tau \Can_{\rm phys} f - \chi_{\rm phys} u_\tau = \int_0^\tau K_{\tau - s} \, [L, \chi_{\rm phys}] \, u_s \dd s .
\]
Multiplying by $w_\Gamma$, and using $T_\Gamma = M_{w_\Gamma} K_\tau$ and \eqref{e:end-exact-tau}, we obtain \eqref{e:end-heat-residual}.
\end{proof}

\begin{proposition}\label{c:end-residual}
Let $|E| \le E_{\rm geom}$, choose the nested radii of \zcref{tab:end-constants} at fixed rank, and let $p \in \R$. Let $A_{\rm sh} \ge 0$ and $p' \ge p$ be the exponent and the body weight defined before \zcref{l:end-heat-residual}, which depend only on $k$, $p$, the orders of the shell jets, and the prescribed local jets of the family. The following holds for all sufficiently large $W$. Let $f = f_{\Gamma, \mathbf V}$ be a family generated from the endpoint that is holomorphic in the normal variables where $\psi_{R_k} \ne 0$, with $\|f\|_{{\rm prep}, \mathcal U_2; p'} < \infty$, and let $\Gamma \in \mathcal U_0$. Then $f$ satisfies the shell bound \eqref{e:end-shell} with $A_{\rm sh}$ and $p'$ in place of $A_{\rm ann}$ and $p_{\rm ann}$, the identity \eqref{e:end-heat-residual}, and, with $u_s$ as in \zcref{l:end-heat-residual},
\begin{equation}\label{e:end-residual-bound}
\begin{aligned}
 &\Bigl\| \int_0^{\tau_{\rm phys}} K_{\tau_{\rm phys} - s} \, [L, \chi_{\rm phys}] \, u_s \dd s \Bigr\|_{\mathcal H_p(\mathcal S)}
 + \bigl\| (T_\Gamma \Can_{\rm phys} - \Can_{\rm phys} A_h) f \bigr\|_{\mathcal H_p(\mathcal S)} \\
 &\qquad \le C_k W^{A_{\rm sh}} e^{-c_{\rm sh} W r_1^2} \|f\|_{{\rm prep}, \mathcal U_2; p'} .
\end{aligned}
\end{equation}
In particular, the heat residual is exponentially small, with the full onsite weight $w_\Gamma$, including its nilpotent terms.
\end{proposition}

\begin{proof}
By the choice of $A_{\rm sh}$ and $p'$ before \zcref{l:end-heat-residual}, the shell bound \eqref{e:end-shell} holds with $A_{\rm sh}$ and $p'$ in place of $A_{\rm ann}$ and $p_{\rm ann}$, and \zcref{l:end-heat-residual} gives \eqref{e:end-heat-residual}.

It remains to prove \eqref{e:end-residual-bound}. By \eqref{e:end-commutator} and \zcref{l:end-shell} in the body weight $p + q$, we have
\[
 \sup_{0 \le s \le \tau} \|[L, \chi_{\rm phys}] u_s\|_{\mathcal H_p(\mathcal S)} \le C_k W^{A_1 + A_{\rm ann}} e^{-c_{\rm sh} W r_1^2} \|f\|_{{\rm prep}, \mathcal U_2; p'} .
\]
By \zcref{l:end-kinetic}, for $0 \le s \le \tau$ we have
\[
 \|K_{\tau - s}\|_{\mathcal H_p(\mathcal S) \to \mathcal H_p(\mathcal S)} \le e^{(C_k W + C_{k, p}) \tau} \le C_k ,
\]
and $\tau \le 1$. Integrating over $s$, we obtain the bound \eqref{e:end-residual-bound} for the first term. By \zcref{l:end-onsite}, multiplication by $w_\Gamma$, including its nilpotent terms, has norm at most $e^{C_k/W + C_k \eta}$ on $\mathcal H_p(\mathcal S)$, and therefore \eqref{e:end-heat-residual} gives the bound for the second term. This completes the proof.
\end{proof}

By \eqref{e:end-prep-block}, the hypotheses of \zcref{c:end-residual} hold for the output $f = A_h^m g$ of a block of $m = \lceil C_0 W \rceil$ adapted steps, for every family $g$ generated from the endpoint with $\|g\|_{\mathcal U_2; p', -a} < \infty$, and then $\|f\|_{{\rm prep}, \mathcal U_2; p'} \le C_k(R_k) \|g\|_{\mathcal U_2; p', -a}$. For a block with finitely many jet steps, the same holds with the changes of \eqref{e:end-jet-constant}. This is how \zcref{s:assembly} uses the proposition. The long partial chains with which these residuals are paired there stay on the real cycle at their original BB damping, and they are bounded there by the $e^{o(W)}$ estimate of \zcref{t:hist-bound} in the opposite body weight.

