==== TABLE OF CONTENTS ====
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//consequences.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//entropy.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//introduction.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//main.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//matrices.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//observation-flow.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//observation.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//references.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//residuals.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//stability.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//style.tex
A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//terminal.tex
===========================

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//consequences.tex ====
\section{Posterior estimates at a stable field}\label{sec:consequences}

The residual rules now yield the two posterior estimates used in
Section~\ref{sec:observation}. We first select a finite collection of
small-residual regions, then compare the primary iterates with the exact
root. Throughout, $J$ satisfies the required fixed-length word diagnostics,
and $h_0$ satisfies the whole stable-field implication through residual
threshold $\rho_0\sqrt n$. Denote the unique zero of $F_{h_0}$ by $r$ and
set $t_*=\tanh r$, $q_*=\av{t_*^2}$. All expectations and forms here are
at $\mu_{h_0}$, and $R_l=m^{l-1}-m^l$ denotes the residual from
Section~\ref{sec:residuals}.

\subsection{Selecting a small-residual region}
The primary recursion need not converge. The finite-depth fact we need is
that two consecutive residuals are small on most of the Gibbs mass, with
a separate bound after square reweighting. Smooth cutoffs make that selection
compatible with discrete differentiation.
\begin{lemma}\label{lem:residual-selection}
For each sufficiently small fixed $\rho>0$, there is a fixed depth $d$ and
functions $C_k$, $2\le k\le d$, with the following properties:
\begin{enumerate}
\item $0\le C_k\le1$, and $C_k$ is supported where
$\max(\norm{R_k},\norm{R_{k-1}})\le\rho\sqrt n$. It equals one where both
norms are at most $\rho\sqrt n/2$, and $\norm{dC_k}\le C_\rho/\sqrt n$
pointwise.
\item Set $\widehat C_k=C_k\prod_{i=2}^{k-1}(1-C_i)$ and
$D=1-\sum_{k=2}^d\widehat C_k$. Then $0\le D\le1$ and
\[
 \Prob_{\mu_{h_0}}(D>0)\le\frac{C_\rho}{n}.
\]
For $N=\E f^2>0$ and $\nu=f^2\mu_{h_0}/N$,
\[
 \Prob_\nu(D>0)\le\frac{C_\rho}{\sqrt n}
                   \left(1+\frac{\D_{h_0}(f)}{N}\right).
\]
\end{enumerate}
All constants are uniform over the field and matrix under the stated
diagnostics. The depth and the required diagnostic length may depend on
$\rho$.
\end{lemma}

\begin{proof}
We use the scalar residual pairing $S_l=R_l^{\mathsf T}m^l/\sqrt n$ to
control the total squared increment. First, the primary derivative bounds
give $\norm{dS_l}\le C_l$ pointwise. In the product rule, the linear
terms are controlled by the operator norms of $\nabla R_l$ and
$\nabla m^l$, multiplied by $\norm{m^l}/\sqrt n$ and
$\norm{R_l}/\sqrt n$. The cross term for coordinate $i$ has size at most
$2\norm{d_i R_l}\norm{d_i m^l}/\sqrt n$; summing its square uses the
bounded column norms and gives the same scale.

Apply Lemma~\ref{lem:residual} with target $m^l/\sqrt n$ and test $G=S_l$.
It gives
\[
 \E S_l^2\le C_l\bigl(\E S_l^2+C_l\bigr)^{1/2},
 \qquad\text{hence}\qquad \E S_l^2\le C_l'.
\]
For the square-density version use $H_0=S_l/\sqrt n$, which is bounded with
bounded difference norm. Dividing the resulting estimate by $N\sqrt n$
gives
\begin{equation}\label{eq:weighted-S}
 \E_\nu\left(\frac{S_l}{\sqrt n}\right)^2
 \le\frac{C_l}{\sqrt n}\left(1+\frac{\D_{h_0}(f)}{N}\right).
\end{equation}
We now use the squared-increment identity
\eqref{eq:residual-telescoping}. Choose a fixed $d$ with
$\lfloor d/2\rfloor\rho^2/4>2$, followed by $\eps>0$ with $2d\eps<1$.
Whenever $\max_{l\le d}|S_l|/\sqrt n\le\eps$, telescoping gives
$\sum_{l=1}^d\norm{R_l}^2/n<2$. If every consecutive pair contained a
residual larger than $\rho\sqrt n/2$, the disjoint pairs
$(1,2),(3,4),\ldots$ would force the sum to exceed this bound. At least
one pair therefore meets the required threshold.

To implement the selection smoothly, apply a scalar cutoff equal to one
on $[0,1/4]$ and zero on $[1,\infty)$ to each of
$\norm{R_k}^2/(\rho^2n)$ and $\norm{R_{k-1}}^2/(\rho^2n)$, and multiply
the results to define $C_k$. The primary operator and column bounds,
with the product rule, give
$\norm{d(\norm{R_l}^2/n)}\le C_l/\sqrt n$, hence the claimed derivative
bound. Finally $D=\prod_{k=2}^d(1-C_k)$: positive deficit implies
$\max_{l\le d}|S_l|/\sqrt n>\eps$. The preceding two moment bounds and
a finite union bound give the ordinary and square-reweighted estimates.
\end{proof}

\subsection{A directional covariance bound}
On each selected region, the implicit test vector converts the residual
identity into a comparison with $t_*$. Its auxiliary scalar error is
estimated first; only then do we test in an arbitrary Euclidean direction.
\begin{proposition}\label{prop:directional}
There is a constant $C$ such that, at every field satisfying the above
stable-field implication,
\begin{equation}\label{eq:directional}
 \norm{\Cov_{\mu_{h_0}}(G,x)}
 \le C\bigl(\Var_{\mu_{h_0}}(G)+\D_{h_0}(G)\bigr)^{1/2}
\end{equation}
for every $G$. The constant is uniform in $n,h_0$, and $G$ on the specified
matrix events; only a fixed finite diagnostic length is required.
\end{proposition}

\begin{proof}
Fix an index $k\ge2$ and work on the small-residual region
\eqref{eq:small-region}, with the buffered implicit construction
\eqref{eq:implicit}. Abbreviate $m=m^k$, $p=m-t_*$, and put
\[
 a=j(1-b_{k-1}-q_*),\qquad
 L_* =\sqrt n\bigl(\chi-(1-q_*)\bigr).
\]
Here $\chi=\av{A_i}$ and $A_i=\Phi(h_i^k;r_i,a)$ are defined globally by
\eqref{eq:phi-data}, even though the implicit vector is only needed locally.
Thus $|L_*|\le C\sqrt n$ and $\norm{dL_*}\le C_k$ pointwise, by the primary
derivative bounds and $\norm{db_{k-1}}=O(n^{-1/2})$.

For either of the two allowed sources $w_0$, the following pointwise identity
holds on the buffered region:
\begin{align}\label{eq:implicit-comparison}
 &\bigl(h^k-h_0-(J-jb_{k-1}I)m\bigr)^{\mathsf T}w
       -j(b_k-b_{k-1})nc_1 \notag\\
 &\quad=\sum_i\bigl[(h_i^k-r_i-am_i)-a\partial_k\bigr]w_{0,i}
       +j(1-q_*-\chi)p^{\mathsf T}w-jc_1R_k^{\mathsf T}p.
\end{align}
Here the zeroth-order term in the first sum acts by multiplication on
$w_{0,i}$. For the verification, subtract the root equation to write the
vector on the left as $h^k-r-am-Jp+j(1-q_*)p$, then substitute
$Jw=y^{\rm imp}+j\chi w+jc_1m^{k-1}$. Apply \eqref{eq:phi-stein} to
the contribution $A(y^{\rm imp}-jc_1t_*)$, holding its auxiliary arguments
and parameters fixed, and use $nc_1=\sum_i\partial_k w_i$. The remaining
scalar terms cancel by
\[
 p^{\mathsf T}(m^{k-1}+t_*)+n(b_k-b_{k-1})
 =n(1-b_{k-1}-q_*)+R_k^{\mathsf T}p,
\]
which follows by expanding $p=m-t_*$ and $R_k=m^{k-1}-m$.

Lemma~\ref{lem:terminal-residual} controls the tested left-hand side of
\eqref{eq:implicit-comparison}. The last term on the right also has tested
expectation at most $C\norm{G}_*$: apply Lemma~\ref{lem:residual} to target
$p/\sqrt n$ and multiplier $\sqrt n c_1$. The latter is bounded with bounded
difference norm on the buffer. Meanwhile the size estimates for the implicit
solution and Lemma~\ref{lem:root} give
\begin{equation}\label{eq:middle-error}
 \abs{j(1-q_*-\chi)p^{\mathsf T}w}\le C_0\rho|L_*|.
\end{equation}
The source size and stability margins determine $C_0$ independently of
$k$ and the selection depth. This lets us choose $\rho$ first so that
$C_0\rho<1/2$ and all stability and flip-buffer conditions hold. The
selection depth in Lemma~\ref{lem:residual-selection} is chosen afterward.
This order will permit absorption of the scalar error without a
self-dependent choice of depth.

First control the averaged coefficient by taking
$w_{0,i}=\partial_{r_0}\Phi(h_i^k;r_i,a)/\sqrt n$.
The second identity in \eqref{eq:implicit-source-actions} makes the first
sum in \eqref{eq:implicit-comparison} equal to $L_*$.
Test with $G=C_k^2L_*$. The cutoff and difference bounds give
\[
 \norm{C_k^2L_*}_*\le C+\norm{C_kL_*}_{L^2(\mu_{h_0})}.
\]
Indeed its difference vector is bounded pointwise: $|L_*|=O(\sqrt n)$
multiplies $\norm{d(C_k^2)}=O(n^{-1/2})$, and $\norm{dL_*}=O(1)$.
Absorbing \eqref{eq:middle-error} now yields
\begin{equation}\label{eq:susceptibility-average}
 \norm{C_kL_*}_{L^2(\mu_{h_0})}\le C_k'.
\end{equation}

Now test a direction by taking $w_0=Ae$, where $e$ is any deterministic
vector with $\norm e\le1$. By the first identity in
\eqref{eq:implicit-source-actions}, the first sum in
\eqref{eq:implicit-comparison} is $p^{\mathsf T}e$.
Test with $\widehat C_kG$. The residual estimates control the left side
and the last term by $C\norm G_*$; the middle term is bounded by
\eqref{eq:susceptibility-average} and Cauchy--Schwarz, since
$0\le\widehat C_k\le C_k$. Thus
\[
 \abs{\E[\widehat C_kG(m^k-t_*)^{\mathsf T}e]}\le C\norm G_*.
\]
Summing Lemma~\ref{lem:residual} with the constant target $e$ over
$l=1,\ldots,k$ gives the same bound for
$\E[\widehat C_kG(x-m^k)^{\mathsf T}e]$.
Sum over $k$. The deficit has contribution at most
\[
 2\sqrt n\,\norm G_{L^2(\mu_{h_0})}
       \Prob_{\mu_{h_0}}(D>0)^{1/2}\le C\norm G_*.
\]
Thus $|\E[G(x-t_*)^{\mathsf T}e]|\le C\norm G_*$ uniformly over
$\norm e\le1$. Centering $G$ and taking the supremum over this seed
class gives \eqref{eq:directional}. The uncentered estimate also records
the ordinary-mean consequence: take $G=1$ to obtain
$\norm{\E_{\mu_{h_0}}x-t_*}\le C$. This norm is the unnormalized
Euclidean norm. It compares the mean itself with $t_*$; no
approximation of the derivative of the mean map is used here.
The local construction required $n$ above a fixed threshold. In the
remaining dimensions, the elementary bounds
$\norm{\Cov(G,x)}^2\le n\Var(G)$ and
$\norm{\E_{\mu_{h_0}}x-t_*}\le2\sqrt n$ extend both conclusions after
enlarging $C$.
\end{proof}

\subsection{Means under square reweighting}
For the change of observation law we need a different estimate, valid for
arbitrary square densities. It uses only ordinary recipes and trades a
small fixed leading error for a constant depending on that tolerance.
\begin{proposition}\label{prop:weighted-mean}
For every sufficiently small fixed $\delta>0$, there is a constant $C_\delta$
such that, for $N=\E_{\mu_{h_0}}f^2>0$ and $\nu=f^2\mu_{h_0}/N$,
\begin{equation}\label{eq:weighted-mean}
 \frac{\norm{\E_\nu x-t_*}}{\sqrt n}
 \le C\delta+\frac{C_\delta}{\sqrt n}
               \left(1+\frac{\D_{h_0}(f)}{N}\right).
\end{equation}
The leading constant $C$ is independent of $\delta$ and of the selection
depth. This estimate requires only ordinary word diagnostics, of a fixed
length that may depend on $\delta$.
\end{proposition}

\begin{proof}
Select the cutoffs with a fixed threshold $\rho\le\delta$ small enough
for Lemma~\ref{lem:root}. On each cutoff support, the residual-to-root
comparison gives
\[
 \norm{m^k-t_*}\le C\delta\sqrt n.
\]
Here the constant comes from the recurrence and the inverse bound in
Lemma~\ref{lem:root}, and is independent of $k$.
For $\norm e\le1$, sum the square-density version of
Lemma~\ref{lem:residual} with target $e$ through $k$, using multiplier
$\widehat C_k$. After division by $N$, this bounds
\[
 \abs{\E_\nu[\widehat C_k(x-m^k)^{\mathsf T}e]}
 \le C_\delta\left(1+\frac{\D_{h_0}(f)}{N}\right).
\]
The cutoff weights sum to at most one, so all root-comparison terms
together cost at most $C\delta\sqrt n$. The leftover mass contributes
at most $2\sqrt n\Prob_\nu(D>0)$, which is controlled by the second
part of Lemma~\ref{lem:residual-selection}. Add the terms, divide by
$\sqrt n$, and take the supremum over $e$ to obtain
\eqref{eq:weighted-mean}. The leading $C$ comes from root stability;
only $C_\delta$ and the finite diagnostic length depend on the selected
tolerance. Any dimensions below a fixed threshold required by this
construction are covered by the elementary bound
$\norm{\E_\nu x-t_*}/\sqrt n\le2$ after increasing $C_\delta$; this
leaves the leading constant unchanged. The argument has used no implicit
test vector.
\end{proof}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//consequences.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//entropy.tex ====
\section{A scalar entropy inequality}
\label{sec:entropy}

The stability argument needs a scalar criterion that forces an
empirical field law toward a Gaussian law.  We first bound relative
entropy below by a squared discrepancy involving the field and its
hyperbolic tangent.  We then identify the Gaussian law compatible
with the scalar consistency equations.  Together with the equality
case of the first estimate, this characterizes the consistent
equality laws.

