\chapter{Proof of the interface theorem}\label{s:assembly}

In this section we prove \zcref{t:interface}. Part III uses this theorem and nothing else from Part IV, and with its proof the reduction of the band matrix to the sigma model of \zcref{s:sigma} is complete. The heuristics behind the argument are explained in \zcref{s:overview}. Here we combine the a priori bounds of \zcref{s:hist}, the normal tube and the normal oscillator of \zcref{s:tube}, the stability estimates of \zcref{s:stab}, the consistency estimates of \zcref{s:cons} and the endpoint estimates of \zcref{s:end}.

Recall from \zcref{p:rep-exact} and \eqref{e:rep-normalization} that the source ratio is, up to an explicit factor, a ratio of two chain integrals,
\begin{equation}\label{e:asm-ratio}
 T_N^{(k)}(\mathbf u, \mathbf b)
 = e^{N \eta_0^2 \Str(\Gamma^2)/2} \, \frac{\mathcal Z_N(\Gamma)}{\mathcal Z_N(\Gamma_0)} ,
\end{equation}
where $\Gamma = \Gamma(\mathbf u, \mathbf b)$ is the source supermatrix of \zcref{s:rep} and $\eta_0 = a_N^{-1}$. Our main task is to show that $\mathcal Z_N(\Gamma)$ converges to the partition function $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ of \zcref{d:sigma-partition}. The chain integral $\mathcal Z_N(\Gamma)$ is a chain of $N$ onsite weights and $N - 1$ heat links on the product of Sch\"afer--Wegner cycles. We compare it with the adapted chain of \zcref{s:tube}, which lives on a tube around the saddle manifold, and which converges to the continuum semigroup by the results of \zcref{s:cons}.

The main difficulty in this comparison is the one of exponential orders explained in \zcref{s:ov-ends}. Every change of integration cycle between the physical chain and the tube produces an error of order $e^{-\gamma W}$ for some $\gamma > 0$, while the physical powers are bounded in operator norm only by $e^{CW}$, and \zcref{t:hist-bound} bounds by $e^{o(W)}$ only the partial chains started from the constant function. We therefore use a one-sided telescoping identity, \zcref{l:asm-telescope}, in which each local error is paired directly with such a partial chain. Every comparison of the physical chain with the adapted chain below is an instance of this lemma. The limit of the adapted chain is then identified by the product approximation of \zcref{s:cons}, after an initial layer in which the normal variables relax.

The section is organized as follows. In \zcref{s:asm-choices}, we fix the order in which the constants are chosen and the two energy intervals of \zcref{t:interface}. In \zcref{s:asm-one-sided}, we write the chain integral in one-sided form. \zcref[S]{s:asm-adapted} collects three properties of the adapted chain, the first of which is proved in \zcref{s:stab-weight-gain}, and \zcref{s:asm-telescope} proves the telescoping lemma. Parts (a), (b), (c) and (d) of \zcref{t:interface} are proved in \zcref{s:asm-a,s:asm-b,s:asm-c,s:asm-d}, respectively. \zcref[S]{f:asm-chain} shows how the chain is decomposed.

\begin{figure}[t]
\centering
\begin{tikzpicture}[x=1cm, y=1cm, font=\small]
 \draw[thick] (0, 0) -- (13, 0);
 \foreach \x in {0, 0.25, ..., 13} \fill (\x, 0) circle (0.9pt);
 \draw[line width=2.2pt] (11.2, 0) -- (13, 0);
 \node[above] at (12.3, 0.12) {right block};
 \node[above] at (12.3, 0.55) {$m_W$ steps};
 \node[left] at (-0.1, 0) {constant $1$};
 \fill[black!30] (10.6, -0.08) rectangle (11.2, 0.08);
 \fill[black!30] (0, -0.08) rectangle (0.6, 0.08);
 \draw[thick] (5.5, -0.15) -- (5.5, 0.15);
 \draw[thick] (6.75, -0.15) -- (6.75, 0.15);
 \node[above] at (6.1, 0.15) {marks};
 \draw[->] (11.1, 1.05) -- (0.3, 1.05);
 \node[above] at (5.7, 1.05) {$N - m_W$ physical steps compared with adapted steps};
 \draw[decorate, decoration={brace, amplitude=5pt, mirror}] (8.6, -0.25) -- (11.2, -0.25)
   node[midway, below=6pt] {right outer portion};
 \draw[decorate, decoration={brace, amplitude=5pt, mirror}] (2.6, -0.25) -- (8.6, -0.25)
   node[midway, below=6pt] {middle portion};
 \draw[decorate, decoration={brace, amplitude=5pt, mirror}] (0, -0.25) -- (2.6, -0.25)
   node[midway, below=6pt] {left outer portion};
 \node[above] at (10.7, 0.08) {$n_h$};
 \node[above] at (0.3, 0.08) {$n_h$};
\end{tikzpicture}
\caption{The decomposition of the chain in \eqref{e:asm-one-sided}. The right block is attached to the tube by \zcref{t:end-entrance}, the remaining $N - m_W$ physical steps are compared with adapted steps by \zcref{l:asm-telescope}, and the left endpoint is the constant function $1$. In the balanced adapted chain, the shaded initial layers of $n_h$ steps (\zcref{l:asm-initial-layer}) move the endpoints to the normal fixed state. The outer portions of length about $\theta N$ suppress the unbalanced signatures and prepare body weights, and the balanced tangent variables converge to the continuum semigroup. The two ticks show sites that carry the local sources of \zcref{t:interface}(b), which \zcref{s:asm-b} calls marks.}
\label{f:asm-chain}
\end{figure}

\section{Conventions and the order of choices}\label{s:asm-choices}

Throughout this section, $k \ge 1$ is the rank, and we use the scales
\[
 h = W^{-1/2}, \qquad \delta_W = h^4 = W^{-2}, \qquad \tau_{\rm phys} = (2 W^2)^{-1}
\]
of Part IV. In one link of the chain, the normal variables relax by a factor $1 - c h^2$, while the tangent variables move by an amount of order $h^2$; their evolution per link has a generator of order $\delta_W$. Let $\eta = \eps \eta_0 = \eps/a_N$ denote the physical imaginary part. The damping $\eps$ and the parameter $X_N(E)$ take values in compact subsets of $(0, \infty)$, and the energy $E$ may depend on $N$ and $W$ within the intervals fixed below. Constants $C, c > 0$ may depend on $k$, on these compact sets, and on the choices listed below, but not on $N$, $W$, $E$, or the sites. Since $X_N(E) = N/((4 - E^2) W^2)$ and $|E| \le 1/2$ below, we have $N \asymp W^2$.

The constants are chosen in the following order. Each choice may depend on all choices made before it.
\begin{enumerate}[label=(\roman*)]
\item\label{i:asm-universal} The universal constants of \zcref{s:end}, in the order of \zcref{tab:end-constants}. Among them are the block length $C_0$, the Gaussian margin $a$ of the prepared families, the coefficient $C_n$ in the exponent of the prepared block bound \eqref{e:end-normal-block}, the coefficient $C_*$ of the shifted onsite bound in \eqref{e:end-strip}, the constant $C_{\rm geo}$ of the canonical normal height in \zcref{l:end-height}, the constants $c_{\rm ann} = a c_Z$ and $C_{\rm ann} = a C_Z$ of the Gaussian factor on the annulus, and the shell-gain constant $c_{\rm sh} = c_{\rm ann}/4$ of \eqref{e:end-shell}. We add to them the energy $E_{\rm sep}$ and the radius $r_{\rm sep}$ of the Gaussian separation in \zcref{l:cons-separation}. None of these constants depends on $k$, and \zcref{r:end-universal} explains why, in particular for the coefficient $C_n$ of $|b|^2$.
\item\label{i:asm-Egeom} The energy threshold $E_{\rm geom} > 0$ of \eqref{e:end-Egeom}. Its two conditions involve the constants of \ref{i:asm-universal} alone.
\item\label{i:asm-Eout} The outer energy bound
\begin{equation}\label{e:asm-Eout}
 E_{\rm out} = \min\bigl( \tfrac12, E_{\rm geom}, E_{\rm sep} \bigr) .
\end{equation}
Here $1/2$ is the energy bound of \zcref{t:hist-bound} and of \zcref{l:end-height}. The estimates of \zcref{s:tube,s:stab} are uniform on every compact set of energies, by \zcref{t:stab-main}, and we use them on $[-1/2, 1/2]$. Since $E_{\rm out} \le 1/2$, \zcref{l:iface-counts} and \zcref{l:iface-density} apply on $[-E_{\rm out}, E_{\rm out}]$.
\item\label{i:asm-Estar} The inner energy bound $E_*$, with $0 < E_* < E_{\rm out}$.
\item\label{i:asm-rank} The choices at rank $k$, in the order of \zcref{tab:end-constants}. These are the radius $r_k$ and the degree-metric parameter $\varkappa_k$ of \zcref{l:cons-sectorial}; the degree-metric parameter $\omega$ and the cutoff radius $r \le r_k$ of \zcref{t:stab-main}; the finitely many graph orders of \zcref{s:cons}; the damping balls $\mathcal U_0 \Subset \mathcal U_1 \Subset \mathcal U_2$ of \zcref{s:end-families}; the Hardy radii $\rho$ and $\rho_{\rm out}$ of \zcref{p:end-escape}; the radius $R_k$ of the root tube, which satisfies
\[
 16 R_k \le r_{\rm out} ,
 \qquad
 4 R_k \le r_{\rm in} \le \tfrac34 r_{\rm out} ,
 \qquad
 2 r_{\rm out} \le \min(r_k, r_{\rm sep})
\]
for the radii $r_{\rm in} < r_{\rm out}$ of \zcref{l:cons-identification}; the collar distance $d_{\rm col} = R_k/8$; the polydisk radius $\rho_{\rm pd}$; the support radius $r_{\rm phys}$ of the physical cutoffs; the inner radius $r_1 < r_{\rm phys}$ of the annulus on which they vary; and the saturation radius $r_{\rm esc}$ of the escape function. Each of them is constrained only as in \zcref{s:cons,s:end}.
\item\label{i:asm-radius} The source radius $r_{k, \eps}$ of \zcref{s:asm-one-sided}.
\item\label{i:asm-W0} The lower threshold $W_0$ on $W$.
\end{enumerate}
Since $E_{\rm out}$ and $E_*$ are chosen in \ref{i:asm-Eout} and \ref{i:asm-Estar}, before the choices at rank $k$ in \ref{i:asm-rank}, they do not depend on $k$. They do not depend on the number $m$ of marks or on the orders of the derivatives in \zcref{t:interface}(b) either. Indeed, a complete jet changes the block estimates of \zcref{s:end} only as \eqref{e:end-jet-constant} states, and we show in \zcref{l:asm-jet-block} that the margins and the shell gain therefore hold for the chain with jets with the same universal constants and under the same conditions \eqref{e:end-Egeom}. The rank, the number of marks, and the orders change only the constants, the tube, the source radius, $W_0$, and the rate of convergence, as \zcref{t:interface} asserts.

Parts (a) and (b) of \zcref{t:interface} are proved for $|E| \le E_{\rm out}$, and parts (c) and (d) for $|E| \le E_*$. The margin between the two intervals absorbs the change of energy $E \mapsto E + u/a_N(E)$ in parts (c) and (d), through the following lemma.

\begin{lemma}\label{l:asm-recenter}
Let $u_0 > 0$, let $|E| \le E_*$ and $|u| \le u_0$, and define
\begin{equation}\label{e:asm-recenter}
 E' = E + \frac{u}{a_N(E)} ,
 \qquad
 \eps' = \eps \, \frac{\rho(E')}{\rho(E)} .
\end{equation}
Then $E + (u + \iu \eps)/a_N(E) = E' + \iu \eps'/a_N(E')$. Moreover, there exist $C > 0$ and $N_0$, depending only on $u_0$, $E_*$, and $E_{\rm out}$, such that for all $N \ge N_0$,
\begin{equation}\label{e:asm-recenter-bounds}
 |E'| \le E_{\rm out} ,
 \qquad
 \Bigl| \frac{\eps'}{\eps} - 1 \Bigr| + \Bigl| \frac{X_N(E')}{X_N(E)} - 1 \Bigr| \le \frac{C}{N} .
\end{equation}
\end{lemma}

\begin{proof}
Recall that $a_N(E) = N \rho(E)$. The identity follows from
\[
 \frac{\eps'}{a_N(E')} = \frac{\eps}{N \rho(E)} = \frac{\eps}{a_N(E)} .
\]
Let $\rho_* > 0$ denote the minimum of $\rho$ on $[-E_{\rm out}, E_{\rm out}]$. Then
\[
 |E' - E| \le \frac{u_0}{N \rho_*} ,
 \qquad
 |E'| \le E_* + \frac{u_0}{N \rho_*} \le E_{\rm out}
 \quad \text{for } N \ge \frac{u_0}{\rho_* (E_{\rm out} - E_*)} .
\]
The functions $\rho$ and $x \mapsto (4 - x^2)^{-1}$ are smooth and positive on $[-E_{\rm out}, E_{\rm out}]$, and the second bound in \eqref{e:asm-recenter-bounds} follows from the first display and the mean value theorem.
\end{proof}

\section{The chain in one-sided form}\label{s:asm-one-sided}

Recall from \zcref{s:rep} the cycle $\mathcal S$, its bilinear pairing $\mathcal B_{\mathcal S}(f, g) = \int_{\mathcal S} f g \dd S$, the onsite weight $w_\Gamma$, the heat factor $K_\tau$, and the transfer operator $T_\Gamma = M_{w_\Gamma} K_{\tau_{\rm phys}}$, where $M_w$ denotes multiplication by $w$. Recall also that $K_\tau 1 = 1$, that the analytic transpose of $T_\Gamma$ for $\mathcal B_{\mathcal S}$ satisfies $T_\Gamma^{\rm tr} = K_{\tau_{\rm phys}} M_{w_\Gamma}$, and that by \zcref{p:rep-exact}, for all integers $1 \le m \le N/2$,
\begin{equation}\label{e:asm-endpoints}
 \mathcal Z_N(\Gamma) = \mathcal B_{\mathcal S}(\ell_m, T_\Gamma^{N - 2m} f_m) ,
 \qquad
 f_m = T_\Gamma^m 1 ,
 \qquad
 \ell_m = (T_\Gamma^{\rm tr})^m 1 = K_{\tau_{\rm phys}} f_m .
\end{equation}
We define the block length to be
\begin{equation}\label{e:asm-block}
 m_W = \lceil C_0 W \rceil ,
\end{equation}
with the constant $C_0$ of \zcref{s:end}. Taking $m = m_W$ in \eqref{e:asm-endpoints} and moving the $m_W$ factors of $\ell_{m_W}$ to the right by analytic transposition, we obtain
\begin{equation}\label{e:asm-one-sided}
 \mathcal Z_N(\Gamma) = \mathcal B_{\mathcal S}(1, T_\Gamma^{N - m_W} f_{m_W}) .
\end{equation}
In this form the chain is attached to the tube only at its right endpoint (see \zcref{f:asm-chain}). By \zcref{t:end-entrance}, the block $f_{m_W}$ is close to a sum of reference vectors on the tube, and the remaining $N - m_W$ steps are compared with adapted steps. The left endpoint is the constant function $1$. The point of this form is that the left factors produced by the comparison are the partial chains $T_\Gamma^j 1$ and their versions with local sources, which \zcref{t:hist-bound} bounds.

