\chapter{A priori bounds on transfer chains}\label{s:hist}

Part IV proves \zcref{t:interface}, which is the only result of Part IV that Part III uses. The proof compares the exact transfer chain of \zcref{s:rep} with the continuum semigroup of the sigma model of \zcref{s:sigma}. Recall from \zcref{p:rep-exact} that the source ratio $T_N^{(k)}$ is, up to an explicit prefactor, the pairing of two endpoints with a power of the transfer operator $T_\Gamma$, which acts on superfunctions on the Sch\"afer--Wegner cycle. In this section we prove a priori bounds on the partial chains $T_\Gamma^j 1$. \zcref{s:tube} introduces the normal tube around the saddle manifold and the normal oscillator, and \zcref{s:stab} proves that the transfer kernel is stable on this tube. In \zcref{s:cons} we prove that the transfer operator agrees with the continuum generator to first order on the invariant normal line, and in \zcref{s:end} we attach the endpoints of the chain to the tube. Finally, \zcref{s:assembly} combines these results into the proof of \zcref{t:interface}. The heuristics behind this strategy are explained in \zcref{s:overview}.

The main result of this section is \zcref{t:hist-bound}. It states that the partial chains $T_\Gamma^j 1$, for $1 \le j \le N$, and their versions with finitely many local source derivatives, have norm $e^{o(W)}$ in a space with a negative polynomial weight. We now explain why a bound of this order is needed. In \zcref{s:assembly}, the physical chain is compared with a chain on the normal tube. Every local error in this comparison is of order $e^{-\gamma W}$ for some $\gamma > 0$, and the telescoping identity of \zcref{l:asm-telescope} pairs it with a partial chain. The errors are therefore harmless only if the partial chains are of size $e^{o(W)}$. One step satisfies the trivial bound $\|T_\Gamma\| \le e^{C/W}$, by \zcref{l:end-kinetic,l:end-onsite}. Over the $N \asymp W^2$ steps of the chain, this bound grows to $e^{C W}$, which is of the same exponential order as the error.

We obtain the bound $e^{o(W)}$ from two estimates of a different nature. Away from the saddle manifold, one step of the chain contracts the norm by a factor $e^{-c (\log W)^2/W}$, by the onsite bound \zcref{l:end-onsite}. Near the saddle manifold, on cores of radius $h \log W$, we use the anchored identity of \zcref{p:rep-anchored}. It writes a partial chain at a point of the cycle as the expectation, over a Gaussian band matrix, of a ratio of a fermionic and a bosonic determinant. We compare the two determinants by Hua's determinant inequality, and we take the expectation by spectral averaging over the diagonal entries of the matrix, in the form of Minami's estimate due to Graf and Vaghi. These tools yield a bound $e^{C \sqrt W (\log W)^2}$ on the cores, and a recurrence combines the two estimates. Since the fermionic determinants can vanish, we only bound them from above and never divide by them.

In \zcref{s:hist-setting} we fix the coordinates on the cycle and the weighted spaces, and in \zcref{s:hist-det} we prove the determinant inequalities. \zcref{s:hist-step} contains the two bounds for one step on the cycle, the kinetic bound \zcref{l:end-kinetic} and the onsite bound \zcref{l:end-onsite}. \zcref{s:hist-averaging,s:hist-coefficients} contain the spectral averaging and the coefficient bounds that we use on the cores, and \zcref{s:hist-bound} proves \zcref{t:hist-bound}.

\section{Setting}\label{s:hist-setting}

Throughout Part IV, the rank $k \ge 1$ is fixed, and we write
\begin{equation}\label{e:hist-scales}
 h = W^{-1/2}, \qquad \delta_W = W^{-2} = h^4, \qquad \tau_{\rm phys} = (2W^2)^{-1}, \qquad r_W = h \log W.
\end{equation}
In this section we assume that $|E| \le 1/2$, so that $|\kappa_E| \le 1/4$ and $s_E \ge \sqrt{15}/4$, and we abbreviate $\kappa = \kappa_E$ and $s = s_E$. Let $\| \cdot \|$ denote the operator norm of a matrix and $\| \cdot \|_{\rm HS}$ its Hilbert--Schmidt norm. In \zcref{s:hist-step}, and only there, we use instead the convention of \zcref{s:rep}, in which $\| \cdot \|$ denotes the Hilbert--Schmidt norm and $\| \cdot \|_{\rm op}$ the operator norm. Constants $C, c > 0$ may depend on $k$ and on the parameters fixed in the statements, but not on $N$, $W$, $E$, $\eta$, the point of the cycle, or the positions of the marks defined below. Their values may change from line to line.

\paragraph{The cycle and its coordinates.}
Recall the Sch\"afer--Wegner cycle $\mathcal S$ and its coordinates $(P, X, M)$ from \zcref{d:rep-cycle}. We set $P_0 = P - \kappa \Id$, so that the BB and FF blocks of a point $S \in \mathcal S$ are
\begin{equation}\label{e:hist-blocks}
 S_{BB} = \kappa \Id + P_0 - \iu s \Lambda_B \mathsf g^2, \qquad S_{FF} = \kappa \Id + \iu M,
\end{equation}
where $\mathsf g = \mathsf g_X$ is the positive definite matrix of \eqref{e:rep-f0}, which we call the \emph{boost} of $S$. The odd blocks $S_{BF}$ and $S_{FB}$ of $S$ have free odd generators as entries. Recall from \zcref{l:rep-cycle}(a) that $\mathsf g \Lambda_B \mathsf g = \Lambda_B$ and $\det \mathsf g = 1$. We let $v_{\mathcal S} = 1 + \Tr f_0(X)$ denote the \emph{orbit height} of $S$. It is the counterpart on $\mathcal S$ of the weight $v$ of \eqref{e:sigma-weight}: at the point of $\mathcal S$ with $P = \kappa \Id$, \zcref{l:rep-kick} gives $v_{\mathcal S} = 1 + 2s(v(Q_X) - 1)$, with $Q_X$ as in \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle}. Since $\|X\| \le \|f_0(X)\| \le \Tr f_0(X)$ and $\Lambda_B \mathsf g^2 \Lambda_B = \mathsf g^{-2}$, we have
\begin{equation}\label{e:hist-boost-props}
 \|\mathsf g^2\| = \|\mathsf g^{-2}\| \le 2 s^{-1} v_{\mathcal S} \le 3 v_{\mathcal S}, \qquad \Tr \mathsf g^2 = s^{-1} \Tr f_0(X) = s^{-1}(v_{\mathcal S} - 1),
\end{equation}
where we used $\Tr X = 0$ in the second identity.

We define the \emph{transported odd coordinates} of $S$ to be the entries of the $2k \times 2k$ matrices
\begin{equation}\label{e:hist-transported}
 O_1 = \mathsf g^{-1} \Lambda_B S_{BF}, \qquad O_2 = S_{FB} \, \mathsf g^{-1}.
\end{equation}
At each point of the underlying manifold of $\mathcal S$, these $8k^2$ entries are odd generators, obtained from the entries of $S_{BF}$ and $S_{FB}$ by an invertible linear map, and we call them the \emph{transported odd frame}. For a superfunction $f$ on $\mathcal S$, let $|f|_{1; h}$ and $|f|_{2; h}$ denote the coefficient norms of \zcref{d:grass-norm} with the weight $h$, taken at each point of the body in the monomials of the transported odd coordinates. For a matrix with such entries, $\| \cdot \|_{1; h}$ is the matrix norm of \zcref{d:grass-norm}.

\paragraph{Weighted spaces.}
We parametrize the underlying manifold of $\mathcal S$ by $(P_0, X, M)$, and we let $\dd P_0 \dd X \dd M$ denote Lebesgue measure in these coordinates, with the Hilbert--Schmidt metric. For $p \in \R$, we define $\mathcal H_p(\mathcal S)$ to be the space of superfunctions $f$ on $\mathcal S$ with measurable coefficients such that
\begin{equation}\label{e:hist-Hp}
 \|f\|_{\mathcal H_p(\mathcal S)}^2 = \int v_{\mathcal S}^{2p} \, |f|_{2; h}^2 \dd P_0 \dd X \dd M < \infty.
\end{equation}
By \zcref{l:grass-norm}(a), we have $|a f|_{2; h} \le |a|_{1; h} |f|_{2; h}$, and therefore multiplication by a superfunction $a$ has norm at most $\sup |a|_{1; h}$ on $\mathcal H_p(\mathcal S)$. We will also need the following bound for the pairing $\mathcal B_{\mathcal S}$ of \zcref{d:rep-chain}, which is used in \zcref{s:assembly}.

\begin{lemma}\label{l:hist-pairing}
For all $p \in \R$, $f \in \mathcal H_{-p}(\mathcal S)$, and $u \in \mathcal H_p(\mathcal S)$, we have
\[
 |\mathcal B_{\mathcal S}(f, u)| \le C_k W^{4k^2} \|f\|_{\mathcal H_{-p}(\mathcal S)} \|u\|_{\mathcal H_p(\mathcal S)},
\]
where $C_k$ depends only on $k$.
\end{lemma}

\begin{proof}
By \zcref{d:rep-cycle} and the normalization \eqref{e:rep-flat-normalization}, $\mathcal B_{\mathcal S}(f, u)$ is a constant $c_k$ times the integral over the body, with respect to $\dd P_0 \dd X \dd M$, of the coefficient of the top monomial of $fu$ in the entries of $S_{BF}$ and $S_{FB}$. The map \eqref{e:hist-transported} from these entries to the transported odd coordinates has determinant
\[
 \det(\mathsf g^{-1} \Lambda_B)^{2k} \det(\mathsf g^{-1})^{2k} = 1,
\]
so by \zcref{l:grass-berezin}(c) the top coefficient is the same in both frames. In the $h$-scaled monomials, the top coefficient of $fu$ is the product of $h^{-8k^2} = W^{4k^2}$ with a signed sum, over complementary monomials, of products of a coefficient of $f$ and a coefficient of $u$. The Cauchy--Schwarz inequality bounds this sum by $|f|_{2; h} |u|_{2; h}$, and we obtain
\[
 |\mathcal B_{\mathcal S}(f, u)| \le |c_k| W^{4k^2} \int v_{\mathcal S}^{-p} |f|_{2; h} \, v_{\mathcal S}^p |u|_{2; h} \dd P_0 \dd X \dd M \le |c_k| W^{4k^2} \|f\|_{\mathcal H_{-p}(\mathcal S)} \|u\|_{\mathcal H_p(\mathcal S)}. \qedhere
\]
\end{proof}

\paragraph{Sources and marks.}
Recall from \zcref{d:rep-sources,d:rep-chain} the source supermatrix $\Gamma = \diag(\eps \Lambda_B \,|\, \Gamma_F)$, the onsite weight $w_\Gamma$, the marked onsite weight $w_{\Gamma, Y}$, and the transfer operator $T_\Gamma = M_{w_\Gamma} K_{\tau_{\rm phys}}$, and recall that $\eta = \eps \eta_0$. These objects depend on $\eta_0$ and $\Gamma$ only through $\eta_0 \Gamma$. We define the \emph{normalized source} $\hat \Gamma = \eps^{-1} \Gamma$, so that
\begin{equation}\label{e:hist-normalized}
 \eta_0 \Gamma = \eta \hat \Gamma, \qquad \hat \Gamma = \diag(\Lambda_B \,|\, \hat \Gamma_F), \qquad
 w_\Gamma(S) = e^{-\Str(S^2)/2 - \iu \eta \Str(S \hat \Gamma)} \, \Sdet(E - S)^{-1}.
\end{equation}
In this section, $\eta > 0$ is arbitrary, and $\hat \Gamma_F$ is a complex $2k \times 2k$ matrix with $\Tr \hat \Gamma_F = 0$ and $\|\hat \Gamma_F\| \le C_\Gamma$. The sources $\Gamma(\mathbf u, \mathbf b)$ of \zcref{d:rep-sources} are of this form, also after complex continuation of $(\mathbf u, \mathbf b)$, since their FF block is traceless. We write the FF spectral matrix as
\begin{equation}\label{e:hist-ff-source}
 \iu \hat \Gamma_F = \gamma_1 + \iu \gamma_2, \qquad \gamma_1 = \gamma_1^*, \quad \gamma_2 = \gamma_2^*, \qquad \Tr \gamma_1 = \Tr \gamma_2 = 0, \qquad \|\gamma_1\|, \|\gamma_2\| \le C_\Gamma.
\end{equation}

Recall that a local source at one site is a supermatrix $Y = \diag(Y^B \,|\, Y^F)$ with zero odd blocks, and that $w_{\Gamma, Y}(S)$ is obtained from $w_\Gamma(S)$ by replacing $\Sdet(E - S)^{-1}$ with $\Sdet(E - S + Y)^{-1}$. A jet of local source derivatives at one site gives the following onsite weights.

\begin{definition}\label{d:hist-marks}
Let $Y_1, \dots, Y_m$ be local sources with $\|Y_l^B\|, \|Y_l^F\| \le 1$. The \emph{onsite jet} of order $m$ with these directions, a mixed derivative of the marked onsite weight, is
\[
 w_\Gamma[Y_1, \dots, Y_m](S) = \frac{\partial^m}{\partial t_1 \cdots \partial t_m} \, w_{\Gamma, t_1 Y_1 + \dots + t_m Y_m}(S) \Big|_{t = 0},
\]
and for $m = 0$ it is $w_\Gamma$. A \emph{marked partial chain} of length $j \ge 1$ and total order $d$ is a superfunction
\begin{equation}\label{e:hist-marked-chain}
 F_j = M_{w_j} K_{\tau_{\rm phys}} M_{w_{j - 1}} K_{\tau_{\rm phys}} \cdots M_{w_2} K_{\tau_{\rm phys}} w_1,
\end{equation}
where each $w_i$ is an onsite jet of some order $m_i$, and $m_1 + \dots + m_j = d$. We call the steps $i$ with $m_i \ge 1$ the \emph{marked steps}, and we set $F_0 = 1$.
\end{definition}

The derivatives at zero of $w_{\Gamma, Y}$ are defined at every point of $\mathcal S$, as noted after \zcref{d:rep-chain}, and every derivative of finite order in the entries of $Y$ is a linear combination of onsite jets. Since $K_{\tau_{\rm phys}} 1 = 1$, the unmarked partial chain of length $j$ satisfies $F_j = T_\Gamma^j 1$. By \zcref{p:rep-exact}(c), the derivatives of the source ratio in local sources are chain integrals of onsite jets, and they are pairings of marked partial chains. In the analytic transpose order of \zcref{d:rep-chain}, the marked partial chains are the functions $K_{\tau_{\rm phys}} F_j$, just as $\ell_m = K_{\tau_{\rm phys}} f_m$ in \zcref{p:rep-exact}(b).

\paragraph{The anchored identity.}
For $n \ge 1$, recall the anchored covariance $J_n$, the boundary profile $\varsigma \in (0, 1)^n$, the number $\sigma_n$, and the Gaussian matrix $H^{(n)}$ of \eqref{e:rep-anchored-cov}. We also write $\varsigma$ for the diagonal matrix $\diag(\varsigma_1, \dots, \varsigma_n)$. For $S \in \mathcal S$, we define the supermatrix $\mathbb M_n(S)$ on $\C^n \otimes \C^{2k|2k}$ and the Gaussian factor $\mathcal G_n(S)$ by
\begin{align}
 \mathbb M_n(S) &= \Id_n \otimes E - H^{(n)} \otimes \Id - \varsigma \otimes S + \iu \eta (\Id_n - \varsigma) \otimes \hat \Gamma, \label{e:hist-Mn}\\
 \mathcal G_n(S) &= \exp\Bigl[ - \frac{\sigma_n}{2} \Str(S^2) - \iu \eta \sigma_n \Str(S \hat \Gamma) - \frac{\eta^2 (n - \sigma_n)}{2} \Str(\hat \Gamma^2) \Bigr]. \label{e:hist-Gn}
\end{align}
By \eqref{e:hist-normalized}, \zcref{p:rep-anchored} states that
\begin{equation}\label{e:hist-anchored}
 (K_{\tau_{\rm phys}} T_\Gamma^n 1)(S) = \mathcal G_n(S) \, \E \Sdet\bigl[\mathbb M_n(S)\bigr]^{-1}.
\end{equation}
Let $\mathbf B$, $\mathbf F$, $\mathbf O_{BF}$, and $\mathbf O_{FB}$ denote the blocks of $\mathbb M_n(S)$. As in \zcref{d:rep-chain}, and by \zcref{l:grass-sdet}(a), the inverse superdeterminant is
\[
 \Sdet\bigl[\mathbb M_n(S)\bigr]^{-1} = \frac{\det(\mathbf F - \mathbf O_{FB} \mathbf B^{-1} \mathbf O_{BF})}{\det \mathbf B},
\]
which only requires $\mathbf B$ to be invertible; we check this in \zcref{l:hist-bb}. In the anchored chain, the site $i$ is linked to $S$ through $i - 1$ other sites, so it carries the step $n + 1 - i$ of the chain.