Throughout this section, $N(d,s)$ denotes the Gaussian law with mean
$d$ and \emph{variance} $s$.  All scalar expectations are taken with
respect to the indicated probability law.

\begin{lemma}[Scalar entropy inequality]
\label{lem:scalar-entropy}
Let $0<j<1$, $d\in\R$, and $s>0$.  Let $P$ be a probability law on $\R$
with finite second moment, and set
\[
 m(y)=\tanh y,\qquad q=\E_P m^2,\qquad b=1-q,\qquad
 a=\E_P[(y-d)m],\qquad S=s+jq.
\]
If $s\ge jq$, then
\begin{equation}
 D(P\Vert N(d,s))\ge \frac{j(a-sb)^2}{2sS}.
 \label{eq:scalar-entropy}
\end{equation}
Equality holds if and only if $P=N(d,s)$.
\end{lemma}

\begin{proof}
The second-moment assumption makes the right-hand side finite, so
it suffices to consider the case $D:=D(P\Vert N(d,s))<\infty$.  In this case $P$ is
absolutely continuous, and hence $q>0$: otherwise $P$ would be the
point mass at zero.  Also $q<1$, because $\tanh^2 y<1$ for every finite
$y$.  We use the following dimensionless parameters:
\begin{equation}
 k=\frac{jq}{s}\in(0,1],\qquad C=\frac{ja}{s},\qquad
 B=jb\in(0,1),\qquad a_0=1-B\in(0,1).
 \label{eq:entropy-scalars}
\end{equation}
In these parameters, multiplying the desired inequality by $j$
gives the target
\begin{equation}
 jD\ge \frac{(C-B)^2}{2(1+k)}.
 \label{eq:entropy-target}
\end{equation}

We will combine two entropy bounds.  A monotone change of variables
handles one range of the discrepancy $C-B$; comparison with a Gaussian
of a different variance handles the remaining range.  We retain
strictness in both arguments to identify all equality cases.

\paragraph{A change-of-variables bound.}
For the change of variables, fix $u<1$.  The map
\[
 T_u(y)=y-u\tanh y
\]
is an increasing $C^1$ diffeomorphism of $\R$ onto itself, with derivative
$T_u'(y)=1-u(1-m(y)^2)>0$.  Change of variables in relative entropy yields
\begin{align*}
 D
 &=D((T_u)_\#P\Vert N(d,s))
   +\frac{2ua-u^2q}{2s}
   +\E_P\log\bigl(1-u(1-m^2)\bigr)\\
 &\ge \frac{2ua-u^2q}{2s}
   +\E_P\log\bigl(1-u(1-m^2)\bigr).
\end{align*}
The entropy identity is well defined: for fixed $u<1$, $\log T_u'$
is bounded, and the second-moment assumption makes the difference
of the Gaussian log densities integrable.  To estimate the logarithmic
term, apply concavity on the segment joining $1$ and $1-u$.  For
$0\le v\le1$, this gives
\[
 \log(1-uv)\ge v\log(1-u).
\]
Consequently, with $\ell(u)=-\log(1-u)-u$,
\begin{equation}
 jD\ge u(C-B)-\frac{k u^2}{2}-B\ell(u),\qquad u<1.
 \label{eq:entropy-transport}
\end{equation}
Completing the square isolates the amount by which this bound can
exceed the target in \eqref{eq:entropy-target}:
\begin{equation}
 jD-\frac{(C-B)^2}{2(1+k)}
 \ge h(u)-\frac{\bigl(C-B-(1+k)u\bigr)^2}{2(1+k)},
 \qquad h(u)=\frac{u^2}{2}-B\ell(u).
 \label{eq:entropy-slack}
\end{equation}

We first locate a range in which the extra term $h(u)$ is strictly
positive:
\begin{equation}
 h(u)>0\quad\text{for }u\le a_0,\ u\ne0,
 \qquad h(a_0)\ge\frac{a_0^3}{6}.
 \label{eq:entropy-slack-positive}
\end{equation}
For the negative range, integrate $v/(1+v)\le v$ over
$0\le v\le -u$ to obtain $\ell(u)\le u^2/2$.  Since $B<1$, this
proves strict positivity for $u<0$.  For the positive range, use
$\ell(u)=\sum_{r\ge2}u^r/r$: it shows that $\ell(u)/u^2$ is increasing
on $0<u<1$.
Moreover, using $B=1-a_0$,
\[
 h(a_0)=\frac{a_0^2}{2}-(1-a_0)\ell(a_0)
       =\sum_{r\ge3}\frac{a_0^r}{r(r-1)}
       \ge\frac{a_0^3}{6}.
\]
It follows that $h(u)/u^2\ge h(a_0)/a_0^2>0$ for $0<u\le a_0$,
proving \eqref{eq:entropy-slack-positive}.

We now choose the transport parameter.  If
$u_*=(C-B)/(1+k)\le a_0$ and $C\ne B$, set $u=u_*$ in
\eqref{eq:entropy-slack}.  The square vanishes and $h(u_*)>0$, so
\eqref{eq:entropy-target} is strict.  If $u_*>a_0$, put
\[
 C_0=1+ka_0,
 \qquad C_1=C_0+\sqrt{\frac{1+k}{3}}\,a_0^{3/2}.
\]
In this case $C>C_0$.  The endpoint choice $u=a_0$ in
\eqref{eq:entropy-slack} still gives
\[
 jD-\frac{(C-B)^2}{2(1+k)}
 \ge \frac{a_0^3}{6}-\frac{(C-C_0)^2}{2(1+k)}.
\]
This proves strictness throughout $C_0<C<C_1$ and leaves only
the range beginning at $C_1$.

\paragraph{The remaining range.}
For $C\ge C_1$, we use the second moment to obtain the remaining
entropy bound.  Put $v=\E_P[(y-d)^2]/s>0$.  Comparison with the
Gaussian $N(d,sv)$ gives the exact identity
\[
 D=D(P\Vert N(d,sv))+\frac{v-1-\log v}{2},
\]
and hence $2D\ge v-1-\log v$.  Cauchy--Schwarz gives
\[
 v\ge\frac{a^2}{sq}=\frac{C^2}{jk}>1,
\]
where the last inequality follows from $C\ge C_1>C_0>1$ and $jk<1$.
The function $v\mapsto v-1-\log v$ is increasing for $v\ge1$, so
\[
 2jD\ge \frac{C^2}{k}-j-j\log\frac{C^2}{jk}
       \ge \frac{C^2}{k}-1-\log\frac{C^2}{k}.
\]
The second inequality removes $j$ from the lower bound.  Indeed,
with $C,k$ held fixed, the derivative of
$C^2/k-\lambda-\lambda\log(C^2/(\lambda k))$ is
$-\log(C^2/(\lambda k))\le0$ for $j\le\lambda\le1$.
Thus the remaining comparison with the target reduces to
\[
 f(C):=\frac{C^2}{k}-1-\log\frac{C^2}{k}
       -\frac{(C-B)^2}{1+k}>0
 \quad(C\ge C_1).
\]
We estimate $f$ at $C_0$ and control its increase from $C_0$ to
$C_1$.  Direct simplification at $C_0$ gives
\begin{align*}
 f(C_0)
 &=\frac1k-1+\log k
   +2a_0-a_0^2-2\log(1+ka_0)\\
 &\ge -\frac{2a_0^3}{3}.
\end{align*}
Indeed $1/k-1+\log k\ge0$ for $0<k\le1$, and
\[
 \log(1+ka_0)\le\log(1+a_0)
 \le a_0-\frac{a_0^2}{2}+\frac{a_0^3}{3}.
\]
The last bound follows by integrating
$(1+x)^{-1}\le1-x+x^2$ from zero to $a_0$.  Differentiation also gives
\begin{align*}
 f'(C_0)&=\frac2k-\frac{2}{1+ka_0}
          \ge\frac{2a_0}{1+a_0},\\
 f''(C)&=\frac{2}{k(1+k)}+\frac{2}{C^2}\ge1
          \qquad(C>0).
\end{align*}
For completeness, the first inequality follows from
\[
 \frac1k-\frac1{1+ka_0}-\frac{a_0}{1+a_0}
 =\frac{(1-k)(1+a_0+ka_0^2)}{k(1+ka_0)(1+a_0)}\ge0.
\]
The derivative bounds compensate for the possible negative value
at $C_0$.  Write $\Delta=C_1-C_0\ge a_0^{3/2}/\sqrt3$ and integrate
them to obtain
\begin{align*}
 f(C_1)
 &\ge -\frac{2a_0^3}{3}
       +\frac{2a_0}{1+a_0}\Delta+\frac{\Delta^2}{2}\\
 &\ge a_0^3\left(
       -\frac23+\frac{2}{\sqrt3(1+a_0)\sqrt{a_0}}+\frac16
       \right)>0.
\end{align*}
The strict inequality uses $0<a_0<1$ and
$1/\sqrt3+1/6>2/3$.  Since $f'$ is positive at $C_0$ and increasing
thereafter, $f(C)>0$ for every $C\ge C_1$.

The two ranges together prove strictness in
\eqref{eq:entropy-target} whenever $C\ne B$.  To finish the equality
case, suppose $C=B$.  The target then has right-hand side zero, so
equality is equivalent to $D=0$, or $P=N(d,s)$.  Conversely, Gaussian
integration by parts gives $a=s\E_P\sech^2 y=sb$ for this law,
which verifies equality.
\end{proof}

We next impose the scalar consistency equations used in the
application.  Here the Gaussian parameters themselves depend on
$P$: for given $t\ge0$ and $\sigma\ge0$, set
\begin{equation}
 M=\E_P\tanh y,\qquad q=\E_P\tanh^2 y,\qquad
 d=t+jM,\qquad s=t+\sigma^2+jq.
 \label{eq:scalar-consistency}
\end{equation}
These equations give $s\ge jq$, the comparison required by
Lemma~\ref{lem:scalar-entropy}.  When $s>0$, that lemma restricts
equality to Gaussian laws.  The next result identifies which
Gaussian laws satisfy the consistency equations when $\sigma=0$,
and also treats the degenerate case at $t=0$.

\begin{lemma}[The consistent Gaussian law]
\label{lem:gaussian-equality}
Fix $0<j<1$.  For each $t\ge0$, the equation
\begin{equation}
 s_*(t)=t+j\E_{N(s_*(t),s_*(t))}\tanh y
 \label{eq:scalar-fixed-point}
\end{equation}
has a unique nonnegative solution, where $N(0,0)$ means the point mass
at zero.  Its associated Gaussian law
$P_*(t)=N(s_*(t),s_*(t))$ is the unique Gaussian law satisfying
\eqref{eq:scalar-consistency} with $\sigma=0$.  Thus, among consistent
laws with $s>0$, equality in \eqref{eq:scalar-entropy} holds precisely at
$P_*(t)$.