\section{The early barrier and the attachment of the endpoint}\label{s:end-attach}

In this section we compare the physical block $f_m = T_\Gamma^m 1$ with local references on the tube, and prove the attachment theorem. For each signature $\alpha \in \{0, \dots, 2k\}$ we choose disjoint normal neighborhoods of the corresponding FF orbit, with a common BB normal radius below the root gap. We work first in the raw realization, before the Gaussian gauge. Let $\varpi_\alpha$ be the half-density and $\Can_\alpha^{\rm raw}$ the canonical pullback with the inverse half-density and frame.

\paragraph{The domains of the comparison.}
In each of these neighborhoods we fix three nested normal neighborhoods of the orbit, defined through the norm $|U|$ of the normal coordinate. They are the \emph{inner region} $\{|U| \le R_k/2\}$, on which we compare the physical chain with the references; the larger \emph{comparison domain} $\{|U| < 7 R_k/8\}$; and the region where the cutoffs $\chi_{\rm inv}$ of the raw steps equal one. Recall from \zcref{s:end-energy} that the last region contains the support of $\psi_{R_k}$, and hence the neighborhood $\{|U| \le R_k\}$ of the comparison domain.

We also use the graph cutoff $\psi_{\bar r}$ of \eqref{e:cons-graph} with $\bar r = 15 R_k/16$. We have $\psi_{\bar r} = 1$ on the neighborhood $\{|U| \le 15 R_k/16\}$ of the comparison domain, and the support of $\psi_{\bar r}$ lies in that of $\psi_{R_k}$, at distance at least $R_k/16$ from the complement of the latter. We choose an invariant normal class cutoff $\xi_\alpha$, with all its nilpotent terms, equal to one on a neighborhood of the support of $\psi_{\bar r}$ and supported in the interior of the support of $\psi_{R_k}$, where $\chi_{\rm inv} = 1$. Its body takes values in $[0, 1]$, and it is fixed in the physical normal variable $U$. In particular, $\xi_\alpha$ differs from one only at distance at least $R_k/16$ from the comparison domain. We also choose a real normal cutoff $\phi_\alpha$, supported in the comparison domain and equal to one on the inner region, whose collar lies in $\{5 R_k/8 \le |U| \le 3 R_k/4\}$; we call it the \emph{barrier cutoff}. Its collar lies in $\{|U| \le R_k\}$, at normal distance at least $d_{\rm col} = R_k/8$ from the inner region.

We define the raw references
\[
 g_{\alpha, 0}^{\rm raw} = \xi_\alpha \varpi_\alpha ,
 \qquad
 g_{\alpha, j}^{\rm raw} = S_{\alpha, \Gamma}^j g_{\alpha, 0}^{\rm raw} ,
\]
where $S_{\alpha, \Gamma}$ is the adapted step with cutoffs of \eqref{e:end-raw-steps} in the signature $\alpha$, in its onsite-after-heat order. Then $\Can_\alpha^{\rm raw} g_{\alpha, 0}^{\rm raw} = 1$ throughout the comparison domain, with the physical orientation. In the symmetric normal gauge, these references become $\gamma_h^{-1} w_0^{-1/2} g_{\alpha, j}^{\rm raw}$; both endpoint factors are transported together. Let $f_{\alpha, j}^{\rm raw}$ denote the partial chain $f_j$ written in the raw chart of the signature $\alpha$ through the canonical map, as in \zcref{p:end-shifted}\ref{i:end-shifted-collar}. For a body weight $p$, we write $\| \cdot \|_{{\rm comparison}, \mathcal H_p}$ and $\| \cdot \|_{{\rm inner}, \mathcal H_p}$ for the raw norms in the body weight $v^p$ of the restrictions to the comparison domain and to the inner region, taken coefficientwise.