\section{Determinant inequalities}\label{s:hist-det}

We now prove the determinant inequalities that compare the bosonic and fermionic blocks of the anchored identity. We begin with elementary bounds for logarithms of determinants, which we will use several times.

\begin{lemma}\label{l:hist-logdet}
Let $A$ be a Hermitian and $Y$ a positive definite matrix of the same size, and set $R = (A + \iu Y)^{-1}$ and $\Im R = (R - R^*)/(2\iu)$. Then $A + \iu Y$ is invertible, and
\begin{equation}\label{e:hist-resolvent}
 \|R\| \le \|Y^{-1}\|, \qquad 0 \le -\Im R = R Y R^* \le Y^{-1}.
\end{equation}
For all positive semidefinite $Z$ of the same size,
\begin{equation}\label{e:hist-height}
 \frac{\dd}{\dd t} \log |\det(A + \iu (Y + t Z))| \Big|_{t = 0} = \Tr\bigl(Z (-\Im R)\bigr) \in \bigl[0, \Tr(Z Y^{-1})\bigr].
\end{equation}
In particular, $Y \mapsto |\det(A + \iu Y)|$ is nondecreasing for the order of positive definite matrices. Moreover, for all Hermitian $Z$ of the same size, the function $\ell(u) = \log |\det(A + u Z + \iu Y)|$ of $u \in \R$ satisfies
\begin{equation}\label{e:hist-shift}
 |\ell'(0)| \le \|Z\| \Tr(Y^{-1}), \qquad |\ell''(0)| \le \|Z\|^2 \|Y^{-1}\| \Tr(-\Im R).
\end{equation}
\end{lemma}

\begin{proof}
Set $A' = Y^{-1/2} A Y^{-1/2}$. Then
\[
 A + \iu Y = Y^{1/2} (A' + \iu \Id) Y^{1/2}, \qquad \|(A' + \iu \Id)^{-1}\| \le 1,
\]
where the second bound holds since $A'$ is Hermitian. The invertibility and the bound $\|R\| \le \|Y^{-1}\|$ follow. Further,
\[
\begin{gathered}
 R - R^* = R \bigl((A - \iu Y) - (A + \iu Y)\bigr) R^* = -2\iu R Y R^*, \\
 Y^{1/2} R Y R^* Y^{1/2} = (A'^2 + \Id)^{-1} \le \Id,
\end{gathered}
\]
and \eqref{e:hist-resolvent} follows.

The derivative in \eqref{e:hist-height} is
\[
 \Re \Tr(\iu R Z) = -\Tr(Z \Im R) = \Tr\bigl(Z^{1/2} (-\Im R) Z^{1/2}\bigr),
\]
and \eqref{e:hist-resolvent} places it in the stated interval. The monotonicity follows by integrating \eqref{e:hist-height} along the segment between two ordered positive definite matrices.

We now prove \eqref{e:hist-shift}. Set $Z' = Y^{-1/2} Z Y^{-1/2}$. We have
\[
 \ell'(0) = \Re \Tr(R Z), \qquad \ell''(0) = -\Re \Tr(R Z R Z), \qquad -\|Z\| Y^{-1} \le Z' \le \|Z\| Y^{-1},
\]
where the last bound follows by conjugating $-\|Z\| \Id \le Z \le \|Z\| \Id$ with $Y^{-1/2}$. Let $\Pi_\pm$ be the spectral projections of $Z'$ onto its positive and negative eigenspaces. Then
\[
\begin{aligned}
 |\Tr(R Z)| = \bigl|\Tr\bigl((A' + \iu \Id)^{-1} Z'\bigr)\bigr| \le \Tr |Z'| &= \Tr(\Pi_+ Z' \Pi_+) - \Tr(\Pi_- Z' \Pi_-) \\
 &\le \|Z\| \Tr(Y^{-1}).
\end{aligned}
\]

For the second derivative, the Cauchy--Schwarz inequality for the Hilbert--Schmidt inner product gives
\[
 |\Tr(R Z R Z)| \le \Tr(Z R^* R Z) = \Tr(R Z^2 R^*) \le \|Z\|^2 \Tr(R R^*) \le \|Z\|^2 \|Y^{-1}\| \Tr(-\Im R),
\]
where in the last step we used $R R^* \le \|Y^{-1}\| R Y R^*$, which follows from $Y \ge \|Y^{-1}\|^{-1} \Id$.
\end{proof}

The next lemma is the main determinant inequality. Let $\Lambda'$ be a Hermitian involution (a Hermitian matrix with $\Lambda'^2 = \Id$, such as $\Lambda_B$). The first part of the lemma says that, if $L$ is Hermitian and commutes with $\Lambda'$, and if $Y$ is positive definite with $\Lambda' Y \Lambda' = Y^{-1}$, then replacing $Y$ by the identity does not increase the modulus of $\det(L + \iu Y)$. The proof rests on Hua's determinant inequality \eqref{e:hist-hua-inequality} below, which Hua derived from a matrix identity \cite{hua1955inequalities}. Paige, Styan, Wang, and Zhang state the identity and the inequality as \cite[(1), (4)]{paige2008hua}, and they locate both in the proof of Theorem~1 of \cite{hua1955inequalities}. Since Hua's paper is not easily available, we include the short derivation, following \cite{paige2008hua}.

\begin{lemma}\label{l:hist-hua}
The following statements hold.
\begin{enumerate}[label=(\alph*)]
\item Let $\Lambda'$ be a Hermitian involution, let $Y$ be positive definite with $\Lambda' Y \Lambda' = Y^{-1}$, and let $L$ be Hermitian with $L \Lambda' = \Lambda' L$, all of the same size. Then
\[
 |\det(L + \iu Y)| \ge |\det(L + \iu \Id)|.
\]
\item Let $\mathsf g$ be a positive definite $2k \times 2k$ matrix with $\mathsf g \Lambda_B \mathsf g = \Lambda_B$, let $Y_0$ be a positive semidefinite $n \times n$ matrix, let $s_0 > 0$, and let $L$ be a Hermitian matrix on $\C^n \otimes \C^{2k}$ that commutes with $\Id_n \otimes \Lambda_B$. Then
\[
 \bigl|\det\bigl(L + \iu (Y_0 \otimes \Id + s_0 \Id_n \otimes \mathsf g^2)\bigr)\bigr| \ge \bigl|\det\bigl(L + \iu (Y_0 + s_0 \Id_n) \otimes \Id\bigr)\bigr|.
\]
\end{enumerate}
\end{lemma}

\begin{proof}
We first recall Hua's inequality. Let $A$ and $B$ be complex $d_1 \times d_2$ matrices with $\|A\|, \|B\| < 1$. Hua's matrix identity \cite[(1)]{paige2008hua}, with $A$ and $B$ replaced by $B^*$ and $A^*$, states that
\[
 (\Id - A B^*)(\Id - B B^*)^{-1}(\Id - B A^*) - (\Id - A A^*) = (A - B)(\Id - B^* B)^{-1}(A - B)^*.
\]
One checks it by multiplying out, using
\[
 (\Id - B B^*)^{-1} = \Id + B(\Id - B^* B)^{-1} B^*.
\]

Since the right side is positive semidefinite, the first term on the left is bounded below by $\Id - A A^* > 0$. Taking determinants, we find
\begin{equation}\label{e:hist-hua-inequality}
 |\det(\Id - A B^*)|^2 \ge \det(\Id - A A^*) \det(\Id - B B^*),
\end{equation}
which is \cite[(4)]{paige2008hua} with $A$ and $B$ replaced by $A^*$ and $B^*$.

We now prove part (a) by Cayley transforms. Set
\[
 Y_c = (Y - \Id)(Y + \Id)^{-1}, \qquad U = (\Id + \iu L)^{-1}(\Id - \iu L).
\]
The matrix $Y_c$ is Hermitian with $\|Y_c\| < 1$, and
\[
 \Lambda' Y_c \Lambda' = (Y^{-1} - \Id)(Y^{-1} + \Id)^{-1} = -Y_c.
\]
The matrix $U$ is unitary and commutes with $\Lambda'$. In an orthonormal basis in which $\Lambda' = \diag(\Id, -\Id)$, we therefore have
\[
 Y_c = \begin{pmatrix} 0 & y \\ y^* & 0 \end{pmatrix}, \qquad U = \diag(U_+, U_-),
\]
with $\|y\| < 1$.

We now compute the determinant. Since $Y = (\Id + Y_c)(\Id - Y_c)^{-1}$ and $(\Id + \iu L) U = \Id - \iu L$, we have
\[
 Y + \iu L = (\Id + \iu L)(\Id + U Y_c)(\Id - Y_c)^{-1}.
\]
Taking determinants and computing $\det(\Id + U Y_c)$ by a Schur complement, we find
\[
 \frac{\det(Y + \iu L)}{\det(\Id + \iu L)} = \frac{\det(\Id + U Y_c)}{\det(\Id - Y_c)} = \frac{\det(\Id - U_+ y U_- y^*)}{\det(\Id - y y^*)}.
\]
We apply \eqref{e:hist-hua-inequality} with $A = U_+ y$ and $B = y U_-^*$, for which
\[
\begin{gathered}
 \det(\Id - A B^*) = \det(\Id - U_+ y U_- y^*), \\
 \det(\Id - A A^*) = \det(\Id - B B^*) = \det(\Id - y y^*),
\end{gathered}
\]
and we conclude that the right side has modulus at least one. Taking moduli in the identities
\[
 L + \iu Y = \iu (Y + \iu L)^*, \qquad L + \iu \Id = \iu (\Id + \iu L)^*,
\]
we obtain part (a).

We now prove part (b). Set $\Lambda' = \Id_n \otimes \Lambda_B$ and
\[
 Y_1 = Y_0 \otimes \Id + s_0 \Id_n \otimes \mathsf g^2, \qquad \Lambda' Y_1 \Lambda' = Y_0 \otimes \Id + s_0 \Id_n \otimes \mathsf g^{-2},
\]
where we used $\Lambda_B \mathsf g^2 \Lambda_B = \mathsf g^{-2}$. The matrices $Y_1$ and $\Lambda' Y_1 \Lambda'$ commute. Let $G_1$ be the positive square root of $Y_1 \, \Lambda' Y_1 \Lambda'$. Then
\begin{equation}\label{e:hist-hua-G}
 G_1^2 = Y_0^2 \otimes \Id + s_0 Y_0 \otimes (\mathsf g^2 + \mathsf g^{-2}) + s_0^2 \Id \ge (Y_0 + s_0 \Id_n)^2 \otimes \Id,
\end{equation}
where we used $\mathsf g^2 + \mathsf g^{-2} \ge 2 \Id$ and $Y_0 \ge 0$.

We have
\[
 \Lambda' G_1^2 \Lambda' = \Lambda' Y_1 \Lambda' \, Y_1 = G_1^2,
\]
so the matrix $G_1$ commutes with $\Lambda'$; it also commutes with $Y_1$. Consider the positive definite matrix
\[
 Y = G_1^{-1/2} Y_1 G_1^{-1/2} = Y_1 G_1^{-1}, \qquad \text{which satisfies} \qquad Y \, \Lambda' Y \Lambda' = Y_1 \, \Lambda' Y_1 \Lambda' \, G_1^{-2} = \Id,
\]
and $L' = G_1^{-1/2} L G_1^{-1/2}$ is Hermitian and commutes with $\Lambda'$. Using part (a) for $L'$ and $Y$, we find
\[
 |\det(L + \iu Y_1)| = \det G_1 \, |\det(L' + \iu Y)| \ge \det G_1 \, |\det(L' + \iu \Id)| = |\det(L + \iu G_1)|.
\]
Since $G_1$ commutes with $(Y_0 + s_0 \Id_n) \otimes \Id$, \eqref{e:hist-hua-G} implies $G_1 \ge (Y_0 + s_0 \Id_n) \otimes \Id$. The monotonicity in \zcref{l:hist-logdet} completes the proof.
\end{proof}

We next bound the modulus of the bosonic part of the onsite weight from above. For $-1 < t < 1$, the energy $E = 2t$ has $\kappa_E = t$ and $s_E = (1 - t^2)^{1/2}$. The following lemma compares the BB part of the onsite weight at a point with BB coordinate $P_0$ with its value at $P_0 = 0$, for every boost and every rank. Its proof shows that the deficit $V_t(P_0)$ defined below, which is minus the logarithm of the ratio of the two moduli, is a concave function of $t$, and it bounds $V_{\pm 1/2}(P_0)$ from below by Hua's inequality.

\begin{lemma}\label{l:hist-coercivity}
Let $\mathsf g$ be a positive definite $2k \times 2k$ matrix with $\mathsf g \Lambda_B \mathsf g = \Lambda_B$, let $P_0$ be a Hermitian $2k \times 2k$ matrix commuting with $\Lambda_B$, and set $L = \mathsf g^{-1} \Lambda_B P_0 \mathsf g^{-1}$. For $-1 < t < 1$, set $s_t = (1 - t^2)^{1/2}$ and $R_t(P_0) = t \Id + P_0 - \iu s_t \Lambda_B \mathsf g^2$, and define the \emph{BB onsite deficit}
\begin{multline}\label{e:hist-deficit-def}
 V_t(P_0) = \Bigl[\tfrac12 \Re \Tr R_t(P_0)^2 + \log |\det(2t \Id - R_t(P_0))|\Bigr] \\
 - \Bigl[\tfrac12 \Re \Tr R_t(0)^2 + \log |\det(2t \Id - R_t(0))|\Bigr].
\end{multline}
Then $L$ is Hermitian,
\begin{equation}\label{e:hist-congruence}
 \mathsf g^{-1} \Lambda_B \bigl(2t \Id - R_t(P_0)\bigr) \mathsf g^{-1} = t \Lambda_B - L + \iu s_t \Id,
\end{equation}
and
\begin{equation}\label{e:hist-deficit}
 V_t(P_0) = \tfrac12 \Tr P_0^2 + t \Tr P_0 + \tfrac12 \Tr \log\bigl[\Id + L^2 - t (\Lambda_B L + L \Lambda_B)\bigr].
\end{equation}
Moreover, for $|t| \le 1/2$,
\begin{equation}\label{e:hist-coercive}
 V_t(P_0) \ge (1 - 2|t|) V_0(P_0),
\end{equation}
and in particular $V_t(P_0) \ge \frac12 V_0(P_0) \ge \frac14 \Tr P_0^2$ for $|t| \le 1/4$.
\end{lemma}

When $t = \kappa_E$, the matrix $R_t(P_0)$ is the BB block \eqref{e:hist-blocks} of the point of $\mathcal S$ with coordinates $(\kappa \Id + P_0, X, M)$, and $e^{-V_t(P_0)}$ is the modulus of the BB part $e^{-\Tr S_{BB}^2/2} \det(E - S_{BB})^{-1}$ of the onsite weight, divided by its value at $P_0 = 0$.

\begin{proof}
The matrix $\Lambda_B P_0$ is Hermitian, since $P_0$ commutes with $\Lambda_B$, and therefore $L = \mathsf g^{-1} (\Lambda_B P_0) \mathsf g^{-1}$ is Hermitian. 

We begin with the Gaussian part. The diagonal blocks of $\Lambda_B \mathsf g^2$ are Hermitian, since $\mathsf g^2$ is, and $t \Id + P_0$ is block diagonal and Hermitian. Therefore $\Tr((t \Id + P_0) \Lambda_B \mathsf g^2)$ is real. Using $(\Lambda_B \mathsf g^2)^2 = \Id$, which follows from $\Lambda_B \mathsf g^2 \Lambda_B = \mathsf g^{-2}$, we find
\[
 \Re \Tr R_t(P_0)^2 = \Tr(t \Id + P_0)^2 - 2k s_t^2 = \Tr P_0^2 + 2t \Tr P_0 + 2k(t^2 - s_t^2).
\]

We now turn to the determinant. The identity $\mathsf g \Lambda_B \mathsf g = \Lambda_B$ implies that
\[
 \mathsf g^{-1} \Lambda_B \mathsf g^{-1} = \Lambda_B, \qquad \mathsf g^{-1} \Lambda_B (\Lambda_B \mathsf g^2) \mathsf g^{-1} = \Id.
\]
Multiplying the identity
\[
 2t \Id - R_t(P_0) = t \Id - P_0 + \iu s_t \Lambda_B \mathsf g^2
\]
by $\mathsf g^{-1} \Lambda_B$ on the left and by $\mathsf g^{-1}$ on the right, we obtain \eqref{e:hist-congruence}. The matrix $t \Lambda_B - L$ is Hermitian, and $\det \mathsf g = 1$ and $t^2 + s_t^2 = 1$. Using \eqref{e:hist-congruence}, we find
\[
\begin{aligned}
 \log |\det(2t \Id - R_t(P_0))| &= \tfrac12 \Tr \log\bigl[(t \Lambda_B - L)^2 + s_t^2 \Id\bigr] \\
 &= \tfrac12 \Tr \log\bigl[\Id + L^2 - t(\Lambda_B L + L \Lambda_B)\bigr].
\end{aligned}
\]
The two parts equal $k(t^2 - s_t^2)$ and $0$ at $P_0 = 0$, and \eqref{e:hist-deficit} follows.

We now prove \eqref{e:hist-coercive} by concavity in $t$, at fixed $\mathsf g$ and $P_0$. Set
\[
 A_L = \Id + L^2, \qquad M_L = A_L^{-1/2} (\Lambda_B L + L \Lambda_B) A_L^{-1/2},
\]
so that
\[
 \Id + L^2 - t(\Lambda_B L + L \Lambda_B) = A_L^{1/2} (\Id - t M_L) A_L^{1/2}.
\]
Then \eqref{e:hist-deficit} becomes
\begin{equation}\label{e:hist-deficit-line}
 V_t(P_0) = V_0(P_0) + t \Tr P_0 + \tfrac12 \Tr \log(\Id - t M_L), \qquad |t| < 1,
\end{equation}
where $M_L$ does not depend on $t$.

The matrix $M_L$ is Hermitian with $\|M_L\| \le 1$. Indeed, the Hermitian matrices $Z_1 = L A_L^{-1/2}$ and $Z_2 = A_L^{-1/2}$ commute with $L$, and they satisfy
\[
 Z_1^2 + Z_2^2 = \Id, \qquad M_L = Z_1 \Lambda_B Z_2 + Z_2 \Lambda_B Z_1.
\]
Using these identities, we find
\[
 |\langle x, M_L x \rangle| = 2 |\Re \langle Z_1 x, \Lambda_B Z_2 x \rangle| \le \|Z_1 x\|^2 + \|Z_2 x\|^2 = \|x\|^2.
\]
Differentiating \eqref{e:hist-deficit-line} twice, we find
\[
 \frac{\dd^2}{\dd t^2} V_t(P_0) = -\tfrac12 \Tr\bigl[\bigl((\Id - t M_L)^{-1} M_L\bigr)^2\bigr] \le 0,
\]
since $(\Id - t M_L)^{-1} M_L$ is Hermitian. Thus $t \mapsto V_t(P_0)$ is concave on $(-1, 1)$.