We now fix the source radius of \ref{i:asm-radius}. Recall from \zcref{s:end} the nested complex balls $\mathcal U_0 \Subset \mathcal U_1 \Subset \mathcal U_2$ of damping parameters around $\Gamma_0$, on which the estimates of that section are uniform. We choose $r_{k, \eps} > 0$ independent of $N$, $W$, and $E$ and so small that $b_j \ge \eps/2$ for all $j$ and $\Gamma(\mathbf u, \mathbf b) \in \mathcal U_0$ whenever $\mathbf u$ and $\mathbf b$ are real, $\sum_j u_j = 0$ and
\begin{equation}\label{e:asm-source-ball}
 \|\mathbf u\| + \|\mathbf b - \eps \1\| < r_{k, \eps} .
\end{equation}
This is possible since $\Gamma$ depends linearly on $(\mathbf u, \mathbf b)$ and $\Gamma(\mathbf 0, \eps \1) = \Gamma_0$. The radius depends on $k$ and on the compact range of $\eps$. The FF block of $\Gamma$ is the part that depends on $(\mathbf u, \mathbf b)$. It is traceless, as \zcref{t:hist-bound} requires.

\section{Three properties of the adapted chain}\label{s:asm-adapted}

We now prove two properties of the adapted transfer operator $A_h$ of \eqref{e:tube-table} that the comparison uses, and we recall a third. The adapted step improves polynomial body weights, by \zcref{l:asm-weight-gain}, it contracts exponentially in the unbalanced root signatures, and in the balanced signature it moves the right endpoint factor $g_R$ to the normal fixed state $\Omega_h$ over a short initial block. Recall the identity \eqref{e:asm-weights} for the weighted spaces $\mathcal H^{\rm tube}_p$.

We also record how the analytic transpose of the chain reduces to the chain itself. By \eqref{e:tube-table} and \eqref{e:cons-steps}, we have $A_h = s_\Gamma T_h$, where the source factor $s_\Gamma$ is a multiplier and $T_h$ is the unsourced symmetric step. Recall the analytic Berezin pairing $\mathcal B_{\rm ad}(f, g) = \int f g \, \gamma_h^2 \dd U \dd \nu$ of the adapted tube, defined in \eqref{e:tube-pairing-ad}. The step $T_h$ is symmetric for $\mathcal B_{\rm ad}$, since $\Delta_{\rm amb}$ is symmetric for the flat Berezin form $\varpi^2 \dd U \dd \nu$ of \eqref{e:tube-volume}, and since $\varpi$, $\gamma_h$, $w_0^{1/2}$, and the cutoffs are multipliers. Let $A_h^{\rm tr}$ denote the transpose of $A_h$ for $\mathcal B_{\rm ad}$. Since multipliers are symmetric, we have $A_h^{\rm tr} = T_h s_\Gamma$, and, as in \zcref{r:cons-transpose},
\begin{equation}\label{e:asm-transpose}
 (A_h^{\rm tr})^n = T_h \, A_h^{n - 1} \, s_\Gamma ,
 \qquad n \ge 1 .
\end{equation}
The factor $T_h$ is bounded on every space $\mathcal H^{\rm tube}_p$ by \zcref{l:cons-short-heat}, and $s_\Gamma$ is a bounded multiplier by the proof of \zcref{l:end-kick}. Every bound on the powers of $A_h$ below therefore holds for the powers of $A_h^{\rm tr}$, up to a constant factor, and every limit of the powers of $A_h$ implies the corresponding limit for $A_h^{\rm tr}$.


\subsection{Unbalanced signatures}

Recall from \zcref{s:tube} that the saddle manifold has $2k + 1$ components, indexed by the signature $p \in \{0, \dots, 2k\}$, and that the signature $p = k$ is balanced. \zcref[S]{t:end-entrance} retains all signatures. We now show that the unbalanced ones are exponentially small after an adapted leg of macroscopic length. We use the following lemma at each place below where an unbalanced signature occurs.

\begin{lemma}\label{l:asm-unbalanced}
Let $p \ne k$ be an unbalanced signature, and let $q \in \R$. There exist $C, c > 0$ such that, in this signature,
\begin{equation}\label{e:asm-unbalanced-gap}
 \|A_h^n\|_{\mathcal H^{\rm tube}_q} \le C e^{-c n h^2} ,
 \qquad n \ge 0 .
\end{equation}
In particular, for all $\theta > 0$ there exists $c' > 0$, depending on $\theta$ and on the compact range of $X_N(E)$, such that $\|A_h^n\|_{\mathcal H^{\rm tube}_q} \le C e^{-c' W}$ for all $n \ge \theta N$.
\end{lemma}

\begin{proof}
In the signature $p$, the reference contraction $C_h$ of \zcref{t:stab-main} on $\mathcal H^{\rm tube}_0$ carries the full scalar factor $\lambda$ of \zcref{p:tube-gap}, including the physical saddle phase $\zeta_p=w_0(R_*)$. The defects depend only on $|\lambda|$, so by \eqref{e:tube-unbalanced-defect} its right defect satisfies $D_R \ge c_1 h^2 \Id$. On $\mathcal H^{\rm tube}_q$ we use the reference $C_{h, q} = v^{-q} C_h v^q$. Since multiplication by $v^q$ is unitary from $\mathcal H^{\rm tube}_q$ onto $\mathcal H^{\rm tube}_0$, the right defect of $C_{h, q}$ in $\mathcal H^{\rm tube}_q$ satisfies
\[
 D_{R, q} = v^{-q} D_R v^q \ge c_1 h^2 \Id .
\]
Using the conclusion of \zcref{t:stab-main} in $\mathcal H^{\rm tube}_q$ and $\delta_W = h^4$, we find for $h$ small
\[
 A_h^* A_h \le (1 + B \delta_W/2) \Id - \tfrac12 D_{R, q} \le \bigl( 1 + B h^4/2 - \tfrac12 c_1 h^2 \bigr) \Id \le \bigl( 1 - \tfrac14 c_1 h^2 \bigr) \Id ,
\]
where the adjoint is taken in $\mathcal H^{\rm tube}_q$, and in the second inequality we used $D_{R, q} \ge c_1 h^2 \Id$. It follows that $\|A_h\|_{\mathcal H^{\rm tube}_q} \le e^{-c_1 h^2/8}$, which implies \eqref{e:asm-unbalanced-gap} with $c = c_1/8$ and $C = 1$.

We now prove the second assertion. Since $X_N(E)$ is bounded below and $|E| \le 1/2$, there exists $c_X > 0$ such that $N \ge c_X W^2$, and for $n \ge \theta N$ we obtain
\[
 n h^2 \ge \frac{\theta N}{W} \ge c_X \theta W . \qedhere
\]
\end{proof}

In our applications, the other factors of a pairing in an unbalanced signature are bounded by fixed powers of $W$. These are the endpoint norms of \zcref{l:end-raw-norms}, the bounds on the preparation in \zcref{l:end-real-family}, and the terminal bound of \zcref{l:asm-terminal}. We carry each unbalanced signature, with its phase and orientation, until an adapted leg of length at least $\theta N$ suppresses it by \zcref{l:asm-unbalanced}.

\subsection{The initial layer}\label{s:asm-initial-layer}

Throughout this subsection, we work in the balanced root signature $p=k$. The unbalanced signatures are treated by \zcref{l:asm-unbalanced}.

Recall the endpoint factors $g_R$ and $g_L$ and the frozen normal transfer $\mathbf F_h$ from \eqref{e:tube-table}. Recall from \zcref{l:cons-fixed-state} that the normal fixed state $\Omega_h$ is the unique closed vector with normalized Gaussian mean one such that $\mathbf F_h \Omega_h = \Omega_h$, that it satisfies the bounds \eqref{e:cons-state-bounds}, that closed vectors relax to multiples of it at rate $h^2$ per step by \eqref{e:cons-relaxation}, and that its analytic Berezin pairing in the normal variables satisfies
\begin{equation}\label{e:asm-normal-pairing}
 \mathcal B_{\rm nor}(\Omega_h, \Omega_h) = 1
\end{equation}
by \eqref{e:cons-pairing}. By \zcref{l:tube-class}, the endpoint factor is a normal class function times the constant function of the tangent variables, and we denote it by $g_R \otimes 1$. Let $c_{\rm lay} \ge 1$, and define the length of the initial layer to be
\begin{equation}\label{e:asm-layer-length}
 n_h = \bigl\lceil c_{\rm lay} h^{-2} \log(1/h) \bigr\rceil .
\end{equation}

\begin{lemma}\label{l:asm-initial-layer}
In the balanced signature $p=k$, let $\xi$ be either $1$ or a fixed smooth invariant normal class cutoff in the physical variable $U$, with its full nilpotent completion and equal to one near $U=0$. This includes the cutoff $\xi_k$ used in \zcref{t:end-entrance}. Let $c_{\rm lay}$ be sufficiently large. Then $n_h \delta_W \to 0$, and
\begin{equation}\label{e:asm-initial-layer}
 A_h^{n_h} ((\xi g_R) \otimes 1) = \Omega_h \otimes 1 + o(1)
\end{equation}
in the base normal norm and a sufficiently negative fixed body weight. The same holds for the analytic transpose of the chain, with $\xi g_L$ in place of $\xi g_R$.
\end{lemma}

We first explain the idea. On the tangent constant, the adapted step without its source factor reduces to the frozen normal transfer $\mathbf F_h \otimes \Id$, which relaxes at rate $h^2$ per step. The source factor changes each step by $O(\delta_W)$, and over $n_h$ steps the source error is $O(n_h \delta_W) = O(h^2 \log(1/h))$, whereas the relaxation error is $O(h^{c c_{\rm lay}})$.

\begin{proof}
We first note that
\[
 n_h \delta_W \le 2 c_{\rm lay} h^2 \log(1/h) \longrightarrow 0 .
\]
Set $u_R = \xi g_R$ and $u_L = \xi g_L$. By \zcref{l:tube-class}, the endpoint factors are invariant normal class functions. The cutoff $\xi$ has the same invariance, including its nilpotent terms, so $u_R$ and $u_L$ are annihilated by the normal charges and are $\mathfrak q_{\rm nor}$-closed. Since $\xi$ equals one near the origin, their body values at $U=0$ are one in the balanced signature. Normal localization \zcref{l:tube-localization} therefore gives $\ell_0(u_R) = \ell_0(u_L) = 1$. The bounds of \zcref{l:end-raw-norms} also hold for $u_R$ and $u_L$: the fixed cutoff has bounded coefficients in the standardized metric, and each of its nonbody terms carries the corresponding positive power of $h$. For an input $f \otimes 1$ with $f$ of this kind, the horizontal part of the decomposition \eqref{e:tube-decomposition} of the ambient generator vanishes, since the horizontal derivatives act on a constant and the normal stabilizer fields annihilate the class function $f$. Therefore the adapted step without its source factor agrees with $\mathbf F_h \otimes \Id$ on these inputs.

A larger normal cutoff changes this identity only by a Gaussian error from the separated boundary, which is $O(h^L)$ for all $L$. Indeed, a Gaussian input with negative margin grows at most like $e^{C h^{-2}}$ on the tube, while a jump to the separated boundary carries a factor $e^{-c h^{-4}}$.

Next, the source factor differs from one by $O(\delta_W v)$. We absorb the factor $v$ in a sufficiently negative body weight $p_0$ such that $v \in \mathcal H^{\rm tube}_{p_0}$. Writing $\|f\|_{p, a'} = \|e^{a' |Z|^2} f\|_{\mathcal H^{\rm tube}_p}$, where $Z$ is the standardized normal coordinate, we obtain
\begin{equation}\label{e:asm-layer-step}
 \bigl\| (A_h - \mathbf F_h \otimes \Id)(\mathbf F_h^j u_R \otimes 1) \bigr\|_{p_0, a_{j + 1}} \le C \delta_W ,
 \qquad j \ge 0 .
\end{equation}
Here $(a_j)$ is the signed Gaussian ramp of \zcref{c:end-ramp}, which increases from $-a$ to $a$ over the first $m_W$ steps, and $a_j = 0$ for $j > m_W$. The frozen path at step $j + 1$ has bounded norm in this signed weight, and the source factor commutes with the weight. This ramp uses real normal inputs, so its application to $u_R$ requires only the negative-margin bound and evaluates $\xi$ only on the real tube.

We now telescope. We have
\[
 A_h^{n_h} (u_R \otimes 1) - (\mathbf F_h^{n_h} u_R) \otimes 1
 = \sum_{j = 0}^{n_h - 1} A_h^{n_h - 1 - j} (A_h - \mathbf F_h \otimes \Id)(\mathbf F_h^j u_R \otimes 1) .
\]
For $j < m_W$, we use the signed weights at all steps. The weighted energy estimate \zcref{l:end-energy}, in the form of \zcref{c:end-ramp}, bounds the powers $A_h^{n_h - 1 - j}$ from the signed weight at step $j + 1$ to the base norm.

For $m_W \le j < n_h$, the inputs $\mathbf F_h^j u_R$ are closed normal class functions with bounded base norm, and we use the power bounds of \zcref{t:stab-main}. Inserting \eqref{e:asm-layer-step} into the telescoping sum, we obtain
\[
 \bigl\| A_h^{n_h} (u_R \otimes 1) - (\mathbf F_h^{n_h} u_R) \otimes 1 \bigr\|
 \le C m_W \delta_W + C n_h \delta_W \le C h^2 \log(1/h) .
\]

It remains to relax the frozen path. The closed-space Ward identity preserves the Gaussian mean one of $u_R$. After the signed block, $\mathbf F_h^{m_W} u_R$ is therefore closed, has Gaussian mean one and has bounded base norm. Using \eqref{e:cons-relaxation}, we obtain
\[
 \|\mathbf F_h^{n_h} u_R - \Omega_h\|
 \le e^{-c (n_h - m_W) h^2} \|\mathbf F_h^{m_W} u_R - \Omega_h\|
 = O(h^{c c_{\rm lay}}) ,
\]
where we used $(n_h - m_W) h^2 \ge c_{\rm lay} \log(1/h) - 2 C_0$. The three estimates together imply \eqref{e:asm-initial-layer}.