The main result of this section is the following attachment theorem.

\begin{theorem}\label{t:end-entrance}
Let $m = \lceil C_0 W \rceil$, let $\chi_\alpha^{\rm phys}$ be real physical normal cutoffs supported in the disjoint tubes of radius $r_{\rm phys}$, and set $\Can_{\alpha, {\rm phys}}^{\rm raw} = \chi_\alpha^{\rm phys} \Can_\alpha^{\rm raw}$. Let $p \in \R$, and let a finite family of derivatives in the damping parameter and in the local sources at zero be given. Then
\begin{equation}\label{e:end-full}
T_\Gamma^m 1 = \sum_{\alpha = 0}^{2k} \Can_{\alpha, {\rm phys}}^{\rm raw} g_{\alpha, m}^{\rm raw} + e_m^R ,
 \qquad
 \|e_m^R\|_{\mathcal H_p(\mathcal S)} \le C_k W^{A_k} e^{-\gamma_k W} ,
\end{equation}
and the same bound holds for the derivatives of $e_m^R$ in the given family, where $\gamma_k > 0$, $A_k$ and $C_k$ may depend on $p$ and on the family. The constants are uniform for $\Gamma$ in the smallest damping ball $\mathcal U_0$ of \eqref{e:end-damping}, for $|E| \le E_{\rm geom}$, and on compact sets of the aspect ratio and of the damping $\eps$.
\end{theorem}

We prove \zcref{t:end-entrance} at the end of this section. Near the saddle manifold, the physical chain and the references satisfy the same local equation and agree initially, so that their difference is created only on the collar of the barrier cutoff, at normal distance $d_{\rm col}$ from the inner region. The Davies--Gaffney estimate \zcref{l:end-davies} shows that over one block this difference reaches the inner region only with a factor $e^{-c_{\rm D} W d_{\rm col}^2/C_0}$. The proof uses five lemmas: the bound on the references near the collar (\zcref{l:end-early}), the first step (\zcref{l:end-first-step}), the barrier (\zcref{l:end-barrier}), the choice of the radii (\zcref{l:end-radii}), and the continuation of the error to the canonical graph (\zcref{l:end-continuation}).

\paragraph{Early annuli.}
The barrier below uses the raw references on the collar, where they need not be small, and it needs a bound there for every $0 \le j \le C_0 W$, including $j = 0$. We call the finitely many normal and tangent derivatives of a fixed order, restricted to the part of an annulus $|U| \ge r$ that lies in the comparison domain, the \emph{annular jets}, and we write $\| \cdot \|_{{\rm raw}, {\rm jets}}$ for the sum of their raw norms in the body weight $v^{p_0}$ of \zcref{l:end-raw-norms}. The annular jets are measured in raw norms of functions on the tube, on the part of $\{|U| \ge r\}$ in the comparison domain, whereas the shell jets of \zcref{s:end-residual} are measured in $\mathcal H_p(\mathcal S)$, for functions on the cycle, on the annulus where $\chi_{\rm phys}$ varies. The comparison domain contains the collar of the barrier cutoff, where the annular jets are used. Let $c_{\rm e} = \min(v_0, a_0/C_0)$.

\begin{lemma}\label{l:end-early}
For all $0 \le j \le C_0 W$ and $r > 0$, the raw references satisfy
\begin{equation}\label{e:end-early-annuli}
 \bigl\| \1_{\{|U| \ge r\}} \, g_{\alpha, j}^{\rm raw} \bigr\|_{{\rm raw}, {\rm jets}} \le C_k W^{A_k} e^{- c_{\rm e} c_Z j r^2} .
\end{equation}
For a positive body weight $v^p$, the same bound holds for $j \ge 1$ with a larger power of $W$.
\end{lemma}

\begin{proof}
The bound is a corollary of \zcref{l:end-energy}. Let $a_i' = c_{\rm e} i h^2$ for $0 \le i \le j$. Then
\[
 a_0' = 0 ,
 \qquad
 0 \le a_i' \le c_{\rm e} C_0 W h^2 \le a_0 ,
 \qquad
 |a_i' - a_{i - 1}'| = c_{\rm e} h^2 \le v_0 h^2 .
\]
We apply \zcref{l:end-energy} with $b = 0$ to the raw steps $S_{\alpha, \Gamma}$, in the body weight $v^{p_0}$. The input $\xi_\alpha \varpi_\alpha$ is bounded and supported in the tube, and its raw norm is at most $C_k W^{A_k}$, since the tube has volume $O(W^{A_k})$ in the standardized coordinates. Therefore
\[
 \|w_{a_j'} v^{p_0} g_{\alpha, j}^{\rm raw}\|_{\rm raw} \le C_k \|v^{p_0} \xi_\alpha \varpi_\alpha\|_{\rm raw} \le C_k W^{A_k} .
\]
Since $|Z|^2 \ge c_Z W |U|^2$ and $a_j' W = c_{\rm e} j$, we have $w_{a_j'} \ge e^{c_{\rm e} c_Z j r^2}$ where $|U| \ge r$, and this proves \eqref{e:end-early-annuli} without derivatives. For a positive body weight $v^p$ and $j \ge 1$, the damping $e^{-c \eta v}$ of the first onsite factor bounds the weight $v^{p - p_0}$, since $v^q e^{-c\eta v} \le C_q \eta^{-q}$ for $q \ge 0$, and $\eta^{-q}$ is a power of $W$.