It remains to show that $V_{\pm 1/2}(P_0) \ge 0$, and we show that $V_t(P_0) \ge 0$ for $|t| \le 1/2$. Note that
\[
 \Lambda_B \bigl(2t \Id - R_t(P_0)\bigr) = t \Lambda_B - \Lambda_B P_0 + \iu s_t \mathsf g^2.
\]
We apply \zcref{l:hist-hua}(a) with $\Lambda' = \Lambda_B$, $Y = \mathsf g^2$, and $L$ replaced by $s_t^{-1}(t \Lambda_B - \Lambda_B P_0)$, which is Hermitian and commutes with $\Lambda_B$. It implies that
\[
 |\det(2t \Id - R_t(P_0))| \ge |\det(t \Lambda_B - \Lambda_B P_0 + \iu s_t \Id)| = \prod_{\nu} (1 - 2t\nu + \nu^2)^{1/2},
\]
where the product is over the eigenvalues $\nu$ of $P_0$, counted with multiplicity. Here we diagonalized $P_0$ by a unitary that commutes with $\Lambda_B$, and we used $(t - \nu)^2 + s_t^2 = 1 - 2t\nu + \nu^2$.

Combining this bound with the formula for the Gaussian part, we obtain
\[
 V_t(P_0) \ge \sum_\nu \phi_t(\nu), \qquad \phi_t(\nu) = \tfrac12 \nu^2 + t \nu + \tfrac12 \log(1 - 2t\nu + \nu^2).
\]
We have $\phi_t(0) = 0$ and
\[
 \phi_t'(\nu) = \frac{\nu \bigl[\nu^2 - t \nu + 2(1 - t^2)\bigr]}{1 - 2t\nu + \nu^2}.
\]
The quadratic polynomial in brackets has discriminant $9t^2 - 8 < 0$. Therefore $\phi_t'(\nu)$ has the sign of $\nu$, so that $\phi_t \ge 0$ and $V_t(P_0) \ge 0$.

Now let $|t| \le 1/2$, and let $\pm$ be the sign of $t$. Since $t$ is the convex combination of $0$ and $\pm 1/2$ with weights $1 - 2|t|$ and $2|t|$, concavity implies that
\[
 V_t(P_0) \ge (1 - 2|t|) V_0(P_0) + 2|t| \, V_{\pm 1/2}(P_0) \ge (1 - 2|t|) V_0(P_0).
\]
Finally, at $t = 0$, the identity \eqref{e:hist-deficit} reads
\[
 V_0(P_0) = \tfrac12 \Tr P_0^2 + \tfrac12 \Tr \log(\Id + L^2) \ge \tfrac12 \Tr P_0^2. \qedhere
\]
\end{proof}

We now apply Hua's inequality to the BB block of the anchored identity. In the following lemma, $\varsigma$ is an arbitrary diagonal matrix with entries in $(0, 1]$; in our application it is the boundary profile.

\begin{lemma}
Let\label{l:hist-bb} $A$ be a Hermitian $n \times n$ matrix, let $\varsigma = \diag(\varsigma_1, \dots, \varsigma_n)$ with $0 < \varsigma_i \le 1$, and let $\eta, s_0 > 0$. Let $\mathsf g$ be a positive definite $2k \times 2k$ matrix with $\mathsf g \Lambda_B \mathsf g = \Lambda_B$, and let $Z$ be a Hermitian $2k \times 2k$ matrix that commutes with $\Lambda_B$ and satisfies $\|Z\| \le r$. Set
\begin{equation}\label{e:hist-B}
 \mathbf B = A \otimes \Id - \varsigma \otimes Z + \iu \eta (\Id_n - \varsigma) \otimes \Lambda_B + \iu s_0 \varsigma \otimes \Lambda_B \mathsf g^2,
 \qquad
 \Pi = \sum_{i = 1}^n \frac{\varsigma_i}{\eta(1 - \varsigma_i) + s_0 \varsigma_i}.
\end{equation}
Then $\mathbf B$ is invertible, and
\begin{align}
 |\det \mathbf B| &\ge e^{-2k r \Pi} \, \bigl|\det\bigl(A + \iu \eta (\Id_n - \varsigma) + \iu s_0 \varsigma\bigr)\bigr|^{2k}, \label{e:hist-B-det}\\
 \|\mathbf B^{-1}\| &\le \max\bigl\{\eta^{-1}, s_0^{-1} \|\mathsf g^2\|\bigr\}. \label{e:hist-B-inverse}
\end{align}
\end{lemma}

\begin{proof}
We first prove \eqref{e:hist-B-inverse}. We write
\[
\begin{gathered}
 (\Id_n \otimes \Lambda_B) \mathbf B = X_1 + \iu Y_1, \\
 X_1 = A \otimes \Lambda_B - \varsigma \otimes \Lambda_B Z, \qquad Y_1 = \eta(\Id_n - \varsigma) \otimes \Id + s_0 \varsigma \otimes \mathsf g^2.
\end{gathered}
\]
The matrices $X_1$ and $Y_1$ are Hermitian, since $\Lambda_B Z$ is Hermitian. Since $\Lambda_B \mathsf g^2 \Lambda_B = \mathsf g^{-2}$, we have $\|\mathsf g^{-2}\| = \|\mathsf g^2\|$, and therefore
\[
 \mathsf g^2 \ge \|\mathsf g^{-2}\|^{-1} \Id = \|\mathsf g^2\|^{-1} \Id, \qquad Y_1 \ge \min\bigl\{\eta, s_0 \|\mathsf g^2\|^{-1}\bigr\} \Id.
\]
The matrix $\Id_n \otimes \Lambda_B$ is unitary. Therefore $\|\mathbf B^{-1}\| = \|(X_1 + \iu Y_1)^{-1}\|$, and the first bound in \eqref{e:hist-resolvent} implies \eqref{e:hist-B-inverse}.

For \eqref{e:hist-B-det}, we set
\[
 A' = \varsigma^{-1/2} A \varsigma^{-1/2}, \qquad Y_0 = \eta \varsigma^{-1}(\Id_n - \varsigma) \ge 0.
\]
Then
\[
\begin{gathered}
 (\varsigma^{-1/2} \otimes \Id)(X_1 + \iu Y_1)(\varsigma^{-1/2} \otimes \Id) = L + \iu (Y_0 \otimes \Id + s_0 \Id_n \otimes \mathsf g^2), \\
 L = A' \otimes \Lambda_B - \Id_n \otimes \Lambda_B Z,
\end{gathered}
\]
and $L$ is Hermitian and commutes with $\Id_n \otimes \Lambda_B$. Since $|\det(\Id_n \otimes \Lambda_B)| = 1$, \zcref{l:hist-hua}(b) implies that
\[
 |\det \mathbf B| \ge (\det \varsigma)^{2k} \bigl|\det\bigl(L + \iu (Y_0 + s_0 \Id_n) \otimes \Id\bigr)\bigr|.
\]

The last matrix is block diagonal in the flavor space. We diagonalize $Z$ by a unitary that commutes with $\Lambda_B$, with eigenvalues $\nu_1, \dots, \nu_{2k}$, and use $|\det(-Y' + \iu Y'')| = |\det(Y' + \iu Y'')|$ for Hermitian $Y'$ and $Y''$. This yields
\[
 \bigl|\det\bigl(L + \iu (Y_0 + s_0 \Id_n) \otimes \Id\bigr)\bigr| = \prod_{a = 1}^{2k} \bigl|\det\bigl(A' - \nu_a \Id_n + \iu (Y_0 + s_0 \Id_n)\bigr)\bigr|.
\]
By \eqref{e:hist-shift} with $Z = \Id_n$, we have
\[
 \Bigl|\frac{\dd}{\dd u} \log \bigl|\det\bigl(A' - u \Id_n + \iu (Y_0 + s_0 \Id_n)\bigr)\bigr|\Bigr| \le \Tr (Y_0 + s_0 \Id_n)^{-1} = \Pi \qquad \text{for all } u \in \R.
\]
Using $|\nu_a| \le r$ and
\[
 \det \varsigma \, |\det(A' + \iu(Y_0 + s_0 \Id_n))| = |\det(A + \iu \eta (\Id_n - \varsigma) + \iu s_0 \varsigma)|,
\]
we obtain \eqref{e:hist-B-det}.
\end{proof}

\section{One step on the cycle}\label{s:hist-step}

One step of the chain is the heat link followed by the onsite weight, $T_\Gamma = M_{w_\Gamma} K_{\tau_{\rm phys}}$, and in this section we bound the two factors separately on the weighted spaces $\mathcal H_p(\mathcal S)$. The kinetic bound \zcref{l:end-kinetic} shows that $K_{\tau_{\rm phys}}$ has norm at most $e^{C/W}$. The onsite bound \zcref{l:end-onsite} shows that $|w_\Gamma|_{1; h} \le e^{C/W} e^{-c \vartheta}$, where the escape function $\vartheta$ of \eqref{e:end-escape} vanishes on the saddle manifold and grows with the distance from it. The proof of \zcref{t:hist-bound} needs both bounds. Together they give the trivial bound $\|T_\Gamma\| \le e^{C/W}$, and away from the cores, where $\vartheta \ge c r_W^2$, they give the contraction by $e^{-c (\log W)^2/W}$. For this reason we prove them here, before \zcref{t:hist-bound}; \zcref{s:end} uses them as well. The proof of the onsite bound rests on the concavity bound \zcref{l:hist-coercivity} and the congruence \eqref{e:hist-congruence}.

We use the following objects of \zcref{s:rep}. Recall from \zcref{d:rep-cycle} that the Sch\"afer--Wegner cycle $\mathcal S$ has the coordinates $(P, X, M)$, where $P$ is Hermitian and block diagonal, $X$ is Hermitian and block off-diagonal, and $M$ is Hermitian, and that the odd entries of $S \in \mathcal S$ are free generators. By \eqref{e:hist-blocks}, the blocks of $S$ are
\[
 S_{BB} = \kappa \Id + P_0 - \iu s \Lambda_B \mathsf g^2, \qquad S_{FF} = \kappa \Id + \iu M, \qquad P_0 = P - \kappa \Id,
\]
where the boost $\mathsf g = \mathsf g_X$ of \eqref{e:rep-f0} satisfies $s \mathsf g^2 = f_0(X) + X$, with $f_0(X) = (s^2 \Id + X^2)^{1/2}$. Recall also from \zcref{d:rep-chain} the onsite weight $w_\Gamma$, the heat operators $K_t$ with the kernels $k_t(S, S') = e^{-\Str(S - S')^2/(4t)}$, and the transfer operator $T_\Gamma$. Finally, recall from \zcref{s:rep-heuristics} the saddle manifold, the union of the $2k + 1$ orbits of the diagonal supermatrices $R_*^{(p)}$, $p \in \{0, \dots, 2k\}$, under conjugation by even invertible supermatrices, where $R_*^{(p)}$ has the BB block $\kappa \Id - \iu s \Lambda_B$ and an FF block that depends on the signature $p$.

From here to the end of this section, $\| \cdot \|$ denotes the Hilbert--Schmidt norm of a matrix and $\| \cdot \|_{\rm op}$ its operator norm, as in \zcref{s:rep}. This replaces the convention of \zcref{s:hist-setting}, since the gradients, the Fourier variable, and the defect below are taken for the Hilbert--Schmidt inner product. Recall also that $|E| \le 1/2$, so that $s^{-1} \le 4/\sqrt{15}$.

\subsection{The cycle near the saddle manifold}\label{s:end-cycle}

We first write the BB block of the cycle in coordinates adapted to the saddle manifold, and we prove the kinetic bound.

\paragraph{The boost.}
By \zcref{l:rep-cycle}\ref{i:rep-cycle-g}, the boost satisfies
\begin{equation}\label{e:end-boost}
 s \mathsf g^2 = f_0(X) + X , \qquad \mathsf g \Lambda_B \mathsf g = \Lambda_B , \qquad \det \mathsf g^2 = 1 .
\end{equation}
Let $\log \mathsf g$ be its Hermitian logarithm, which is off-diagonal, and define the operators $\operatorname{ch}_{\mathsf g} = \cosh(\operatorname{ad}_{\log \mathsf g})$ and $\operatorname{th}_{\mathsf g} = \tanh(\operatorname{ad}_{\log \mathsf g})$ on $2k \times 2k$ matrices. Since $\operatorname{ad}_{\log \mathsf g}$ is selfadjoint for the Hilbert--Schmidt inner product and exchanges block-diagonal and off-diagonal matrices, $\operatorname{ch}_{\mathsf g}$ preserves both classes and satisfies $\operatorname{ch}_{\mathsf g} \ge \Id$, while $\operatorname{th}_{\mathsf g}$ exchanges them and has norm at most one. A direct computation gives
\begin{equation}\label{e:end-boost-conj}
 \mathsf g S_{BB} \mathsf g^{-1} = R_{*, B} + \widetilde P + \operatorname{th}_{\mathsf g} \widetilde P ,
 \qquad R_{*, B} = \kappa \Id - \iu s \Lambda_B , \quad \widetilde P = \operatorname{ch}_{\mathsf g} P_0 ,
\end{equation}
where $R_{*, B}$ is the BB block of $R_*^{(p)}$, which is the same for all $p$, and we used $S_{BB} = \kappa \Id + P_0 - \iu s \Lambda_B \mathsf g^2$, $\mathsf g \Lambda_B \mathsf g = \Lambda_B$, and $\operatorname{Ad}_{\mathsf g} = \operatorname{ch}_{\mathsf g} + \operatorname{th}_{\mathsf g} \operatorname{ch}_{\mathsf g}$. We call the block-diagonal matrix $\widetilde P$ the \emph{physical normal} of the point. Since $\operatorname{ch}_{\mathsf g} \ge \Id$, we have $\|\widetilde P\| \ge \|P_0\|$.

\paragraph{Divided differences.}
We diagonalize $X$ at a given point, with eigenvalues $x_a = s \sinh r_a$. Differentiating $f_0(X)^2 = s^2 \Id + X^2$ in the direction $V$ gives the Sylvester equation $f_0 \, Df_0(V) + Df_0(V) f_0 = X V + V X$, and therefore
\begin{equation}\label{e:end-divided}
 \bigl(Df_0(X)V\bigr)_{ab} = c_{ab} V_{ab} ,
 \qquad
 c_{ab} = \frac{x_a + x_b}{(s^2 + x_a^2)^{1/2} + (s^2 + x_b^2)^{1/2}} = \tanh\frac{r_a + r_b}{2} ,
\end{equation}
where repeated eigenvalues use the continuous value. For a block-diagonal Hermitian $\Xi$, we set $\phi_\Xi(X) = \langle \Xi, \Lambda_B f_0(X)\rangle$ and define the \emph{defect}
\begin{equation}\label{e:end-defect}
 d_X(\Xi) = \|\Xi\|^2 - \|\nabla \phi_\Xi(X)\|^2 = \sum_{a, b} (1 - c_{ab}^2) |(\Lambda_B \Xi)_{ab}|^2 \ge 0 .
\end{equation}
Since $|c_{ab}| < 1$, the map $X \mapsto -\Lambda_B f_0(X)$ is Lipschitz with constant one, and by \zcref{d:rep-cycle}, the BB block of $S$ differs from that of the flat cycle $\mathcal S_\flat$ by $\iu$ times this map. In the proof of \zcref{l:end-kinetic}, we take the Fourier transform of the heat kernel in $P_0$, with Fourier variable $\Xi$, and the bound $d_X(\Xi) \ge 0$ then shows that the transformed generator is dissipative. For the logarithmic derivative of the boost, with $\bar r = (r_a + r_b)/2$ and $\delta r = (r_a - r_b)/2$, the same calculation gives
\begin{equation}\label{e:end-logg}
 \bigl(\mathsf g^{-1} D\mathsf g(V)\bigr)_{ab} = \frac{e^{-\delta r/2}}{2 s \cosh \bar r \cosh(\delta r/2)} V_{ab} ,
\end{equation}
and the right logarithmic derivative has $e^{\delta r/2}$ in place of $e^{-\delta r/2}$. It follows that
\begin{equation}\label{e:end-connection}
\begin{aligned}
 \|\mathsf g^{-1} D\mathsf g(V)\| + \|D\mathsf g(V) \mathsf g^{-1}\| &\le C s^{-1} \|V\| , \\
 \|\mathsf g^{-1} D\mathsf g(\nabla \phi_\Xi)\| + \|D\mathsf g(\nabla \phi_\Xi) \mathsf g^{-1}\| &\le C s^{-1} d_X(\Xi)^{1/2} ,
\end{aligned}
\end{equation}
where the second bound uses $\cosh^{-2} \bar r = 1 - \tanh^2 \bar r$ and \eqref{e:end-defect}. The constant $C$ in \eqref{e:end-connection} is universal; in particular, it does not depend on $k$.