For the analytic transpose we use \eqref{e:asm-transpose}, which gives
\[
 (A_h^{\rm tr})^{n_h} (u_L \otimes 1) = T_h \, A_h^{n_h - 1} (s_\Gamma u_L \otimes 1) .
\]
The factor $u_L$ is a closed invariant normal class function with Gaussian mean one, like $u_R$, it has the negative Gaussian margins proved above, and $s_\Gamma - 1 = O(\delta_W v)$. The argument above therefore shows that $A_h^{n_h - 1} (s_\Gamma u_L \otimes 1) = \Omega_h \otimes 1 + o(1)$. Finally, $T_h$ is bounded, and $T_h (\Omega_h \otimes 1) = \Omega_h \otimes 1 + O(\delta_W h)$ by \eqref{e:cons-kinetic}, since $L_{H, 0} 1 = 0$.
\end{proof}

\section{The telescoping comparison}\label{s:asm-telescope}

We now compare the physical chain on the cycle with the adapted chain on the tube. Every comparison has the same three steps. We attach the right endpoint by \zcref{t:end-entrance}, we telescope the chain from the right so that all left factors are physical partial chains started from the constant function, and we return the final pairing with the constant function to the tube. The main result of this section is \zcref{l:asm-telescope}, which carries out the second step; the first and the third enter through its hypothesis (H4). We state it after introducing the objects that it uses, and we prove it after the lemmas that we use to verify its hypotheses.

Recall from \zcref{s:end-canonical} the physical cutoff $\chi_{\rm phys}$ of \eqref{e:asm-cutoff}, with the radial profile $\chi$, the decomposition $U = U^{\rm re} + \iu U^{\rm im}$, and the cutoff $\chi_{\rm tube} = \chi(|U|)$ on the real tube, and the map $\Can_{\rm phys} = \chi_{\rm phys} \Can$ of \zcref{s:end-residual}. We let $g_L^{\rm cut} = \chi_{\rm tube} \, g_L$ denote the left endpoint factor multiplied by the cutoff. Recall finally the families generated from $g_R$ of \zcref{d:end-generated}, their parameter-family norm $\| \cdot \|_{\mathcal U_2; p, a}$ of \eqref{e:end-family-norm}, with body weight $p$ and Gaussian margin $a$, and their prepared norm $\| \cdot \|_{{\rm prep}, \mathcal U_2; p}$ of \eqref{e:end-prep-norm}. The prepared norm is the supremum over the heights $|b| \le c_1 R_k$ of the family norms of the values on the complex normal graphs of height $b$, with the Gaussian margin $a$ of \zcref{p:end-coupled} and the factor $e^{-C_{\rm prep} W |b|^2}$ of \eqref{e:end-prep-norm}. Its term with $b = 0$ is the family norm, so that $\|f\|_{\mathcal U_2; p, a} \le \|f\|_{{\rm prep}, \mathcal U_2; p}$. The shell bound and the residual bound of \zcref{c:end-residual} are stated in the prepared norm, since their proofs use the values of the family on these graphs.

We now write the difference between a physical step and an adapted step. Let $T_i = M_{w_i} K_{\tau_{\rm phys}}$ denote a physical step, where $w_i$ is either the onsite weight $w_\Gamma$ or, at finitely many sites, a derivative of the onsite weight with respect to a local source, as in \zcref{s:asm-b}. The pullback to the tube and the gauge of \eqref{e:tube-table} give an adapted step $A_i = M_{w_i, {\rm ad}} K_{\rm ad}$ such that $M_{w_i} \Can_{\rm phys} = \Can_{\rm phys} M_{w_i, {\rm ad}}$, where $K_{\rm ad}$ is the gauged heat factor of the adapted tube. Therefore
\[
 T_i \Can_{\rm phys} - \Can_{\rm phys} A_i = M_{w_i} (K_{\tau_{\rm phys}} \Can_{\rm phys} - \Can_{\rm phys} K_{\rm ad}) .
\]
On the support of the cutoffs, the canonical map intertwines the two heat generators, and the difference of the heat factors comes only from the commutator of the physical heat generator $L_{\mathcal S}$ with the cutoff. On the families generated from $g_R$ of finite prepared norm, Duhamel's formula then yields the exact identity \eqref{e:end-heat-residual} of \zcref{l:end-heat-residual}, which we write as
\begin{equation}\label{e:asm-residual}
\begin{gathered}
 T_i \Can_{\rm phys} - \Can_{\rm phys} A_i = M_{w_i} \mathcal R , \\
 \mathcal R f = \int_0^{\tau_{\rm phys}} K_{\tau_{\rm phys} - s} \, [L_{\mathcal S}, \chi_{\rm phys}] \, \Can K_{{\rm ad}, s} (\chi_{\rm inv} f) \dd s ,
\end{gathered}
\end{equation}
where $\chi_{\rm inv}$ is the invariant normal input cutoff of the adapted step. We call $\mathcal R$ the heat residual. Its Duhamel forcing $[L_{\mathcal S},\chi_{\rm phys}]\,\Can K_{{\rm ad},s}(\chi_{\rm inv}f)$ is supported on the annulus where $\chi_{\rm phys}$ varies; the outer heat operator propagates this forcing beyond the annulus.

\begin{lemma}\label{l:asm-telescope}
Let $1 \le n \le N$, let $T_1, \dots, T_n$ and $A_1, \dots, A_n$ be physical and adapted steps as above, let $g$ be a family generated from $g_R$, and set $g_j = A_j \cdots A_1 g$ for $0 \le j \le n$. Let $f = \Can_{\rm phys} g + e$. Let $p$ be at least the body weight of \zcref{t:hist-bound} for the rank $k$ and the total order $d$ of the local derivatives among the steps, and let $A_{\rm sh}$ and $p'$ be as in \zcref{c:end-residual} for $p$. Suppose that there exist $A \ge 0$ and $C, c > 0$ such that the following hold, with these $p$ and $p'$ in all four hypotheses.
\begin{enumerate}
\item[(H1)] Every marked partial chain $F$ of \zcref{d:hist-marks} of length $1 \le j \le N$ whose onsite weights are the weights of $j$ consecutive given steps, in reversed order, satisfies $\|F\|_{\mathcal H_{-p}(\mathcal S)} + \|K_{\tau_{\rm phys}} F\|_{\mathcal H_{-p}(\mathcal S)} \le e^{o(W)}$.
\item[(H2)] For all $0 \le j \le n$, the state $g_j$ is a family generated from $g_R$ that is holomorphic in the normal variables where $\psi_{R_k} \ne 0$, and $\|g_j\|_{{\rm prep}, \mathcal U_2; p'} \le W^A$.
\item[(H3)] For all $0 \le j < n$, we have $\|\mathcal R g_j\|_{\mathcal H_p(\mathcal S)} \le C W^{A_{\rm sh}} e^{-c W} \|g_j\|_{{\rm prep}, \mathcal U_2; p'}$.
\item[(H4)] We have $\|e\|_{\mathcal H_p(\mathcal S)} \le W^A e^{-c W}$ and $c_{\rm sh} r_1^2 \ge c$, and
\[
 \bigl| \mathcal B_{\mathcal S}(1, \Can_{\rm phys} g_n) - \mathcal B_{\rm ad}(g_L^{\rm cut}, g_n) \bigr|
 \le C_k W^{A_{\rm sh} + 4 k^2} e^{-c_{\rm sh} r_1^2 W} \|g_n\|_{{\rm prep}, \mathcal U_2; p'} .
\]
\end{enumerate}
Then
\begin{equation}\label{e:asm-telescope}
 \bigl| \mathcal B_{\mathcal S}(1, T_n \cdots T_1 f) - \mathcal B_{\rm ad}(g_L^{\rm cut}, g_n) \bigr|
 \le C N W^{A_{\rm sh} + A} e^{-c W + o(W)} .
\end{equation}
\end{lemma}

Hypothesis (H1) is the a priori bound of \zcref{t:hist-bound}, which remains true when $p$ is increased, since $v_{\mathcal S} \ge 1$. In our applications, we take $p$ so large that \zcref{l:asm-return} below applies, and (H4) then follows from \zcref{t:end-entrance} and \zcref{l:asm-terminal}. There we also have $N \asymp W^2$, and the right side of \eqref{e:asm-telescope} tends to zero. The proof of the lemma is elementary. The work is in the verification of (H1)--(H4), which uses \zcref{s:hist,s:end}. The bound (H3) is available only on families generated from $g_R$, which is why (H2) asks for this structure and not only for a bound on the norm. We prove \zcref{l:asm-telescope} at the end of this section, and we begin with the return of the final pairing to the tube, in two steps.

The first step returns the cycle of integration from $\mathcal S$ to the real tube, by Stokes' theorem along the homotopy $\Phi_t$ of body cycles of \eqref{e:asm-homotopy}, defined in \zcref{s:end-canonical} on the parameter domain $\mathcal Y$ of the coordinates $y$ of $\mathcal S$, with the properties stated in \zcref{l:asm-homotopy}.

We use the orientation of the real tube fixed before \zcref{l:tube-volume}. Near each component of the saddle manifold, the domain of $\Phi_0$ is connected, so that $\Phi_0$ either preserves or reverses this orientation there. A reversal would multiply the integral in \eqref{e:asm-returned} below by $-1$, for all sources, and with it the term of that signature in each comparison below. Near the balanced component, $\Phi_0$ preserves the orientation, as we show in \zcref{s:asm-calibration} by comparing the limit of $\mathcal Z_N(\Gamma_0)$ with \eqref{e:rep-normalization}. The terms of the unbalanced signatures tend to zero, so that their signs play no role. In these signatures, the orientation of $G_0/K_0$ fixed before \zcref{l:tube-volume} is arbitrary, and by \eqref{e:tube-volume} the orientation of the real tube reverses with it. We take it such that $\Phi_0$ preserves the orientation.

Recall from \eqref{e:asm-cutoff} that $\chi_{\rm phys} = \chi_\Phi(1, \cdot)$, where $\chi_\Phi(t, y) = \chi_{\rm tube}(\Phi_0(y))$ coefficientwise in the ambient odd coordinates, and that the nilpotent coefficients of $\chi_\Phi$, and the derivatives of every coefficient of $\chi_\Phi$, vanish outside the annulus where $\chi_{\rm phys}$ varies. We denote this annulus by
\[
 \mathcal A = \bigl\{ y : r_1 \le |U^{\rm re}_{\rm can}(y)| \le r_{\rm phys} \bigr\} .
\]
We now state the first step.

\begin{lemma}\label{l:asm-return}
Let $p$ be sufficiently large, and let $A_{\rm sh}$ and $p' \ge p$ be the exponent and the body weight of \zcref{c:end-residual} for $p$. Let $f = f_{\Gamma, \mathbf V}$ be a family generated from $g_R$ in a fixed signature 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
\begin{equation}\label{e:asm-returned}
 \mathcal B_{\mathcal S}(1, \Can_{\rm phys} f)
 = \int \chi_{\rm tube} \, \Can f \, \varpi^2 \dd U \dd \nu
 + O\bigl( C_k W^{A_{\rm sh} + 4 k^2} e^{-c_{\rm sh} W r_1^2} \bigr) \|f\|_{{\rm prep}, \mathcal U_2; p'} ,
\end{equation}
and the integral on the right converges absolutely.
\end{lemma}

\begin{proof}
The function $\Can f$ is holomorphic, but the cutoff is not, and we apply Stokes' theorem to their product along the homotopy $\Phi$. The exterior derivative then falls only on the cutoff, which varies only on the annulus $\mathcal A$, and there the bound \eqref{e:end-shell} applies along the homotopy.

We begin by reducing to body cycles. We write both integrals in the ambient odd coordinates. On $\mathcal S$ these are free generators, and $\mathcal B_{\mathcal S}(1, \Can_{\rm phys} f)$ is the integral over the body of $\mathcal S$ of the top odd coefficient of $\Can_{\rm phys} f$. On the real tube we pass from the tubular to the ambient coordinates by Berezin's formula \zcref{t:grass-change}, which is exact for integrands with compact support; we apply it only after an exhaustion cutoff has been inserted. Let $U_{\rm can}$ be the canonical normal coordinate of \zcref{s:end-canonical} with its nilpotent part, which is obtained by solving \eqref{e:end-canonical-equation} in successive odd degree. It depends holomorphically on the body point, and on the real tube it is the tubular coordinate $U$. Consequently, by \zcref{l:end-ward}\ref{i:end-ward-b} and the normal holomorphy of $f$, the coefficients of $F = \Can f$ in the ambient odd coordinates are holomorphic functions of the body point $z$ on the region swept by $\Phi$.

We now apply Stokes' theorem with an exhaustion. Let $\varphi$ be a smooth function on $[0, \infty)$ that equals one on $[0, 1]$ and zero on $[2, \infty)$, and set
\[
 \varphi_R(y) = \varphi\bigl( \log v_{\mathcal S}(X)/R \bigr) ,
 \qquad R \ge 1 .
\]
Let $\dd z$ be the flat holomorphic volume form of the body of the ambient space, normalized as the body part of $\dd S$. Consider the form
\[
 \omega_R = \varphi_R \, \chi_\Phi \, (F \circ \Phi) \, \Phi^* \dd z
\]
on $[0, 1] \times \mathcal Y$, coefficientwise in the ambient odd coordinates. It has compact support, since $\chi_\Phi$ vanishes where $|U^{\rm re}_{\rm can}| \ge r_{\rm phys}$ and $\varphi_R$ vanishes where $v_{\mathcal S} \ge e^{2R}$. Since $F$ is holomorphic, $\dd F \wedge \dd z = 0$, and therefore $\dd \omega_R = \dd(\varphi_R \chi_\Phi) \wedge (F \circ \Phi) \, \Phi^* \dd z$. Stokes' theorem gives
\begin{equation}\label{e:asm-stokes}
 \int_{\{1\} \times \mathcal Y} \omega_R - \int_{\{0\} \times \mathcal Y} \omega_R
 = \int_{[0, 1] \times \mathcal Y} \bigl( \varphi_R \, \dd \chi_\Phi + \chi_\Phi \, \dd \varphi_R \bigr) \wedge (F \circ \Phi) \, \Phi^* \dd z
 = I_R + I\!I_R .
\end{equation}
The integrand of the second term on the left has compact support on the real tube. Using the definition \eqref{e:asm-cutoff} of $\chi_{\rm phys}$ for the first term, and \zcref{t:grass-change} and \eqref{e:tube-volume} for the second, we obtain
\[
 \int_{\{1\} \times \mathcal Y} \omega_R = \int_{\mathcal S} \varphi_R \, \Can_{\rm phys} f \dd S ,
 \qquad
 \int_{\{0\} \times \mathcal Y} \omega_R = \int \bigl( \varphi_R \circ \Phi_0^{-1} \bigr) \, \chi_{\rm tube} \, \Can f \, \varpi^2 \dd U \dd \nu .
\]
Since $\chi_\Phi$ and $\varphi_R$ do not depend on $t$, only the component $\dd t \wedge \iota_{\partial_t \Phi} \dd z$ of $\Phi^* \dd z$ contributes to $I_R$ and $I\!I_R$. The integrand of $I_R$ is supported in $[0, 1] \times \mathcal A$, and that of $I\!I_R$ where $e^R \le v_{\mathcal S} \le e^{2R}$.