The function $t\mapsto s_*(t)$ is Lipschitz with constant $(1-j)^{-1}$,
and
\begin{equation}
 t\le s_*(t)\le\frac{t}{1-j},\qquad
 q_*(t):=\E_{P_*(t)}\tanh^2 y
       =\frac{s_*(t)-t}{j}\le\frac{t}{1-j}.
 \label{eq:scalar-fixed-point-bounds}
\end{equation}
In particular $s_*(0)=q_*(0)=0$, and $s_*(t),q_*(t)>0$ for $t>0$.
\end{lemma}

\begin{proof}
We first show that a consistent Gaussian must have equal mean and
variance.  We then solve the resulting one-variable fixed-point
equation and prove the bounds in the statement.

Two Gaussian symmetry identities provide the first step.  For $s>0$,
the density $\rho_s$ of $N(s,s)$ satisfies
\[
 \frac{\rho_s(y)-\rho_s(-y)}{\rho_s(y)+\rho_s(-y)}=\tanh y.
\]
Pairing the integrals over $y$ and $-y$ therefore gives
\begin{equation}
 \E_{N(s,s)}\tanh y=\E_{N(s,s)}\tanh^2 y.
 \label{eq:gaussian-nishimori}
\end{equation}
Second, if $d\ge0$ and $s>0$, pairing the positive and negative
half-lines shows that
\begin{equation}
 \E_{N(d,s)}[\tanh y\,\sech^2 y]\ge0.
 \label{eq:gaussian-odd-sign}
\end{equation}
Here the integrand is odd and nonnegative for $y\ge0$, while the density
at $y$ is at least its value at $-y$ when $d\ge0$.

Now suppose a Gaussian law $N(d,s)$ with $s>0$ satisfies
\eqref{eq:scalar-consistency} with $\sigma=0$.  We begin by locating
its mean.  For fixed $s$, the map
\[
 d\longmapsto t+j\E_{N(d,s)}\tanh y
\]
is a contraction with Lipschitz constant at most $j$, because its
derivative is $j\E_{N(d,s)}\sech^2 y\in[0,j]$.  Iteration from zero
stays nonnegative, so its unique fixed point $d$ is nonnegative.

To compare this nonnegative mean with the variance, keep $s$ fixed
and define
\[
 H_s(d)=\E_{N(d,s)}[\tanh y-\tanh^2 y].
\]
For $d\ge0$, Gaussian differentiation and
\eqref{eq:gaussian-odd-sign} give
\[
 H_s'(d)=\E_{N(d,s)}[(1-m^2)(1-2m)]\in[-1,1].
\]
Indeed the integrand is bounded below by $-1$, and its expectation is
at most $\E(1-m^2)\le1$ after subtracting
$2\E[m(1-m^2)]\ge0$.  Thus $H_s$ is 1-Lipschitz on $[0,\infty)$.
By \eqref{eq:gaussian-nishimori}, $H_s(s)=0$.  On the other hand,
consistency says $d-s=j(M-q)=jH_s(d)$.  Consequently
\[
 |H_s(d)|\le |d-s|=j|H_s(d)|,
\]
forcing $H_s(d)=0$ and $d=s$.  The consistent Gaussian law must therefore
satisfy \eqref{eq:scalar-fixed-point}.

It remains to solve \eqref{eq:scalar-fixed-point}.  Set
\[
 F(s)=\E_{N(s,s)}\tanh y\quad(s>0),\qquad F(0)=0.
\]
It is continuous at zero by bounded convergence.  For $s>0$,
differentiating both the Gaussian mean and variance gives
\begin{align*}
 F'(s)
 &=\E_{N(s,s)}\left[(\tanh y)'
                       +\tfrac12(\tanh y)''\right]\\
 &=\E_{N(s,s)}[(1-m^2)(1-m)]\in[0,1].
\end{align*}
The lower bound is pointwise; the upper bound follows from
\eqref{eq:gaussian-odd-sign}.  Hence $F$ is nonnegative and
1-Lipschitz on $[0,\infty)$, with $F(s)\le s$.
Consequently, $s\mapsto t+jF(s)$ is a contraction of
$[0,\infty)$ into itself and has exactly one fixed point $s_*(t)$.
The identity \eqref{eq:gaussian-nishimori} then verifies both
consistency equations for $N(s_*(t),s_*(t))$.
If a consistent law has $s=0$, then $0=t+jq$ forces $t=q=0$,
and hence the law is the point mass at zero and $d=0$; this agrees
with the solution just constructed.

Existence and uniqueness are now established, including the
degenerate case.  To obtain the dependence on time, compare the
fixed-point equations at $t,t'$:
\[
 |s_*(t)-s_*(t')|
 \le |t-t'|+j|s_*(t)-s_*(t')|,
\]
which proves the asserted Lipschitz bound.  The size estimates follow
from $0\le F(s)\le s$, which gives $t\le s_*(t)\le t/(1-j)$, and
\eqref{eq:gaussian-nishimori} gives
$q_*(t)=F(s_*(t))=(s_*(t)-t)/j\le t/(1-j)$.
For $t>0$, the variance $s_*(t)$ is positive, so its Gaussian law is
nondegenerate and $q_*(t)>0$.  The equality assertion follows from
Lemma~\ref{lem:scalar-entropy}.
\end{proof}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//entropy.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//introduction.tex ====
\section{The theorem and the structure of the proof}\label{sec:introduction}

How quickly does local resampling remove fluctuations in a densely coupled
spin system? For the Sherrington--Kirkpatrick model~\cite{SK1975}, the signs
of the interactions matter: each spin interacts weakly with every other spin,
but the sum of the absolute interaction strengths grows with the dimension.
We study heat-bath dynamics, which updates a spin from its conditional Gibbs
law, in the tradition of Glauber's local stochastic dynamics~\cite{Glauber1963}.
Our result is a dimension-independent Poincar\'e inequality throughout the
high-temperature phase, at each fixed inverse temperature below one. It controls every real observable on a single
event of disorder probability tending to one.

Fix $0<\beta<1$ and put $j=\beta^2$. Let $J$ be a real symmetric
$n\times n$ matrix with zero diagonal and independent off-diagonal entries
$J_{ij}\sim N(0,j/n)$, $i<j$. For a field $h\in\R^n$, write
\[
 \mu_h(x)=\frac{1}{Z_h}\exp\left\{\frac12x^{\mathsf T}Jx+h^{\mathsf T}x\right\},
 \qquad x\in\{-1,1\}^n.
\]
Let $P_i^h$ denote conditional expectation under $\mu_h$ given all spins
except the $i$th. Define the unscaled Dirichlet form
\begin{equation}\label{eq:dirichlet}
 \D_h(f)=\sum_{i=1}^n\E_{\mu_h}(f-P_i^h f)^2.
\end{equation}

We use a rate-one clock at each site in continuous time, giving the
generator $\mathcal L_h=\sum_i(P_i^h-I)$. A discrete attempt selects one
site uniformly and has transition operator $P_h=n^{-1}\sum_iP_i^h$.
Thus $\mathcal L_h=n(P_h-I)$, and the discrete gap is the continuous gap
divided by $n$. The gap of a reversible generator is the infimum of its
Dirichlet form divided by the variance, over all nonconstant functions.

\begin{theorem}\label{thm:main}
There is a finite constant $C_\beta$, depending only on $\beta$, such that
\[
 \Prob_J\left\{\Var_{\mu_0}(f)\le C_\beta\D_0(f)
       \text{ for every }f:\{-1,1\}^n\longrightarrow\R\right\}
 \longrightarrow1.
\]
Consequently, the discrete-time heat-bath chain that chooses a uniform site
has spectral gap at least $1/(C_\beta n)$ with probability tending to one.
\end{theorem}

The parameter $\beta$ is fixed before $n$ tends to infinity; the constant
$C_\beta$ may depend on that fixed choice. External fields enter the proof
through posterior Gibbs laws, while the theorem concerns $\mu_0$.
The diagonal convention is immaterial to the Gibbs law: because $x_i^2=1$,
adding any diagonal matrix to $J$ adds a constant to the Hamiltonian.
This observation will let us introduce a Gaussian diagonal for matrix
calculations and remove it again.

\subsection{Earlier results and methods}
The interval $\beta<1$ is the classical zero-field high-temperature
range. Aizenman, Lebowitz, and Ruelle~\cite[Proposition~2.1]{ALR1987}
proved that $n^{-1}\log Z_0$ converges in probability and in mean to
$\log2+\beta^2/4$, the annealed free-energy value, throughout this interval.
The dynamical question is whether local resampling also removes fluctuations at a rate independent
of the system size.

A basic route to such a bound controls conditional influences.
Wu~\cite{Wu2006} proved a Poincar\'e inequality under a
Dobrushin spectral-radius condition on a matrix of nonnegative
conditional influences.
In dense SK systems, stronger fixed-temperature estimates become possible
by exploiting the signed random interactions. Bauerschmidt and
Bodineau~\cite{BB2019} used a Gaussian field decomposition to prove a
high-temperature logarithmic Sobolev inequality; its full spin-flip energy
differs from the heat-bath form~\eqref{eq:dirichlet}.
Eldan, Koehler, and Zeitouni~\cite{EKZ2022} established a spectral
condition for the heat-bath gap, yielding the SK range $\beta<1/4$.
Adhikari, Brennecke, Xu, and Yau~\cite{ABXY2024} obtained constant
unscaled gaps for mixed $p$-spin models under a sufficiently small
weighted condition on their interaction coefficients.

For mixing in single-site discrete attempts, Anari, Jain, Koehler, Pham,
and Vuong~\cite{AJKPV2022} used entropic independence to prove an
$O(n\log n)$ bound in the SK range $\beta<1/4$.
Anari, Koehler, and Vuong~\cite{AKV2024} extended this order to an
inverse-temperature threshold approximately equal to $0.295$.
More recently, Wang~\cite[Theorem~1.1]{Wang2026} proved worst-start
mixing in $O_\beta(n\log(n/\varepsilon))$ attempts for every fixed
$\beta<1/2$, simultaneously over external fields. Boban, Li, and Oveis
Gharan~\cite[Theorem~1.5 and Corollary~1.6]{BLO2026} obtained a
constant unscaled gap and $O(n^2)$ zero-field mixing for
$\beta<1/2+\varepsilon_0$, with an absolute
$\varepsilon_0\ge5\cdot10^{-5}$. Theorem~\ref{thm:main} reaches
every fixed $\beta<1$ at zero field.

Equilibrium covariance estimates already cover this entire range.
El Alaoui and Gaitonde~\cite[Theorem~1.1]{EAG2024} proved that the
expected operator norm of the zero-field spin covariance matrix stays
bounded for every fixed $\beta<1$. Brennecke, Schertzer, Xu, and
Yau~\cite[Theorem~1.1]{BSXY2024} proved the operator-norm approximation
\[
 \left\|\Cov_{\mu_0}(x)-\bigl((1+\beta^2)I-J\bigr)^{-1}\right\|
 \longrightarrow0
 \qquad\text{in probability}.
\]
For $a\in\R^n$, the variance of the linear observable $a^{\mathsf T}x$
is $a^{\mathsf T}\Cov_{\mu_0}(x)a$. These covariance results therefore
control linear fluctuations. The all-functions heat-bath inequality
requires additional estimates; our proof obtains them at the random
posterior fields generated by Gaussian observations.

Our argument transfers a functional inequality along stochastic
localization, introduced by Eldan~\cite{Eldan2013}.
We use its Gaussian-observation representation developed by El Alaoui
and Montanari~\cite{ElAlaouiMontanari2021}. The variance-retention and
Dirichlet-form framework of Chen and Eldan~\cite{CE2025} explains how
posterior estimates can imply an inequality for the initial law.
Here an observation event of exponentially small probability can still
carry substantial variance for a particular test function. We control
its probability after weighting paths by the square of that function.
The entropy calculation is a stopped finite-mixture version of the
drift-energy representation; see Lehec~\cite{Lehec2013} and the work
of F\"ollmer discussed there.

Gaussian observations, planted laws, and approximate message passing
(AMP) are also combined in the SK sampling method of El Alaoui,
Montanari, and Sellke~\cite{EAMS2022}. Celentano~\cite{Celentano2024}
proved local convexity of the TAP free energy used by that sampler near
sufficiently late AMP iterates. Our argument needs invertibility at every
sufficiently accurate solution of the field equation along the observation
path. A scalar entropy inequality and a local-volume estimate give this
uniformity in Section~\ref{sec:stability}. The field recursions retain the
Onsager correction of the
Thouless--Anderson--Palmer equations~\cite{TAP1977} and are related to
Bolthausen's iterative construction~\cite{Bolthausen2014}. We initialize
them at a Gibbs spin sample and use exact conditional-spin identities
through a fixed finite depth, without requiring convergence of the
recursion. Finally, the terminal gap adapts Wang's signed two-spin
method~\cite{Wang2026}; Section~\ref{sec:terminal} proves the variant
with an entrywise-square cost and a quartic row error.

\subsection{Notation and elementary identities}
For a vector $v$, set $\av{v}=n^{-1}\sum_i v_i$ and let $D_v$ be the diagonal
matrix with diagonal $v$. Vector products and scalar functions of vectors are
understood coordinatewise. Unless another subscript is displayed, $\norm{\cdot}$
is Euclidean norm for vectors and operator norm for matrices;
$\norm{\cdot}_{\Frob}$ is Frobenius norm. A centered Gaussian orthogonal ensemble (GOE) matrix of scale $j$
has variances $j/n$ off diagonal and $2j/n$ on the diagonal.