For the normal derivatives we use the same graph at every node, with the graph cutoff $\psi_{\bar r}$, $\bar r = 15 R_k/16$, of the domains of the comparison in place of $\psi_{R_k}$. The proof of \zcref{l:end-energy} applies verbatim to this cutoff, for heights $|b| \le c_1 \bar r$. Indeed, since $R_k/2 \le \bar r \le R_k$, the support of $\psi_{\bar r}$ is contained in that of $\psi_{R_k}$, where the cutoffs equal one, and the bound on the graph Jacobian satisfies $C|b|/\bar r \le 2C|b|/R_k$. Let $b$ be a constant real normal direction with $|b| \le c/W$, and apply \zcref{l:end-energy} in this form with $\lambda_i = 1$ for all $i$, including the input, and with the same weights $a_i'$. This is allowed since $\xi_\alpha \varpi_\alpha$ is holomorphic in the normal variables on the support of $\psi_{\bar r}$, where $\xi_\alpha = 1$. There are no velocity terms, and the exponent in \eqref{e:end-energy} is at most
\[
 (C_1 + C_k R_k) C_0 W |b|^2 \le (C_1 + C_k R_k) \, C_0 c^2 / W = O(1) .
\]
Where $\psi_{\bar r} = 1$, the weight on such a graph satisfies
\[
 |w_{a_j'}(U_\lambda)| = e^{a_j'(|Z|^2 - |Z_b|^2)} ,
 \qquad
 a_j' |Z_b|^2 \le a_0 C_Z c^2/W ,
\]
and the last quantity is small. For $W$ large, the polydisks of radius $c/W$ about the real points of the annulus in the comparison domain lie where $\psi_{\bar r} = 1$, and Cauchy's estimates on them bound the normal derivatives of order $\ell$ with an additional factor $C W^\ell$. Shifting the annulus by $c/W$ changes $e^{-c_{\rm e} c_Z j r^2}$ only by a bounded factor, since $j \le C_0 W$. The tangent derivatives are derivatives in the parameters, by the equivariance \eqref{e:end-covariance}, since the references are generated from the endpoint. By Cauchy's formula on the larger damping ball, they are bounded in fixed stronger body weights. The comparison with the Dirichlet heat is contained in the proof of \zcref{l:end-energy}.
\end{proof}

At small $j$ the bound \eqref{e:end-early-annuli} gives no decay, since the raw references have no positive Gaussian margin at early times, and at $j = m$ it is exponentially small on each fixed annulus.

\paragraph{The first step.}
The partial chain and the reference satisfy the same local heat equation between steps and have the same onsite factor, including its nilpotent terms, and their initial data agree on the comparison domain. In a negative body weight this is literal agreement at time zero. For a positive body weight we start after the first step.

\begin{lemma}\label{l:end-first-step}
For every body weight $p$, coefficientwise, including every nilpotent term of the cutoff,
\begin{equation}\label{e:end-first-step}
 \|f_{\alpha, 1}^{\rm raw} - g_{\alpha, 1}^{\rm raw}\|_{{\rm comparison}, \mathcal H_p} \le C_k W^{A_k} e^{-c W^2} .
\end{equation}
\end{lemma}

\begin{proof}
The local heat of the constant function is one, and the input cutoff $\xi_\alpha$ differs from one only at a fixed distance from the comparison domain. Since the heat kernel of one link is a Gaussian of variance of order $W^{-2}$, the part of the input where $\xi_\alpha \ne 1$ contributes at most $C_k W^A e^{-c W^2}$ on the comparison domain. The two chains are then multiplied by the same onsite factor. In a positive body weight $v^p$, the fresh damping $e^{-c \eta v}$ of this factor bounds the weight, since $v^p e^{-c\eta v} \le C_p \eta^{-p}$, and $\eta^{-p}$ is a power of $W$. These two bounds imply \eqref{e:end-first-step}.
\end{proof}

\paragraph{The barrier.}
We now compare $f_{\alpha, j}^{\rm raw}$ and $g_{\alpha, j}^{\rm raw}$ on the inner region. Since the comparison takes place on the real raw chart, the difference is forced only on the collar of the barrier cutoff, through two collar norms. For the partial chain, which is pulled to the chart by the complex canonical map, the collar norm is bounded by \eqref{e:end-actual-collar}. For the reference, only its real collar norm enters, and \zcref{l:end-early} bounds it, including at $j = 0$. Neither bound needs a Gaussian margin, since the decay comes from the distance $d_{\rm col}$.