We now bound $I_R$. Let $y \in \mathcal A$ and $0 \le t \le 1$. By \zcref{l:asm-homotopy}\ref{i:asm-homotopy-frame} and \zcref{l:end-ward}\ref{i:end-ward-b}, the value $F(\Phi_t(y))$ is the family at the conjugated damping $e^{t Y_{\mathsf g}} \Gamma e^{-t Y_{\mathsf g}}$, with the conjugated jet directions, evaluated at the real frame $\mathsf g^{-1}$ and at the normal coordinate $U^{\rm re} + \iu t U^{\rm im}$. These dampings lie in $\mathcal U_1$ for the tube of \zcref{s:end-canonical}, since $\|S_{\mathsf g}^{\pm t} - \Id\| \le C \|n_B\|$ by \zcref{l:end-height}, and on the shell $\Omega_s$ of the proof of \zcref{l:end-shell} the heights satisfy $t |U^{\rm im}| \le B_s$. That proof applies \zcref{l:tools-submean} jointly in the damping and in the complex normal coordinate on each shell, to selections of the damping in $\mathcal U_1$ and of heights at most $B_s$, and it bounds the selected values by the prepared norm. It therefore applies verbatim to the selections $(e^{t Y_{\mathsf g}} \Gamma e^{-t Y_{\mathsf g}}, \iu t U^{\rm im})$, which yields, uniformly in $t$,
\begin{equation}\label{e:asm-shell-homotopy}
 \int_{\mathcal A} v^{2p} \bigl| F(\Phi_t(y)) \bigr|^2 \dd y
 \le C_k W^{2A_{\rm sh}} e^{-2 c_{\rm sh} W r_1^2} \|f\|_{{\rm prep}, \mathcal U_2; p'}^2 ,
\end{equation}
where $| \cdot |$ is the coefficient norm and $v$ is the weight of the family at the real frame $\mathsf g^{-1}$, which is the weight of $Q_X$.

The remaining factors in $I_R$ are polynomial in $v_{\mathcal S}$. Indeed, \zcref{l:asm-homotopy}\ref{i:asm-homotopy-bounds} bounds $|\partial_t \Phi_t|$ and $\|D \Phi_t\|^{\dim - 1}$ by $C_k v_{\mathcal S}^{A_\Phi}$. The coefficients of $\dd \chi_\Phi$ are bounded at fixed rank, and the passage from the top coefficient in the ambient odd coordinates to the coefficient norm loses at most a factor $C_k W^{4 k^2}$, as in \zcref{l:hist-pairing}. Finally, $v_{\mathcal S} = 1 + 2 s (v(Q_X) - 1)$ by \zcref{s:hist}, so that $v_{\mathcal S}$ and $v$ are comparable. By the Cauchy--Schwarz inequality, with the weight $v_{\mathcal S}^{-p}$ on the first factor and $p$ so large that $v_{\mathcal S}^{2 A_\Phi - 2p}$ is integrable over $\mathcal A$, and by \eqref{e:asm-shell-homotopy}, we obtain
\[
 |I_R| \le C_k W^{A_{\rm sh} + 4 k^2} e^{-c_{\rm sh} W r_1^2} \|f\|_{{\rm prep}, \mathcal U_2; p'} ,
\]
uniformly in $R$.

We next show that $I\!I_R \to 0$ as $R \to \infty$. The same selection argument, without the smallness of the annulus and with heights at most $B_{\max} = C_{\rm geo}(|E| r_{\rm phys} + r_{\rm phys}^2)$ on the support of $\chi_\Phi$, shows that $v^p \, F \circ \Phi_t$ is square integrable there, uniformly in $t$; here a bound that depends on $W$ suffices. Since $|\dd \varphi_R| \le C/R$ and the integrand of $I\!I_R$ is supported where $v_{\mathcal S} \ge e^R$, the Cauchy--Schwarz inequality and dominated convergence give $I\!I_R \to 0$.

It remains to let $R \to \infty$ on the left of \eqref{e:asm-stokes}. By the case $t = 1$ of the selection argument, $\Can_{\rm phys} f \in \mathcal H_p(\mathcal S)$. We pair it with a compact normal cutoff rather than require the constant function to belong to the opposite weighted space. Indeed, \zcref{l:end-height} on the chosen small canonical chart bounds $n_B$ and $n_F$ uniformly on the support of $\chi_{\rm phys}$. Since $P_0 = \operatorname{ch}_{\mathsf g}^{-1}n_B$ with $\|\operatorname{ch}_{\mathsf g}^{-1}\| \le 1$, and $M = V(-s\Sigma_F+n_F)V^*$ with $V$ unitary, the physical normal coordinates $(P_0,M)$ lie in a fixed compact set, independently of $X$. The nilpotent coefficients of $\chi_{\rm phys}$ have the same body support. Choose a smooth scalar cutoff $\Theta(P_0,M)$, compactly supported in the physical normal variables and equal to one on a neighborhood of this set. Then $\Theta\Can_{\rm phys}f=\Can_{\rm phys}f$, and $\Theta\in\mathcal H_{-p}(\mathcal S)$ for $p$ sufficiently large: its normal fiber volume is bounded, while $v_{\mathcal S}(X)\ge c(1+|X|)$ and the real dimension of the $X$ space is $2k^2$.

Dominated convergence gives $(\varphi_R-1)\Can_{\rm phys}f\to0$ in $\mathcal H_p(\mathcal S)$. Applying \zcref{l:hist-pairing} with $\Theta$ therefore gives
\[
 \int_{\mathcal S} \varphi_R \, \Can_{\rm phys} f \dd S \longrightarrow \mathcal B_{\mathcal S}(1, \Can_{\rm phys} f) ,
 \qquad R \to \infty .
\]
In the tubular coordinates, $\varphi_R \circ \Phi_0^{-1} - 1$ and its nilpotent coefficients are bounded and supported where $v \ge e^R/C$, and the integrand $\chi_{\rm tube} \Can f \varpi^2$ is integrable. Indeed, on the adapted tube we have $\Can f = \varpi^{-1} w_0^{1/2} \gamma_h f$, and $g_L = \varpi w_0^{1/2} \gamma_h^{-1}$ by \eqref{e:tube-table}, so that
\begin{equation}\label{e:asm-returned-density}
 \chi_{\rm tube} \, \Can f \, \varpi^2 = g_L^{\rm cut} \, f \, \gamma_h^2 .
\end{equation}
The right side is integrable by the Cauchy--Schwarz inequality in the opposite body weights $-p$ and $p$ and the bound of \zcref{l:end-raw-norms} on $g_L$. Letting $R \to \infty$ in \eqref{e:asm-stokes}, we obtain \eqref{e:asm-returned}.
\end{proof}

We now identify the returned integral with the analytic Berezin pairing.

\begin{lemma}\label{l:asm-terminal}
Let $p$, $A_{\rm sh}$, $p'$, $f$, and $\Gamma$ be as in \zcref{l:asm-return}. There exists $C$ such that
\begin{equation}\label{e:asm-terminal}
\begin{gathered}
 \mathcal B_{\mathcal S}(1, \Can_{\rm phys} f)
 = \mathcal B_{\rm ad}(g_L^{\rm cut}, f) + O\bigl( C_k W^{A_{\rm sh} + 4 k^2} e^{-c_{\rm sh} W r_1^2} \bigr) \|f\|_{{\rm prep}, \mathcal U_2; p'} , \\
 |\mathcal B_{\rm ad}(g_L^{\rm cut}, f)| \le C \|f\|_{\mathcal U_2; p, a} .
\end{gathered}
\end{equation}
In the balanced signature $C$ is independent of $W$, and in an unbalanced signature $C \le C_k W^{A_k}$.
\end{lemma}

\begin{proof}
By \zcref{l:asm-return}, it remains to identify the returned integral and to bound it. Integrating \eqref{e:asm-returned-density}, we obtain
\[
 \int \chi_{\rm tube} \, \Can f \, \varpi^2 \dd U \dd \nu
 = \int g_L^{\rm cut} \, f \, \gamma_h^2 \dd U \dd \nu
 = \mathcal B_{\rm ad}(g_L^{\rm cut}, f) .
\]
Together with \eqref{e:asm-returned}, this gives the identity in \eqref{e:asm-terminal}.

It remains to bound $\mathcal B_{\rm ad}(g_L^{\rm cut}, f)$. The analytic Berezin pairing is represented, relative to the positive coefficient metric, by a bounded signed matrix on the exterior coefficients. The Cauchy--Schwarz inequality therefore implies that
\[
 |\mathcal B_{\rm ad}(g_L^{\rm cut}, f)|
 \le C \, \|e^{-a|Z|^2} v^{-p} \widetilde\chi g_L\| \, \|e^{a|Z|^2} v^{p} f_\Gamma\| ,
\]
where $\widetilde\chi$ is a real invariant normal cutoff as in \zcref{l:end-raw-norms} that equals one on the support of $\chi_{\rm tube}$.

Once $p$ is so large that $-p \le p_0$, \zcref{l:end-raw-norms} bounds the first factor by a constant in the balanced signature and by $C_k W^{A_k}$ otherwise. By \zcref{l:tools-submean}, applied with $\zeta = \Gamma$ and the constant selection $\zeta_* = \Gamma \in \mathcal U_0$, the second factor is bounded by $C \|f\|_{\mathcal U_2; p, a}$, where $C$ depends on the distance from $\mathcal U_0$ to the complement of $\mathcal U_2$.
\end{proof}

The hypotheses (H2) and (H3) of \zcref{l:asm-telescope} bound the states of the adapted chain in the prepared norm, since \zcref{c:end-residual} bounds the heat residual by it. Every state that we use is the output of a block of $m_W$ adapted steps, and \eqref{e:end-prep-block} bounds the prepared norm of such an output in the body weight of the input. The input of the first block is $(\xi_\mu g_R) \otimes 1$, with the invariant cutoff of \zcref{t:end-entrance}; \zcref{l:end-raw-norms} bounds it only in a negative body weight. The following lemma shows that a block also raises the body weight, at a cost polynomial in $W$, through the damping of its last source factor.

\begin{lemma}\label{l:asm-prep-gain}
Let $p \in \R$ and $q \ge 0$. There exists $C$, depending on $k$, $p$, and $q$, such that for every family $g$ generated from the endpoint,
\begin{equation}\label{e:asm-prep-gain}
 \|A_h^{m_W} g\|_{{\rm prep}, \mathcal U_2; p + q} \le C W^{2q} \|g\|_{\mathcal U_2; p, -a} .
\end{equation}
\end{lemma}

\begin{proof}
The source factor of the adapted step carries the BB damping $|s_\Gamma|_1 \le e^{C_k \eta - c_0 \eta v}$ of \zcref{l:end-coercivity}, for some $c_0 > 0$. As in the proof of \zcref{l:asm-weight-gain}, let $\mathbf S_\flat = e^{-c \delta_W v/2}$, where $c > 0$ is so small that $c \delta_W \le c_0 \eta/2$. The last of the $m_W$ steps is $A_h = s_\Gamma T_h$, with the multiplier $s_\Gamma$ on the left, and we write
\[
 A_h^{m_W} = \mathbf S_\flat^2 B^\flat ,
 \qquad
 B^\flat = (\mathbf S_\flat^{-2} s_\Gamma) \, T_h \, A_h^{m_W - 1} .
\]
The block $B^\flat$ differs from $A_h^{m_W}$ only in the source factor of its last step, and $\mathbf S_\flat^{-2} s_\Gamma = e^{c \delta_W v} s_\Gamma$ keeps at least half of the BB damping of $s_\Gamma$. The proof of \zcref{p:end-coupled} uses the source factor only through \zcref{l:end-kick}. The proof of \zcref{l:end-kick} shows that \eqref{e:end-kick} also holds with $s_\Gamma$ replaced by $e^{c' \eta v} s_\Gamma$, for every $0 \le c' \le c_0/2$. Applied with $c' = c \delta_W/\eta$, which is at most $c_0/2$ by our choice of $c$, this gives \eqref{e:end-kick} for the source factor $\mathbf S_\flat^{-2} s_\Gamma$. Hence \eqref{e:end-prep-block} holds for $B^\flat$,
\[
 \|B^\flat g\|_{{\rm prep}, \mathcal U_2; p} \le C_k(R_k) \|g\|_{\mathcal U_2; p, -a} .
\]
The multiplier $v^q \mathbf S_\flat^2$ depends only on the tangent variables, which the normal graphs of \eqref{e:end-gauge-graph-norm} leave unchanged, and, as in the derivation of \eqref{e:asm-weight-crude},
\[
 \sup_{x \ge 1} x^q e^{-c \delta_W x} \le C_q \delta_W^{-q} = C_q W^{2q} .
\]
Therefore $\|\mathbf S_\flat^2 F\|_{b; p + q, a} \le C_q W^{2q} \|F\|_{b; p, a}$ for all $|b| \le c_1 R_k$ and all $F$, and \eqref{e:asm-prep-gain} follows from the bound for $B^\flat$.
\end{proof}

\begin{proof}[Proof of \zcref{l:asm-telescope}]
By \eqref{e:asm-residual}, we have the telescoping identity
\[
 T_n \cdots T_1 \Can_{\rm phys} g - \Can_{\rm phys} g_n
 = \sum_{j = 1}^{n} T_n \cdots T_{j + 1} M_{w_j} \mathcal R g_{j - 1} .
\]
We pair it with $1$ and move all physical factors to the left by analytic transposition. Since $T_i^{\rm tr} = K_{\tau_{\rm phys}} M_{w_i}$ and multiplication is symmetric for the pairing, we obtain
\[
 \mathcal B_{\mathcal S}(1, T_n \cdots T_{j + 1} M_{w_j} \mathcal R g_{j - 1})
 = \mathcal B_{\mathcal S}(F_j, \mathcal R g_{j - 1}) ,
 \qquad
 F_j = M_{w_j} K_{\tau_{\rm phys}} M_{w_{j + 1}} \cdots K_{\tau_{\rm phys}} M_{w_n} 1 .
\]

Note that each $F_j$ is a marked partial chain as in (H1), with $n - j + 1$ onsite factors in reversed chronological order (this is where the one-sided form of the pairing is used). Without local sources, \eqref{e:asm-endpoints} gives
\[
 F_j = M_{w_\Gamma} (T_\Gamma^{\rm tr})^{n - j} 1 = M_{w_\Gamma} K_{\tau_{\rm phys}} T_\Gamma^{n - j} 1 = T_\Gamma^{n - j + 1} 1 .
\]
In particular, the last term also uses a partial chain with one onsite factor. Similarly, $\mathcal B_{\mathcal S}(1, T_n \cdots T_1 e) = \mathcal B_{\mathcal S}(K_{\tau_{\rm phys}} F_1, e)$.