If $x^{(i)}$ is obtained by flipping the $i$th spin, put
\[
 d_i f(x)=\frac{f(x)-f(x^{(i)})}{2x_i},\qquad
 (\nabla u)_{ij}=d_j u_i.
\]
Thus $d_i f$ is independent of $x_i$ and
\begin{equation}\label{eq:discrete-product}
 d_i(fg)=f\,d_i g+g\,d_i f-2x_i(d_i f)(d_i g).
\end{equation}
At a fixed field $h_0$ set
\[
 h^1=h_0+Jx,\qquad m^1=\tanh h^1,\qquad v^1=\sech^2 h^1.
\]
Since $J_{ii}=0$, $m_i^1$ and $v_i^1$ are respectively the conditional mean
and variance of $x_i$. In particular,
\begin{equation}\label{eq:dirichlet-gradient}
 \D_{h_0}(f)=\E_{\mu_{h_0}}\sum_i v_i^1(d_i f)^2,\qquad
 \norm{G}_*^2=\E_{\mu_{h_0}}G^2+\D_{h_0}(G).
\end{equation}
For probability laws $\nu,\mu$, their relative entropy is
$D(\nu\Vert\mu)=\int\log(\dd\nu/\dd\mu)\,\dd\nu$ when $\nu\ll\mu$,
and $+\infty$ otherwise.

Constants are independent of $n$; dependence on a fixed temperature,
construction depth, or positive stability margin is recorded at its use.
Their values may change between displays. The prerequisites are Gaussian
concentration and comparison, Gaussian integration by parts, and stochastic
calculus. We prove the model-specific estimates in the sections below.

\subsection{The proof mechanism}
There are two parts to the argument: prove useful inequalities for
posterior Gibbs laws, then transfer them to the unobserved law. A long but
fixed observation makes most individual spins predictable. The corresponding
small conditional variances, combined with a two-spin curvature inequality,
give a terminal gap. To recover the initial variance, we need a directional
covariance estimate along the path and a mean estimate under square
reweighting. The former bounds the loss of variance; the latter prevents
that variance from accumulating on exceptional paths.

The posterior estimates come from finite Onsager-corrected recursions.
Exact conditional-spin identities control their residuals. A telescoping
identity for the squared increments then selects two consecutive small
increments without requiring convergence of the recursion. Uniform matrix
estimates handle
the resulting adaptive diagonal coefficients under an explicit inverse
margin. To compare these approximate fields with an exact root, stability
must hold throughout the small-residual region and persist when the
diagonal variance matrix is replaced by the nearby response coefficients
used in the comparison. A scalar entropy estimate controls a Gaussian
integral, and
a local-volume lower bound turns that integral control into exclusion of
unstable approximate roots. All tolerances and finite depths are chosen
before the large-$n$ limit.

\subsection{The three inputs}
We observe a spin sample in independent Gaussian noise:
\begin{equation}\label{eq:observation-process}
 X\sim\mu_0,\qquad Y(t)=tX+B(t),\quad 0\le t\le T,
\end{equation}
where $B$ is standard $n$-dimensional Brownian motion independent of $X$.
Conditional on the observations up to $t$, the spin law is $\mu_{Y(t)}$.
The proof uses the following three inputs, established in the later sections.

\begin{enumerate}
\item \textbf{Stability along ordinary observation paths.}
Proposition~\ref{prop:stable-fields} gives a deterministic notion of a good
field such that, for typical $J$, all fields $Y(t)$ are good except on an event
of observation probability at most $e^{-a n}$, where $a>0$ is fixed.

\item \textbf{Two estimates at a good field.}
For every $G$, Proposition~\ref{prop:directional} proves
\begin{equation}\label{eq:input-directional}
 \norm{\Cov_{\mu_h}(G,x)}^2
 \le C\bigl(\Var_{\mu_h}(G)+\D_h(G)\bigr).
\end{equation}
For $N=\E_{\mu_h}f^2>0$ and $\nu=f^2\mu_h/N$,
Proposition~\ref{prop:weighted-mean} proves, for each sufficiently small fixed $\delta>0$,
\begin{equation}\label{eq:input-weighted}
 \frac{\norm{\E_\nu x-t_*(h)}}{\sqrt n}
 \le C\delta+\frac{C_\delta}{\sqrt n}
                 \left(1+\frac{\D_h(f)}{N}\right).
\end{equation}
Here $t_*(h)=\tanh r(h)$, where $r(h)$ is the unique solution of
\[
 r-h+\{j(1-n^{-1}\|\tanh r\|^2)I-J\}\tanh r=0.
\]
Existence and uniqueness at the fields under consideration are part of
Lemma~\ref{lem:root}. The constant $C$ is independent of $\delta$; $C_\delta$ may depend
on it. Taking $f=1$ gives the same comparison for $\E_{\mu_h}x$.

\item \textbf{A terminal gap.}
For a sufficiently large fixed $T$, Proposition~\ref{prop:terminal-gap}
gives $\Var_{\mu_{Y(T)}}f\le2\D_{Y(T)}(f)$ outside an event of ordinary
observation probability at most $e^{-a n}$.
\end{enumerate}

The transfer uses the two estimates for different purposes. Directional
covariance controls posterior variance loss, while square-reweighted means
bound the entropy of the changed observation law. Their combination controls
the contribution of the bad paths before the terminal gap is applied.
Section~\ref{sec:observation} proves this implication first, so the later
technical sections have a precise target.

\subsection{Organization}
After the observation transfer in Section~\ref{sec:observation},
Section~\ref{sec:matrices} constructs the matrix diagnostics used by the
adaptive field calculations. Sections~\ref{sec:entropy}
and~\ref{sec:stability} identify the scalar equality law and establish
stability along observations. Section~\ref{sec:residuals} develops the
finite recursions and their weak residual rules; Section~\ref{sec:consequences}
uses them to prove the two posterior estimates. Section~\ref{sec:terminal}
closes the argument with the fixed-time terminal gap.
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//introduction.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//main.tex ====
\documentclass[11pt]{article}
\input{style}
\title{A spectral gap throughout the high-temperature\texorpdfstring{\\}{ }Sherrington--Kirkpatrick phase}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
For every fixed inverse temperature $0<\beta<1$, we prove that the unscaled
spectral gap of single-site heat-bath dynamics for the zero-field Gaussian
Sherrington--Kirkpatrick model is bounded away from zero with probability
tending to one over the disorder. Equivalently, the Gibbs law satisfies a
dimension-free Poincar\'e inequality for all functions.
\end{abstract}
{\small\tableofcontents}
\clearpage
\input{sections/introduction}
\input{sections/observation}
\input{sections/matrices}
\input{sections/entropy}
\input{sections/stability}
\input{sections/residuals}
\input{sections/consequences}
\input{sections/terminal}
\input{references}
\end{document}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//main.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//matrices.tex ====
\section{Uniform random-matrix estimates}\label{sec:matrices}

The field analysis uses random-matrix estimates in two forms.  Finite
field recursions produce products whose diagonal coefficients may be
chosen after the disorder is observed.  Lemma~\ref{lem:words} controls
such products of bounded length, subject to a separate positivity
margin whenever an inverse occurs.  We also obtain fixed-vector
bilinear estimates and a simultaneous determinant bound for the
stability argument in Section~\ref{sec:stability}.  Their expectation
calculations use the same inverse identity.  We begin with fixed
deterministic coefficients and then use concentration and finite meshes
to obtain the required simultaneous estimates.

Throughout this section, $j\in(0,1)$ is fixed.  A centered GOE matrix
$W$ has independent entries on and above the diagonal, with
\[
 \E W_{ab}^{2}=j/n\quad(a<b),\qquad
 \E W_{aa}^{2}=2j/n.
\]
We couple $J$ to $W$ by putting $J=W-D_{\diag W}$.  In particular,
the off-diagonal law of $J$ is the law used in the theorem.  Fix
$\eta>0$ so that, with $a_+=1+\eta$,
\begin{equation}\label{eq:matrix-subcritical}
 j a_+^2<1.
\end{equation}
For a nonnegative diagonal matrix $A\le a_+I$, put
\begin{equation}\label{eq:K}
 \begin{gathered}
 \chi_A=\frac1n\Tr A,\qquad B_A=j\chi_A,\qquad
 K_A(M)=K_A(1;M),\\
 K_A(z;M)=(I-zAM+z^2B_AA)^{-1}.
 \end{gathered}
\end{equation}
The associated symmetric matrix is
\[
 S_A(z;M)=I+z^2B_AA-zA^{1/2}MA^{1/2}.
\]
The symmetric matrix lets us control the inverse without imposing a
positive lower bound on the individual diagonal entries of $A$.
Whenever $S_A(z;M)$ is invertible, the following identities hold:
\begin{align}
 K_A(z;M)
 &=I+A^{1/2}S_A(z;M)^{-1}A^{1/2}(zM-z^2B_AI),
 \label{eq:K-symmetric-representation}\\
 K_A(z;M)A&=A^{1/2}S_A(z;M)^{-1}A^{1/2}.
 \label{eq:KA-symmetric-representation}
\end{align}
To verify the first identity, apply
$(I-UV)^{-1}=I+U(I-VU)^{-1}V$.  For the second, multiply the two
sides using $S_A=I-A^{1/2}(zM-z^2B_AI)A^{1/2}$.  These formulas
will also let us truncate the inverse while retaining bounded
matrix factors.

\subsection{Words and their predictions}

A \emph{word} is an ordered product of diagonal matrices, copies of a
symmetric matrix $M$, and at most one factor $K_A(M)$.  Its length is
the number of factors.  Consecutive diagonal factors may always be
combined, and an absent diagonal factor is interpreted as $I$.
To each word $Q$ we associate a diagonal matrix $\mathsf P(Q)$ by
the contraction rule below.  The estimates will compare the diagonal
of $Q$ with this explicitly defined matrix.

Begin with a word without $K_A$.  Consider every complete
noncrossing pairing of its occurrences of $M$, in their linear
order.  For each pairing, remove an innermost pair by the rule
\begin{equation}\label{eq:noncrossing-contraction}
 MDM\ \longmapsto\ j\frac{\Tr D}{n}I,
\end{equation}
where $D$ is diagonal after previous removals.  Combine adjacent
diagonals and continue.  Sum the resulting diagonal matrices over
all complete noncrossing pairings.  If there is no complete pairing,
the prediction is zero; if there is no occurrence of $M$, it is the
word itself.  The result for a fixed pairing does not depend on the
order of removal: two available innermost pairs are disjoint, and
their removals commute.
For words without an inverse factor, this is the operator-valued
semicircular moment rule with covariance map
$D\mapsto jn^{-1}\Tr(D)I$; see Speicher~\cite{Speicher1998}.

For example, four occurrences of $M$ have two complete noncrossing
pairings, giving
\[
 \begin{split}
 \mathsf P(MD_1MD_2MD_3M)
 ={}&j^2\frac{\Tr D_1}{n}\frac{\Tr D_3}{n}D_2\\
 &+j^2\frac{\Tr D_2}{n}\frac{\Tr(D_1D_3)}n I.
 \end{split}
\]
The first term pairs the first two and last two occurrences; the second
pairs the outside occurrences around the middle pair.

For a word containing $K_A$, replace that factor by the formal series
\begin{equation}\label{eq:K-formal-series}
 K_A(z;M)=\sum_{r\ge0}(zAM-z^2B_AA)^r,
\end{equation}
and apply the preceding rule coefficient by coefficient.  Ordinary
occurrences of $M$ outside this factor are left unscaled.  We prove
below that the resulting series $\mathsf P(Q;z)$ is a polynomial;
we define $\mathsf P(Q)=\mathsf P(Q;1)$.

The corresponding trace prediction is
$\mathsf p(Q)=n^{-1}\Tr\mathsf P(Q)$.  To see its cyclic invariance,
draw an ordinary trace word on a circle, with a chord for each
pair in a noncrossing pairing.  Its contribution is $j$ for each chord,
multiplied by, for each face, the average of the product of the
diagonal matrices on that face's boundary arcs.  Removing an
innermost chord gives exactly \eqref{eq:noncrossing-contraction}, and
the average over the last face is the final normalized trace.  This
description is unchanged by rotating the circle.  For words with
$K_A(z;M)$, apply this argument coefficient by coefficient in
\eqref{eq:K-formal-series}.

\begin{lemma}[Simultaneous word estimates]\label{lem:words}
Fix $L\in\mathbb N$, $M_0<\infty$, and $c>0$.  There are constants
$C,c_1>0$ and $n_0$, depending only on $j,a_+,L,M_0,c$, such that the
following event has probability at least $1-Ce^{-c_1n}$ for
$n\ge n_0$.  Simultaneously for all words $Q$ of length at most $L$,
all their real diagonal factors of norm at most $M_0$, and all
$0\le A\le a_+I$, one has
\begin{equation}\label{eq:word-estimates}
 \norm{Q}\le C,\qquad
 \left(\sum_{a\ne b}|Q_{ab}|^4\right)^{1/4}\le C,\qquad
 \norm{\diag(Q-\mathsf P(Q))}_2\le C.
\end{equation}
For a word containing $K_A(J)$, the assertion is restricted to
parameters satisfying $S_A(1;J)\succeq cI$.  The same conclusion
holds with $W$ in place of $J$, with the corresponding stability
restriction.  Thus all diagonal parameters in this assertion may
be chosen after the random matrix has been observed.