\begin{lemma}\label{l:end-barrier}
Let $p \in \R$, and suppose that $R_k \le R_{\rm s}$, with $R_{\rm s}$ as in \zcref{p:end-shifted}. Then, uniformly for $\Gamma \in \mathcal U_0$,
\begin{equation}\label{e:end-barrier}
 \|f_{\alpha, m}^{\rm raw} - g_{\alpha, m}^{\rm raw}\|_{{\rm inner}, \mathcal H_p}
 \le C_k W^{A_k} \exp\Bigl[- \frac{c_{\rm D} W d_{\rm col}^2}{C_0} + 16 C_{\rm geo}^2 C_* C_0 W (E^2 R_k^2 + R_k^4)\Bigr] .
\end{equation}
\end{lemma}

\begin{proof}
We write $\tau = \tau_{\rm phys}$, $L = L_\alpha^{\rm raw}$, and $\phi = \phi_\alpha$. Let $f^{\rm raw}(t)$ and $g^{\rm raw}(t)$ be the two chains on the heat intervals, so that $f^{\rm raw}(t) = e^{(t - j\tau) L} f_{\alpha, j}^{\rm raw}$ for $j\tau \le t < (j + 1)\tau$, and similarly for $g^{\rm raw}(t)$, and at the end of each heat interval both are multiplied by the onsite factor $w$ of the step. By \zcref{l:cons-identification}, the heat of the chains agrees with the Dirichlet heat $e^{tL}$ on the support of the cutoffs up to an error $C_k W^{C_k} e^{-c R_k^2 W^2}$ times the input norm. The input norms are at most $e^{C W}$ on the tube, and there are $O(W)$ steps, so that these errors are superexponentially small, and we omit them below.

We first derive the Duhamel formula. Let $\delta(t) = f^{\rm raw}(t) - g^{\rm raw}(t)$. On each heat interval we have $\partial_t \delta = L \delta$ on the comparison domain, and since $\phi$ is supported there,
\[
 \partial_t (\phi \delta) = L (\phi \delta) - [L, \phi] \delta .
\]
The commutator $[L, \phi]$ is of first order, and its coefficients vanish outside the collar of $\phi$. By Duhamel's formula on the heat interval of step $j + 1$, we have
\[
 \phi \delta\bigl((j + 1)\tau^-\bigr) = e^{\tau L} \phi \delta(j\tau) - \int_{j\tau}^{(j + 1)\tau} e^{((j + 1)\tau - t) L} [L, \phi] \delta(t) \dd t .
\]
At the end of the interval, both chains are multiplied by the same onsite factor $w$, which commutes with $\phi$. Let $S_{\rm loc} = w e^{\tau L}$ be the local step. Iterating from $j = 1$ to $j = m - 1$, we obtain
\begin{equation}\label{e:end-barrier-duhamel}
 \phi \, \delta_m = S_{\rm loc}^{m - 1} \phi \, \delta_1 - \sum_{j = 1}^{m - 1} \int_{j\tau}^{(j + 1)\tau} S_{\rm loc}^{m - j - 1} \, w \, e^{((j + 1)\tau - t) L} [L, \phi] \delta(t) \dd t ,
\end{equation}
where $\delta_j = f_{\alpha, j}^{\rm raw} - g_{\alpha, j}^{\rm raw}$. Since $\phi = 1$ on the inner region, we have $\phi \, \delta_m = \delta_m$ there.