\zcref[S]{l:hist-pairing} bounds the pairing $\mathcal B_{\mathcal S}$ on $\mathcal H_{-p}(\mathcal S) \times \mathcal H_p(\mathcal S)$ by $C_k W^{4k^2}$, and this factor is absorbed in $e^{o(W)}$. Hypotheses (H1), (H2) and (H3) bound the $n$ residual terms, and (H1) and (H4) bound the term with $e$. We find
\[
 \bigl| \mathcal B_{\mathcal S}(1, T_n \cdots T_1 f) - \mathcal B_{\mathcal S}(1, \Can_{\rm phys} g_n) \bigr|
 \le e^{o(W)} \bigl( n W^{A_{\rm sh} + A} e^{-c W} + W^A e^{-c W} \bigr) .
\]
Finally, (H4) and (H2) imply that
\[
\begin{aligned}
 |\mathcal B_{\mathcal S}(1, \Can_{\rm phys} g_n) - \mathcal B_{\rm ad}(g_L^{\rm cut}, g_n)|
 &\le C_k W^{A_{\rm sh} + 4 k^2} e^{-c_{\rm sh} r_1^2 W} \|g_n\|_{{\rm prep}, \mathcal U_2; p'} \\
 &\le C_k W^{4 k^2} W^{A_{\rm sh} + A} e^{-c W} ,
\end{aligned}
\]
where in the last inequality we used $c_{\rm sh} r_1^2 \ge c$. As above, the factor $W^{4 k^2}$ is absorbed in $e^{o(W)}$. Combining the last two displays, we obtain \eqref{e:asm-telescope}.
\end{proof}

\section{The source limit: proof of part (a)}\label{s:asm-a}

We prove \zcref{t:interface}(a) in two steps. The first step replaces $\mathcal Z_N(\Gamma)$ by a sum of adapted pairings, one for each signature, and the second identifies the limit of these pairings. In this section $\mathbf u$ and $\mathbf b$ are real, $\sum_j u_j = 0$, and \eqref{e:asm-source-ball} holds.

\subsection{Replacement of the physical chain}

Recall from \zcref{t:end-entrance} that the right block has the decomposition
\begin{equation}\label{e:asm-entrance}
 f_{m_W} = \sum_{\mu = 0}^{2k} \Can_{\mu, {\rm phys}} \, g_\mu + e_{m_W} ,
 \qquad
 \|e_{m_W}\|_{\mathcal H_p(\mathcal S)} \le C_k W^{A_k} e^{-\gamma_k W} ,
\end{equation}
for all required body weights $p$. Here $\mu$ runs over the signatures, $\Can_{\mu, {\rm phys}}$ is the canonical map of the component $\mu$ with its physical cutoff, and $g_\mu$ is the reference endpoint of that component, transported to the gauge \eqref{e:tube-table}. Let $A_{h, \mu}$ denote the adapted step in the signature $\mu$, and $g_{L, \mu}^{\rm cut}$ the corresponding left endpoint factor. The raw input in \zcref{s:end-attach} is $\xi_\mu \varpi_\mu$, so the exact similarity \eqref{e:end-similarity} gives
\[
 g_\mu = A_{h, \mu}^{m_W} ((\xi_\mu g_R) \otimes 1) .
\]
The invariant normal cutoff $\xi_\mu$ is retained in this input throughout the comparison and the initial layer. In particular, $g_\mu$ is a family generated from $g_R$ by \zcref{d:end-generated}.

\begin{proposition}\label{p:asm-physical}
Uniformly on compact subsets of the set \eqref{e:asm-source-ball},
\begin{equation}\label{e:asm-physical}
 \mathcal Z_N(\Gamma)
 = \sum_{\mu = 0}^{2k} \mathcal B_{\rm ad}(g_{L, \mu}^{\rm cut}, A_{h, \mu}^{N - m_W} g_\mu) + o(1) .
\end{equation}
\end{proposition}

\begin{proof}
By \eqref{e:asm-one-sided} and \eqref{e:asm-entrance}, we have
\[
 \mathcal Z_N(\Gamma) = \sum_\mu \mathcal B_{\mathcal S}\bigl( 1, T_\Gamma^{N - m_W} (\Can_{\mu, {\rm phys}} g_\mu + e_{\mu}) \bigr) ,
\]
where we split $e_{m_W}$ into $2k + 1$ pieces $e_\mu$ with the same bound. For each $\mu$, we apply \zcref{l:asm-telescope} without local sources, with $T_i = T_\Gamma$, $A_i = A_{h, \mu}$, $n = N - m_W$, $g = g_\mu$, and $e = e_\mu$. We now verify its hypotheses.

Hypothesis (H1) is \zcref{t:hist-bound} without local sources. Its hypotheses hold, since $N \asymp W^2$, $|E| \le E_{\rm out} \le 1/2$, $\eta = \eps/a_N \asymp W^{-2}$, and the FF block of the source is traceless. The factor $(1 + \eta^{-1})^{A'}$ in that theorem, where $A'$ denotes its exponent, is therefore polynomial in $W$.

For (H2), we set $g_{-m_W} = (\xi_\mu g_R) \otimes 1$, with the invariant cutoff used to define $g_\mu$, and $g_j = A_{h, \mu}^{m_W + j} g_{-m_W}$ for $-m_W < j \le n$. Then $g_0 = g_\mu$ exactly, the states of \zcref{l:asm-telescope} are the $g_j$ with $0 \le j \le n$, and
\[
 g_j = A_{h, \mu}^{m_W} \, g_{j - m_W} ,
 \qquad 0 \le j \le n .
\]
These states are families generated from $g_R$, since intermediate powers, invariant normal cutoffs, and partial pure heats are among the operations of \zcref{d:end-generated}. For $j \ge 0$ they are outputs of adapted steps, and hence holomorphic in the normal variables where $\psi_{R_k} \ne 0$. Let $p_0$ be the body weight of \zcref{l:end-raw-norms}. For $-m_W \le j < 0$, \zcref{l:end-raw-norms} and the constant Gaussian weights of \zcref{c:end-ramp} bound $\|g_j\|_{\mathcal U_2; p_0, -a}$ by a constant in the balanced signature and by $C_k W^{A_k}$ otherwise, uniformly on $\mathcal U_2$ by \zcref{l:end-real-family}. Applying \zcref{l:asm-prep-gain} with the body weight $p_0$ and $q = p' - p_0$, we obtain
\[
 \|g_j\|_{{\rm prep}, \mathcal U_2; p'} \le C W^{2(p' - p_0)} \|g_{j - m_W}\|_{\mathcal U_2; p_0, -a} ,
 \qquad 0 \le j \le n .
\]
For $j < m_W$, the right side is bounded by a power of $W$. For $j \ge m_W$, the margin $-a$ is weaker than the base norm, and the power bounds of \zcref{t:stab-main}, which are uniform on $\mathcal U_2$ by \zcref{l:end-real-family}, bound the base norm of $g_{j - m_W} = A_{h, \mu}^{j - m_W} g_0$ in the body weight $p_0$ by a constant times $\|g_0\|_{{\rm prep}, \mathcal U_2; p'}$. Hence (H2) holds for all $0 \le j \le n$.

For (H3), \zcref{c:end-residual} applies to every $g_j$ with $0 \le j < n$, since $\Gamma \in \mathcal U_0$ and $g_j$ has finite prepared norm by (H2). Its residual bound \eqref{e:end-residual-bound} implies (H3) with the rate $c = c_{\rm sh} r_1^2$. The continuation of the families to the canonical body coordinates is made inside \zcref{c:end-residual}, by \zcref{l:end-ward}. Only the right states are continued analytically; the left factors are evaluated on the real cycle at the original real sources.

Hypothesis (H4) is \eqref{e:asm-entrance} together with \zcref{l:asm-terminal}, which applies by (H2). Applying \zcref{l:asm-telescope} with $c = \min(c_{\rm sh} r_1^2, \gamma_k)$ in each signature and summing over $\mu$, we obtain
\[
 \Bigl| \mathcal Z_N(\Gamma) - \sum_{\mu = 0}^{2k} \mathcal B_{\rm ad}(g_{L, \mu}^{\rm cut}, A_{h, \mu}^{N - m_W} g_\mu) \Bigr|
 \le C (2k + 1) N W^{A_{\rm sh} + A} e^{-c W + o(W)} ,
\]
and the right side is $o(1)$ since $N \asymp W^2$. The bounds are uniform on compact parameter sets by \zcref{l:end-real-family}.
\end{proof}

\subsection{The limit of the adapted chain}

In the balanced signature, recall from \zcref{l:asm-time} that $N \delta_W L^{\rm tan}_h = - \mathsf G_{X_N(E), \Gamma}$, where $L^{\rm tan}_h = \tfrac12 L_{H, 0} - V_0$ is the tangent generator of one adapted step and $\mathsf G_{X, \Gamma}$ is written as in \eqref{e:asm-generator}. Thus $j$ steps of this balanced chain correspond to the macroscopic time $j/N$ of the semigroup $e^{-t \mathsf G_{X_N(E), \Gamma}}$.

\begin{proposition}\label{p:asm-adapted}
Uniformly on compact subsets of the set \eqref{e:asm-source-ball},
\begin{equation}\label{e:asm-adapted}
 \sum_{\mu = 0}^{2k} \mathcal B_{\rm ad}(g_{L, \mu}^{\rm cut}, A_{h, \mu}^{N - m_W} g_\mu)
 \longrightarrow \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) .
\end{equation}
\end{proposition}

\begin{proof}
The terms with $\mu \ne k$ are $O(W^A e^{-c W})$, by \zcref{l:asm-unbalanced} and the polynomial bounds of \zcref{l:asm-terminal} and \zcref{l:end-raw-norms}. We consider the balanced term and drop the index $\mu = k$. Fix $0 < \theta < 1/4$ (the limit will not depend on $\theta$), and split the $N - m_W$ adapted steps into a right outer portion, a middle portion, and a left outer portion, of lengths $n_R = \lfloor \theta N \rfloor$, $n_M$, and $n_L = \lfloor \theta N \rfloor$, so that $n_M = (1 - 2\theta) N + O(W)$. We treat the left portion in the analytic transpose order, so that
\begin{equation}\label{e:asm-three-pieces}
 \mathcal B_{\rm ad}(g_L^{\rm cut}, A_h^{N - m_W} g)
 = \mathcal B_{\rm ad}\bigl( (A_h^{\rm tr})^{n_L} g_L^{\rm cut}, A_h^{n_M} A_h^{n_R} g \bigr) .
\end{equation}

We begin with the right state. Since $n_h > m_W$ for $W$ large, \zcref{l:asm-initial-layer}, applied with $\xi = \xi_k$, implies that
\[
 A_h^{n_h - m_W} g = A_h^{n_h} ((\xi_k g_R) \otimes 1) = \Omega_h \otimes 1 + o(1)
\]
in the base normal norm and a sufficiently negative body weight. Its tangent time $n_h/N$ tends to zero. A base-norm error suffices, since the power bounds of \zcref{t:stab-main} propagate base-norm errors.

Next, we use the product limit \zcref{t:cons-limit} (the Trotter--Kurtz theorem \cite{trotter1958approximation,kurtz1969extensions}, in the form of \zcref{l:cons-product}), whose hypotheses are the power bounds of \zcref{t:stab-main} and the consistency relation of \zcref{p:cons-consistency}. It implies that
\[
 \sup_{n \delta_W \le T} \bigl\| A_h^n \iota_h f - \iota_h e^{n \delta_W L^{\rm tan}_h} f \bigr\| \longrightarrow 0
\]
for all $T > 0$ and all $f$ in the base space, uniformly on compact parameter sets. Recall that $e^{j \delta_W L^{\rm tan}_h} = e^{-(j/N) \mathsf G_{X_N(E), \Gamma}}$, by \zcref{l:asm-time}. Since $X_N(E) \to X$, the generators $\mathsf G_{X_N(E), \Gamma}$ converge to $\mathsf G_{X, \Gamma}$ on the common core, and the parameter version \zcref{c:cons-product-param} of the product limit gives
\begin{equation}\label{e:asm-segment-limit}
 \sup_{0 \le j \le T N} \bigl\| A_h^j (\Omega_h \otimes f) - \Omega_h \otimes e^{-(j/N) \mathsf G_{X, \Gamma}} f \bigr\| \longrightarrow 0 .
\end{equation}
The tangent constant $1$ belongs to the base space once its negative body weight is strong enough.

We now combine the two steps. Using the power bounds once more to propagate the error of the initial layer, together with \eqref{e:asm-segment-limit} on the right outer portion and then on the middle portion, we find
\[
 A_h^{n_R} g = \Omega_h \otimes e^{-\theta \mathsf G_{X, \Gamma}} 1 + o(1) ,
 \qquad
 A_h^{n_M} A_h^{n_R} g = \Omega_h \otimes e^{-(1 - \theta) \mathsf G_{X, \Gamma}} 1 + o(1) ,
\]
in a negative body weight, where we used the semigroup property and $(n_M + n_R)/N \to 1 - \theta$. Since $n_R \ge \theta N/2$, \zcref{l:asm-weight-gain} bounds $A_h^{n_R} g$ and $A_h^{n_M} A_h^{n_R} g$ in all positive body weights, and interpolation between the weighted $L^2$ norms promotes both convergences to every body weight.