If a word has $r$ ordinary occurrences of $M$ outside its one
$K_A(z;M)$ factor, then $\mathsf P(Q;z)$ has degree at most $r$.
\end{lemma}

We prove the simultaneous estimate in three steps.  For a fixed
deterministic diagonal, Gaussian comparison gives stability along
$0\le z\le1$.  We then calculate the expected diagonal of a
truncated word.  Finally, concentration makes the word estimates
simultaneous in the diagonal parameters.  The first step supplies
stability for each deterministic choice; it does not remove the
separate stability restriction for adaptive $A$ in
Lemma~\ref{lem:words}.

\begin{lemma}[Stability for a deterministic diagonal]
\label{lem:matrix-path-stability}
There is $c_0>0$, depending only on $j,a_+$, with the following
property.  For every fixed deterministic $0\le A\le a_+I$, and every
fixed $R>2\sqrt j$, there are constants $C,c_2>0$ and $n_0$,
uniform in $A$, such that
\begin{equation}\label{eq:deterministic-path-stability}
 \Prob\left\{\norm W\le R\ \hbox{and}\quad
 S_A(z;W)\succeq c_0I\text{ for all }0\le z\le1\right\}
 \ge1-Ce^{-c_2n}\qquad(n\ge n_0).
\end{equation}
In addition, for every fixed $\varepsilon>0$,
$\Prob\{\norm W>2\sqrt j+\varepsilon\}\le
C_\varepsilon e^{-c_\varepsilon n}$ for sufficiently large $n$.
\end{lemma}

\begin{proof}
We first bound the expected quadratic process that defines the
smallest eigenvalue of $S_A$.  Concentration will give a margin at
each fixed $z$, and a finite mesh will extend it along the path.

We use two standard Gaussian facts.  A real-valued
$L$-Lipschitz function of a standard Gaussian vector satisfies
$\Prob\{|F-\E F|>u\}\le2\exp(-u^2/(2L^2))$.
Also, if two centered Gaussian processes have ordered squared
increments, the process with larger squared increments has the
larger expected supremum (Sudakov--Fernique). The latter comparison remains valid
after the same deterministic function is added to both processes;
see the deterministic-shift comparison in Vitale~\cite{Vitale2000}.
Both facts apply by finite nets and continuity to the compact
index sets below.  The linear map from the independent standard
Gaussian coordinates to $W$, in Frobenius norm, has norm
$\sqrt{2j/n}$.

For the comparison, write $w=A^{1/2}p$ for $\norm p=1$ and consider
\[
 X_p=w^TWw,\qquad
 Y_p=2\sqrt{j/n}\,\norm w\,g^Tw,
\]
where $g$ is standard Gaussian in $\R^n$.  If $w'=A^{1/2}p'$, direct
calculation gives
\begin{align*}
 &\E(Y_p-Y_{p'})^2-\E(X_p-X_{p'})^2\\
 &\qquad=\frac{2j}{n}
 \left[(\norm w^2-\norm{w'}^2)^2
       +2(\norm w\norm{w'}-w^Tw')^2\right]\ge0.
\end{align*}
Consequently, for each $z\in[0,1]$,
\[
 \E\sup_{\norm p=1}\{zX_p-z^2j\chi_A\norm w^2\}
 \le \E\sup_{\norm p=1}\{zY_p-z^2j\chi_A\norm w^2\}.
\]
Set $T_g=\sqrt{j/n}\norm{A^{1/2}g}$.  Since
$g^Tw=p^TA^{1/2}g\le\norm{A^{1/2}g}$, the last supremum is at most
\[
 \sup_{0\le v\le\sqrt{a_+}}
       (2zT_gv-z^2j\chi_Av^2).
\]
The remaining supremum depends on the Gaussian vector only through
$T_g$.  The variable $T_g^2$ has mean $j\chi_A$ and variance
$2j^2\Tr(A^2)/n^2\le C/n$.  The inequality
$|\sqrt x-\sqrt y|\le\sqrt{|x-y|}$ therefore shows, uniformly in
$A$, that $\E|T_g-\sqrt{j\chi_A}|=o(1)$.  On replacing $T_g$ by
$\sqrt{j\chi_A}$ in the preceding display, the expression becomes
$2d-d^2$, with
$0\le d=z\sqrt{j\chi_A}v\le\sqrt j\,a_+<1$.
Thus, with $\delta=(1-\sqrt j\,a_+)^2>0$,
\begin{equation}\label{eq:quadratic-process-margin}
 \E\sup_{\norm p=1}
 \{zp^TA^{1/2}WA^{1/2}p-z^2j\chi_Ap^TAp\}
 \le1-\delta+o(1),
\end{equation}
uniformly in $A,z$.
The supremum in \eqref{eq:quadratic-process-margin} is
$a_+$-Lipschitz in $W$ for Frobenius norm.  Gaussian concentration
gives an exponentially small probability that it exceeds
$1-\delta/2$, uniformly for each fixed $A,z$.

To pass from a fixed $z$ to the entire interval, we also need a
bound on $\norm W$.  Taking $A=I$ in the comparison without
deterministic offsets gives
\[
 \E\lambda_{\max}(W)\le2\sqrt{j/n}\E\norm g\le2\sqrt j.
\]
The same holds for $-W$.  Concentration of these two eigenvalues
proves the stated norm tail.  On $\norm W\le R$, the function of
$z$ inside the supremum in \eqref{eq:quadratic-process-margin} is
uniformly Lipschitz, with constant at most $a_+R+2ja_+^2$.
A sufficiently fine constant mesh of $[0,1]$, followed by a union
bound, therefore gives \eqref{eq:deterministic-path-stability}, for
example with $c_0=\delta/4$.
\end{proof}

\subsection{Truncation and the loop recursion}

The inverse in a word need not exist for every realization of the
matrix.  We therefore replace each word by a globally defined
bounded function of $W$, agreeing with the original word whenever
the required norm and stability conditions hold.  This permits
concentration and integration by parts before those conditions
are imposed.

Fix a sufficiently large constant $R$.  Let $\Pi_R$ be metric
projection, in the Hilbert space of real symmetric matrices with
Frobenius norm, onto $\{M:\norm M\le R\}$.  This is a nonexpansive
map and is equivariant under orthogonal conjugation.  Put
$\widehat W=\Pi_R(W)$.  Choose a smooth compactly supported real
function $f$ which equals $x^{-1}$ on the interval
\[
 \left[\tfrac12\min(c,c_0),\,2+c+c_0+ja_+^2+a_+R\right].
\]
This interval contains, with slack, every stable spectrum needed
both for deterministic-parameter estimates and for the restriction
$S_A(1;M)\succeq cI$ in Lemma~\ref{lem:words}.  Replace $K_A(z;W)$
by
\begin{equation}\label{eq:truncated-K}
 \widehat K_A(z)
 =I+A^{1/2}f(S_A(z;\widehat W))A^{1/2}
       (z\widehat W-z^2B_AI).
\end{equation}
Replace every ordinary $W$ in a word by $\widehat W$ as well; denote
the resulting word by $\widehat Q_\theta(z)$, where $\theta$ lists
all its diagonal parameters.  These truncated factors equal the
original factors on the stability events where they will be used.

For each fixed length and fixed bounds on the diagonal parameters,
there is a constant $C$ such that
\begin{align}
 \norm{\widehat Q_\theta(z)}&\le C,
 &\norm{\widehat Q_\theta(z;W)-\widehat Q_\theta(z;W')}_{\Frob}
 &\le C\norm{W-W'}_{\Frob},
 \label{eq:truncated-word-lipschitz}\\
 \Lip_{\Frob}\bigl(\widehat Q_\theta(z;\cdot)
                   -\widehat Q_{\theta'}(z;\cdot)\bigr)
 &\le C\sqrt\Delta
 &&\text{if }\norm{\theta-\theta'}_\infty\le\Delta\le1.
 \label{eq:truncated-word-increment-lipschitz}
\end{align}
The constants are uniform for $z\in[0,1]$ and for all matrix sizes.
Here the Lipschitz constants concern maps into matrices with
Frobenius norm.

We verify both estimates with constants independent of dimension.
On the operator-norm ball, all factors are bounded, and their
differentials in a perturbation $E$ are bounded in Frobenius norm
by $C\norm E_{\Frob}$.  The only functional-calculus factor is
handled by writing $f$ as a Fourier integral and using
\[
 \dd(e^{itS})[E]
 =it\int_0^1 e^{it(1-u)S}E e^{ituS}\,\dd u.
\]
The Fourier transform of $f$ decays faster than any power, so its
first two weighted absolute moments are finite.  This formula and
the corresponding difference formula give
\[
 \norm{\dd f(S)[E]}_{\Frob}\le C\norm E_{\Frob},\qquad
 \norm{(\dd f(S)-\dd f(S'))[E]}_{\Frob}
 \le C\norm{S-S'}\norm E_{\Frob}.
\]
Changing the diagonal parameters by at most $\Delta$ changes their
square roots in norm by at most $\sqrt\Delta$.  Consequently the
factors, and their matrix differentials, change by respectively
$C\sqrt\Delta$ and $C\sqrt\Delta\norm E_{\Frob}$.  The product rule
proves the same differential assertions for a word.  Integrating
along the line segment between two matrices in the convex
operator-norm ball proves \eqref{eq:truncated-word-lipschitz} and
\eqref{eq:truncated-word-increment-lipschitz} there.  Composition with
the nonexpansive projection $\Pi_R$ proves them everywhere.

These Lipschitz estimates first give the moment bounds needed for
the expectation calculation.  For a fixed parameter tuple,
\begin{equation}\label{eq:word-entry-and-trace-variance}
 \Var(\widehat Q_{ab})\le C/n,\qquad
 \Var\left(\frac1n\Tr\widehat Q\right)\le C/n^2.
\end{equation}
The normalized trace gains a factor $n^{-1/2}$ in its Frobenius
Lipschitz constant.  The same estimates for a difference of two
parameter tuples gain a factor $\Delta$ on the right-hand sides.
Fourth centered entry moments are bounded by $C/n^2$, or
$C\Delta^2/n^2$ for such a difference, by integrating the Gaussian
concentration tail.

\begin{lemma}[Fixed-parameter diagonal expectations]
\label{lem:word-bias}
For the truncated words just defined, uniformly in deterministic
diagonal parameters in the indicated bounded ranges, in
$z\in[0,1]$, and in the diagonal index $i$,
\begin{equation}\label{eq:word-bias}
 \E\widehat Q_\theta(z)_{ii}
 =\mathsf P(Q_\theta;z)_{ii}+O(n^{-1}).
\end{equation}
For words without $K_A$, the prediction has no $z$ dependence.
The formal prediction for a word with $r$ ordinary $W$ letters is a
polynomial of degree at most $r$.
\end{lemma}

\begin{proof}
The calculation has two parts.  We first justify integration by
parts for the truncated words and compute the inverse factor
alone.  We then obtain a recursion in the number of ordinary
$W$ factors; the same recursion determines the formal
prediction exactly and the expectation up to $O(n^{-1})$.

All constants in this proof depend only on the fixed length and
parameter bounds.  Lemma~\ref{lem:matrix-path-stability} shows that,
for every fixed deterministic parameter tuple, the truncations
agree with the actual matrices for all $z\in[0,1]$, except on an
event of probability $Ce^{-c_2n}$.  Gaussian integration by parts
can be applied to the globally Lipschitz truncated factors, using
weak derivatives.  More explicitly, for a scalar Lipschitz
function $H$,
\begin{equation}\label{eq:GOE-integration-by-parts}
 \E[W_{ab}H(W)]
 =\frac jn\E\bigl[\dd H(W)[E_{ab}+E_{ba}]\bigr].
\end{equation}
When $a=b$, the elementary direction on the right is doubled,
as required by the diagonal variance $2j/n$.