Next, we insert a Davies weight. Let $d_{\rm n}$ be the normal distance to the collar of $\phi$, truncated at $d_{\rm col}$. It has Lipschitz constant one, it vanishes on the collar, and it equals $d_{\rm col}$ on the inner region. Since the collar lies in $\{5 R_k/8 \le |U| \le 3 R_k/4\}$, the gradient of $d_{\rm n}$ vanishes outside $\{R_k/2 \le |U| \le 7 R_k/8\}$, and \zcref{l:end-davies} therefore applies to $d_{\rm n}$. Let $a_{\rm D} \ge 0$. A function $F$ supported on the collar satisfies $e^{-a_{\rm D} d_{\rm n}} F = F$, and on the inner region $1 \le e^{-a_{\rm D} d_{\rm col}} e^{a_{\rm D} d_{\rm n}}$. By \zcref{l:end-davies}, the conjugated local step satisfies
\[
 \|e^{a_{\rm D} d_{\rm n}} S_{\rm loc} e^{-a_{\rm D} d_{\rm n}}\| \le e^{C_k/W + (C_k W + Ca_{\rm D}^2)\tau} ,
\]
and the conjugated heat satisfies the same bound without the factor $e^{C_k/W}$. Since $m \le 2 C_0 W$ and $m\tau \le C_0/W$, it follows that for $0 \le s \le \tau$ and $1 \le j < m$,
\[
 \bigl\| \1_{\rm inner} \, S_{\rm loc}^{m - j - 1} w \, e^{s L} F \bigr\|
 \le e^{-a_{\rm D} d_{\rm col}} \, e^{2 C_k C_0 + (C_k W + C a_{\rm D}^2) m\tau} \|F\|
 \le C_k \exp\Bigl[-a_{\rm D} d_{\rm col} + \frac{C C_0}{W} a_{\rm D}^2\Bigr] \|F\| ,
\]
where the terms $C_k W$ of \zcref{l:end-davies} contributed the factor $e^{C_k W m\tau} \le e^{C_k C_0}$.

We now bound the forcing. At each time $\tau \le t \le m\tau$, the function $[L, \phi] \delta(t)$ is bounded by $C_k W^{A_k}$ times the collar norms of $f^{\rm raw}(t)$ and $g^{\rm raw}(t)$ and of their derivatives of order one. For the partial chain, recall from the choice of the domains at the start of this section that the collar lies in $\{|U| \le R_k\}$. By the choice of $\rho$ in \zcref{tab:end-constants}, the closure of $\mathcal U_0$ is a compact subset of the open parameter ball of radius $\rho_{\rm out}$ about $\Lambda$ of \zcref{p:end-escape}, and $R_k \le R_{\rm s}$. Hence \zcref{p:end-shifted}\ref{i:end-shifted-collar} with $R = R_k$ applies uniformly for $\Gamma \in \mathcal U_0$. Together with the kinetic bound on the heat intervals, it bounds the collar norms of $f^{\rm raw}(t)$ and $\nabla f^{\rm raw}(t)$ by $C_k W^{A_k} \exp[16 C_{\rm geo}^2 C_* C_0 W (E^2 R_k^2 + R_k^4)]$. For the reference, let $j\tau \le t < (j + 1)\tau$ with $j \ge 1$. We apply \zcref{l:end-early} at the step $j$, the heat-step bound \eqref{e:end-heat-step} with $\lambda_i = 1$ on the interval $[j\tau, t]$, and Cauchy's estimates on polydisks of radius $c/W$, as in the proof of \zcref{l:end-early}. They bound the collar norms of $g^{\rm raw}(t)$ and $\nabla g^{\rm raw}(t)$ by $C_k W^{A_k}$. Therefore
\[
 \|[L, \phi] \delta(t)\| \le C_k W^{A_k} \exp\bigl[16 C_{\rm geo}^2 C_* C_0 W (E^2 R_k^2 + R_k^4)\bigr] .
\] For the first term of \eqref{e:end-barrier-duhamel}, the bound of \zcref{l:end-davies} with $a_{\rm D} = 0$ and \zcref{l:end-first-step} give $\|S_{\rm loc}^{m - 1} \phi \, \delta_1\| \le C_k W^{A_k} e^{-c W^2}$.

Inserting these bounds into \eqref{e:end-barrier-duhamel}, and using $m\tau \le 1$ for the total length of the heat intervals, we obtain
\[
 \|\delta_m\|_{{\rm inner}, \mathcal H_p}
 \le C_k W^{A_k} e^{-c W^2} + C_k W^{A_k} \exp\Bigl[-a_{\rm D} d_{\rm col} + \frac{C C_0}{W} a_{\rm D}^2 + 16 C_{\rm geo}^2 C_* C_0 W (E^2 R_k^2 + R_k^4)\Bigr] .
\]
We now choose the Davies parameter. The function $a_{\rm D} \mapsto -a_{\rm D} d_{\rm col} + C C_0 a_{\rm D}^2/W$ is minimal at
\[
 a_{\rm D} = \frac{W d_{\rm col}}{2 C C_0} ,
 \qquad
 \text{with the value} \quad - \frac{W d_{\rm col}^2}{4 C C_0} = - \frac{c_{\rm D} W d_{\rm col}^2}{C_0} ,
\]
since $c_{\rm D} = 1/(4C)$. The first term $e^{-c W^2}$ is smaller than the second for $W$ large, and the finite jets and the number of steps change only $W^{A_k}$. This proves \eqref{e:end-barrier}.
\end{proof}