For the left state, we first replace $g_L^{\rm cut}$ by $g_L$. The two differ only where $|U| \ge r_1$. There the right state $A_h^{N - m_W} g$ is exponentially small, by the positive Gaussian margin of \zcref{p:end-coupled}, and this replacement changes the pairing by $O(e^{-c W r_1^2})$. We next reduce the transposed chain to the chain itself. By \eqref{e:asm-transpose},
\[
 (A_h^{\rm tr})^{n_L} g_L = T_h \, A_h^{n_L - 1} (s_\Gamma g_L) .
\]
The factor $g_L$ has the structure of $g_R$ that the initial layer uses, with a constant tangent part, and $s_\Gamma - 1 = O(\delta_W v)$. The argument for the right state, with \zcref{l:asm-initial-layer} for $g_L$, therefore shows that $A_h^{n_L - 1} (s_\Gamma g_L) = \Omega_h \otimes e^{-\theta \mathsf G_{X, \Gamma}} 1 + o(1)$ in a negative body weight. The final factor $T_h$ is bounded by \zcref{l:cons-short-heat}, and by \eqref{e:cons-kinetic} it changes $\Omega_h \otimes F$ only by $O(\tau_{\rm phys})$ for $F$ in the operator core of \zcref{p:sigma-regular}. Hence
\[
 (A_h^{\rm tr})^{n_L} g_L = \Omega_h \otimes e^{-\theta \mathsf G_{X, \Gamma}} 1 + o(1)
\]
in a negative body weight, and, by \zcref{l:asm-weight-gain} applied to $A_h^{n_L - 1}$, in every body weight.

In the balanced signature, \eqref{e:tube-pairing-factor} with $u = u' = \Omega_h$, which is even, factors the adapted pairing of two vectors $\Omega_h \otimes f$ and $\Omega_h \otimes f'$ into the normal pairing $\mathcal B_{\rm nor}(\Omega_h, \Omega_h)$ and $\int f f' \dd \nu$. In the tubular measure of \zcref{s:tube-gauge}, the tangent measure $\dd \nu$ is the invariant measure of $\mathcal M_k$ with the orientation of $\mathcal B_k$, so that $\int f f' \dd \nu = \mathcal B_k(f, f')$. By \eqref{e:asm-three-pieces}, the convergence just proved in opposite body weights, and \eqref{e:asm-normal-pairing}, we conclude that
\[
 \mathcal B_{\rm ad}(g_L^{\rm cut}, A_h^{N - m_W} g)
 \longrightarrow \mathcal B_{\rm nor}(\Omega_h, \Omega_h) \, \mathcal B_k\bigl( e^{-\theta \mathsf G_{X, \Gamma}} 1, e^{-(1 - \theta) \mathsf G_{X, \Gamma}} 1 \bigr)
 = \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) ,
\]
where the last equality is \zcref{d:sigma-partition}, and the pairing is finite and independent of $\theta$ by \zcref{p:sigma-regular}(d). Since every estimate above holds uniformly on compact parameter sets, so does the convergence.
\end{proof}

\subsection{Normalization}\label{s:asm-calibration}

By \zcref{c:sigma-normalization}, we have $\mathcal Z^{(k)}_{X, \eps}(\mathbf 0, \eps \1) = 1$, and \zcref{p:asm-physical,p:asm-adapted}, applied at $\Gamma = \Gamma_0$, give
\begin{equation}\label{e:asm-baseline}
 \mathcal Z_N(\Gamma_0) \longrightarrow 1 .
\end{equation}
This agrees with the exact identity $\mathcal Z_N(\Gamma_0) = 1$ of \eqref{e:rep-normalization}. The agreement also shows that $\Phi_0$ preserves the orientation of the real tube near the balanced component, as stated before \zcref{l:asm-return}. To see this, suppose that $\Phi_0$ reversed it there. Then the proofs of \zcref{l:asm-return}, \zcref{p:asm-physical} and \zcref{p:asm-adapted} would apply with the balanced term multiplied by $-1$, while the unbalanced terms would still tend to zero, and we would obtain $\mathcal Z_N(\Gamma_0) \to -1$. The proof of \zcref{t:interface} uses only that the limit is nonzero. Indeed, the source ratio \eqref{e:asm-ratio}, and its versions with local sources in \zcref{s:asm-b}, are chain integrals divided by $\mathcal Z_N(\Gamma_0)$. A constant factor $\lambda \ne 0$ in the normalizations of the adapted chain would multiply the limits of the numerator and of $\mathcal Z_N(\Gamma_0)$ alike and would cancel in every ratio. Such a factor would not depend on the sources, since the orientations, the half-weight $w_0^{1/2}$, the gauge $\gamma_h$, and the half-density $\varpi$ are built from the source-free weight.

\begin{proof}[Proof of \zcref{t:interface}(a)]
By \zcref{p:asm-physical} and \zcref{p:asm-adapted}, we have $\mathcal Z_N(\Gamma) \to \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$, uniformly on compact subsets of the set \eqref{e:asm-source-ball}, and in particular \eqref{e:asm-baseline} holds. The source supermatrix $\Gamma$ is bounded on this set, and since
\[
 N \eta_0^2 = \frac{1}{N \rho(E)^2} \longrightarrow 0 ,
\]
the prefactor in \eqref{e:asm-ratio} tends to one uniformly. Inserting these limits into \eqref{e:asm-ratio}, we obtain
\[
 T_N^{(k)}(\mathbf u, \mathbf b) \longrightarrow \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) ,
\]
uniformly on compact subsets of the set \eqref{e:asm-source-ball}, for $|E| \le E_{\rm out}$. This is part (a), with the radius $r_{k, \eps}$ of \eqref{e:asm-source-ball}.
\end{proof}

\section{Local sources: proof of part (b)}\label{s:asm-b}

We now add finitely many local sources to the chain. Let $m \ge 1$, $\delta \in (0, 1/2)$, sites $i_1 < \dots < i_m$ with $\delta \le i_a/N \le 1 - \delta$, local sources $\mathbf Y = (Y_1, \dots, Y_m)$ and multi-indices $\alpha = (\alpha_1, \dots, \alpha_m)$ be as in \zcref{t:interface}(b). We call the sites $i_a$ the marks. We label the sites so that the right block of \eqref{e:asm-one-sided} consists of the sites $N - m_W + 1, \dots, N$, and the $j$-th step after it carries the onsite weight of the site $N - m_W + 1 - j$.

Recall from \zcref{s:iface-theorem} that the local source function of \zcref{d:iface-local} is the function $T_N^{(k)}(\Gamma; \mathbf Y)$ of \zcref{l:rep-dictionary}, in which the local sources are added to $E + \iu \eta_0 \Gamma$. By \eqref{e:rep-insertion}, the local source $Y_a$ multiplies the onsite weight at the site $i_a$ by
\[
 \Sdet\bigl( \Id + (E - S)^{-1} Y_a \bigr)^{-1} ,
\]
where $S$ is the chain variable at that site. We define the complete local jet at the mark $i_a$ and the differentiated chain integral by
\begin{equation}\label{e:asm-jet}
\begin{gathered}
 w^{(\alpha_a)}_\Gamma = \partial_{Y_a}^{\alpha_a} w_{\Gamma, Y_a} \big|_{Y_a = 0} = \partial_{Y_a}^{\alpha_a} \bigl( w_\Gamma \Sdet(\Id + (E - S)^{-1} Y_a)^{-1} \bigr) \big|_{Y_a = 0} , \\
 \mathcal Z_N(\Gamma; \alpha) = \mathcal Z_N\Bigl( \Gamma; \prod_{a = 1}^m \frac{w^{(\alpha_a)}_\Gamma(S_{i_a})}{w_\Gamma(S_{i_a})} \Bigr) ,
\end{gathered}
\end{equation}
in the notation \eqref{e:rep-partition}. In words, $\mathcal Z_N(\Gamma; \alpha)$ is the chain integral with the jets $w^{(\alpha_a)}_\Gamma$ in place of the onsite weights at the marks, and the jet $w^{(\alpha_a)}_\Gamma$ contains all the derivatives at the site $i_a$. By \eqref{e:rep-exact-marks} and \eqref{e:rep-normalization}, we have
\begin{equation}\label{e:asm-local-ratio}
 \partial_{\mathbf Y}^\alpha T_N^{(k)}(\mathbf u, \mathbf b; \mathbf Y) \big|_{\mathbf Y = 0}
 = e^{N \eta_0^2 \Str(\Gamma^2)/2} \, \frac{\mathcal Z_N(\Gamma; \alpha)}{\mathcal Z_N(\Gamma_0)} .
\end{equation}

We obtain the orbit value of a jet from the expansion of $(E - S)^{-1}$ on the tube. Recall from \eqref{e:tube-map} the tubular coordinates $S = T(x)(R_* + U) T(x)^{-1}$, where $R_* = \kappa_E \Id - \iu s_E \Sigma$, and set $Q = T(x) \Sigma T(x)^{-1}$, so that $\kappa_E \Id - \iu s_E Q$ runs over the saddle manifold as $x$ varies. Recall also the orbit resolvent $\mathcal G(Q) = -\kappa_E \Id + \iu s_E Q$ of \eqref{e:iface-orbit-resolvent} and the orbit multipliers $\mathfrak j^{(\alpha_a)}$ of \eqref{e:iface-multiplier}.

\begin{proposition}\label{p:asm-jets}
Let $k$, $\eps$, and the local sources be as above.
\begin{enumerate}[label=(\alph*)]
\item On a fixed root-separated tube, there exist $C$ and $M_0$ such that
\begin{equation}\label{e:asm-projection}
 (E - S)^{-1} = -\mathcal G(Q) + \mathcal E_h(S) ,
 \qquad
 \|v^{-1} \mathcal E_h\|_{\rm coeff} \le C h \langle Z \rangle^{M_0} ,
\end{equation}
as a multiplier from a space with finitely many stronger normal moments and body weights to the required space. Consequently, the adapted jet $w^{(\alpha_a)}_{\Gamma, {\rm ad}}/w_{\Gamma, {\rm ad}}$ and the orbit multiplier $\mathfrak j^{(\alpha_a)}$ differ by $O(h v^{d_a} \langle Z \rangle^{M_a})$ between such spaces, where $d_a = |\alpha_a|$.
\item In each signature, the chain with the jets satisfies the hypotheses (H1)--(H4) of \zcref{l:asm-telescope}, and
\[
 \mathcal Z_N(\Gamma; \alpha) = \mathcal B_{\rm ad}\bigl( g_L^{\rm cut}, A_{N - m_W} \cdots A_1 g \bigr) + o(1) ,
\]
where the right side is the term of the balanced signature, $g$ is its reference endpoint from \eqref{e:asm-entrance}, and $A_j$ is the adapted step of the $j$-th step after the right block, with the adapted jet at the marks.
\item As $N \to \infty$ with $i_a/N \to x_a$, the adapted pairing in (b) converges to
\[
 \mathcal B_k \bigl( 1, \, e^{-x_1 \mathsf G_{X, \Gamma}} \mathfrak j^{(\alpha_1)} e^{-(x_2 - x_1) \mathsf G_{X, \Gamma}} \mathfrak j^{(\alpha_2)} \cdots \mathfrak j^{(\alpha_m)} e^{-(1 - x_m) \mathsf G_{X, \Gamma}} 1 \bigr) ,
\]
uniformly on compact parameter sets. This holds also when some of the differences $x_{a + 1} - x_a$ vanish.
\end{enumerate}
\end{proposition}

We prove the three parts in turn. The order of the two replacements matters. The physical comparison in (b) retains the exact jets, and the algebraic $O(h)$ error of (a) is used only afterward, inside the stable adapted chain. In this way an algebraic error is never paired with a physical partial chain, whose norm is only $e^{o(W)}$.

\begin{proof}[Proof of \zcref{p:asm-jets}(a)]
Recall that $S$ is the shifted chain variable of \zcref{s:rep}. In particular, the inverse $(E - S)^{-1}$ contains no $\eta$. We first compute it on the saddle manifold. Using $E = 2 \kappa_E$, $\Sigma^2 = \Id$, and $\kappa_E^2 + s_E^2 = 1$, we find
\[
 (E - R_*)(\kappa_E \Id - \iu s_E \Sigma)
 = (\kappa_E \Id + \iu s_E \Sigma)(\kappa_E \Id - \iu s_E \Sigma)
 = (\kappa_E^2 + s_E^2) \Id = \Id ,
\]
so that $(E - R_*)^{-1} = -\mathcal G(\Sigma)$. Conjugation by $T(x)$ gives the leading term $-\mathcal G(Q)$ in \eqref{e:asm-projection}.

We now bound the remainder. We expand the uniformly invertible matrix $E - R_* - U$ on a smaller root-separated tube. Every term of the difference $(E - R_* - U)^{-1} - (E - R_*)^{-1}$ contains $U = hZ$, and the finite nilpotent part of the inverse retains the full normal and tangent exterior algebra.

Conjugation by the odd part of $T(x)$ is bounded in the degree metric, conjugation by the BB part of $T(x)$ is bounded by $C v$, and the FF part of $T(x)$ is unitary. The normal Gaussian margin bounds $\|\langle Z \rangle^{M_0} f\|$ uniformly on the prepared families, and only normal multiplication moments are needed. We conclude that \eqref{e:asm-projection} holds.

The jet $w^{(\alpha_a)}_{\Gamma, {\rm ad}}/w_{\Gamma, {\rm ad}}$ is the derivative $\partial_{Y_a}^{\alpha_a} \Sdet(\Id + (E - S)^{-1} Y_a)^{-1}|_{Y_a = 0}$, a polynomial of degree $d_a$ in the entries of $(E - S)^{-1}$ formed before this expansion, so that the terms that couple its several derivatives are retained. Its orbit value is
\[
 \partial_{Y_a}^{\alpha_a} \Sdet\bigl( \Id - \mathcal G(Q) Y_a \bigr)^{-1} \big|_{Y_a = 0} = \mathfrak j^{(\alpha_a)}(Q) ,
\]
by \eqref{e:iface-multiplier}. The bound on the difference follows from \eqref{e:asm-projection} and the product rule.
\end{proof}

\subsection{The chain with jets}\label{s:asm-post-mark}

We now prove \zcref{p:asm-jets}(b). Recall that the physical comparison retains the exact jets. At the mark $i_a$ we set $T_{i_a} = M_{w^{(\alpha_a)}_\Gamma} K_{\tau_{\rm phys}}$, and the pullback and the gauge give an adapted step $A_{i_a} = M_{w^{(\alpha_a)}_{\Gamma, {\rm ad}}} K_{\rm ad}$ such that $M_{w^{(\alpha_a)}_\Gamma} \Can_{\rm phys} = \Can_{\rm phys} M_{w^{(\alpha_a)}_{\Gamma, {\rm ad}}}$. Since the similarity of the normal metric is conjugated into $w^{(\alpha_a)}_{\Gamma, {\rm ad}}$, a step at a mark has the same heat residual $\mathcal R$ as in \eqref{e:asm-residual}, and the jet becomes part of the left factor $F_j$ in the proof of \zcref{l:asm-telescope}. Hypothesis (H1) is \zcref{t:hist-bound} with the given local derivatives, and (H4) is verified as in the proof of \zcref{p:asm-physical}. It remains to verify (H2) and (H3) for the states of the adapted chain with the jets.