For \eqref{eq:GOE-integration-by-parts}, remove the projection
from the distinguished ordinary $W$ factor.  The resulting
expectation error is exponentially small, since the other factors
are bounded and Gaussian norm tails control the removed factor.
On the event of agreement, the differential of the inverse is
\begin{equation}\label{eq:inverse-differential}
 \dd K_A(z;W)[E]=zK_A(z;W)AEK_A(z;W).
\end{equation}
We may use the expressions on the right, with truncated factors,
on the whole probability space.  The error is exponentially small
even after the index sums: all factors and their true truncated
differentials are bounded, there are at most a fixed polynomial
number of indices, and the disagreement event has exponentially
small probability.  More precisely, the elementary direction
$E_{ab}+E_{ba}$ has Frobenius norm at most $2$.  Both the true weak
differential and its formal replacement are therefore bounded by
$C$ in each entry; their difference vanishes on the agreement event.
Each such differential error has expected absolute value at most
$Ce^{-c_2n}$.  Expanding the diagonal entry of a length-$L$ word and
applying the product rule uses at most
$(L-1)n^{L-1}$ scalar contributions before any index identifications;
the factor $j/n$ in integration by parts only improves this bound.
Thus their total error is at most a fixed polynomial in $n$ times
$e^{-c_2n}$, hence at most $C'e^{-c'n}$.  Subsequent words have
bounded lengths depending only on the original length.  We
therefore suppress hats during the loop calculation, with this
convention understood.

We now compute the base case, in which the only nondiagonal
factor is $K=K_A(z;W)$.  Write
$a_i=A_{ii}$, $k_i(z)=\E K_{ii}$, and
$r(z)=n^{-1}\sum_i a_i k_i(z)$.  Differentiating $K_{bi}$ in
\eqref{eq:GOE-integration-by-parts} gives
\begin{align*}
 \E(WK)_{ii}
 &=\frac{zj}{n}\E\left[(\Tr KA)K_{ii}\right]
   +\frac{zj}{n}\sum_b\E[(KA)_{bi}K_{bi}]+O(e^{-c'n})\\
 &=zjr(z)k_i(z)+O(n^{-1}).
\end{align*}
For the second sum, the absolute value of the sum over $b$ is at
most $\norm{KAe_i}\norm{Ke_i}\le C$.  The factorization in the
first term follows from \eqref{eq:word-entry-and-trace-variance};
its covariance is at most $C n^{-3/2}$, which is more than enough.
The inverse identity $K=I+zAWK-z^2B_AAK$ now yields
\begin{equation}\label{eq:K-loop-equations}
 \E(WK)_{ii}=zjr k_i+O(n^{-1}),\qquad
 k_i=1+z^2ja_i(r-\chi_A)k_i+O(n^{-1}).
\end{equation}

The loop equations must select the solution near $k_i=1$.
We establish this by continuation from $z=0$, where the inverse
is the identity.  Put
$h(z)=r(z)-\chi_A$.  The truncated $k_i$ are uniformly bounded and
continuous in $z$.  If $|h(z)|\le\delta_1$, the second equation in
\eqref{eq:K-loop-equations} gives
$\max_i|k_i(z)-1|\le C\delta_1+C/n$.  Choose $\delta_1>0$ small,
using \eqref{eq:matrix-subcritical}, so that throughout this region
and for large $n$,
\[
 \left|z^2j\frac1n\sum_i a_i^2 k_i(z)\right|
 \le1-\delta_2
\]
for some fixed $\delta_2>0$.  Averaging the second equation in
\eqref{eq:K-loop-equations} with weights $a_i$ gives
\[
 h(z)=z^2j\left(\frac1n\sum_i a_i^2k_i(z)\right)h(z)+O(n^{-1}),
\]
and hence $|h(z)|\le C/n$ while $|h(z)|\le\delta_1$.  Since
$h(0)=0$, continuity rules out a first exit from that region for
large $n$.  It follows uniformly in $z$ that
\begin{equation}\label{eq:K-and-WK-bias}
 \E K_{ii}=1+O(n^{-1}),\qquad
 \E(WK)_{ii}=zB_A+O(n^{-1}).
\end{equation}

To complete the base case, we verify the formal identity
$\mathsf P(K_A(z;W))=I$ independently of the expectation calculation.
The cancellation takes place at each coefficient of the series.
In the expansion
\eqref{eq:K-formal-series}, a block $-z^2B_AA$ may be regarded as
the negative contraction of two adjacent blocks $zAW,zAW$, since
their contraction is $z^2j\chi_AA=z^2B_AA$.
Fix a completely noncrossing-paired chain $(AW)^{2m}$ with $m>0$.
Let $\mathcal A$ be the nonempty set of its pairs which join two
originally adjacent occurrences of $W$.  It is nonempty because
every nonempty noncrossing pairing of a linear order has an
adjacent pair.  All pairs in $\mathcal A$ are disjoint.  The terms
of \eqref{eq:K-formal-series} which give this paired chain are in
bijection with subsets $T\subseteq\mathcal A$: contract the pairs
in $T$ into the negative deterministic blocks, and pair the
remaining occurrences normally.  The magnitude of the final
diagonal contribution is unchanged, and its sign is $(-1)^{|T|}$.
Thus the total multiplier is
$\sum_{T\subseteq\mathcal A}(-1)^{|T|}=0$.  This argument is finite
at each power of $z$ and proves the asserted formal identity.
Multiplication by surrounding diagonal matrices proves both
claims of the lemma for words with no ordinary $W$ letters.

For the induction step, select an ordinary occurrence of $W$ in
a general word and apply integration by parts at that occurrence.
We describe the resulting finite recursion directly, without
requiring convergence of the formal series.  In each product-rule
term, mark the partner of the selected occurrence: for a derivative
falling on an ordinary factor, this is the $W$ being differentiated.
If the derivative falls on $K_A$, replace that factor by
$zK_AA\,\boxed W\,K_A$ and take the boxed occurrence as the
partner.  In either case, list the two marked positions in their
linear order, so the marked product reads
\[
 U\,\boxed W\,V\,\boxed W\,Q.
\]
The two orientations in \eqref{eq:GOE-integration-by-parts}
contribute respectively
\begin{equation}\label{eq:word-loop-contractions}
 \frac jn\E\bigl[(UQ)_{ii}\Tr V\bigr],\qquad
 \frac jn\E(UV^TQ)_{ii}.
\end{equation}
There is an additional factor $z$ when the derivative fell on
$K_A$.  The second expression in
\eqref{eq:word-loop-contractions} is $O(n^{-1})$ by operator-norm
bounds, including when it contains two copies of $K_A$.
The first expression is
\begin{equation}\label{eq:word-loop-leading}
 j\left(\frac1n\E\Tr V\right)\E(UQ)_{ii}+O(n^{-1}),
\end{equation}
again by \eqref{eq:word-entry-and-trace-variance}.  The index sums
explain the two terms.  The opposite orientation
identifies the two interior indices and gives $\Tr V$; the equal
orientation reverses the interior path and gives $V^T$.

Consequently $\E\widehat Q_{ii}$ is the finite sum of the leading
products \eqref{eq:word-loop-leading} over all ordinary partners,
plus $z$ times the corresponding leading product for the one
$K_A$ factor if present, with an $O(n^{-1})$ remainder.  Each word
$V$ and $UQ$ in this recursion has fewer ordinary $W$ letters than
the original word.  Each contains at most one $K_A$: if the
derivative fell on $K_A$, the splitting places one of its two
copies in each word.  Extra diagonal factors $A$ cause no problem,
since at most one is added per decrease of the induction index.

The decrease in the number of ordinary letters holds whether the
selected occurrence lies before or after the inverse.  The two
possible orders can be checked explicitly.  If the original word is
$P K_A R W T$, the contraction with $K_A$ contributes
\[
 zj\,\E\left[(P K_A A T)_{ii}\frac{\Tr(K_A R)}n\right].
\]
If it is $P W R K_A T$, that contribution is
\[
 zj\,\E\left[(P K_A T)_{ii}\frac{\Tr(R K_A A)}n\right].
\]
Here $P,R,T$ contain no inverse factor.  The displayed outer and
inner words each have one inverse factor, even though the original
distinguished occurrence may lie anywhere among the ordinary letters.

It remains to compare this expectation recursion with the formal
prediction.  Sort the noncrossing pairings by the partner of the
selected ordinary occurrence.  The inside and outside may then be paired
independently; their predictions are respectively
$\mathsf p(V)$ and $\mathsf P(UQ)$.  If the partner occurs inside
the series for $K_A(z;W)$, marking that occurrence in the formal
series is exactly the formal differentiation identity
$\dd K_A[E]=zK_AAEK_A$.  Thus the entire sum over positions in that
series is the single $z$-weighted product just described.

We can now induct on the number $r$ of ordinary $W$ letters.
An ordinary partner leaves a total of $r-2$ such letters in the
two smaller words; a partner in $K_A$ leaves $r-1$.  The base
prediction is constant.  The recursion therefore proves that
the formal prediction is a polynomial of degree at most $r$.
It also proves \eqref{eq:word-bias}: use the induction hypothesis
for the two expectations in each leading product, and average
the diagonal induction hypothesis to handle its normalized
trace.  There are only finitely many products and all factors
are uniformly bounded.  The same induction, with no inverse
factor, proves the statement for ordinary words.
\end{proof}

\subsection{Making the word estimates simultaneous}

\begin{proof}[Proof of Lemma~\ref{lem:words}]
We first make the centered estimates uniform over the diagonal
parameters, then add the expectation estimate from
Lemma~\ref{lem:word-bias}.  Fix a letter pattern and let $\theta$
range over its parameter box, which has at most $C_Ln$ real
coordinates.  Work initially with the truncated words at $z=1$.  Conjugation by any
diagonal sign matrix preserves the law of $W$ and conjugates
$\widehat Q_\theta$ by that same sign matrix.  The diagonal
parameters commute with the conjugation, and both the projection
and the functional calculus respect it.  Thus
$\E(\widehat Q_\theta)_{ab}=0$ whenever $a\ne b$.

Define the two matrix seminorms
\[
 N_2(M)=\norm{\diag M}_2,\qquad
 N_4(M)=\left(\sum_{a\ne b}|M_{ab}|^4\right)^{1/4}.
\]
Both are $1$-Lipschitz with respect to Frobenius norm.
The entry moment bounds following
\eqref{eq:word-entry-and-trace-variance} give
\begin{equation}\label{eq:centered-word-norm-expectations}
 \E N_k(\widehat Q_\theta-\E\widehat Q_\theta)\le C
 \quad(k=2,4).
\end{equation}
Indeed, sum the $n$ second moments for $k=2$, and the at most
$n^2$ fourth moments for $k=4$, then apply Jensen's inequality.
For two parameter tuples at distance at most $\Delta$, the same
argument, using \eqref{eq:truncated-word-increment-lipschitz}, gives
\[
 \E N_k\bigl((\widehat Q_\theta-\E\widehat Q_\theta)
             -(\widehat Q_{\theta'}-\E\widehat Q_{\theta'})\bigr)
 \le C\sqrt\Delta.
\]
The random norms in this last display concentrate around their
means at Gaussian scale $C\sqrt{\Delta/n}$, again by
\eqref{eq:truncated-word-increment-lipschitz}.  The centered
expectations are deterministic and do not change Lipschitz
constants.

To control every parameter tuple, approximate it successively
on finer meshes.  For each $\ell\ge0$, choose a sup-norm mesh
$\mathcal N_\ell$ of spacing $2^{-\ell}$ in the parameter box.  It may be chosen with
\[
 |\mathcal N_\ell|\le\exp(C_L(\ell+1)n).
\]
For a tuple $\theta$, choose nearest mesh points
$\theta_\ell\in\mathcal N_\ell$.  Then
$\norm{\theta_\ell-\theta_{\ell-1}}_\infty\le C2^{-\ell}$.
For all pairs of mesh points at this distance, concentration and
a union bound show that their centered increments have both
$N_2$ and $N_4$ norm at most
\[
 C'\sqrt{\ell+1}\,2^{-\ell/2},
\]
except with probability $Ce^{-c(\ell+1)n}$, provided $C'$ is
chosen sufficiently large.  Specifically, the squared ratio
of this threshold to the Gaussian scale is a constant multiple
of $(\ell+1)n$, and its constant can be chosen to dominate the
logarithm of the number of mesh pairs.  The initial mesh is
bounded in the same way using
\eqref{eq:centered-word-norm-expectations}.  Summing the failure
probabilities over $\ell$, and summing the convergent increment
bounds along each chain, proves, with probability
$1-Ce^{-c_1n}$,
\begin{equation}\label{eq:simultaneous-centered-words}
 \sup_\theta N_k(\widehat Q_\theta-\E\widehat Q_\theta)\le C
 \quad(k=2,4).
\end{equation}
At each fixed $n$, continuity of the truncated word and its
expectation in $\theta$ justifies taking the limit of the mesh
chain.  There are only finitely many letter patterns of length
at most $L$, so their events may be intersected.