\paragraph{The radii.}
The barrier decays at the rate $c_{\rm D} d_{\rm col}^2/C_0$, while the canonical heights over the collar produce the growth $16 C_{\rm geo}^2 C_* C_0 (E^2 R_k^2 + R_k^4)$. The second condition in \eqref{e:end-Egeom} and the choice of $R_k$ make the growth a small fraction of the decay. We set
\[
 \gamma_k^{\rm bar} = \frac{c_{\rm D} R_k^2}{128 \, C_0} .
\]

\begin{lemma}\label{l:end-radii}
Let $|E| \le E_{\rm geom}$. If $R_k$ is sufficiently small, then the exponent in \eqref{e:end-barrier} is at most $-3\gamma_k^{\rm bar} W/2$. This choice of $R_k$ at fixed rank does not change the energy threshold $E_{\rm geom}$.
\end{lemma}

\begin{proof}
Since $d_{\rm col} = R_k/8$, the decay term of \eqref{e:end-barrier} satisfies $c_{\rm D} W d_{\rm col}^2/C_0 = c_{\rm D} W R_k^2/(64 C_0) = 2\gamma_k^{\rm bar} W$. By the second condition in \eqref{e:end-Egeom},
\[
 16 C_{\rm geo}^2 C_* C_0 W E^2 R_k^2 \le \frac{16 \, c_{\rm D}}{8192 \, C_0} W R_k^2 = \frac{c_{\rm D} W R_k^2}{512 \, C_0} = \frac{\gamma_k^{\rm bar} W}{4} ,
\]
which is an eighth of the decay term. If $R_k^2 \le c_{\rm D}/(8192 \, C_{\rm geo}^2 C_* C_0^2)$, then the quartic term $16 C_{\rm geo}^2 C_* C_0 W R_k^4$ is at most $\gamma_k^{\rm bar} W/4$ as well. Adding these bounds, we see that the exponent is at most $-2\gamma_k^{\rm bar} W + \gamma_k^{\rm bar} W/2 = -3\gamma_k^{\rm bar} W/2$. The energy threshold was chosen in \eqref{e:end-Egeom} before the radii, and the condition on $R_k$ above does not involve it.
\end{proof}

\paragraph{Continuation to the canonical graph.}
The barrier bounds the error on the real chart. We also need it on the complex canonical graph and at the canonical frame, and we obtain it from the real bound and a rough bound on a polydisk by the two-constants theorem \cite{ransford1995potential}.

\begin{lemma}\label{l:end-continuation}
Let $|E| \le E_{\rm geom}$, let $R_k \le R_{\rm s}$ be as in \zcref{l:end-radii}, and let $p \in \R$. If $r_{\rm phys}/R_k$ is sufficiently small at fixed rank, then there exists $\gamma_k' > 0$ such that for every signature $\alpha$,
\[
 \bigl\| \chi_\alpha^{\rm phys} \bigl(f_m - \Can_\alpha^{\rm raw} g_{\alpha, m}^{\rm raw}\bigr) \bigr\|_{\mathcal H_p(\mathcal S)} \le C_k W^{A_k} e^{-\gamma_k' W} ,
\]
uniformly for $\Gamma \in \mathcal U_0$.
\end{lemma}

\begin{proof}
On the inner region the output cutoffs equal one, so the physical endpoint and the reference have holomorphic normal continuations to the polydisk of radius $\rho_{\rm pd}$ about each point of the inner region. Here $\rho_{\rm pd}$ is a fixed fraction of $R_k$, chosen so that the real parts of the points of the polydisk lie at distance less than $3 R_k/8$ from the inner region, hence in the comparison domain, so that the polydisk lies in the strip of \zcref{p:end-shifted}, and so that its heights satisfy $|b| \le c_1 \bar r$, with $\bar r = 15 R_k/16$ as in the domains of the comparison. The physical support lies in the inner region, since $r_{\rm phys}/R_k$ is small. Since there are finitely many normal coordinates, $\|D\|^2 + \|D_F\|^2 \le C_k R_k^2$ on the polydisk, and by \eqref{e:end-rough-strip} with $j \le m \le 2 C_0 W$, the physical endpoint is bounded there by $W^{A_k} e^{B_k W}$, with $B_k \le C_* C_0 C_k R_k^2$. For the reference we use the same constant normal graph at every node, with the graph cutoff $\psi_{\bar r}$, including the initial datum $\xi_\alpha \varpi_\alpha$, which is holomorphic on the support of $\psi_{\bar r}$, where $\xi_\alpha = 1$. Then \zcref{l:end-energy}, in the form with the cutoff $\psi_{\bar r}$ of the proof of \zcref{l:end-early}, applied to the raw steps $S_{\alpha, \Gamma}$ with $\lambda_j = 1$ for all $j$ and without Gaussian weight, gives a bound of the same form.