\paragraph{Norms on the cycle.}
Recall the transported odd frame of \eqref{e:hist-transported}, the orbit height $v_{\mathcal S} = 1 + \Tr f_0(X)$, and the weighted spaces $\mathcal H_p(\mathcal S)$ of \eqref{e:hist-Hp}, whose norms are built from the coefficient norm $| \cdot |_{2; h}$ of \zcref{d:grass-norm}, taken in the transported odd frame. We let $\rho(\mathsf g)$ denote the linear map \eqref{e:hist-transported} from the odd entries of $S$ to the transported odd frame. Recall from \zcref{l:grass-norm}(b) that multiplication by an $h$-scaled monomial is a partial isometry for $| \cdot |_{2; h}$, and from \zcref{s:hist-setting} that multiplication by a coefficient function $a$ has norm at most $\sup |a|_{1; h}$ on $\mathcal H_p(\mathcal S)$.

\begin{lemma}\label{l:end-kinetic}
Let $K_t$ be the heat operators of \zcref{d:rep-chain}\ref{i:rep-chain-heat}, acting on superfunctions on $\mathcal S$ whose coefficients are taken in the transported odd frame. For all $p \in \R$,
\[
 \|K_t\|_{\mathcal H_p(\mathcal S) \to \mathcal H_p(\mathcal S)} \le \exp\bigl[(C_k W + C_{k, p}) t\bigr] ,
 \qquad 0 \le t \le \tau_{\rm phys} ,
\]
and the same bound holds for the adjoint of $K_t$ in the Hilbert space $\mathcal H_p(\mathcal S)$. Moreover, the operators $K_t$ satisfy the semigroup law and are strongly continuous at $t = 0$ on $\mathcal H_p(\mathcal S)$.
\end{lemma}

\begin{proof}
We first treat the BB block without weight and without odd variables. In the coordinates $(P_0, X)$, we have
\[
 S_{BB} - S_{BB}' = \bigl[(P_0 - P_0') - \iu \Lambda_B (f_0(X) - f_0(X'))\bigr] - \iu \Lambda_B (X - X') ,
\]
where the first term is block diagonal and the second is off-diagonal, so that the cross term of the two parts has zero trace. Since $\Lambda_B Y \Lambda_B = -Y$ for off-diagonal $Y$, the off-diagonal part contributes $\Tr (X - X')^2$ to $\Tr (S_{BB} - S_{BB}')^2$. The block-diagonal part is a Gaussian in $P_0 - P_0'$ shifted by the imaginary amount $\iu \Lambda_B(f_0(X) - f_0(X'))$, and partial Fourier transformation in $P_0$, with the Fourier variable $\Xi$, turns the BB kernel into
\[
 T_{t, \Xi} = e^{-t \|\Xi\|^2} \, e^{\phi_\Xi} \, e^{t \Delta_X} \, e^{-\phi_\Xi} ,
\]
where $\Delta_X$ is the flat Laplacian in $X$. Its generator is
\[
 \mathcal L_\Xi = \sum_j (\partial_j - \partial_j \phi_\Xi)^2 - \|\Xi\|^2 .
\]
Since $\phi_\Xi$ is real, the operator $\sum_j (\partial_j \phi_\Xi \, \partial_j + \partial_j \, \partial_j \phi_\Xi)$ is skew-adjoint. Integration by parts on a bounded Dirichlet domain, together with \eqref{e:end-defect}, gives
\[
 \Re \langle u, \mathcal L_\Xi u\rangle = - \|\nabla u\|^2 + \int \bigl(\|\nabla \phi_\Xi\|^2 - \|\Xi\|^2\bigr) |u|^2 = - \|\nabla u\|^2 - \int d_X(\Xi) |u|^2 \le 0 .
\]

Next, we add the odd variables. In the transported frame the fibre generator is conjugated by $\rho(\mathsf g)$. On a bounded Dirichlet domain we set
\[
 C_j = \rho(\mathsf g)^{-1} \partial_j \rho(\mathsf g) = A_j + S_j ,
\]
where $A_j$ and $S_j$ are the skew-adjoint and selfadjoint parts of $C_j$ for the inner product of $| \cdot |_{2; h}$. With the skew-adjoint operators $D_j^0 = \partial_j + A_j$, the fibre generator becomes
\[
 \mathcal L_\Xi = \sum_j \bigl(D_j^0 + S_j - \partial_j \phi_\Xi\bigr)^2 - \|\Xi\|^2 .
\]
Since the anticommutator of a skew-adjoint operator with a selfadjoint multiplication is skew-adjoint, integration by parts yields the identity
\[
 \Re \langle u, \mathcal L_\Xi u\rangle = - \sum_j \|D_j^0 u\|^2 + \sum_j \|(S_j - \partial_j \phi_\Xi) u\|^2 - \|\Xi\|^2 \|u\|^2 .
\]
The operators $C_j$ act on the finite-dimensional Grassmann algebra of the transported odd coordinates through the logarithmic derivatives of the boost, and \eqref{e:end-connection} therefore implies that
\[
 \Bigl\|\sum_j (\partial_j \phi_\Xi) S_j\Bigr\| \le C_k d_X(\Xi)^{1/2} ,
 \qquad
 \sum_j \|S_j\|^2 \le C_k .
\]
Expanding $\|(S_j - \partial_j \phi_\Xi) u\|^2$ and using \eqref{e:end-defect}, we see that the last two terms in the identity for $\Re \langle u, \mathcal L_\Xi u\rangle$ are bounded by
\[
 \int \bigl(-d_X(\Xi) + C_k d_X(\Xi)^{1/2} + C_k\bigr) |u|^2 \le C_k' \|u\|^2 .
\]
The odd part of the Laplacian of the heat kernel is preserved by the congruence actions, since $\mathsf g \Lambda_B \mathsf g = \Lambda_B$ preserves the odd bilinear form, and by \zcref{l:grass-norm}(c), two odd derivatives have norm at most $h^{-2} = W$ for $| \cdot |_{2; h}$. Therefore the odd part of the heat has norm at most $e^{C_k W t}$, and the energy inequality yields the bound $e^{(C_k W + C_k) t}$ for the fibre operator.

We now remove the Dirichlet domain. For compactly supported input, the Dirichlet heat kernels on an exhaustion of the Cartesian domain converge pointwise to the free Gaussian kernel. The frame factors are polynomially bounded and $\phi_\Xi$ grows at most linearly. Fatou's lemma then implies the same norm bound for the actual fibre kernel, and by density the bound extends to all inputs. Plancherel's theorem in $P_0$ yields the BB bound. The FF block of the cycle is the flat Cartesian space of Hermitian matrices $M$, on which the FF part of the heat kernel defines the flat heat $e^{t \Delta_M}$, which is a contraction. In a product of two kernels, the frame changes cancel at the middle point, so that the semigroup law follows from \zcref{l:rep-heat}. Strong continuity holds on compactly supported smooth inputs, by local heat approximation, and it extends to all inputs by the uniform bound.

Finally, we add the weight $v_{\mathcal S}^p$. Conjugation by $v_{\mathcal S}^p$ adds first- and zeroth-order terms with coefficients given by the derivatives of $p \log v_{\mathcal S}$. The only potentially unbounded cross term in the Fourier variable is aligned with the defect. Indeed, in the eigenbasis of $X$ we have
\[
 |\langle \nabla \phi_\Xi, \nabla \log v_{\mathcal S}\rangle|
 \le d_X(\Xi)^{1/2} \, \frac{(\sum_a \sinh^4 r_a/\cosh^2 r_a)^{1/2}}{v_{\mathcal S}} \le s^{-1} d_X(\Xi)^{1/2} .
\]
The other derivatives of $\log v_{\mathcal S}$ and their frame terms are bounded. Retaining part of the negative defect, we obtain the weighted bound by the same exhaustion. The adjoint of $K_t$ in $\mathcal H_p(\mathcal S)$ has the same norm.
\end{proof}

\subsection{The escape function and the onsite weight}\label{s:end-onsite}

In this subsection we define the escape function, which measures the distance from the saddle manifold, and we prove that the onsite weight gains a universal fraction of it, in the form of an $\ell^1$ bound on its coefficients.

Fix a nondecreasing smooth function $\chi_0$ on $[0, \infty)$ with $\chi_0(0) = 0$ and $\chi_0'(0) > 0$, which is linear near zero and constant on $[1, \infty)$, and fix a radius $0 < r_{\rm esc} \le 1$. Then $\chi_0(x)/\min(x, 1)$ is continuous and positive on $(0, \infty)$, with the positive limit $\chi_0'(0)$ at zero and the constant value $\chi_0(1) > 0$ on $[1, \infty)$, so there exist $c, C > 0$ such that
\begin{equation}\label{e:end-chi0}
 c \min(x, 1) \le \chi_0(x) \le C \min(x, 1) , \qquad x \ge 0 .
\end{equation}
We define the \emph{escape function}
\begin{equation}\label{e:end-escape}
 \vartheta = \vartheta_B + \vartheta_F ,
 \qquad
 \vartheta_B = r_{\rm esc}^2 \, \chi_0\bigl(\|\widetilde P\|^2/r_{\rm esc}^2\bigr) ,
 \qquad
 \vartheta_F = \Tr \frac{(M^2 - s^2 \Id)^2}{\Id + M^2} .
\end{equation}
By \eqref{e:end-chi0}, we have
\[
 c \min(\|\widetilde P\|^2, r_{\rm esc}^2) \le \vartheta_B \le C \min(\|\widetilde P\|^2, r_{\rm esc}^2) .
\]
Moreover, with $x = \|\widetilde P\|^2/r_{\rm esc}^2$,
\[
 |\nabla_{\widetilde P} \vartheta_B|^2 = 4 r_{\rm esc}^2 \chi_0'(x)^2 x \le C \vartheta_B ,
\]
since $\chi_0'(x)^2 x \le C \chi_0(x)$ near zero and $\chi_0'$ vanishes on $[1, \infty)$. All BB derivatives of $\vartheta_B$ vanish where $\|\widetilde P\| \ge r_{\rm esc}$. Writing $\vartheta_F = \sum_i \theta(\mu_i)$ over the eigenvalues $\mu_i$ of $M$, with $\theta(\mu) = (\mu^2 - s^2)^2/(1 + \mu^2)$, we have
\begin{equation}\label{e:end-escape-gradient}
 |\nabla_M \vartheta_F|^2 = \sum_i \theta'(\mu_i)^2 \le C_F \vartheta_F ,
 \qquad
 \frac{\theta'(\mu)^2}{\theta(\mu)} = \frac{4 \mu^2 (\mu^2 + 2 + s^2)^2}{(1 + \mu^2)^3} \le C_F ,
\end{equation}
with a universal constant $C_F$. Since $\operatorname{ch}_{\mathsf g} \ge \Id$, the escape function vanishes exactly at the points where $P_0 = 0$ and $M$ has only the eigenvalues $\pm s$. By \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle}, these are the points of $\mathcal S$ with BB block $\kappa \Id - \iu s Q_X$ and with FF block of eigenvalues $\kappa \pm \iu s$, and they lie on the saddle manifold, in the $2k + 1$ signatures.

We first bound the FF part of the onsite weight. For $D = 0$ and $d = 2k$, the function of the following lemma satisfies
\[
 \mathcal W_\kappa(M, 0) = e^{k(s^2 - \kappa^2)} \bigl|e^{\Tr S_{FF}^2/2} \det(E - S_{FF})\bigr| ,
\]
where $e^{\Tr S_{FF}^2/2} \det(E - S_{FF})$ is the FF part of the onsite weight, and $\mathcal W_\kappa(M, 0) = 1$ when $M$ has only the eigenvalues $\pm s$. The lemma also treats a shift by a Hermitian matrix $D$, which \zcref{s:end-canonical} needs; here we use only the case $D = 0$.

\begin{lemma}\label{l:end-ff-strip}
Let $|\kappa| \le \kappa_0 < 1$, $s^2 = 1 - \kappa^2$, and let $M$ and $D$ be Hermitian $d \times d$ matrices with $\|D\|_{\rm op} \le d_0$. Define
\[
 \mathcal W_\kappa(M, D) = e^{(d s^2 - \Tr M^2)/2 + \kappa \Tr D + \Tr D^2/2} \, |\det(\kappa \Id - D + \iu M)| .
\]
There exist $c, C > 0$, depending only on $\kappa_0$ and $d_0$ and not on $d$, such that
\[
 \mathcal W_\kappa(M, D) \le \exp\Bigl\{ C \|D\|^2 - c \Tr \frac{(M^2 - s^2 \Id)^2}{\Id + M^2} \Bigr\} .
\]
\end{lemma}

\begin{proof}
We diagonalize $M$ but not $D$. Let $\lambda_i$ be the eigenvalues of $M$, and in the eigenbasis set
\[
 d_i = D_{ii} \in \R, \qquad t_i = \sum_j |D_{ji}|^2 .
\]
By Hadamard's inequality for the columns, we have
\[
 |\det(\kappa \Id - D + \iu M)| \le \prod_i y_i^{1/2} ,
 \qquad
 y_i = \lambda_i^2 + \kappa^2 - 2 \kappa d_i + t_i \ge 0 .
\]
We may assume that every $y_i$ is nonzero, since otherwise the corresponding column vanishes and the conclusion is immediate. Set $x_i = \lambda_i^2 + \kappa^2$. Using $s^2 = 1 - \kappa^2$, we find that the logarithm of the resulting bound for $\mathcal W_\kappa(M, D)$ is
\[
 \sum_i \Bigl\{ - \frac{x_i - 1}{2} + \kappa d_i + \frac{t_i}{2} + \frac12 \log y_i \Bigr\}
 = \sum_i \Bigl\{ t_i - \frac12 (y_i - 1 - \log y_i) \Bigr\} .
\]
The terms linear in $D$ cancel in this identity, because of the factor $e^{\kappa \Tr D}$ in the definition of $\mathcal W_\kappa$.

For $y > 0$ we have $y - 1 - \log y \ge (y - 1)^2/(2(1 + y))$, as one checks by differentiating the difference on $(0, 1)$ and on $(1, \infty)$. The function $x \mapsto (x - 1)(1 + x)^{-1/2}$ has bounded derivative on $[0, \infty)$, and
\[
 |x_i - y_i|^2 = |2 \kappa d_i - t_i|^2 \le 8 \kappa_0^2 t_i + 2 t_i^2 \le C_{\kappa_0, d_0} t_i ,
\]
since $t_i \le \|D\|_{\rm op}^2 \le d_0^2$. It follows that
\[
 \frac{(x_i - 1)^2}{1 + x_i} \le C (y_i - 1 - \log y_i) + C t_i .
\]
The quantities $1 + x_i$ and $1 + \lambda_i^2$ are comparable with constants independent of $d$, and $x_i - 1 = \lambda_i^2 - s^2$. Substituting the last inequality into the logarithmic bound, we obtain
\[
 \log \mathcal W_\kappa(M, D) \le C \sum_i t_i - c \sum_i \frac{(\lambda_i^2 - s^2)^2}{1 + \lambda_i^2} = C \|D\|^2 - c \Tr \frac{(M^2 - s^2 \Id)^2}{\Id + M^2} ,
\]
where we used $\sum_i t_i = \|D\|^2$.
\end{proof}

We now state the onsite bound. We write the damping of the onsite weight as $e^{-\iu \eta \Str(S \Gamma)}$, with a matrix $\Gamma$ whose BB block is close to $\Lambda_B$, as in \eqref{e:hist-normalized}, and we allow complex values of the damping, which \zcref{s:stab,s:end} use.

\begin{lemma}\label{l:end-onsite}
Let $|E| \le 1/2$ and $0 < \eta \le C_X W^{-2}$. Let $\Gamma$ be a constant even $(2k|2k)$ supermatrix with zero odd blocks such that $\|\Gamma_{BB} - \Lambda_B\|_{\rm op} \le 1/4$ and $\|\Gamma_{FF}\|_{\rm op} \le C_\Gamma$, and set
\[
 w_\Gamma(S) = e^{-\Str(S^2)/2 - \iu \eta \Str(S \Gamma)} \, \Sdet(E - S)^{-1} .
\]
There exists a universal $c > 0$ such that for all sufficiently large $W$ and all $S \in \mathcal S$,
\begin{equation}\label{e:end-onsite}
 |w_\Gamma|_{1; h} \le e^{C_k/W + C_k \eta} \, e^{-c \vartheta - c \eta v_{\mathcal S}} .
\end{equation}
More precisely, the body of $w_\Gamma$ has modulus at most $e^{C_k \eta - c \vartheta - c \eta v_{\mathcal S}}$, and the absolute values of its other coefficients sum to at most $C_k h^2 e^{C_k \eta - c \vartheta - c \eta v_{\mathcal S}}$. The constants $C_k$ and the threshold on $W$ depend on $k$, $C_X$, and $C_\Gamma$.
\end{lemma}

The hypothesis on $\Gamma$ covers the normalized sources $\hat \Gamma$ of \eqref{e:hist-normalized}, for which $\hat \Gamma_{BB} = \Lambda_B$ and $\|\hat \Gamma_F\|_{\rm op} \le C_\Gamma$, and the parameter balls of \eqref{e:end-damping} below. For $\Gamma = \hat \Gamma$, the function defined in the lemma is the onsite weight of \zcref{d:rep-chain} for the source supermatrix $\eps \hat \Gamma$, by \eqref{e:hist-normalized}. The body bound in \eqref{e:end-onsite} carries no constant factor, and the factor $e^{C_k/W}$ comes only from the nonbody coefficients. This is the form that the contraction away from the saddle manifold in the proof of \zcref{t:hist-bound} needs, since a constant factor larger than one would be raised to the power of the number of steps.

\begin{proof}
We first bound the body. Its modulus is the product of the BB factor $e^{-V_\kappa(P_0)}$, where $V_\kappa$ is the BB onsite deficit \eqref{e:hist-deficit-def} of \zcref{l:hist-coercivity} (the real part of $\frac12 \Tr S_{BB}^2 + \log \det(E - S_{BB})$ minus its value at $P_0 = 0$), of the FF factor $\mathcal W_\kappa(M, 0)$ of \zcref{l:end-ff-strip}, and of the damping $|e^{-\iu \eta \Str(S \Gamma)}|$. Indeed, the BB part of $\log |w_0|$ is $-V_\kappa(P_0) - k(\kappa^2 - s^2)$, since its value at $P_0 = 0$ is $k(\kappa^2 - s^2)$, and the FF part is
\[
 \tfrac12 \Re \Tr(\kappa \Id + \iu M)^2 + \log|\det(\kappa \Id - \iu M)| = \log \mathcal W_\kappa(M, 0) + k(\kappa^2 - s^2) .
\]
The two constants cancel, so that the product of the BB and FF parts equals one at the saddle points.