The block estimates of \zcref{s:end} hold for blocks that contain jets, with the changes stated in \eqref{e:end-jet-constant}, and every state of the chain with jets has finite prepared norm \eqref{e:end-prep-norm}, as in the chain without marks, whatever the distance between the marks. We define the adapted jet ratio at the mark $i_a$ by
\begin{equation}\label{e:asm-jet-ratio}
 J_{\alpha_a} = \frac{w^{(\alpha_a)}_{\Gamma, {\rm ad}}}{w_{\Gamma, {\rm ad}}} = \partial_{Y_a}^{\alpha_a} \Sdet\bigl( \Id + (E - S)^{-1} Y_a \bigr)^{-1} \big|_{Y_a = 0} .
\end{equation}
It does not depend on $\Gamma$, and in the raw realization the step at the mark is the jet step $S^{(\alpha_a)}$ of \eqref{e:end-jet-step}. Recall that $d_a = |\alpha_a|$, and let $d = d_1 + \dots + d_m$ denote the total order. By \zcref{l:end-kick}, we have $|J_{\alpha_a}|_1 \le C_k v^{d_a}$ at the complex points of the tube, with no growth in $\langle Z \rangle$.

\begin{lemma}\label{l:asm-jet-block}
Let $p \in \R$, and let $B$ be the product of $m_W$ consecutive adapted steps of the chain with jets, in one signature, some of which may be steps at marks.
\begin{enumerate}[label=(\alph*)]
\item\label{i:asm-jet-block-a} The bounds of \zcref{c:end-ramp}, \eqref{e:end-normal-block}, and \eqref{e:end-prep-block} hold for $B$ in place of $A_h^{m_W}$, with the same universal constants $a$, $C_n$, and $C_0$, with an additional factor $C_k^m$, and with the body weight of the input raised by $d$. In particular, $B$ maps a family with the Gaussian margin $-a$ to a family with the Gaussian margin $a$, and for every family $g$ generated from the endpoint,
\begin{equation}\label{e:asm-jet-block}
 \|B g\|_{{\rm prep}, \mathcal U_2; p} \le C_k^{1 + m} \|g\|_{\mathcal U_2; p + d, -a} .
\end{equation}
\item\label{i:asm-jet-block-b} For all $q \ge 0$, the bound \eqref{e:asm-prep-gain} holds for $B$ in place of $A_h^{m_W}$, with an additional factor $C_k^m$ and with the body weight of the input raised by $d$.
\end{enumerate}
\end{lemma}

\begin{proof}
In the raw realization of $B$, the steps at the marks are the jet steps $S^{(\alpha_a)}$ of \eqref{e:end-jet-step}, and there are at most $m$ of them. Therefore \eqref{e:end-jet-constant} bounds $B$ with the exponent of \eqref{e:end-energy}, up to a factor $C_k^m$, and with the body weight of the input raised by the total degree of these jet steps. Since $v \ge 1$, we may raise it by $d$ instead. The proofs of \zcref{c:end-ramp} and \zcref{p:end-coupled} use the steps only through \eqref{e:end-energy}, the similarity \eqref{e:end-similarity}, and the conversion \eqref{e:end-conversion}, and their universal constants $a$, $C_n$, and $C_0$ are those of \eqref{e:end-energy}. For \eqref{e:end-normal-block} and \eqref{e:end-prep-block}, this is the last assertion of \zcref{p:end-coupled}, in the graph norm \eqref{e:end-gauge-graph-norm}, as stated after \eqref{e:end-prep-block}. This proves \ref{i:asm-jet-block-a}.

For \ref{i:asm-jet-block-b}, the last step of $B$ has the multiplier $s_\Gamma$ on the left, also when it is a step at a mark, whose onsite factor is $s_\Gamma w_0 J_{\alpha_a}$. The proof of \zcref{l:asm-prep-gain} therefore applies, with \ref{i:asm-jet-block-a} in place of \eqref{e:end-prep-block}.
\end{proof}

Let $g_j$, $0 \le j \le n$, be the states of \zcref{l:asm-telescope} for the chain with jets, so that $g_0 = g_\mu$ is the image of $(\xi_\mu g_R) \otimes 1$ under the $m_W$ adapted steps of the right block, and set $g_{-m_W} = (\xi_\mu g_R) \otimes 1$ and $g_j = A_h^{m_W + j} g_{-m_W}$ for $-m_W < j < 0$, as in the proof of \zcref{p:asm-physical}. Then, for all $0 \le j \le n$,
\begin{equation}\label{e:asm-every-block}
 g_j = A_j A_{j - 1} \cdots A_{j - m_W + 1} \, g_{j - m_W} ,
\end{equation}
where $A_i$ for $i \le 0$ are the steps of the right block. Some of the $m_W$ steps in \eqref{e:asm-every-block} may be steps at marks. The right block contains no mark, since the marks are interior; in part (c) we place it at the endpoint farther from the mark.

\begin{proof}[Proof of \zcref{p:asm-jets}(b)]
It remains to verify (H2) and (H3). Every $g_j$ with $0 \le j \le n$ is a family generated from $g_R$, since \zcref{d:end-generated} includes complete local jets, and it is the output of an adapted step, hence holomorphic in the normal variables where $\psi_{R_k} \ne 0$. Let $p_0$ be the body weight of \zcref{l:end-raw-norms}. By \eqref{e:asm-every-block} and \zcref{l:asm-jet-block}\ref{i:asm-jet-block-b}, applied with the body weight $p_0 - d$ and $q = p' - p_0 + d$, we have
\[
 \|g_j\|_{{\rm prep}, \mathcal U_2; p'} \le C \, C_k^{1 + m} W^{2(p' - p_0 + d)} \|g_{j - m_W}\|_{\mathcal U_2; p_0, -a} ,
 \qquad 0 \le j \le n ,
\]
uniformly on $\mathcal U_2$ by \zcref{l:end-real-family}. For $j < m_W$, the input $g_{j - m_W} = A_h^j g_{-m_W}$ carries no jet, since the right block contains no mark, and \zcref{l:end-raw-norms} and the constant Gaussian weights of \zcref{c:end-ramp} bound its norm on the right by a power of $W$. For $j \ge m_W$, the margin $-a$ is weaker than the base norm. The adapted legs are bounded in every body weight by \zcref{t:stab-main}, and each jet maps $\mathcal H^{\rm tube}_{q + d_a}$ to $\mathcal H^{\rm tube}_q$ with norm at most $C_k$, by \zcref{l:end-kick}. Hence the base norm of $g_{j - m_W}$ in the body weight $p_0$ is at most $C \, C_k^m$ times that of $g_0$ in the body weight $p_0 + d$. This is at most $\|g_0\|_{{\rm prep}, \mathcal U_2; p_0 + d}$, which \zcref{l:asm-prep-gain} bounds by a power of $W$. This proves (H2).

For (H3), \zcref{c:end-residual} applies to every $g_j$ with $0 \le j < n$, since $\Gamma \in \mathcal U_0$ and $g_j$ has finite prepared norm by (H2). Its residual bound \eqref{e:end-residual-bound}, with the same shell gain $c_{\rm sh}$ and the original conditions \eqref{e:end-Egeom}, implies (H3) with the rate $c_{\rm sh} r_1^2$, for all states, including those immediately after a mark and those between marks that are fewer than $m_W$ steps apart. The constants depend on the number $m$ of marks and on the orders $d_a$ only through the factor $C_k^{1 + m}$, the exponent $A_{\rm sh}$, and the body weights.

We now apply \zcref{l:asm-telescope} in each signature. By \zcref{t:hist-bound} with local derivatives, the bound $e^{o(W)}$ of (H1) takes the form $W^{A'} (1 + \eta^{-1})^{A'} e^{\mathcal E_W(\eta)}$, where $A'$ denotes the exponent of that theorem. Then \eqref{e:asm-telescope} bounds the total replacement error by $C N W^{A_{\rm sh} + A} e^{-\gamma W + o(W)} = o(1)$, with $\gamma = \min(c_{\rm sh} r_1^2, \gamma_k)$. The unbalanced signatures are $O(W^A e^{-c W})$ by \zcref{l:asm-unbalanced}, since the leg before the first mark has length at least $\delta N/2$.
\end{proof}

The same argument bounds the Gaussian moments of the states at macroscopic times independently of $W$. We will use this in the proof of \zcref{p:asm-jets}(c).

\begin{lemma}\label{l:asm-incoming}
Let $g_j$, $0 \le j \le N - m_W$, be the states of the chain with jets in the balanced signature, and let $\delta' > 0$ and $p \in \R$. There exists $C_{p, a, \delta'}$, which does not depend on $W$ and depends on the marks only through their number $m$ and the orders $d_a$, such that
\begin{equation}\label{e:asm-incoming}
 \sup_{\delta' N \le j \le N - m_W} \bigl\| e^{a |Z|^2} g_j \bigr\|_{\mathcal H^{\rm tube}_p} \le C_{p, a, \delta'} .
\end{equation}
The same bound holds for the states of the chain with jets in the analytic transpose order.
\end{lemma}

\begin{proof}
Let $W$ be so large that $m_W \le \delta' N/2$, and let $\delta' N \le j \le N - m_W$. By \eqref{e:asm-every-block} and \zcref{l:asm-jet-block}\ref{i:asm-jet-block-a}, and since $e^{-a|Z|^2} \le 1$, we have
\[
 \bigl\| e^{a|Z|^2} g_j \bigr\|_{\mathcal H^{\rm tube}_p}
 \le C_k^{1 + m} \bigl\| e^{-a|Z|^2} g_{j - m_W} \bigr\|_{\mathcal H^{\rm tube}_{p + d}}
 \le C_k^{1 + m} \|g_{j - m_W}\|_{\mathcal H^{\rm tube}_{p + d}} ,
\]
with $j - m_W \ge \delta' N/2$. It suffices to bound the base norms $\|g_i\|_{\mathcal H^{\rm tube}_{p + d}}$ for $i \ge \delta' N/2$ independently of $W$.

Let $p_0$ be the body weight of \zcref{l:end-raw-norms}. In the balanced signature, \zcref{l:end-raw-norms} and \zcref{c:end-ramp}, combined as in the proof of \zcref{c:end-margin}, bound $\|e^{a|Z|^2} g_0\|_{\mathcal H^{\rm tube}_{p_0}}$ independently of $W$, and the power bounds of \zcref{t:stab-main} keep the base norm of $A_h^i g_0$ in $\mathcal H^{\rm tube}_{p_0}$ bounded for $i \le N$. The leg before the first mark has length $n_0 \ge \delta N/2$. For $\min(\delta, \delta') N/2 \le n \le N$, we have $n \delta_W \ge c$ for some $c > 0$, since $N \asymp W^2$, and \zcref{l:asm-weight-gain} bounds $A_h^n$ from $\mathcal H^{\rm tube}_{p_0}$ to $\mathcal H^{\rm tube}_{p + 2d}$ independently of $W$. This bounds $\|g_i\|_{\mathcal H^{\rm tube}_{p + 2d}}$ for $\delta' N/2 \le i \le n_0$. After the first mark, the jets map $\mathcal H^{\rm tube}_{q + d_a}$ to $\mathcal H^{\rm tube}_q$ with norm at most $C_k$, by the bound on $|J_{\alpha_a}|_1$, and the adapted legs between them are bounded in every fixed body weight by \zcref{t:stab-main}. The total loss of body weight is at most $d$, which bounds $\|g_i\|_{\mathcal H^{\rm tube}_{p + d}}$ for $i > n_0$.

For the analytic transpose order, recall that the multipliers commute and that $T_h$ is symmetric for $\mathcal B_{\rm ad}$. As in \eqref{e:asm-transpose}, the transposed chain with jets started from $g_L$ is therefore the chain with the jets in the reversed order, started from $s_\Gamma g_L$ and followed by one step $T_h$. The same argument applies, with \zcref{l:end-raw-norms} for $g_L$ and \zcref{l:cons-short-heat} for $T_h$.
\end{proof}

\begin{proof}[Proof of \zcref{p:asm-jets}(c)]
We work with the balanced pairing in (b). Recall that we use \eqref{e:asm-projection} only after the exact physical replacement of (b), inside the stable adapted chain. \zcref[S]{l:asm-incoming} supplies all the stronger body weights and normal moments with $O(1)$ norms for the finitely many $O(h)$ errors of \zcref{p:asm-jets}(a). Observe that the orbit multiplier $\mathfrak j^{(\alpha_a)}$ acts as the identity on the normal factor, so that it preserves $\Omega_h$.

The terms that a jet creates in the normal complement of $\Omega_h$ have real Hilbert norm $O(h)$, and they remain $O(h)$ under the power bounds of \zcref{t:stab-main}. We may therefore replace each jet by its orbit multiplier with an error $o(1)$.

It remains to identify the limit of the adapted chain with the orbit multipliers inserted. After the right initial layer, the state has the form $\Omega_h \otimes 1 + o(1)$ in the base norm, by \zcref{l:asm-initial-layer} with $\xi = \xi_k$. On a segment between two marks, \eqref{e:asm-segment-limit} holds, and we extend it to the inputs produced by the preceding segments. Suppose that $f_N \to f$ in a weighted base space. The power bounds imply that
\[
 \bigl\| A_h^j (\Omega_h \otimes f_N) - \Omega_h \otimes e^{-(j/N) \mathsf G_{X, \Gamma}} f \bigr\|
 \le C \|f_N - f\| + \bigl\| A_h^j (\Omega_h \otimes f) - \Omega_h \otimes e^{-(j/N) \mathsf G_{X, \Gamma}} f \bigr\| ,
\]
and both terms tend to zero uniformly for $j \le T N$.