The real comparison \eqref{e:end-barrier} holds also after fixed finite real normal derivatives, by the same collar equation, and Sobolev embedding on a slightly smaller box makes its Hilbert-valued supremum on the real axis at most $W^{A_k} e^{-\gamma_k^{\rm bar} W}$, by \zcref{l:end-radii}.

We pass from the real box to the canonical graph by the two-constants theorem \cite{ransford1995potential}. In one normal coordinate, we slit a rectangle of size comparable to $\rho_{\rm pd}$ along its inner real interval. The logarithm of the norm of a holomorphic vector-valued function is subharmonic. Suppose that the norm is at most $M_0$ on the slit and at most $M_1$ on the rectangle. Then
\[
 \|f(z)\| \le M_0^{\omega(z)} M_1^{1 - \omega(z)} ,
\]
where $\omega$ is the harmonic measure of the slit; for vector-valued functions this follows by applying the scalar inequality to every unit dual functional and taking the supremum. Above every smaller real interval we have $1 - \omega(z) \le C|\Im z|/\rho_{\rm pd}$. We apply this successively to the finitely many normal coordinates, shrinking the real interval once each time, with $M_0 = W^{A_k} e^{-\gamma_k^{\rm bar} W}$ and $M_1 = W^{A_k} e^{B_k W}$. By \zcref{l:end-height}, on the physical support the canonical graph has height at most $C_{\rm geo}(|E| r_{\rm phys} + r_{\rm phys}^2)$, and the product $\omega$ of the harmonic measures satisfies $1 - \omega \le C_k r_{\rm phys}/R_k$. Since $\gamma_k^{\rm bar}$ and $B_k$ are both of order $R_k^2$, the condition
\[
 \omega \gamma_k^{\rm bar} > (1 - \omega) B_k
\]
holds once $r_{\rm phys}/R_k$ is small enough at fixed rank, and then the continued error is at most $W^{A_k} e^{-\gamma_k' W}$ on that graph, where, since $1 - \omega \le C_k r_{\rm phys}/R_k$,
\[
 \gamma_k' = \gamma_k^{\rm bar} - \frac{C_k r_{\rm phys}}{R_k} \bigl(\gamma_k^{\rm bar} + B_k\bigr) > 0 .
\]

It remains to pass from the real frame to the canonical frame. The partial chain and the raw reference satisfy the same equivariance \eqref{e:end-covariance}, by \zcref{l:end-ward}, and so does their difference. We apply \zcref{l:tools-submean} with $\zeta = \Gamma$ to the difference, with the selection given by \eqref{e:end-canonical-frame}, as in the proof of \zcref{l:end-ward}\ref{i:end-ward-c}. It keeps the exponential error on the body of the canonical image, and all nilpotent canonical corrections are retained as finite coefficient jets.
\end{proof}

\begin{proof}[Proof of \zcref{t:end-entrance}]
The exact identity
\[
 f_m - \sum_\alpha \chi_\alpha^{\rm phys} \Can_\alpha^{\rm raw} g_{\alpha, m}^{\rm raw}
 = \Bigl(1 - \sum_\alpha \chi_\alpha^{\rm phys}\Bigr) f_m + \sum_\alpha \chi_\alpha^{\rm phys} \bigl(f_m - \Can_\alpha^{\rm raw} g_{\alpha, m}^{\rm raw}\bigr)
\]
splits the error into two parts. The first term is supported away from the union of the normal cores, and the global tail of $f_m$ after \zcref{p:end-escape} bounds it by $C_k W^{A_k} e^{-c W r_0^2}$, where $r_0 > 0$ is the radius of these cores. By \zcref{l:end-continuation}, each remaining term is at most $C_k W^{A_k} e^{-\gamma_k' W}$, and their finite number changes only the constant. We obtain \eqref{e:end-full} with $\gamma_k = \min(\gamma_k', c r_0^2)$.

Transporting by $\gamma_h^{-1} w_0^{-1/2}$ once converts the raw references to the symmetric normal gauge without changing the identity; its endpoint factors are those of \eqref{e:tube-table}. Parameter derivatives follow from Cauchy's formula on the larger parameter ball. Local BB jets, if needed later, are taken as finite coefficients at zero before the return to the contour.
\end{proof}