By the concavity bound \eqref{e:hist-coercive}, for $|E| \le 1/2$ we have
\[
 V_\kappa \ge (1 - 2|\kappa|) V_0 \ge V_0/2 ,
\]
where
\[
 V_0 = \tfrac12 \Tr P_0^2 + \tfrac12 \Tr \log(\Id + L^2) ,
 \qquad L = \mathsf g^{-1} \Lambda_B P_0 \mathsf g^{-1} = \Lambda_B \, \mathsf g P_0 \mathsf g^{-1} ,
\]
and we used $\mathsf g^{-1} \Lambda_B = \Lambda_B \mathsf g$. Since $\mathsf g P_0 \mathsf g^{-1} = \widetilde P + \operatorname{th}_{\mathsf g} \widetilde P$ is an orthogonal sum, we have $\|L\| \ge \|\widetilde P\|$. Let $k_j$ be the eigenvalues of the Hermitian matrix $L$. Then
\[
\begin{aligned}
 \tfrac12 \Tr \log(\Id + L^2) &= \tfrac12 \log \prod_j (1 + k_j^2) \ge \tfrac12 \log\Bigl(1 + \sum_j k_j^2\Bigr) \\
 &= \tfrac12 \log\bigl(1 + \|L\|^2\bigr) \ge \tfrac12 \log\bigl(1 + \|\widetilde P\|^2\bigr) .
\end{aligned}
\]
Using also $\log(1 + x) \ge (\log 2) \min(x, 1)$ and $\vartheta_B \le C \min(\|\widetilde P\|^2, r_{\rm esc}^2)$ with $r_{\rm esc} \le 1$, we obtain
\begin{equation}\label{e:end-BB-deficit}
 V_\kappa \ge \tfrac14 \Tr P_0^2 + \tfrac14 \log\bigl(1 + \|\widetilde P\|^2\bigr) \ge \tfrac14 \Tr P_0^2 + c_B \vartheta_B
\end{equation}
for a universal $c_B > 0$. By \zcref{l:end-ff-strip} with $D = 0$, we also have $\mathcal W_\kappa(M, 0) \le e^{-c_F \vartheta_F}$.

Next, we bound the damping. Write $\Gamma_{BB} = \Lambda_B + \gamma$. Since $X$ is off-diagonal, $\Tr X = 0$, and therefore
\[
\begin{gathered}
 \Tr(S_{BB} \Lambda_B) = \Tr(P_0 \Lambda_B) - \iu \Tr f_0(X) , \\
 \Re\bigl(-\iu \eta \Tr(S_{BB} \Lambda_B)\bigr) = -\eta \Tr f_0(X) = -\eta(v_{\mathcal S} - 1) .
\end{gathered}
\]
The positive matrix $f_0(X) + X = s \mathsf g^2$ has trace norm $\Tr f_0(X)$, and therefore
\[
 |\eta \Tr(S_{BB} \gamma)| \le \frac{\eta}{4} \bigl(2k|\kappa| + \|P_0\|_1 + \Tr f_0(X)\bigr)
 \le C_k \eta + \tfrac18 \Tr P_0^2 + \tfrac14 \eta \Tr f_0(X) ,
\]
where we used $\|P_0\|_1 \le 2k + \Tr P_0^2$ and $\eta \le 1/2$. For the FF block, the bounds $\|\kappa \Id + \iu M\|_1 \le 2k + \|M\|_1$ and $|\mu| \le 2 + 2\theta(\mu)$ for real $\mu$ give
\[
 |\eta \Tr(S_{FF} \Gamma_{FF})| \le C_k \eta + 2 C_\Gamma \eta \vartheta_F ,
\]
and we take $W$ so large that $2 C_\Gamma \eta \le c_F/2$. Combining these bounds with \eqref{e:end-BB-deficit}, we find that the modulus of the body is at most
\[
 \exp\bigl\{C_k \eta - c_B \vartheta_B - c_F \vartheta_F/2 - 3\eta(v_{\mathcal S} - 1)/4\bigr\} .
\]

We now bound the other coefficients. By \eqref{e:hist-congruence}, congruence by $\mathsf g^{-1}$ turns the BB block of $\Lambda_B(E - S)$ into $\kappa \Lambda_B - L + \iu s \Id$, whose inverse has operator norm at most $s^{-1}$, and the congruence changes the superdeterminant only by a sign, since $\det \mathsf g^2 = 1$. In the transported frame, the odd blocks of $E - S$ are the transported odd coordinates, and the Schur complement of the BB block yields
\[
 \Sdet(E - S)^{-1} = \pm \frac{\det(E - S_{FF} - \mathcal N)}{\det(\kappa \Lambda_B - L + \iu s \Id)} ,
\]
where the entries of $\mathcal N$ are quadratic in the transported odd coordinates, with coefficients bounded by $C/s$. The congruence preserves the odd part of $\Str(S^2)$, since $\mathsf g \Lambda_B \mathsf g = \Lambda_B$, and that part has bounded coefficients. Every nonbody coefficient therefore contains at least two $h$-scaled odd coordinates, and hence a factor $h^2$. We expand $\det(E - S_{FF} - \mathcal N)$ by complementary minors, without inverting $E - S_{FF} = \kappa \Id - \iu M$. Let $\mu_1, \dots, \mu_{2k}$ be the eigenvalues of $M$. After a unitary diagonalization of $M$, whose minors have modulus at most one, each minor of $E - S_{FF}$ is bounded by $C_k$ times a product $\prod_{i \in I} |\kappa + \iu \mu_i|$ over a set $I \subset \{1, \dots, 2k\}$. The proof of \zcref{l:end-ff-strip}, with $D = 0$, shows that
\[
 |\kappa + \iu \mu_i| e^{(s^2 - \mu_i^2)/2} \le e^{-c\theta(\mu_i)} \quad (i \in I), \qquad e^{(s^2 - \mu_i^2)/2} \le C e^{-c\theta(\mu_i)} \quad (i \notin I) .
\]
Each minor, times the FF Gaussian, is thus bounded by $C_k e^{-c \vartheta_F}$. Multiplying by the bounded coefficients of $\mathcal N$, the BB body and the damping, and summing over the finitely many monomials, we obtain the bound on the nonbody coefficients. Adding the body bound and using $1 + C_k h^2 \le e^{C_k/W}$, we obtain \eqref{e:end-onsite}.
\end{proof}

\paragraph{Parameter balls.}
In \zcref{s:stab,s:end}, the damping of the onsite weight takes complex values. We call a constant even $(2k|2k)$ supermatrix $\Gamma$ with zero odd blocks, used in the damping $e^{-\iu \eta \Str(S \Gamma)}$ as in \zcref{l:end-onsite}, a \emph{linear damping direction}. For the sources of \zcref{d:rep-sources}, it is the normalized source $\hat \Gamma$ of \eqref{e:hist-normalized}, whose BB and FF blocks are the normalized damping and source matrices. We fix nested complex balls $\mathcal U_0 \Subset \mathcal U_1 \Subset \mathcal U_2$ of linear damping directions around the normalized base point $\Lambda = \eps^{-1} \Gamma_0$ of \zcref{d:rep-sources}, with uniformly bounded FF blocks, such that
\begin{equation}\label{e:end-damping}
 \Re(\Gamma_{BB} \Lambda_B) \ge c_\eps \Id , \qquad \Gamma \in \mathcal U_2 ,
\end{equation}
for some $c_\eps > 0$, and such that $\mathcal U_2$ lies inside the ball $\|\Gamma_{BB} - \Lambda_B\|_{\rm op} \le 1/4$ of \zcref{l:end-onsite}. The order in which these balls are chosen, relative to the other parameters of Part IV, is given in \zcref{tab:end-constants}.

\section{Spectral averaging}\label{s:hist-averaging}

The determinant inequalities reduce the anchored identity to exponential moments of traces of imaginary parts of resolvents of the Gaussian matrix $H^{(n)}$. We bound these by averaging over the diagonal entries. The following lemma is a consequence of Minami's estimate \cite{minami1996local}, in the form proved by Graf and Vaghi \cite[Lemma~2]{graf2007remark}, which applies to determinants of all orders and does not need the symmetry of the matrix. (We use the numbering of the arXiv version of \cite{graf2007remark}.)

\begin{lemma}\label{l:hist-averaging}
Let $A$ be a random Hermitian $n \times n$ matrix whose diagonal entries are, conditionally on the off-diagonal entries, independent real random variables with densities bounded by $\mathsf d_1, \dots, \mathsf d_n$. Let $\eta, c > 0$, let $Y$ be a deterministic positive definite matrix with $Y \ge c \eta \Id$, let $Z_0$ be a deterministic real diagonal matrix, and set $R = (A + Z_0 + \iu Y)^{-1}$. Then for all $t \ge 0$,
\begin{equation}\label{e:hist-averaging}
 \E e^{t \eta \Tr(-\Im R)} \le \exp\Bigl(\pi c \bigl(e^{t/c} - 1\bigr) \eta \sum_{i = 1}^n \mathsf d_i\Bigr).
\end{equation}
\end{lemma}

\begin{proof}
Since $A + Z_0$ satisfies the same hypothesis as $A$, with the same bounds $\mathsf d_i$, we may assume that $Z_0 = 0$. Then \eqref{e:hist-resolvent} implies that
\[
 0 \le c \eta (-\Im R) \le c \eta Y^{-1} \le \Id.
\]

For $0 \le x \le 1$ and $\lambda \ge 0$, we have $e^{\lambda x} \le 1 + (e^\lambda - 1) x$, by convexity. We apply this bound to the eigenvalues of $c \eta(-\Im R)$, with $\lambda = t/c$, and we expand the resulting determinant into principal minors. We obtain
\[
 e^{t \eta \Tr(-\Im R)} \le \det\bigl[\Id + (e^{t/c} - 1) c \eta (-\Im R)\bigr] = \sum_{I \subset [n]} \bigl((e^{t/c} - 1) c \eta\bigr)^{|I|} \det\bigl[(-\Im R)_{II}\bigr],
\]
where $(-\Im R)_{II}$ is the principal submatrix with rows and columns in $I$, and the term with $I = \emptyset$ is one. We will show that
\begin{equation}\label{e:hist-minami}
 \E \det\bigl[(-\Im R)_{II}\bigr] \le \pi^{|I|} \prod_{i \in I} \mathsf d_i \qquad \text{for all } I \subset [n].
\end{equation}
Inserting \eqref{e:hist-minami} into the sum, we then find
\[
 \E e^{t \eta \Tr(-\Im R)} \le \prod_{i = 1}^n \bigl[1 + \pi (e^{t/c} - 1) c \eta \mathsf d_i\bigr] \le \exp\Bigl(\pi c (e^{t/c} - 1) \eta \sum_{i = 1}^n \mathsf d_i\Bigr).
\]

To prove \eqref{e:hist-minami}, let $I \ne \emptyset$, and condition on all entries of $A$ except the diagonal entries $a_i$ with $i \in I$; let $V_I = \diag(a_i)_{i \in I}$. Recall from \eqref{e:hist-resolvent} that $\Im R^* = -\Im R = R Y R^*$, where $R^* = (A - \iu Y)^{-1}$, and this matrix is positive definite. It follows that the principal submatrix $C_I = (R^*)_{II}$ has positive definite imaginary part as well. By the Schur complement formula, $C_I = (V_I - Z_I)^{-1}$, where the matrix $Z_I$ is a function of the entries we conditioned on. Moreover,
\[
 \Im Z_I = -\Im(C_I^{-1}) = C_I^{-1} (\Im C_I) (C_I^{-1})^*
\]
is positive definite.

By \cite[Lemma~2]{graf2007remark}, we have
\[
 \int_{\R^{|I|}} \det\bigl[\Im (V_I - Z_I)^{-1}\bigr] \prod_{i \in I} \dd a_i \le \pi^{|I|}
\]
for every square matrix $Z_I$ with positive definite imaginary part $\Im Z_I = (Z_I - Z_I^*)/(2\iu)$. Recall that $\Im C_I = (-\Im R)_{II}$, so that the integrand is
\[
 \det \Im (V_I - Z_I)^{-1} = \det \Im C_I = \det(-\Im R)_{II} \ge 0.
\]
The conditional density of $V_I$ is bounded by $\prod_{i \in I} \mathsf d_i$, and \eqref{e:hist-minami} follows.
\end{proof}

We now record the properties of the anchored covariance and of the boundary profile that the averaging uses.

\begin{lemma}\label{l:hist-profile}
Let $n, W \ge 1$, and define $\theta_W > 0$ by $\cosh \theta_W = 1 + (2W^2)^{-1}$, so that $\theta_W = 2 \operatorname{arsinh}(1/(2W))$ as in \zcref{s:intro}. Then the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item For $1 \le i \le n$, we have $\varsigma_i = \cosh(\theta_W(n + \frac12 - i))/\cosh(\theta_W(n + \frac12))$, and in particular $0 < \varsigma_i \le \min(1, 2 e^{-\theta_W i})$. Moreover, $1/(2W) \le \theta_W \le 1/W$ and $\sigma_n \le 4W$.
\item For $1 \le i \le n$, we have $(J_n)_{ii} \ge \min(i, W)/(8W^2)$. Consequently, the diagonal entries of $H^{(n)}$ are independent of the other entries and have densities bounded by $\mathsf d_i = (2\pi (J_n)_{ii})^{-1/2}$, with
\[
 \sum_{i = 1}^n \mathsf d_i \le 3 \bigl(W^{3/2} + n \sqrt W\bigr).
\]
\item For all $\eta > 0$ and $s_0 \in (0, 1]$,
\[
 \Pi_n = \sum_{i = 1}^n \frac{\varsigma_i}{\eta(1 - \varsigma_i) + s_0 \varsigma_i} \le \frac{6W}{s_0} \bigl(1 + \log(1 + \eta^{-1})\bigr).
\]
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with (a). Set $\varsigma_0 = 1$ and $\varsigma_{n + 1} = \varsigma_n$. Recall that
\[
 \varsigma = W^2 J_n e_1, \qquad J_n^{-1} = \Id_n - W^2 \Delta_n + W^2 e_1 e_1^*.
\]
The equation $J_n^{-1} \varsigma = W^2 e_1$ then says that
\[
 \varsigma_{i + 1} + \varsigma_{i - 1} = (2 + W^{-2}) \varsigma_i = 2 \cosh(\theta_W) \varsigma_i, \qquad 1 \le i \le n.
\]

The solutions of this recurrence are linear combinations of $e^{\pm \theta_W i}$, and the conditions $\varsigma_0 = 1$ and $\varsigma_{n + 1} = \varsigma_n$ determine the stated formula. This formula implies that $0 < \varsigma_i \le 1$ and
\[
 \varsigma_i \le \frac{e^{\theta_W(n + 1/2 - i)}}{e^{\theta_W(n + 1/2)}/2} = 2 e^{-\theta_W i}.
\]

Comparing the Taylor series term by term, we have
\[
 \frac{\theta_W^2}{2} \le \cosh \theta_W - 1 \le \frac12 \theta_W^2 \cosh \theta_W.
\]
The first inequality implies that $\theta_W \le 1/W \le 1$. Using $\cosh 1 \le 8/5$, we deduce from the second inequality that
\[
 \frac{1}{2W^2} = \cosh \theta_W - 1 \le \frac45 \theta_W^2,
\]
so that $\theta_W \ge 1/(2W)$. Finally,
\[
 \sigma_n = \sum_{i = 1}^n \varsigma_i \le \sum_{i \ge 1} 2 e^{-\theta_W i} \le \frac{2}{\theta_W} \le 4W.
\]

For (b), let $\ell = \min(i, W)$, and define $f \in \R^n$ by
\[
 f_j = \max\bigl(0, 1 - |j - i|/\ell\bigr), \qquad 1 \le j \le n.
\]
By the variational characterization of the diagonal entries of the inverse of a positive definite matrix, we have
\[
 (J_n)_{ii} \ge \frac{f_i^2}{\langle f, J_n^{-1} f \rangle}, \qquad
 \langle f, J_n^{-1} f \rangle = \sum_{j = 1}^n f_j^2 + W^2 \sum_{j = 1}^{n - 1} (f_{j + 1} - f_j)^2 + W^2 f_1^2.
\]
We have $f_i = 1$, and $f$ vanishes outside an interval of length $2\ell$ about $i$. In particular, the first sum is bounded by $2\ell + 1 \le 3\ell$, and the second sum has at most $2\ell + 2 \le 4\ell$ nonzero terms, each bounded by $\ell^{-2}$.

Moreover, $f_1 \le 1/\ell$. Indeed, if $\ell = i$, then $f_1 = 1/i$, and if $\ell = W < i$, then $(i - 1)/W > 1 - 1/W$ and $f_1 < 1/W$. Using $1 \le \ell \le W$, we find
\[
 \langle f, J_n^{-1} f \rangle \le 3\ell + \frac{4W^2}{\ell} + \frac{W^2}{\ell^2} \le \frac{8 W^2}{\ell},
\]
which implies the lower bound on $(J_n)_{ii}$. The matrix $H^{(n)}$ has independent entries up to the Hermitian symmetry, and $H^{(n)}_{ii}$ is a real centered Gaussian variable with variance $(J_n)_{ii}$, whose density is bounded by $\mathsf d_i$. We conclude that
\[
 \sum_{i = 1}^n \mathsf d_i \le \frac{2W}{\sqrt \pi} \sum_{i = 1}^n \min(i, W)^{-1/2} \le \frac{2W}{\sqrt \pi} \bigl(2 \sqrt W + n W^{-1/2}\bigr) \le 3\bigl(W^{3/2} + n \sqrt W\bigr).
\]

For (c), set
\[
 i_0 = \bigl\lceil \theta_W^{-1} \log(4(1 + \eta^{-1})) \bigr\rceil.
\]
Each term of $\Pi_n$ is bounded by $s_0^{-1}$. Therefore the terms with $i \le i_0$ contribute at most
\[
 \frac{i_0}{s_0} \le \frac{2W \log(4(1 + \eta^{-1})) + 1}{s_0} \le \frac{W}{s_0} \bigl(4 + 2 \log(1 + \eta^{-1})\bigr).
\]

For $i > i_0$, part (a) gives
\[
 \varsigma_i \le 2 e^{-\theta_W i_0} \le \frac{\eta}{2(1 + \eta)} \le \frac12,
\]
so that $1 - \varsigma_i \ge 1/2$ and
\[
 \frac{\varsigma_i}{\eta(1 - \varsigma_i) + s_0 \varsigma_i} \le \frac{2 \varsigma_i}{\eta} \le \frac{4 e^{-\theta_W i}}{\eta}.
\]
These terms therefore contribute at most
\[
 \frac{4}{\eta} \sum_{i > i_0} e^{-\theta_W i} \le \frac{4}{\eta} \cdot \frac{e^{-\theta_W i_0}}{e^{\theta_W} - 1} \le \frac{4}{\eta} \cdot \frac{\eta}{4 \theta_W} \le 2W \le \frac{2W}{s_0}.
\]
Adding the two contributions proves (c).
\end{proof}

\section{Coefficient bounds}\label{s:hist-coefficients}

The anchored identity contains the odd coordinates of the free site $S$ and, after differentiation, the local sources. The next lemma bounds the coefficients of determinants and inverses whose body is controlled, relative to a determinant that may vanish. We state it for a finite commutative algebra with a monomial basis, which covers the even part of a Grassmann algebra tensored with truncated polynomials in the source variables.