An orbit multiplier of degree $d_a$ is bounded from $\mathcal H^{\rm tube}_{q + d_a}$ to $\mathcal H^{\rm tube}_q$, and there are finitely many marks. We start from convergence in a sufficiently strong body weight, which the leg before the first mark supplies by \zcref{l:asm-weight-gain}, and we lose $d_a$ weights at the mark $i_a$. A segment of length $o(N)$, including length one, acts as the identity in the limit, since $e^{-t \mathsf G_{X, \Gamma}} f \to f$ as $t \downarrow 0$.

Distinct sites remain separate jets, even when they are adjacent, while the complete jet at one site also contains the terms that couple its several derivatives, such as the term $\mathcal G^{\mathrm{BB}}_{\mathrm{RA}} \mathcal G^{\mathrm{BB}}_{\mathrm{AR}}$ of \zcref{r:iface-onsite}. Pairing with the left state as in the proof of \zcref{p:asm-adapted}, and using \eqref{e:asm-normal-pairing} and the symmetry of $e^{-t \mathsf G_{X, \Gamma}}$ for $\mathcal B_k$, we obtain the limit stated in \zcref{p:asm-jets}(c).
\end{proof}

\begin{proof}[Proof of \zcref{t:interface}(b)]
By \zcref{p:asm-jets}(b) and (c), the numerator $\mathcal Z_N(\Gamma; \alpha)$ converges to the pairing of semigroups and multipliers in \zcref{p:asm-jets}(c). The denominator converges to one by \eqref{e:asm-baseline}, and the prefactor tends to one, as in the proof of part (a). Inserting these limits into \eqref{e:asm-local-ratio}, we obtain \eqref{e:iface-jets} for $|E| \le E_{\rm out}$, uniformly for $(\mathbf u, \mathbf b)$ in compact subsets of \eqref{e:asm-source-ball}.
\end{proof}

\section{All-site moments: proof of part (c)}\label{s:asm-c}

We now bound the moments of the diagonal resolvent entries at all sites, including the sites near the endpoints of the chain. There is a single complete jet, at a site $i$ that may be anywhere in $\{1, \dots, N\}$, and the argument uses only the boundedness of the adapted pairing, not its limit.

\begin{proof}[Proof of \zcref{t:interface}(c)]
We first take $u = 0$ and $|E| \le E_{\rm out}$, and we fix $p \ge 1$. We take $k = 1$, $\mathbf u = 0$, and $\mathbf b = \eps$, so that $Z_{\rm F} = Z_{\rm B}$ in \zcref{d:iface-local}, and the local source $Y = \diag(Y^{\rm B} \mid 0)$ at the site $i$, with $Y^{\rm B} = \diag(y, y')$. Using
\[
 \det(z - H + y e_i e_i^*) = \det(z - H) \bigl( 1 - y G_{ii}(z) \bigr) ,
 \qquad
 G_{ii}(\bar z) = \overline{G_{ii}(z)} ,
\]
we obtain
\begin{equation}\label{e:asm-moment-identity}
\begin{aligned}
 \partial_y^p \partial_{y'}^p T_N^{(1)}(0, \eps; Y) \big|_{y = y' = 0}
 &= \partial_y^p \partial_{y'}^p \E \bigl[ (1 - y G_{ii}(z^{\rm B}_+))^{-1} (1 - y' G_{ii}(z^{\rm B}_-))^{-1} \bigr] \big|_{y = y' = 0} \\
 &= (p!)^2 \, \E |G_{ii}(E + \iu \eta)|^{2p} .
\end{aligned}
\end{equation}

We now bound the left side of \eqref{e:asm-moment-identity}. By the reflection $i \mapsto N + 1 - i$ of the chain, we may assume that $i \le (N + 1)/2$, and we write the chain in the one-sided form \eqref{e:asm-one-sided} with its right block at the farther endpoint. The leg from the right block to the site $i$ then carries no local source. Its length is at least $(N - 1)/2 - m_W$. By \zcref{l:end-real-family}, the signed preparation, the BB damping on this leg of macroscopic length and one final block of $m_W$ steps put the state arriving at $i$ in all prescribed positive body weights with a positive Gaussian margin, with an $O(1)$ norm, uniformly on $\mathcal U_2$. After this leg, the unbalanced signatures are $O(W^A e^{-c W})$, by \zcref{l:asm-unbalanced}.

The complete jet of \eqref{e:asm-moment-identity} has the multiplier bound $|J_\alpha|_1 \le C_p v^{2p}$ of \zcref{l:end-kick}, and the incoming state supplies these body weights. After the jet, adapted propagation over an arbitrary number of steps has bounded base norms, by \zcref{t:stab-main}. Every state after the jet, including the state immediately after it, is the image of the $m_W$ adapted steps that precede it, one of which may be the jet step, as in \eqref{e:asm-every-block}. \zcref[S]{l:asm-jet-block} therefore implies that it has the positive Gaussian margin $a$, with a norm bounded independently of $W$.

By \zcref{p:asm-jets}(b), with (H1)--(H4) verified as there for this single jet, the differentiated chain integral differs from $\mathcal B_{\rm ad}(g_L^{\rm cut}, g_{\rm out})$ by $o(1)$, where $g_{\rm out}$ denotes the adapted state after the last step. \zcref[S]{l:asm-terminal} implies that
\[
 |\mathcal B_{\rm ad}(g_L^{\rm cut}, g_{\rm out})| \le C \|g_{\rm out}\|_{\mathcal U_2; \bar p, a} \le C' ,
\]
where $\bar p$ denotes the body weight of \zcref{l:asm-terminal}, since $p$ is the order of the moment here.
For $i = 1$ there is no step after the jet, and the left factor of the last residual term in the proof of \zcref{l:asm-telescope} is the jet times the constant function. This is a partial chain of length one with a local derivative, which \zcref{t:hist-bound} covers.

By \eqref{e:asm-local-ratio}, the left side of \eqref{e:asm-moment-identity} is the differentiated chain integral divided by $\mathcal Z_N(\Gamma_0)$, at $\Gamma = \Gamma_0$, where the prefactor is one. Since $\mathcal Z_N(\Gamma_0) \to 1$ by \eqref{e:asm-baseline}, we conclude that $(p!)^2 \E |G_{ii}(E + \iu \eta)|^{2p} = O(1)$, uniformly in $i$. This is part (c) for $u = 0$, for all $|E| \le E_{\rm out}$.

The bound is uniform in the critical regime of \zcref{d:intro-critical}, with $E_{\rm out}$ in place of $E_*$. Fix $0 < X_- \le X_+ < \infty$ and $\eps$ in a compact subset of $(0, \infty)$. If the bound failed uniformly over all $(N, W, E)$ with $|E| \le E_{\rm out}$ and $X_- \le X_N(E) \le X_+$, there would be sequences of such parameters, of dampings, and of sites $i$ along which the left side of \eqref{e:asm-moment-identity} tends to infinity. Along a subsequence, $X_N(E)$ and $\eps$ converge, and $N \asymp W^2$. The argument above applies along this subsequence, with the energy depending on $N$ as \zcref{t:interface} allows, and shows that the left side stays bounded, which is a contradiction.

Now let $u_0 > 0$ be the bound on $|u|$ in \zcref{t:interface}(c), let $|u| \le u_0$ and $|E| \le E_*$, and let $E'$ and $\eps'$ be as in \eqref{e:asm-recenter}. By \zcref{l:asm-recenter}, we have
\[
 G_{ii}\bigl( E + (u + \iu \eps)/a_N(E) \bigr) = G_{ii}\bigl( E' + \iu \eps'/a_N(E') \bigr) ,
\]
and for $N \ge N_0$ we have $|E'| \le E_{\rm out}$, while $\eps'$ and $X_N(E')$ lie in slightly larger compact ranges, by \eqref{e:asm-recenter-bounds}. The case $u = 0$ at the energy $E'$ with the damping $\eps'$, which holds uniformly over these ranges, therefore implies \eqref{e:iface-allsite}, uniformly in the critical regime.
\end{proof}

\section{One-point normalization: proof of part (d)}\label{s:asm-d}

The last part of \zcref{t:interface} identifies the mean of the diagonal resolvent. It is the case of one site and one first derivative of part (b), evaluated by the localization lemma \zcref{l:sigma-localization}.

\begin{proof}[Proof of \zcref{t:interface}(d)]
We first take $u = 0$ and $|E| \le E_{\rm out}$. We take $k = 1$, $\mathbf u = 0$, and $\mathbf b = \eps$, and the local source $Y = \diag(0 \mid t E_{\rm RR})$ at the site $i$, where $E_{\rm RR}$ is the matrix unit of the retarded flavor. By \eqref{e:iface-examples}, we have
\[
 -\partial_t T_N^{(1)}(0, \eps; Y) \big|_{t = 0} = \E G_{ii}(E + \iu \eta) .
\]
The supermatrix $\Id - \mathcal G Y$ is block upper triangular with FF block $\Id - t \mathcal G^{\rm FF} E_{\rm RR}$, so that $\Sdet(\Id - \mathcal G Y)^{-1} = \det(\Id - t \mathcal G^{\rm FF} E_{\rm RR})$, and the corresponding orbit multiplier satisfies
\[
 -\partial_t \Sdet\bigl( \Id - \mathcal G(Q) Y \bigr)^{-1} \big|_{t = 0} = \mathcal G(Q)^{\rm FF}_{\rm RR} = -\kappa_E + \iu s_E Q^{\rm FF}_{\rm RR} .
\]
Let $i/N \to x \in [\delta, 1 - \delta]$, and write $\mathsf G = \mathsf G_{X, \Gamma_0}$ at rank one. By part (b) with $m = 1$, applied to $-\partial_t$, we have
\[
 \E G_{ii}(E + \iu \eta)
 = -\kappa_E \, \mathcal B_1\bigl( 1, e^{-\mathsf G} 1 \bigr)
 + \iu s_E \, \mathcal B_1\bigl( 1, e^{-x \mathsf G} Q^{\rm FF}_{\rm RR} e^{-(1 - x) \mathsf G} 1 \bigr) + o(1) .
\]
By \zcref{c:sigma-normalization} for the first pairing, and by the symmetry of $e^{-x \mathsf G}$ for $\mathcal B_1$ in \zcref{p:sigma-regular}(c) for the second, we have
\[
\begin{gathered}
 \mathcal B_1\bigl( 1, e^{-\mathsf G} 1 \bigr) = \mathcal Z^{(1)}_{X, \eps}(0, \eps) = 1 , \\
 \mathcal B_1\bigl( 1, e^{-x \mathsf G} Q^{\rm FF}_{\rm RR} e^{-(1 - x) \mathsf G} 1 \bigr) = \mathcal B_1\bigl( e^{-x \mathsf G} 1, Q^{\rm FF}_{\rm RR} e^{-(1 - x) \mathsf G} 1 \bigr) .
\end{gathered}
\]

We now evaluate the second pairing by localization. The two factors $e^{-x \mathsf G} 1$ and $e^{-(1 - x) \mathsf G} 1$ are invariant under $\mathfrak k$, by \zcref{c:sigma-normalization}, and $Q_{\rm RR}$ is advanced-invariant, by \eqref{e:sigma-action}. The integrand $(e^{-x \mathsf G} 1) \, Q^{\rm FF}_{\rm RR} \, (e^{-(1 - x) \mathsf G} 1)$ is therefore invariant under $\mathfrak k_{\rm A}$, hence $\mathfrak q_{\rm A}$-closed by \zcref{l:sigma-supercharge}, and its coefficients are polynomially integrable by \zcref{p:sigma-regular}. By \zcref{l:sigma-localization}, its integral is the value of its body at $Q = \Lambda$, where $Q^{\rm FF}_{\rm RR} = 1$. The two factors also have the body value one at $\Lambda$. Indeed, for $t \in \{x, 1 - x\}$, the superfunction $e^{-t \mathsf G} 1$ is invariant under $\mathfrak k$, hence $\mathfrak q_{\rm A}$-closed, and has polynomially integrable coefficients, by the same results. \zcref[S]{l:sigma-localization}, applied with $A = e^{-t \mathsf G} 1$, and \zcref{c:sigma-normalization}, applied with $z = 2 \pi \eps$ and $T = t$, since $\mathsf G = \mathsf G_{X, 2 \pi \eps}$, therefore give
\[
 \bigl( e^{-t \mathsf G} 1 \bigr)_\emptyset(\Lambda) = \int e^{-t \mathsf G} 1 = 1 .
\]
Therefore the second pairing is one, and
\[
 \E G_{ii}(E + \iu \eta) - (-\kappa_E + \iu s_E) \longrightarrow 0 .
\]

This convergence is uniform for $\delta \le i/N \le 1 - \delta$, for $\eps$ in compact subsets of $(0, \infty)$, and over all $(N, W, E)$ with $|E| \le E_{\rm out}$ and $X_- \le X_N(E) \le X_+$, for every $0 < X_- \le X_+ < \infty$. Indeed, otherwise there would exist $c > 0$ and sequences of sites and parameters along which the difference stays above $c$. Along a subsequence of these, $i/N$, $\eps$, and $X_N(E)$ converge, and part (b) applies along it, since the energy may depend on $N$ there. This contradicts the convergence just proved.

Now let $u_0 > 0$ be the bound on $|u|$ in \zcref{t:interface}(d), let $|u| \le u_0$ and $|E| \le E_*$, and let $E'$ and $\eps'$ be as in \eqref{e:asm-recenter}. By \zcref{l:asm-recenter} and the case $u = 0$ at the energy $E'$, we obtain
\[
 \E G_{ii}\bigl( E + (u + \iu \eps)/a_N \bigr) = \E G_{ii}\bigl( E' + \iu \eps'/a_N(E') \bigr) = -\kappa_{E'} + \iu s_{E'} + o(1) ,
\]
uniformly for $\delta \le i/N \le 1 - \delta$ and $|u| \le u_0$, and uniformly in the critical regime of \zcref{d:intro-critical}, since $|E'| \le E_{\rm out}$ and, by \eqref{e:asm-recenter-bounds}, $X_N(E')$ and $\eps'$ lie in slightly larger compact ranges. Since $|E' - E| \le C/N$, we have $-\kappa_{E'} + \iu s_{E'} = -\kappa_E + \iu s_E + o(1)$, which is the first limit in \eqref{e:iface-onepoint}.

Recall that $\ell_{i, N}^\eps(u) = s_E^{-1} \Im G_{ii}(E + (u + \iu \eps)/a_N)$. Taking imaginary parts, we conclude that $\E \ell_{i, N}^\eps(u) \to 1$. This is part (d), and the proof of \zcref{t:interface} is complete.
\end{proof}