The centered words are now controlled on one event for every
parameter tuple.  Lemma~\ref{lem:word-bias} supplies the remaining
bias estimate,
$N_2(\E\widehat Q_\theta-\mathsf P(Q_\theta))\le C/\sqrt n$,
uniformly in $\theta$.  Sign symmetry makes the off-diagonal of
$\E\widehat Q_\theta$ identically zero.  Combining these facts
with \eqref{eq:simultaneous-centered-words} proves the desired
diagonal and off-diagonal assertions for the truncated words.
Their operator norms are already bounded by construction.
On $\norm W\le R$ and at parameters with $S_A(1;W)\succeq cI$,
the truncations equal the exact words, proving the GOE version.

The GOE estimate transfers to the zero-diagonal model through
$J=W-D_{\diag W}$.  This last step uses a Frobenius bound on the
removed diagonal.  Since
\[
 \norm{D_{\diag W}}_{\Frob}^{2}
   \ \stackrel{\mathrm{law}}=\ \frac{2j}{n}\chi_n^2,
\]
it is bounded by a fixed constant except with exponentially
small probability.  The norm tail for $W$, and a sufficiently
large choice of $R$, ensure also that both $W$ and $J$ lie in the
operator-norm ball except with exponentially small probability.
For every parameter tuple, \eqref{eq:truncated-word-lipschitz}
then bounds the Frobenius norm of the difference of the truncated
word evaluated at $W$ and at $J$ by a fixed constant.  Both
$N_2$ and $N_4$ of this difference are bounded by its Frobenius
norm.  The prediction is unchanged by this substitution.  At
parameters with $S_A(1;J)\succeq cI$, the truncated factors
evaluated at $J$ equal the exact factors.  This proves all of
\eqref{eq:word-estimates}.  The polynomial assertion was proved
in Lemma~\ref{lem:word-bias}.
\end{proof}

\subsection{Bilinear forms and regularized determinants}

We next extract two forms of the preceding estimates.  The first
controls bilinear forms for a prescribed finite family of
deterministic vectors.  The second controls a regularized
determinant simultaneously over its diagonal parameters and
bounded finite-rank perturbations.  Their quantifiers differ,
and we keep the two statements separate.

\begin{lemma}[Bilinear deterministic equivalents]\label{lem:bilinear}
Fix an integer $m\ge1$, a vector bound $M_1<\infty$, a tolerance
$\varepsilon>0$, and $R>2\sqrt j$.  For every deterministic
$0\le A\le a_+I$ and deterministic vectors
$v_1,\ldots,v_m\in\R^n$ with $\norm{v_r}\le M_1$, there is an
event of probability at least $1-Ce^{-c_3n}$ on which
\eqref{eq:deterministic-path-stability} holds and, with
\[
 G_A=A^{1/2}S_A(1;W)^{-1}A^{1/2}=K_A(W)A,
\]
one has, for every $r,s$,
\begin{align}
 |v_r^T(G_A-A)v_s|&\le\varepsilon,
 \label{eq:bilinear-G}\\
 |v_r^T(G_AW-B_AA)v_s|&\le\varepsilon,
 \label{eq:bilinear-GW}\\
 |v_r^T(WG_AW-B_AI-B_A^2A)v_s|&\le\varepsilon.
 \label{eq:bilinear-WGW}
\end{align}
Here $C,c_3,n_0$ may depend on $j,a_+,m,M_1,\varepsilon,R$ but
are uniform in $A$, the vectors, and $n\ge n_0$.  The assertion
is for each deterministic choice of these parameters; it does
not claim simultaneity over all vector choices.
\end{lemma}

\begin{proof}
For each of the specified bilinear forms, we calculate its
expectation and then apply concentration.  Use the truncations
above with a cutoff agreeing with the inverse on the path-stability
event.  Sign conjugation makes the off-diagonal expectations of
each relevant matrix vanish.
Equation~\eqref{eq:K-and-WK-bias} at $z=1$ gives
\[
 \E\widehat K_{ii}=1+O(n^{-1}),\qquad
 \E(\widehat W\widehat K)_{ii}=B_A+O(n^{-1}).
\]
The identities, valid on the event of agreement,
\[
 G_AW=K_A(W)(I+B_AA)-I,
 \qquad
 WG_AW=WK_A(W)(I+B_AA)-W
\]
therefore give the three diagonal expectation predictions
$A$, $B_AA$, and $B_AI+B_A^2A$, with entrywise errors
$O(n^{-1})$.  Their use with truncated factors changes
expectations only exponentially little, by the same argument
as in Lemma~\ref{lem:word-bias}.  In particular, their errors
in any of the specified bilinear forms are $O(n^{-1})$:
for a diagonal error matrix $E$, one has
$|v_r^TEv_s|\le\norm E\norm{v_r}\norm{v_s}$.

The expectation calculation has identified the three required
predictions.  To control fluctuations, each truncated bilinear form
is $C M_1^2$-Lipschitz in $W$ for Frobenius norm, by
\eqref{eq:truncated-word-lipschitz}.  Gaussian
concentration gives failure probability $Ce^{-c_3n}$ at any
fixed positive tolerance.  Intersect these events for the
$3m^2$ forms and with Lemma~\ref{lem:matrix-path-stability}.
For large $n$ the expectation errors are less than half the
specified tolerance, proving the result.
\end{proof}

\begin{lemma}[A simultaneous regularized determinant bound]
\label{lem:logdet}
Fix $\varepsilon>0$, an integer $r_0\ge0$, and $M_2<\infty$.
There is $\gamma_0>0$, depending only on $j,\varepsilon$, such
that for every fixed $0<\gamma\le\gamma_0$ and all sufficiently
large $n$, an event of probability at least $1-Ce^{-c_4n}$ has
the following property.  Simultaneously for all diagonal
$0\le V\le I$ and all real matrices $E$ with
$\rank E\le r_0$ and $\norm E\le M_2$, put
\[
 b=\frac1n\Tr V,\qquad B=jb,\qquad
 L=I+(BI-W)V+E.
\]
Then
\begin{equation}\label{eq:logdet-upper-bound}
 \frac1n\log\det(L^TL+\gamma I)\le jb^2+\varepsilon.
\end{equation}
The constants $C,c_4$ and the lower threshold on $n$ may depend
on $j,\varepsilon,\gamma,r_0,M_2$.  In particular, $V$ and $E$
may depend on $W$.
\end{lemma}

\begin{proof}
We first compute the mean log determinant with $E=0$ and a fixed
deterministic $V$.  Regularization will then provide a global
Lipschitz bound, allowing a finite mesh in $V$ and finally the
finite-rank perturbation.  Set
\[
 L_z=I+z^2BV-zWV,
 \qquad K(z)=(I-zVW+z^2BV)^{-1}.
\]
Since $V\le I$, we use the proof of
Lemma~\ref{lem:matrix-path-stability} with the diagonal upper bound
equal to $1$; its stability and inverse-norm constants here depend
only on $j$.
On the path-stability event from
Lemma~\ref{lem:matrix-path-stability} with $A=V$, the determinant
of $L_z$ equals the positive determinant of $S_V(z;W)$ by
$\det(I-UV)=\det(I-VU)$, and both
$L_z^{-1}$ and $K(z)$ have uniformly bounded operator norm.
Indeed $L_z=K(z)^{-T}$, and
\eqref{eq:K-symmetric-representation} applies.  Differentiating
the normalized log determinant on this event gives
\begin{align}
 \frac{\dd}{\dd z}\frac1n\log\det L_z
 &=\frac1n\Tr\bigl[L_z^{-1}(2zBV-WV)\bigr]\notag\\
 &=2zB\frac1n\Tr(K(z)V)
      -\frac1n\Tr(WK(z)V).
 \label{eq:logdet-derivative}
\end{align}
Here $VL_z^{-1}=K(z)V$, which follows by multiplying
$(I-zVW+z^2BV)V=V L_z$, and cyclicity of trace was used.

The derivative is expressed in the two matrices whose diagonal
expectations were computed in \eqref{eq:K-and-WK-bias}.
Let $\mathcal E_V$ be the path-stability event intersected with
$\norm W\le R$, for a fixed $R>2\sqrt j$.  This one event applies
to the entire interval of $z$.  Equations~\eqref{eq:K-and-WK-bias}, and
replacement by the truncated factors off $\mathcal E_V$, show
uniformly in $z,V$ that the expectation of the right-hand side
of \eqref{eq:logdet-derivative}, multiplied by
$\one_{\mathcal E_V}$, is
\[
 2zBb-zBb+O(n^{-1})=zjb^2+O(n^{-1}).
\]
All differentiated quantities are bounded on $\mathcal E_V$,
so integration and expectation may be interchanged.  Since
$L_0=I$, it follows that
\begin{equation}\label{eq:unregularized-logdet-mean}
 \E\left[\one_{\mathcal E_V}\frac2n\log\det L_1\right]
 =jb^2+O(n^{-1}).
\end{equation}
On $\mathcal E_V$, the singular values of $L_1$ are bounded
below by a positive constant.  Therefore
\[
 0\le\frac1n\log\det(L_1^TL_1+\gamma I)
       -\frac2n\log\det L_1
 \le C\gamma.
\]
Off $\mathcal E_V$, the absolute value of the regularized
normalized log determinant is bounded by
\[
 |\log\gamma|+C+2\log(1+\norm W),
\]
for $0<\gamma\le1$.  The norm tails of $W$ and
\eqref{eq:deterministic-path-stability} make its expected
contribution exponentially small, for each fixed $\gamma$.
Consequently, uniformly in deterministic $V$,
\begin{equation}\label{eq:regularized-logdet-mean}
 \E\left[\frac1n\log\det(L_1^TL_1+\gamma I)\right]
 \le jb^2+C\gamma+O(n^{-1})+O_\gamma(e^{-c n}).
\end{equation}

The mean bound is uniform in deterministic $V$.  To obtain a
single event for all $V$, we use the concentration supplied by
regularization.  Define on all real matrices
\[
 F_\gamma(M)=\frac1n\log\det(M^TM+\gamma I).
\]
Its Frobenius gradient is
$2n^{-1}M(M^TM+\gamma I)^{-1}$.  Each singular value of this
gradient is at most $1/(n\sqrt\gamma)$, and hence
\begin{equation}\label{eq:regularized-logdet-lipschitz}
 \Lip_{\Frob}(F_\gamma)\le\frac1{\sqrt{n\gamma}}.
\end{equation}
For fixed $V$, the map $W\mapsto L_1$ has Frobenius Lipschitz
constant at most one.  Gaussian concentration therefore gives
\begin{equation}\label{eq:regularized-logdet-concentration}
 \Prob\{|F_\gamma(L_1)-\E F_\gamma(L_1)|>u\}
 \le2\exp\left(-\frac{\gamma n^2u^2}{4j}\right).
\end{equation}

Choose $\gamma_0\le1$ small enough that $C\gamma_0$ in
\eqref{eq:regularized-logdet-mean} is at most
$\varepsilon/8$.  For fixed $\gamma\le\gamma_0$ and sufficiently
large $n$, the expectation bound and
\eqref{eq:regularized-logdet-concentration} bound all points
of any fixed sup-norm mesh of the cube $[0,1]^n$ by
$jb^2+\varepsilon/2$, with failure $\exp(-c_\gamma n^2)$.
Indeed a mesh of spacing $\Delta>0$ has at most
$(C/\Delta)^n$ points, whose logarithm is only of order $n$.
On $\norm W\le R$, if $\norm{V-V'}\le\Delta$, then
\[
 |b-b'|\le\Delta,\qquad
 \norm{(I+(BI-W)V)-(I+(B'I-W)V')}_{\Frob}
 \le\sqrt n(2j+R)\Delta.
\]
By \eqref{eq:regularized-logdet-lipschitz}, the corresponding
$F_\gamma$ values differ by at most
$(2j+R)\Delta/\sqrt\gamma$, whereas $|jb^2-j(b')^2|\le2j\Delta$.
Choose $\Delta$, depending only on
$j,R,\gamma,\varepsilon$, so that these errors sum to at most
$\varepsilon/4$.  Including the exponentially likely norm
event proves, simultaneously for all $V$,
\[
 F_\gamma(I+(BI-W)V)\le jb^2+3\varepsilon/4.
\]