Let $\mathcal A$ be a finite-dimensional commutative algebra over $\C$ with a basis of \emph{monomials} that contains $1$, such that the product of two monomials is a monomial, minus a monomial, or zero, and such that every product of $m_{\mathcal A} + 1$ monomials different from $1$ vanishes. For $a \in \mathcal A$, let $|a|_1$ be the sum of the absolute values of the coefficients of $a$ in this basis. For a matrix $\mathcal N = \sum_\mu \mu \, \mathcal N_\mu$ with entries in $\mathcal A$, where $\mu$ runs over the monomials and the $\mathcal N_\mu$ are complex matrices, let $\|\mathcal N\|_1 = \sum_\mu \|\mathcal N_\mu\|$. These norms are submultiplicative, by the argument of \zcref{l:grass-norm}(a). We say that $\mathcal N$ has no constant term if $\mathcal N_1 = 0$.

\begin{lemma}\label{l:hist-coefficients}
In this setting, let $\mathcal N$ be a $d \times d$ matrix with entries in $\mathcal A$ and no constant term.
\begin{enumerate}[label=(\alph*)]
\item Let $A$ and $Y$ be Hermitian $d \times d$ matrices, and let $Y_+$ be a positive definite $d \times d$ matrix with $-Y_+ \le Y \le Y_+$. Then
\begin{equation}\label{e:hist-coef-det}
 |\det(A + \iu Y + \mathcal N)|_1 \le |\det(A + \iu Y_+)| \bigl(1 + d \|\mathcal N\|_1 \|Y_+^{-1}\|\bigr)^{m_{\mathcal A}}.
\end{equation}
\item Let $\mathbf B$ be an invertible complex $d \times d$ matrix, and set $\alpha = \|\mathbf B^{-1}\| \, \|\mathcal N\|_1$. Then $\mathbf B - \mathcal N$ is invertible over $\mathcal A$, and
\begin{equation}\label{e:hist-coef-inverse}
 \|(\mathbf B - \mathcal N)^{-1}\|_1 \le \|\mathbf B^{-1}\| (1 + \alpha)^{m_{\mathcal A}}.
\end{equation}
If, moreover, every $\mathcal N_\mu$ has rank at most $m_0$, then
\begin{equation}\label{e:hist-coef-ratio}
 \Bigl|\frac{\det \mathbf B}{\det(\mathbf B - \mathcal N)}\Bigr|_1 \le \bigl(1 + (m_0 + m_{\mathcal A}) \alpha\bigr)^{m_{\mathcal A}}.
\end{equation}
\end{enumerate}
\end{lemma}

The case $\mathcal N = 0$ of \eqref{e:hist-coef-det} is the bound $|\det(A + \iu Y)| \le |\det(A + \iu Y_+)|$, which holds also when $\det(A + \iu Y) = 0$.

\begin{proof}
Each of the three quantities is obtained by substituting $t_\mu = \mu$ into a formal power series in commuting variables $t_\mu$, one for each monomial $\mu \ne 1$, whose constant term is the value at $\mathcal N = 0$. This substitution is a ring homomorphism from formal power series to $\mathcal A$, since only the terms of total degree at most $m_{\mathcal A}$ survive it. Since every product of monomials has norm at most one, the norm $| \cdot |_1$ of the result is bounded by the sum of the absolute values of the coefficients of total degree at most $m_{\mathcal A}$.

We begin with (a). Set
\[
\begin{gathered}
 A' = Y_+^{-1/2} A Y_+^{-1/2}, \\
 Y' = Y_+^{-1/2} Y Y_+^{-1/2}, \\
 Q_+ = \tfrac12 (\Id + Y'), \\
 U = (A' + \iu \Id)^{-1}(A' - \iu \Id).
\end{gathered}
\]
Then $0 \le Q_+ \le \Id$, $U$ is unitary, and
\[
 (A' + \iu \Id)^{-1}(A' + \iu Y') = Q_+ + U(\Id - Q_+).
\]
Let $(e_a)$ be an orthonormal eigenbasis of $Q_+$, with eigenvalues $q_a \in [0, 1]$. In this basis, the $a$th column of this matrix is
\[
 q_a e_a + (1 - q_a) U e_a,
\]
which has norm at most one.

The matrices
\[
 \mathcal N_\mu' = (A' + \iu \Id)^{-1} Y_+^{-1/2} \mathcal N_\mu Y_+^{-1/2}
\]
have norm at most $\|\mathcal N_\mu\| \|Y_+^{-1}\|$, and we have
\[
 \det\Bigl(A + \iu Y + \sum_\mu t_\mu \mathcal N_\mu\Bigr) = \det(A + \iu Y_+) \det\Bigl(Q_+ + U(\Id - Q_+) + \sum_\mu t_\mu \mathcal N_\mu'\Bigr),
\]
where we used the identity
\[
 \det Y_+ \det(A' + \iu \Id) = \det\bigl(Y_+^{1/2} (A' + \iu \Id) Y_+^{1/2}\bigr) = \det(A + \iu Y_+) .
\]

We expand the last determinant by columns in the basis $(e_a)$. Let $\nu = (\nu_\mu)$ be a multi-index of total degree $r$. The coefficient of $\prod_\mu t_\mu^{\nu_\mu}$ is a sum of $d!/((d - r)! \prod_\mu \nu_\mu!)$ determinants, in which $\nu_\mu$ of the columns are columns of $\mathcal N_\mu'$ and the others are columns of $Q_+ + U(\Id - Q_+)$. By Hadamard's inequality, each of them is bounded in modulus by $\prod_\mu (\|\mathcal N_\mu\| \|Y_+^{-1}\|)^{\nu_\mu}$. Summing over $\nu$ with $|\nu| = r$, and using the multinomial theorem and $d!/(d - r)! \le d^r$, we find that the coefficients of degree $r$ contribute at most
\[
 |\det(A + \iu Y_+)| \, \frac{(d \|\mathcal N\|_1 \|Y_+^{-1}\|)^r}{r!}.
\]
Summing over $r \le m_{\mathcal A}$ and using $1/r! \le \binom{m_{\mathcal A}}{r}$, we obtain \eqref{e:hist-coef-det}.

We now prove (b). Set $\widetilde{\mathcal N}_\mu = \mathbf B^{-1} \mathcal N_\mu$, so that
\[
 \sum_\mu \|\widetilde{\mathcal N}_\mu\| \le \|\mathbf B^{-1}\| \sum_\mu \|\mathcal N_\mu\| = \alpha.
\]

Every entry of $(\mathbf B^{-1} \mathcal N)^{m_{\mathcal A} + 1}$ is a sum of products of $m_{\mathcal A} + 1$ monomials different from $1$, and therefore
\[
 (\mathbf B^{-1} \mathcal N)^{m_{\mathcal A} + 1} = 0.
\]
It follows that the finite geometric series
\[
 (\mathbf B - \mathcal N)^{-1} = \sum_{p = 0}^{m_{\mathcal A}} (\mathbf B^{-1} \mathcal N)^p \mathbf B^{-1}
\]
is the inverse. Its norm satisfies
\[
 \|(\mathbf B - \mathcal N)^{-1}\|_1 \le \|\mathbf B^{-1}\| \sum_{p = 0}^{m_{\mathcal A}} \alpha^p \le \|\mathbf B^{-1}\| (1 + \alpha)^{m_{\mathcal A}},
\]
which is \eqref{e:hist-coef-inverse}.

Next, we prove \eqref{e:hist-coef-ratio}. As formal power series,
\[
 \frac{\det \mathbf B}{\det\bigl(\mathbf B - \sum_\mu t_\mu \mathcal N_\mu\bigr)} = \det\Bigl(\Id - \sum_\mu t_\mu \widetilde{\mathcal N}_\mu\Bigr)^{-1} = \exp\Bigl(\sum_{l \ge 1} \frac1l \Tr\Bigl(\sum_\mu t_\mu \widetilde{\mathcal N}_\mu\Bigr)^l\Bigr).
\]

A product $\widetilde{\mathcal N}_{\mu_1} \cdots \widetilde{\mathcal N}_{\mu_l}$ has rank at most $m_0$, and therefore its trace has modulus at most $m_0 \prod_i \|\widetilde{\mathcal N}_{\mu_i}\|$. The coefficients of the exponent are therefore dominated in modulus by those of the series
\[
 -m_0 \log\Bigl(1 - \sum_\mu \|\widetilde{\mathcal N}_\mu\| t_\mu\Bigr) = m_0 \sum_{l \ge 1} \frac1l \Bigl(\sum_\mu \|\widetilde{\mathcal N}_\mu\| t_\mu\Bigr)^l .
\]
Since the exponential has nonnegative Taylor coefficients, the coefficients of the left side are dominated by those of $(1 - \sum_\mu \|\widetilde{\mathcal N}_\mu\| t_\mu)^{-m_0}$. For $p \le m_{\mathcal A}$, the coefficients of total degree $p$ of the latter series sum to
\[
 \binom{m_0 + p - 1}{p} \Bigl(\sum_\mu \|\widetilde{\mathcal N}_\mu\|\Bigr)^p \le \frac{((m_0 + m_{\mathcal A}) \alpha)^p}{p!}.
\]
Summing over $p \le m_{\mathcal A}$ as in part (a), we obtain \eqref{e:hist-coef-ratio}.
\end{proof}

We now apply this lemma to the onsite jets. The following bound is far from optimal near the saddle manifold, where \zcref{l:end-onsite} is sharper. However, it holds on the whole cycle. We use it to bound the first step $F_1 = w_1$ of a chain, whose norm is an integral over all of $\mathcal S$.

\begin{lemma}\label{l:hist-marked-onsite}
Let $m \ge 0$ and $\eta \in (0, 1]$, and let $w = w_\Gamma[Y_1, \dots, Y_m]$ be an onsite jet. Set $A_1 = 2(4k^2 + m)^2$. Then at every point of $\mathcal S$,
\begin{equation}\label{e:hist-marked-onsite}
 |w|_{1; h} \le C v_{\mathcal S}^{A_1} \exp\Bigl[-\frac14 \|P_0\|_{\rm HS}^2 - \frac18 \|M\|_{\rm HS}^2 - \eta v_{\mathcal S}\Bigr],
\end{equation}
where $C$ depends only on $k$, $m$, and $C_\Gamma$. In particular,
\[
 \sup_{\mathcal S} |w|_{1; h} \le C \eta^{-A_1}, \qquad \|w\|_{\mathcal H_{-p}(\mathcal S)} \le C \eta^{-A_1 - k^2} \quad \text{for all } p \ge 0.
\]
\end{lemma}

\begin{proof}
Let $S \in \mathcal S$. We work in the algebra $\mathcal A$ obtained from the even part of the Grassmann algebra generated by the transported odd coordinates \eqref{e:hist-transported} by adjoining commuting variables $t_1, \dots, t_m$ with $t_l^2 = 0$. Its monomials are the products of $h$-scaled monomials of even degree with monomials in the $t_l$, so that $| \cdot |_1 = | \cdot |_{1; h}$ on the even Grassmann part, and $m_{\mathcal A} = 4k^2 + m$. The onsite jet $w$ is the coefficient of $t_1 \cdots t_m$ in
\[
 \tilde w = e^{-\Str(S^2)/2 - \iu \eta \Str(S \hat \Gamma)} \, \Sdet\Bigl(E - S + \sum_l t_l Y_l\Bigr)^{-1},
\]
so that $|w|_{1; h} \le |\tilde w|_1$. We will bound the superdeterminant and the exponential factor in turn.

\textit{The BB block.} Recall from \eqref{e:hist-blocks} that
\[
 E - S_{BB} = \kappa \Id - P_0 + \iu s \Lambda_B \mathsf g^2.
\]
By \eqref{e:hist-congruence} with $t = \kappa$, we have
\[
 E \Id - S_{BB} = \Lambda_B \mathsf g B_0 \mathsf g, \qquad B_0 = \kappa \Lambda_B - L + \iu s \Id, \qquad L = \mathsf g^{-1} \Lambda_B P_0 \mathsf g^{-1}.
\]
Therefore the BB block of $E - S + \sum_l t_l Y_l$ is
\[
 \mathbf B_t = \Lambda_B \mathsf g (B_0 - \mathcal N_B) \mathsf g, \qquad \mathcal N_B = -\sum_{l = 1}^m t_l \, \mathsf g^{-1} \Lambda_B Y_l^B \mathsf g^{-1}.
\]

Since $\kappa \Lambda_B - L$ is Hermitian, the first bound in \eqref{e:hist-resolvent} implies that $\|B_0^{-1}\| \le s^{-1} \le 2$. By \eqref{e:hist-boost-props}, $\|\mathcal N_B\|_1 \le 3 m v_{\mathcal S}$, and every coefficient of $\mathcal N_B$ has rank at most $2k$. We apply \zcref{l:hist-coefficients}(b) with $\mathbf B = B_0$ and $\mathcal N = \mathcal N_B$, for which $\alpha \le 6 m v_{\mathcal S}$, and the lemma yields
\begin{equation}\label{e:hist-marked-BB}
 \Bigl|\frac{\det B_0}{\det(B_0 - \mathcal N_B)}\Bigr|_1 \le (1 + C v_{\mathcal S})^{m_{\mathcal A}}, \qquad \|(B_0 - \mathcal N_B)^{-1}\|_1 \le 2(1 + 6m v_{\mathcal S})^{m_{\mathcal A}}.
\end{equation}

\textit{The Schur complement.} By \eqref{e:hist-transported}, we have $S_{BF} = \Lambda_B \mathsf g O_1$ and $S_{FB} = O_2 \mathsf g$. By \zcref{l:grass-sdet}(a),
\[
 \Sdet\Bigl(E - S + \sum_l t_l Y_l\Bigr)^{-1} = \frac{\det\bigl(\mathbf F_t - O_2 \mathsf g \mathbf B_t^{-1} \Lambda_B \mathsf g O_1\bigr)}{\det \mathbf B_t}, \qquad \mathbf F_t = \kappa \Id - \iu M + \sum_l t_l Y_l^F.
\]
Here
\[
 \mathsf g \mathbf B_t^{-1} \Lambda_B \mathsf g = (B_0 - \mathcal N_B)^{-1}, \qquad \det \mathbf B_t = \det(\Lambda_B) \det(B_0 - \mathcal N_B),
\]
where we used $\det \mathsf g = 1$. Thus the numerator is the determinant of $\kappa \Id - \iu M + \mathcal N_F$, where
\[
 \mathcal N_F = \sum_l t_l Y_l^F - O_2 (B_0 - \mathcal N_B)^{-1} O_1
\]
has no constant term.

Each odd term of $\mathcal N_F$ is the product of two transported odd coordinates, each of norm $h$, with a matrix of norm at most $\|(B_0 - \mathcal N_B)^{-1}\|_1$. By \eqref{e:hist-marked-BB}, we have
\[
 \|\mathcal N_F\|_1 \le m + (2k)^4 h^2 \|(B_0 - \mathcal N_B)^{-1}\|_1 \le C (1 + v_{\mathcal S})^{m_{\mathcal A}}.
\]
We apply \zcref{l:hist-coefficients}(a) to the matrix
\[
 M + \iu \kappa \Id + \iu \mathcal N_F = \iu(\kappa \Id - \iu M + \mathcal N_F),
\]
with $A = M$, $Y = \kappa \Id$, and $Y_+ = \Id$. Writing $\mu_1, \dots, \mu_{2k}$ for the eigenvalues of $M$, we find
\[
 \bigl|\det(\kappa \Id - \iu M + \mathcal N_F)\bigr|_1 \le |\det(M + \iu \Id)| \bigl(1 + 2k \|\mathcal N_F\|_1\bigr)^{m_{\mathcal A}} \le C v_{\mathcal S}^{m_{\mathcal A}^2} \prod_{j = 1}^{2k} (1 + \mu_j^2)^{1/2}.
\]
Combining this with \eqref{e:hist-marked-BB} and $|\det B_0| = |\det(E - S_{BB})|$, we obtain
\begin{equation}\label{e:hist-marked-sdet}
 \Bigl|\Sdet\Bigl(E - S + \sum_l t_l Y_l\Bigr)^{-1}\Bigr|_1 \le \frac{C v_{\mathcal S}^{A_1}}{|\det(E - S_{BB})|} \prod_{j = 1}^{2k} (1 + \mu_j^2)^{1/2}.
\end{equation}

\textit{The exponential factor.} We have
\begin{equation}\label{e:hist-odd-trace}
\begin{gathered}
 \Str(S^2) = \Tr S_{BB}^2 - \Tr S_{FF}^2 + 2 \Tr(S_{BF} S_{FB}), \\
 \Tr(S_{BF} S_{FB}) = \Tr(\Lambda_B \mathsf g O_1 O_2 \mathsf g) = \Tr(\Lambda_B O_1 O_2),
\end{gathered}
\end{equation}
where we used $\mathsf g \Lambda_B \mathsf g = \Lambda_B$ in the last identity. The odd trace has norm at most $4k^2 h^2 \le 4k^2$, and \zcref{l:grass-norm}(e) shows that the odd factor $e^{-\Tr(S_{BF} S_{FB})}$ has norm at most $e^{4k^2}$. The damping involves only the even blocks of $S$, since $\hat \Gamma$ has zero odd blocks.

We now bound the body. The computation in the proof of \zcref{l:hist-coercivity} gives
\[
 \frac{|e^{-\Tr S_{BB}^2/2}|}{|\det(E - S_{BB})|} = e^{-V_\kappa(P_0) - k(\kappa^2 - s^2)}, \qquad |e^{\Tr S_{FF}^2/2}| = e^{k \kappa^2 - \Tr M^2/2}.
\]
For the damping, we have
\[
 \Str(S \hat \Gamma) = \Tr(S_{BB} \Lambda_B) - \Tr(S_{FF} \hat \Gamma_F), \qquad \Tr(\Lambda_B \mathsf g^2 \Lambda_B) = \Tr \mathsf g^2 = s^{-1}(v_{\mathcal S} - 1),
\]
where we used \eqref{e:hist-boost-props}. Since $\Tr(P_0 \Lambda_B)$ is real, we find
\[
 \Re\bigl(-\iu \eta \Tr(S_{BB} \Lambda_B)\bigr) = -\eta (v_{\mathcal S} - 1), \qquad |\eta \Tr(S_{FF} \hat \Gamma_F)| \le \eta C_\Gamma (2k + \|M\|_1).
\]

Using these bounds, \eqref{e:hist-marked-sdet}, $\eta \le 1$, and \zcref{l:hist-coercivity} in the form $V_\kappa(P_0) \ge \frac14 \|P_0\|_{\rm HS}^2$, we conclude that
\[
 |\tilde w|_1 \le C v_{\mathcal S}^{A_1} \exp\Bigl[-\frac14 \|P_0\|_{\rm HS}^2 - \eta v_{\mathcal S} + C_\Gamma \|M\|_1\Bigr] \prod_{j = 1}^{2k} e^{(s^2 - \mu_j^2)/2} (1 + \mu_j^2)^{1/2}.
\]
Finally, for real $\mu$ we have
\[
 e^{-\mu^2/2 + C_\Gamma |\mu|} (1 + \mu^2)^{1/2} \le C e^{-\mu^2/8},
\]
and \eqref{e:hist-marked-onsite} follows.

The supremum bound follows from \eqref{e:hist-marked-onsite} and
\[
 \sup_{v_{\mathcal S} \ge 1} v_{\mathcal S}^{A_1} e^{-\eta v_{\mathcal S}} \le \Bigl(\frac{A_1}{e \eta}\Bigr)^{A_1}.
\]
To bound the norm, we use $|w|_{2; h} \le |w|_{1; h}$ and $v_{\mathcal S}^{-2p} \le 1$, we integrate the Gaussian factors in $P_0$ and $M$, and we use $v_{\mathcal S} \ge 1 + \|X\|_{\rm HS}$, which follows from \zcref{l:rep-cycle}(a). Since $X$ ranges over a real vector space of dimension $2k^2$, we obtain
\[
 \|w\|_{\mathcal H_{-p}(\mathcal S)}^2 \le C \int v_{\mathcal S}^{2A_1} e^{-2\eta v_{\mathcal S}} \dd X \le C \eta^{-2A_1 - 2k^2}. \qedhere
\]
\end{proof}

\section{The a priori bound on partial chains}\label{s:hist-bound}

We now state the main result of this section. Recall that $r_W = h \log W$, as in \eqref{e:hist-scales}. For a constant $C_1 > 0$, we define
\begin{equation}\label{e:hist-EW}
 \mathcal E_W(\eta) = C_1 \Bigl(\sqrt W + r_W W \bigl(1 + \log(1 + \eta^{-1})\bigr) + \eta W + N \eta^2\Bigr).
\end{equation}

\begin{theorem}\label{t:hist-bound}
Fix integers $k \ge 1$ and $d \ge 0$, and fix $C_\Gamma, C_X \ge 1$. There exist $p, A, C_1$ and $W_0$, depending only on $k$, $d$, $C_\Gamma$, and $C_X$, such that the following holds for all $W \ge W_0$. Let $N \le C_X W^2$, $|E| \le 1/2$, and $0 < \eta \le C_X W^{-2}$, and let $\hat \Gamma$ be a normalized source as in \eqref{e:hist-normalized}, with $\Tr \hat \Gamma_F = 0$ and $\|\hat \Gamma_F\| \le C_\Gamma$. Then every marked partial chain $F_j$ of length $1 \le j \le N$ and total order at most $d$ satisfies
\begin{equation}\label{e:hist-bound}
 \|F_j\|_{\mathcal H_{-p}(\mathcal S)} + \|K_{\tau_{\rm phys}} F_j\|_{\mathcal H_{-p}(\mathcal S)} \le W^A (1 + \eta^{-1})^A e^{\mathcal E_W(\eta)},
\end{equation}
uniformly in the positions and the directions of the marks.
\end{theorem}

We have $r_W W = \sqrt W \log W$ and $N \eta^2 \le C_X^3 W^{-2}$. If moreover $\eta \ge W^{-C}$ for some constant $C$, then $\log(1 + \eta^{-1}) \le (C + 1) \log W$, and the right side of \eqref{e:hist-bound} is at most $\exp(C \sqrt W (\log W)^2) = e^{o(W)}$. This is the case in the applications of \zcref{s:assembly}, where $\eta = \eps/a_N \asymp W^{-2}$, so that $(1 + \eta^{-1})^A$ is a power of $W$. Every marked chain in the analytic transpose order of \zcref{d:rep-chain} is of the form $K_{\tau_{\rm phys}} F_j$, so the theorem also covers that order.

The proof combines two estimates. The first is a bound on the coefficients of the anchored marked chains on the \emph{core}
\begin{equation}\label{e:hist-core}
 \mathcal K_W = \bigl\{ S \in \mathcal S : \|P_0\|_{\rm HS} \le r_W, \ \bigl||\mu| - s\bigr| \le r_W \text{ for every eigenvalue } \mu \text{ of } M \bigr\},
\end{equation}
which contains a neighborhood of the BB saddle orbit in every FF signature of the saddle manifold. The second is the contraction of one step away from the core. In the proofs below we assume that $W$ is so large that $\eta \le C_X W^{-2} \le s/2$ and $r_W \le 1/2$.

\subsection{Local sources in the anchored identity}

To bound marked chains on the core, we need the anchored identity \eqref{e:hist-anchored} with local sources inserted, which is the second part of \zcref{p:rep-anchored}. We restate it in the form used below. Let $F_n$ be a marked partial chain whose marked step $j_a$ carries the directions $Y_{a, 1}, \dots, Y_{a, m_a}$, and let $i_a = n + 1 - j_a$ be the corresponding site of the anchored chain. We introduce commuting variables $t_{a, l}$ with $t_{a, l}^2 = 0$, and we set
\begin{equation}\label{e:hist-Mn-marked}
 \mathbb M_n^t(S) = \mathbb M_n(S) + \sum_{a, l} t_{a, l} \, e_{i_a} e_{i_a}^* \otimes Y_{a, l},
\end{equation}
where $e_i$ is the $i$th standard basis vector of $\C^n$.

\begin{lemma}\label{l:hist-jets}
Let $n \ge 1$, let $F_n$ be a marked partial chain, and let $S \in \mathcal S$. Then $(K_{\tau_{\rm phys}} F_n)(S)$ is the coefficient of $\prod_{a, l} t_{a, l}$ in
\begin{equation}\label{e:hist-jet-HS}
 \mathcal G_n(S) \, \E \Sdet\bigl[\mathbb M_n^t(S)\bigr]^{-1},
\end{equation}
where the expectation is taken coefficientwise.
\end{lemma}

\begin{proof}
For a complex vector $\zeta = (\zeta_{a, l})$, let $\mathbb M_n^\zeta(S)$ be defined by \eqref{e:hist-Mn-marked} with $\zeta$ in place of $t$. The function $K_{\tau_{\rm phys}} F_n$ is the function $\ell_n^{(\alpha)}$ of \zcref{p:rep-anchored}, for the derivatives $\partial_{\zeta_{a, 1}} \cdots \partial_{\zeta_{a, m_a}}$ at $\zeta = 0$ in the directions $Y_{a, 1}, \dots, Y_{a, m_a}$ at the site $i_a$, which are linear combinations of the derivatives in the entries of $Y_{i_a}$. By \zcref{p:rep-anchored}, we therefore have
\[
 (K_{\tau_{\rm phys}} F_n)(S) = \partial_\zeta^{\mathbf 1} \Bigl[\mathcal G_n(S) \, \E \Sdet\bigl[\mathbb M_n^\zeta(S)\bigr]^{-1}\Bigr]\Big|_{\zeta = 0} ,
\]
where $\partial_\zeta^{\mathbf 1}$ is the mixed derivative of first order in each $\zeta_{a, l}$.

It remains to identify this derivative with the coefficient of $\prod t_{a, l}$ in \eqref{e:hist-jet-HS}. After multiplication by $\Id_n \otimes \Lambda_B$, the BB block of $\mathbb M_n(S)$ has imaginary part at least $\min_i \eta (1 - \varsigma_i) > 0$, uniformly in $H^{(n)}$, by the remark after \zcref{p:rep-anchored}. Consider the polydisk
\[
 \mathcal D_\zeta = \Bigl\{ \zeta : \sum_l |\zeta_{a, l}| \le \tfrac12 \min_i \eta(1 - \varsigma_i) \text{ for all } a \Bigr\}.
\]

The local BB sources keep half of this margin on $\mathcal D_\zeta$. There, $\Sdet[\mathbb M_n^\zeta(S)]^{-1}$ is holomorphic in $\zeta$ for every $H^{(n)}$, the inverse of its BB block is bounded uniformly in $H^{(n)}$, and its coefficients are bounded by a polynomial in $\|H^{(n)}\|$. Dominated convergence therefore allows us to differentiate under the expectation. Finally, for a function holomorphic near $\zeta = 0$, the mixed derivative $\partial_\zeta^{\mathbf 1}$ at zero is the coefficient of $\prod t_{a, l}$ in its Taylor expansion evaluated at $\zeta = t$, since $t_{a, l}^2 = 0$. This completes the proof.
\end{proof}

\subsection{The bound on the core}

\begin{lemma}\label{l:hist-core}
In the setting of \zcref{t:hist-bound}, there exist $A$, $p_0$, and $C_1$, depending only on $k$, $d$, $C_\Gamma$, and $C_X$, such that for all marked partial chains $F_n$ with $0 \le n \le N - 1$ and total order at most $d$, and all $S \in \mathcal K_W$,
\begin{equation}\label{e:hist-core-bound}
 |(K_{\tau_{\rm phys}} F_n)(S)|_{1; h} \le W^A (1 + \eta^{-1})^A v_{\mathcal S}^{p_0} e^{\mathcal E_W(\eta)},
\end{equation}
with $\mathcal E_W(\eta)$ given by \eqref{e:hist-EW}.
\end{lemma}

\begin{proof}
For $n = 0$ we have $K_{\tau_{\rm phys}} F_0 = 1$, and there is nothing to prove. Let $n \ge 1$ and $S \in \mathcal K_W$. Let $\mathcal A$ be the algebra of the proof of \zcref{l:hist-marked-onsite}, with the variables $t_{a, l}$ of \eqref{e:hist-Mn-marked} in place of the $t_l$, so that $m_{\mathcal A} \le 4k^2 + d$. By \zcref{l:hist-jets}, it suffices to bound the norm $| \cdot |_1$ in $\mathcal A$ of \eqref{e:hist-jet-HS}. Since the expectation is taken coefficientwise, this norm is at most
\[
 |\mathcal G_n(S)|_{1; h} \, \E \bigl|\Sdet[\mathbb M_n^t(S)]^{-1}\bigr|_1.
\]
We bound the second factor in four steps, and the first factor at the end. Throughout the proof, we write $D_\varsigma = \eta(\Id_n - \varsigma)$, and we recall that $\|D_\varsigma\| \le \eta$.

\textit{The blocks.} Using \eqref{e:hist-blocks} and $\hat \Gamma_{BB} = \Lambda_B$, we see that the BB block of $\mathbb M_n(S)$ is the matrix $\mathbf B$ of \eqref{e:hist-B}, with
\begin{equation}\label{e:hist-core-A}
 A = E \Id_n - \kappa \varsigma - H^{(n)}, \qquad Z = P_0, \qquad s_0 = s, \qquad r = r_W.
\end{equation}

We write the FF block of $\mathbb M_n(S)$ as $\mathbf F = X_F + \iu Y_F$, with the Hermitian matrices
\[
 X_F = A \otimes \Id + D_\varsigma \otimes \gamma_1, \qquad Y_F = D_\varsigma \otimes \gamma_2 - \varsigma \otimes M,
\]
where $\gamma_1$ and $\gamma_2$ are as in \eqref{e:hist-ff-source}, and the odd blocks of $\mathbb M_n(S)$ are given by $-\varsigma \otimes S_{BF}$ and $-\varsigma \otimes S_{FB}$. In $\mathbb M_n^t(S)$, the blocks $\mathbf B$ and $\mathbf F$ are replaced by
\[
 \mathbf B_t = \mathbf B + \sum_{a, l} t_{a, l} \, e_{i_a} e_{i_a}^* \otimes Y_{a, l}^B, \qquad \mathbf F_t = \mathbf F + \sum_{a, l} t_{a, l} \, e_{i_a} e_{i_a}^* \otimes Y_{a, l}^F.
\]

By \zcref{l:hist-bb} and \eqref{e:hist-boost-props}, $\mathbf B$ is invertible, uniformly in $H^{(n)}$, with
\begin{equation}\label{e:hist-core-Binv}
 \|\mathbf B^{-1}\| \le \max\{\eta^{-1}, 3 s^{-1} v_{\mathcal S}\} \le 4v_{\mathcal S}/\eta.
\end{equation}

\textit{Odd coefficients and marks.} We apply \zcref{l:hist-coefficients}(b) to $\mathbf B_t = \mathbf B - \mathcal N_B$, where $\|\mathcal N_B\|_1 \le d$ and every coefficient of $\mathcal N_B$ has rank at most $2k$. By \eqref{e:hist-core-Binv}, we have $\alpha \le 4d v_{\mathcal S}/\eta$, and the lemma yields
\begin{equation}\label{e:hist-core-marks}
 \Bigl|\frac{\det \mathbf B}{\det \mathbf B_t}\Bigr|_1 \le \Bigl(1 + \frac{C v_{\mathcal S}}{\eta}\Bigr)^{m_{\mathcal A}}, \qquad \|\mathbf B_t^{-1}\|_1 \le \frac{4v_{\mathcal S}}{\eta} \Bigl(1 + \frac{4d v_{\mathcal S}}{\eta}\Bigr)^{m_{\mathcal A}}.
\end{equation}

By \eqref{e:hist-transported}, we may write the odd blocks as
\[
 -\varsigma \otimes S_{BF} = -(\Id_n \otimes \Lambda_B \mathsf g)(\varsigma \otimes O_1), \qquad -\varsigma \otimes S_{FB} = -(\varsigma \otimes O_2)(\Id_n \otimes \mathsf g) .
\] Then \zcref{l:grass-sdet}(a) yields
\[
 \Sdet\bigl[\mathbb M_n^t(S)\bigr]^{-1} = \frac{\det(\mathbf F + \mathcal N_F)}{\det \mathbf B_t},
\]
where
\[
 \mathcal N_F = \mathbf F_t - \mathbf F - (\varsigma \otimes O_2)(\Id_n \otimes \mathsf g) \mathbf B_t^{-1} (\Id_n \otimes \Lambda_B \mathsf g)(\varsigma \otimes O_1)
\]
has no constant term. Its odd part is a sum of $(2k)^4$ terms, each the product of two transported odd coordinates with a matrix of norm at most $\|\mathsf g\|^2 \|\mathbf B_t^{-1}\|_1$. The two coordinates contribute $h^2 \le 1$, and \eqref{e:hist-boost-props} implies that
\[
 \|\mathsf g\|^2 = \|\mathsf g^2\| \le 3 v_{\mathcal S}.
\]
Using \eqref{e:hist-core-marks}, we find
\begin{equation}\label{e:hist-core-NF}
 \|\mathcal N_F\|_1 \le d + (2k)^4 \cdot 3v_{\mathcal S} \cdot \frac{4v_{\mathcal S}}{\eta} \Bigl(1 + \frac{4d v_{\mathcal S}}{\eta}\Bigr)^{m_{\mathcal A}} \le C \Bigl(\frac{v_{\mathcal S}}{\eta}\Bigr)^{m_{\mathcal A} + 2}.
\end{equation}

\textit{The fermionic determinant.} Let $c_+ = \max(1, \|\gamma_2\|)$, and set
\[
 Y_+ = c_+ D_\varsigma + (s + r_W) \varsigma.
\]
Every eigenvalue of $M$ has modulus at most $s + r_W$ on $\mathcal K_W$. Using also $\eta \le s$, we find
\[
 -Y_+ \otimes \Id \le Y_F \le Y_+ \otimes \Id, \qquad Y_+ \ge D_\varsigma + s\varsigma \ge \min(\eta, s) \Id = \eta \Id.
\]

We apply \zcref{l:hist-coefficients}(a) to $X_F + \iu Y_F + \mathcal N_F$, with the envelope $Y_+ \otimes \Id$ and the matrix size $2kn \le 2k C_X W^2$. Using \eqref{e:hist-core-NF}, we find
\[
 |\det(\mathbf F + \mathcal N_F)|_1 \le \bigl|\det(X_F + \iu Y_+ \otimes \Id)\bigr| \Bigl(1 + C W^2 \Bigl(\frac{v_{\mathcal S}}{\eta}\Bigr)^{m_{\mathcal A} + 3}\Bigr)^{m_{\mathcal A}}.
\]

We diagonalize $\gamma_1$ by a unitary in the flavor space, with eigenvalues $u_1, \dots, u_{2k}$, which satisfy
\[
 |u_a| \le C_\Gamma, \qquad \sum_{a = 1}^{2k} u_a = \Tr \gamma_1 = 0.
\]
In this basis, we have
\begin{equation}\label{e:hist-ff-body}
 \bigl|\det(X_F + \iu Y_+ \otimes \Id)\bigr| = \prod_{a = 1}^{2k} |\det(A + u_a D_\varsigma + \iu Y_+)|.
\end{equation}

\textit{Comparison of the determinants.} We compare each factor in \eqref{e:hist-ff-body} with $|\det(A + \iu(D_\varsigma + s\varsigma))|$, which appears in \eqref{e:hist-B-det}, in three steps. First, for $u \in \R$, set
\[
 \ell(u) = \log |\det(A + u D_\varsigma + \iu Y_+)|, \qquad R_u = (A + u D_\varsigma + \iu Y_+)^{-1}.
\]
Since $\|D_\varsigma\| \le \eta$ and $\|Y_+^{-1}\| \le \eta^{-1}$, the second bound in \eqref{e:hist-shift} implies that
\[
 |\ell''(u)| \le \|D_\varsigma\|^2 \|Y_+^{-1}\| \Tr(-\Im R_u) \le \eta \Tr(-\Im R_u), \qquad u \in \R.
\]
Using Taylor's formula and the tracelessness of $\gamma_1$, we find
\[
 \sum_{a = 1}^{2k} \ell(u_a) - 2k \ell(0) = \sum_{a = 1}^{2k} u_a^2 \int_0^1 (1 - x) \ell''(x u_a) \dd x \le C_\Gamma^2 \eta \sum_{a = 1}^{2k} \int_0^1 \Tr(-\Im R_{x u_a}) \dd x.
\]

This is the step that uses the tracelessness of the FF source. Without it, the left side would contain the linear term $\ell'(0) \sum_a u_a$, which \eqref{e:hist-shift} bounds only by $C \eta \Tr(Y_+^{-1}) \le C n$, and the resulting factor $e^{C n}$ could be as large as $e^{C W^2}$.

Second, for $1 \le \tau \le c_+$, set
\[
 R'_\tau = \bigl(A + \iu(\tau D_\varsigma + (s + r_W)\varsigma)\bigr)^{-1}.
\]
By \eqref{e:hist-height}, we have
\[
 \frac{\dd}{\dd \tau} \log \bigl|\det\bigl(A + \iu(\tau D_\varsigma + (s + r_W)\varsigma)\bigr)\bigr| = \Tr\bigl(D_\varsigma (-\Im R'_\tau)\bigr) \le \eta \Tr(-\Im R'_\tau).
\]
Integrating from $1$ to $c_+$, we obtain
\[
 \ell(0) \le \log |\det(A + \iu(D_\varsigma + (s + r_W)\varsigma))| + \eta \int_1^{c_+} \Tr(-\Im R'_\tau) \dd \tau.
\]

Third, the upper bound in \eqref{e:hist-height} shows that the derivative in $\sigma \ge s$ of $\log |\det(A + \iu(D_\varsigma + \sigma \varsigma))|$ is bounded by
\[
 \Tr\bigl(\varsigma (D_\varsigma + \sigma\varsigma)^{-1}\bigr) \le \Tr\bigl(\varsigma(D_\varsigma + s\varsigma)^{-1}\bigr) = \Pi_n,
\]
where $\Pi_n$ is as in \zcref{l:hist-profile}(c) with $s_0 = s$. Integrating from $s$ to $s + r_W$, we find
\[
 \log |\det(A + \iu(D_\varsigma + (s + r_W)\varsigma))| \le \log |\det(A + \iu(D_\varsigma + s\varsigma))| + r_W \Pi_n.
\]

Combining the three steps with \eqref{e:hist-ff-body} and \eqref{e:hist-B-det}, we find
\begin{equation}\label{e:hist-quotient}
\begin{gathered}
 \frac{|\det(X_F + \iu Y_+ \otimes \Id)|}{|\det \mathbf B|} \le e^{4k r_W \Pi_n} e^{\eta \mathcal T_W}, \\
 \mathcal T_W = 2k \int_1^{c_+} \Tr(-\Im R'_\tau) \dd \tau + C_\Gamma^2 \sum_{a = 1}^{2k} \int_0^1 \Tr(-\Im R_{x u_a}) \dd x.
\end{gathered}
\end{equation}

\textit{Expectation.} The quantity $\mathcal T_W$ is a sum of $2k + 1$ integrals, over parameter intervals of length at most $c_+$, of $\lambda \Tr(-\Im R)$ with $\lambda \le 2k + C_\Gamma^2$. Each resolvent $R$ here is of the form of \zcref{l:hist-averaging}, with a deterministic real diagonal shift $Z_0 = u D_\varsigma$ or $Z_0 = 0$ and a deterministic diagonal $Y \ge \eta \Id$. We use H\"older's inequality for the $2k + 1$ terms, Jensen's inequality for the normalized parameter integrals, and Fubini's theorem. They show that $\E e^{\eta \mathcal T_W}$ is bounded by the supremum over the parameter of $\E e^{t_* \eta \Tr(-\Im R)}$, where
\[
 t_* = (2k + 1)(2k + C_\Gamma^2) c_+.
\]

The diagonal entries of $A = E \Id_n - \kappa \varsigma - H^{(n)}$ have densities bounded by the $\mathsf d_i$ of \zcref{l:hist-profile}(b). Applying \zcref{l:hist-averaging} with $c = 1$ and $t = t_*$, and using \zcref{l:hist-profile}(b), $n \le N \le C_X W^2$, and $\eta \le C_X W^{-2}$, we obtain
\[
 \E e^{\eta \mathcal T_W} \le \exp\Bigl(\pi (e^{t_*} - 1) \eta \sum_{i = 1}^n \mathsf d_i\Bigr) \le \exp\bigl(C \eta (W^{3/2} + N \sqrt W)\bigr) \le e^{C \sqrt W}.
\]
By \zcref{l:hist-profile}(c), we also have
\[
 4k r_W \Pi_n \le C r_W W \bigl(1 + \log(1 + \eta^{-1})\bigr).
\]
Combining these bounds with \eqref{e:hist-core-marks} and the bound on the fermionic determinant, we conclude that
\begin{equation}\label{e:hist-sdet-bound}
 \E \bigl|\Sdet\bigl[\mathbb M_n^t(S)\bigr]^{-1}\bigr|_1 \le C W^{2m_{\mathcal A}} \Bigl(\frac{v_{\mathcal S}}{\eta}\Bigr)^{C m_{\mathcal A}^2} \exp\Bigl(C \sqrt W + C r_W W \bigl(1 + \log(1 + \eta^{-1})\bigr)\Bigr).
\end{equation}

\textit{The Gaussian factor.} By \eqref{e:hist-odd-trace}, the odd part of the exponent of \eqref{e:hist-Gn} is the product of $-\sigma_n$ with $\Tr(\Lambda_B O_1 O_2)$. Using \zcref{l:hist-profile}(a), we bound its norm by
\[
 \sigma_n (2k)^2 h^2 \le 4W \cdot 4k^2 W^{-1} = 16 k^2.
\]
For the body, \zcref{l:rep-cycle}(c) with $P = \kappa \Id + P_0$ shows that
\[
 \Re \Str(S^2) = \Tr P_0^2 + 2\kappa \Tr P_0 + \Tr M^2 - 2k s^2 \qquad \text{on the body of } \mathcal S.
\]

On $\mathcal K_W$, every eigenvalue $\mu$ of $M$ and the trace of $P_0$ satisfy
\[
 s^2 - \mu^2 = (s - |\mu|)(s + |\mu|) \le 3 r_W, \qquad |\Tr P_0| \le \sqrt{2k} \|P_0\|_{\rm HS} \le \sqrt{2k} \, r_W.
\]
Using also $\sigma_n \le 4W$, we find
\[
 -\tfrac12 \sigma_n \Re \Str(S^2) \le C \sigma_n r_W \le C W r_W \qquad \text{on } \mathcal K_W.
\]
The computation of the damping in the proof of \zcref{l:hist-marked-onsite} and the bound $|\Str(\hat \Gamma^2)| \le C$ give
\[
 \Re\bigl(-\iu \eta \sigma_n \Str(S \hat \Gamma)\bigr) \le \eta \sigma_n C_\Gamma (2k + \|M\|_1) \le C \eta W, \qquad \frac{\eta^2 (n - \sigma_n)}{2} |\Str(\hat \Gamma^2)| \le C N \eta^2
\]
on $\mathcal K_W$. By \zcref{l:grass-norm}(e), we conclude that
\[
 |\mathcal G_n(S)|_{1; h} \le \exp\bigl(16k^2 + C W r_W + C \eta W + C N \eta^2\bigr) \qquad \text{for } S \in \mathcal K_W.
\]
Combining this bound with \eqref{e:hist-sdet-bound}, we conclude that \eqref{e:hist-core-bound} holds, where $C_1$ is the largest of the constants in the exponents.
\end{proof}

\subsection{Proof of the theorem}

\begin{proof}[Proof of \zcref{t:hist-bound}]
Let $\| \cdot \|$ denote the norm of $\mathcal H_{-p}(\mathcal S)$, where $p$ will be chosen below, and let $\1_{\mathcal K_W}$ and $\1_{\mathcal K_W^c}$ denote the indicator functions of $\mathcal K_W$ and of its complement. Set
\[
 a_j = \|\1_{\mathcal K_W^c} F_j\|, \qquad b_j = \|\1_{\mathcal K_W} F_j\|, \qquad 1 \le j \le N.
\]
Let $w_j$ be the onsite weight of the step $j$ in \eqref{e:hist-marked-chain}, so that
\[
 F_j = w_j K_{\tau_{\rm phys}} F_{j - 1}, \qquad 1 \le j \le N.
\]
Let $A_1$ be the largest of the constants $A_1 = 2(4k^2 + m)^2$ of \zcref{l:hist-marked-onsite} for $m \le d$. By that lemma, there exists a constant $C_2$ such that
\begin{equation}\label{e:hist-Qm}
 \sup_{\mathcal S} |w|_{1; h} \le Q_m, \qquad Q_m = C_2 (1 + \eta^{-1})^{A_1},
\end{equation}
for all onsite jets $w$ of order at most $d$, including the unmarked weight $w_\Gamma$.

\textit{The core.} By \zcref{l:grass-norm}(a), we have the pointwise bound
\[
 |F_j|_{1; h} \le |w_j|_{1; h} \, |K_{\tau_{\rm phys}} F_{j - 1}|_{1; h}.
\]
Using \eqref{e:hist-Qm} and \zcref{l:hist-core}, we find
\[
 |F_j|_{1; h} \le Q_m W^A (1 + \eta^{-1})^A v_{\mathcal S}^{p_0} e^{\mathcal E_W(\eta)} \qquad \text{on } \mathcal K_W.
\]

The coordinates $P_0$ and $M$ range over bounded sets on $\mathcal K_W$, while $X$ ranges over all off-diagonal Hermitian matrices. We choose $p = p_0 + k^2 + 1$. Using $|F_j|_{2; h} \le |F_j|_{1; h}$ and $v_{\mathcal S} \ge 1 + \|X\|_{\rm HS}$, we find
\begin{equation}\label{e:hist-core-norm}
 b_j \le Q_m W^A (1 + \eta^{-1})^A e^{\mathcal E_W(\eta)} \Bigl(C \int v_{\mathcal S}^{2p_0 - 2p} \dd X\Bigr)^{1/2} \le C Q_m W^A (1 + \eta^{-1})^A e^{\mathcal E_W(\eta)}.
\end{equation}

\textit{The far region.} Let $S \notin \mathcal K_W$. Then either $\|P_0\|_{\rm HS} > r_W$, or $M$ has an eigenvalue $\mu$ with $||\mu| - s| > r_W$. We claim that the escape function $\vartheta = \vartheta_B + \vartheta_F$ of \eqref{e:end-escape} satisfies $\vartheta(S) \ge c_\vartheta r_W^2$ in both cases, once $r_W \le r_{\rm esc}$, where $c_\vartheta > 0$ depends only on the choice of $\chi_0$ in \eqref{e:end-escape}. Both $\vartheta_B$ and $\vartheta_F$ are nonnegative.

In the first case, recall that the physical normal $\widetilde P = \operatorname{ch}_{\mathsf g} P_0$ of \eqref{e:end-boost-conj} satisfies $\|\widetilde P\|_{\rm HS} \ge \|P_0\|_{\rm HS}$. Recall also the lower bound in \eqref{e:end-chi0}, which holds since $\chi_0(0) = 0$ and $\chi_0'(0) > 0$. We let $c_0 > 0$ denote the constant $c$ of that bound, so that $\chi_0(x) \ge c_0 \min(x, 1)$ for all $x \ge 0$. Using this bound and the definition of $\vartheta_B$ in \eqref{e:end-escape}, we find
\[
 \vartheta(S) \ge \vartheta_B(S) = r_{\rm esc}^2 \chi_0\bigl(\|\widetilde P\|_{\rm HS}^2/r_{\rm esc}^2\bigr) \ge c_0 \min\bigl(\|\widetilde P\|_{\rm HS}^2, r_{\rm esc}^2\bigr) \ge c_0 r_W^2.
\]

In the second case, $\vartheta_F$ is the sum of the nonnegative terms $(\mu'^2 - s^2)^2/(1 + \mu'^2)$ over the eigenvalues $\mu'$ of $M$, and for the given $\mu$,
\[
 \frac{(\mu^2 - s^2)^2}{1 + \mu^2} = (|\mu| - s)^2 \, \frac{(|\mu| + s)^2}{1 + \mu^2} \ge s^2 r_W^2 \ge \tfrac12 r_W^2,
\]
where we used $(x + s)^2 \ge s^2 (1 + x^2)$ for $x \ge 0$ and $s \le 1$. This proves the claim, with $c_\vartheta = \min(c_0, 1/2)$.

We now bound one unmarked step on the far region. We apply \zcref{l:end-onsite} with $\Gamma = \hat \Gamma$, for which $\hat \Gamma_{BB} = \Lambda_B$ and $\|\hat \Gamma_{FF}\| \le C_\Gamma$. Since $\vartheta \ge c_\vartheta r_W^2$ off the core, the lemma implies that
\[
 |w_\Gamma|_{1; h} \le e^{C/W + C\eta - c c_\vartheta r_W^2} \qquad \text{on the complement of } \mathcal K_W.
\]
By \zcref{l:end-kinetic} with the weight $-p$, we have $\|K_{\tau_{\rm phys}}\| \le e^{C/W}$ on $\mathcal H_{-p}(\mathcal S)$.

Recall that $r_W^2 = (\log W)^2/W$ and $\eta \le C_X W^{-2}$. Therefore, for all sufficiently large $W$,
\begin{equation}\label{e:hist-far}
 \|\1_{\mathcal K_W^c} M_{w_\Gamma} K_{\tau_{\rm phys}}\| \le q_W, \qquad q_W = \exp\bigl[-c_2 (\log W)^2/W\bigr],
\end{equation}
with $c_2 = c c_\vartheta/2$. Note that the indicator acts on the output of the step, after the heat.

At a marked step $j$, we use \eqref{e:hist-Qm} and \zcref{l:end-kinetic} instead, and we find
\[
 \|\1_{\mathcal K_W^c} M_{w_j} K_{\tau_{\rm phys}}\| \le 2 Q_m.
\]

\textit{The recurrence.} Set $q_j = q_W$ if the step $j$ is unmarked and $q_j = 2 Q_m$ if it is marked; at most $d$ steps are marked. Using $\|F_j\| \le a_j + b_j$ and
\[
 \1_{\mathcal K_W^c} F_{j + 1} = \1_{\mathcal K_W^c} M_{w_{j + 1}} K_{\tau_{\rm phys}} F_j,
\]
we obtain
\[
 a_{j + 1} \le q_{j + 1} (a_j + b_j), \qquad 1 \le j \le N - 1.
\]
Iterating this inequality, we find
\[
 a_j \le \Bigl(\prod_{l = 2}^{j} q_l\Bigr) a_1 + \sum_{i = 1}^{j - 1} \Bigl(\prod_{l = i + 1}^{j} q_l\Bigr) b_i.
\]
Each product contains at most $d$ marked factors, and all its other factors equal $q_W$. Using $q_W^{-d} \le 2$ for large $W$ and summing the geometric series, we find
\begin{equation}\label{e:hist-far-norm}
 a_j \le 2 (2Q_m)^d \Bigl[a_1 + (1 - q_W)^{-1} \max_{i < j} b_i\Bigr], \qquad (1 - q_W)^{-1} \le C W.
\end{equation}

The recurrence starts at $j = 1$, since $F_0 = 1$ has no finite norm in $\mathcal H_{-p}(\mathcal S)$. Instead, $F_1 = w_1$ is an onsite jet, and \zcref{l:hist-marked-onsite} implies that
\[
 a_1 \le \|w_1\| \le C \eta^{-A_1 - k^2}.
\]

\textit{Conclusion.} Combining \eqref{e:hist-core-norm} and \eqref{e:hist-far-norm}, and recalling \eqref{e:hist-Qm}, we obtain \eqref{e:hist-bound} for $\|F_j\| \le a_j + b_j$, with a larger constant $A$. By \zcref{l:end-kinetic}, the final heat in $K_{\tau_{\rm phys}} F_j$ multiplies the norm by at most $e^{C/W} \le 2$. All the estimates above are uniform in the positions and the directions of the marks, and so is the bound.
\end{proof}