We have obtained the simultaneous bound for $E=0$ with room for
the perturbation.  The same Lipschitz estimate
\eqref{eq:regularized-logdet-lipschitz} gives, for every allowed $E$,
\[
 |F_\gamma(M+E)-F_\gamma(M)|
 \le\frac{\norm E_{\Frob}}{\sqrt{n\gamma}}
 \le M_2\sqrt{\frac{r_0}{n\gamma}}.
\]
This is at most $\varepsilon/4$ for large $n$, uniformly even
when $E$ is chosen from the observed matrix.  It proves
\eqref{eq:logdet-upper-bound} and the lemma.
\end{proof}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//matrices.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//observation-flow.tex ====
\begin{figure}[htbp]
\centering
\small
\setlength{\tabcolsep}{5pt}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{c@{\quad}c}
\fbox{\begin{minipage}[c]{0.36\linewidth}\centering
Ordinary rarity\\[3pt]
$\Prob(\mathrm{bad})\le2e^{-an}$
\end{minipage}}
&\fbox{\begin{minipage}[c]{0.51\linewidth}\centering
Weighted-mean estimate\\[3pt]
$\displaystyle\frac{\mathcal H_\tau}{n}
 \le2C_{\mathrm w}T\delta
       +\frac{C_\delta T}{\sqrt n}(2+d_0)$
\end{minipage}}\\[4pt]
\multicolumn{2}{c}{\shortstack{\(\Downarrow\)\quad entropy inequality;
 choose $\delta$, then $n$}}\\[3pt]
\multicolumn{2}{c}{\fbox{\begin{minipage}{0.89\linewidth}\centering
Variance on bad paths\\[3pt]
$\displaystyle
 \E[\one_{\mathrm{bad}}V_\tau]
 \le\E[\one_{\mathrm{bad}}N_\tau]
 =\Prob^f(\mathrm{bad})\le v/4$
\end{minipage}}}\\[9pt]
&\shortstack{\(\Downarrow\)\quad terminal gap on good paths\\[2pt]
 $\E[\one_{\mathrm{good}}V_\tau]\le2d_0$}\\[4pt]
\fbox{\begin{minipage}[c]{0.36\linewidth}\centering
Directional covariance estimate\\[3pt]
$\E V_\tau\ge v-(1-v)d_0$
\end{minipage}}
&\fbox{\begin{minipage}[c]{0.51\linewidth}\centering
Total remaining variance\\[3pt]
$\E V_\tau\le2d_0+v/4$
\end{minipage}}\\[4pt]
\multicolumn{2}{c}{$\displaystyle
 v-(1-v)d_0\le2d_0+v/4\qquad\Longrightarrow\qquad d_0\ge v/4$}
\end{tabular}
\caption{Variance retention and control of exceptional paths, for a fixed
typical matrix. Normalize $f$ by $\E_{\mu_0}f=0$ and
$\E_{\mu_0}f^2=1$, and put $d_0=\D_0(f)\le1$.
Along $Y(t)=tX+B(t)$, let $V_t=\Var_{\mu_{Y(t)}}f$,
$N_t=\E_{\mu_{Y(t)}}f^2$, and $\tau=T\wedge\tau_0$.
Here $\mathcal H_\tau$ is the relative entropy of the stopped observation
law induced by the spin law $f^2\mu_0$, relative to the ordinary law,
and $v=e^{-C_{\mathrm d}T}$.
The density $N_\tau$ converts its bad-path probability into an upper
bound on exceptional variance. If $d_0>1$, the final lower bound is automatic.}
\label{fig:observation-flow}
\end{figure}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//observation-flow.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//observation.tex ====
\section{From posterior estimates to a spectral gap}\label{sec:observation}

We now prove the transfer announced in the introduction. Its inputs are
a covariance bound, a square-reweighted mean bound, and a terminal gap.
The stopping rule and good-disorder event must be independent of the test
function, so we keep the exceptional-path estimate explicit. The argument
uses the Gaussian-channel formulation of localization
\cite{ElAlaouiMontanari2021} and the variance/Dirichlet-form framework
of~\cite{CE2025}; the finite-spin identities are derived here.

\subsection{Posterior martingales and Dirichlet forms}
Let $\mathcal F_t=\sigma(Y(s):s\le t)$ and write $\E_t$ for expectation under
$\mu_{Y(t)}$. Bayes' formula gives this posterior: the likelihood of an
observation path given $X=x$ is proportional to
$\exp(x^{\mathsf T}Y(t)-tn/2)$. The process
\[
 I(t)=Y(t)-\int_0^t\E_s x\,\dd s
\]
is the innovation Brownian motion in $\mathcal F_t$: conditional
expectation makes it a continuous martingale, and its quadratic
covariation is $tI_n$. The Brownian martingale characterization therefore
applies. Applying It\^o's formula to the finitely many posterior weights
gives, for every spin function $g$,
\begin{equation}\label{eq:filtering}
 \dd(\E_t g)=\Cov_t(g,x)^{\mathsf T}\dd I(t).
\end{equation}
To verify the cancellation, differentiate the posterior expectation with
respect to $Y$; its gradient is $\Cov_t(g,x)$. Each likelihood weight
$\exp(x^{\mathsf T}Y(t)-tn/2)$ satisfies a stochastic exponential
equation. The quotient rule for their weighted sums then gives
\eqref{eq:filtering}: subtracting $\E_t x\,\dd t$ from $\dd Y$ removes
the finite-variation term.

At every deterministic time,
\begin{equation}\label{eq:dirichlet-monotonicity}
 \E\D_{Y(t)}(f)\le\D_0(f).
\end{equation}
This is conditional-variance contraction: the $i$th summand on the left
is $\E\Var(f(X)\mid X_{-i},\mathcal F_t)$, and the one on the right is
$\E\Var(f(X)\mid X_{-i})$. The conditional variance decomposition gives
the inequality after averaging over observations. In this section, an
expectation without a field subscript refers to that experiment, with
$J$ held fixed.

\subsection{Stopping before stability fails}
Stop before the first possible loss of the field estimates. More precisely,
fix the terminal horizon $T$, take the observation stopping time $\tau_0$
from Corollary~\ref{cor:stopping}, and set $\tau=T\wedge\tau_0$. This
rule does not use the test function. Before $\tau$, stability holds through
residual threshold $\rho_0\sqrt n$, so both posterior estimates apply.
On the specified matrix event, $\Prob(\tau_0\le T)\le e^{-an}$.

For a fixed $J$, choose the canonical terminal good set
\[
 \mathcal T_n(J)=\{h:\Var_{\mu_h}(g)\le2\D_h(g)\text{ for every }g\}.
\]
Membership means positivity of a finite-dimensional symmetric quadratic
form with coefficients continuous in $h$, so this set is Borel.
Proposition~\ref{prop:terminal-gap} controls the probability that $Y(T)$
misses it. Declare the stopped path bad if $\tau_0\le T$, or if
$\tau_0>T$ and $Y(T)\notin\mathcal T_n(J)$. The event is
$\mathcal F_\tau$-measurable: before $T$ only stopping can make it bad,
and at $T$ the remaining condition is determined by $Y(T)$. The two
failure probabilities give, on the typical matrix events,
\begin{equation}\label{eq:ordinary-bad}
 \Prob(\mathrm{bad})\le 2e^{-a n}
\end{equation}
for a fixed $a>0$, after decreasing the rate if necessary.

Suppose $\E_{\mu_0}f=0$ and $\E_{\mu_0}f^2=1$. Set
\[
 M_t=\E_t f,\quad N_t=\E_t f^2,\quad V_t=N_t-M_t^2,
 \quad d_0=\D_0(f).
\]
By \eqref{eq:filtering} and optional stopping, $u(t)=\E V_{t\wedge\tau}$ is
absolutely continuous and satisfies, almost everywhere,
\[
 u'(t)=-\E\bigl[\one_{\{t<\tau\}}\norm{\Cov_t(f,x)}^2\bigr].
\]
The directional estimate bounds the loss term before stopping. Together
with \eqref{eq:dirichlet-monotonicity}, applied at the deterministic time
$t$, it gives $u'(t)\ge-C_{\mathrm d}(u(t)+d_0)$, with fixed
$C_{\mathrm d}$. Solving this scalar inequality from $u(0)=1$ yields
\begin{equation}\label{eq:variance-lower}
 \E V_\tau\ge v-(1-v)d_0,\qquad v=e^{-C_{\mathrm d}T}>0.
\end{equation}
All uses of optional stopping here are at bounded times and for bounded
random variables at each fixed $n$.

\subsection{Changing the law of the observations}
The variance lower bound alone does not control how much variance lies on
bad paths. For that purpose change the initial spin law to $f^2\mu_0$,
which is a probability law by normalization, and call the resulting
observation law $\Prob^f$. Its density on $\mathcal F_t$ is $N_t$, and
its spin posterior is $\nu_t=f^2\mu_{Y(t)}/N_t$. Since $\mu_{Y(t)}$ has
full support and $f$ is nonzero, $N_t>0$. Bounded stopping gives the
density $N_\tau$ on $\mathcal F_\tau$.
Let
\[
 a_t=\E_{\nu_t}x-\E_t x.
\]
Equation~\eqref{eq:filtering} with $g=f^2$ gives
$\dd N_t/N_t=a_t^{\mathsf T}\dd I(t)$. Under $\Prob^f$, $I$ has drift $a_t$.
The drift-energy entropy identity, in the form discussed
by Lehec~\cite{Lehec2013}, has a direct finite-mixture proof here.
It\^o's formula for $\log N_t$ yields
\begin{equation}\label{eq:stopped-entropy}
 D\bigl(\Prob^f|_{\mathcal F_\tau}\,\Vert\,
        \Prob|_{\mathcal F_\tau}\bigr)
 =\E^f\log N_\tau
 =\frac12\E^f\int_0^\tau\norm{a_t}^2\,\dd t.
\end{equation}
The stopped identity needs no limiting integrability argument. At fixed
$n$, the drift satisfies $\norm{a_t}\le2\sqrt n$ and $\tau\le T$.
Novikov's condition therefore holds, and the stochastic integral in the
log-density formula has zero mean.

Before stopping, applying \eqref{eq:input-weighted} to $f$ and to $1$ gives
\[
 b_t:=\frac{\norm{a_t}}{\sqrt n}
 \le2C_{\mathrm w}\delta+\frac{C_\delta}{\sqrt n}
                   \left(2+\frac{\D_{Y(t)}(f)}{N_t}\right).
\]
Since $b_t\le2$, we may use $b_t^2\le2b_t$. At deterministic $t$, the density
identity and \eqref{eq:dirichlet-monotonicity} imply
\begin{equation}\label{eq:weighted-dirichlet}
 \E^f\frac{\D_{Y(t)}(f)}{N_t}
 =\E\D_{Y(t)}(f)\le d_0.
\end{equation}
Write $\mathcal H_\tau$ for the entropy in \eqref{eq:stopped-entropy}.
The preceding estimates give the quantitative bound
\begin{equation}\label{eq:entropy-bound}
 \frac{\mathcal H_\tau}{n}
 \le 2C_{\mathrm w}T\delta
       +\frac{C_\delta T}{\sqrt n}(2+d_0).
\end{equation}
The separation of constants is essential: $C_{\mathrm w}$ is independent
of $\delta$, whereas $C_\delta$ may grow when $\delta$ decreases.
We next convert this entropy bound into a bound on the bad-path probability.
If an event has probabilities $p$ and $q$ under two laws, the log-sum
inequality for the event and its complement bounds relative entropy below
by $p\log(1/q)-\log2$. With $p=\Prob^f(\mathrm{bad})$,
\eqref{eq:ordinary-bad} therefore gives, for large $n$,
\[
 p\le\frac{\mathcal H_\tau+\log2}{an-\log2}.
\]
If the ordinary bad probability is zero, absolute continuity gives $p=0$.
For $d_0\le1$, \eqref{eq:entropy-bound} therefore implies
\begin{equation}\label{eq:weighted-bad-bound}
 p\le\frac{4C_{\mathrm w}T}{a}\delta
       +\frac{6C_\delta T}{a\sqrt n}
       +\frac{2\log2}{an}.
\end{equation}
Choose $\delta>0$ once so that the first term is at most $v/8$.
With this choice fixed, enlarge $n$ until the last two terms sum to at
most $v/8$. The resulting bound holds simultaneously for all normalized
$f$ with $d_0\le1$:
\begin{equation}\label{eq:weighted-bad}
 \Prob^f(\mathrm{bad})\le v/4.
\end{equation}

Figure~\ref{fig:observation-flow} records how the two posterior estimates
will control the stopped variance and its exceptional contribution.
\input{figures/observation-flow}

\subsection{Completion of the theorem}
We can now close the argument, keeping the order of choices explicit.
First choose $T$ from the terminal estimate. Next choose the stable-field
constants, the ordinary failure rate $a$, and the finite diagnostic length
for the directional estimate; these choices determine $C_{\mathrm d}$
and $v$. Root stability and the operator bound on $J$ determine the
leading weighted-mean constant $C_{\mathrm w}$ independently of selection
depth. We may therefore select $\delta$ in
\eqref{eq:weighted-bad-bound}, then its diagnostic length and
$C_\delta$, and finally $n_0$. Every depth and tolerance is fixed before
$n$ grows.

With the choices made, the proof needs a finite intersection of matrix
events. Include the directional word event and the word events for the
selected weighted-mean construction, each with its fixed length and
coefficient bounds. Lemma~\ref{lem:words} accommodates each fixed
$M_0$, and Lemma~\ref{lem:implicit-construction} supplies an actual
inverse margin independent of depth. Add the events of
Propositions~\ref{prop:stable-fields} and~\ref{prop:terminal-gap}.
Their intersection still has probability tending to one, and all constants
and the threshold $n_0$ are uniform on it and over the test functions.

For such a matrix and a normalized $f$ with $d_0\le1$, use $V_\tau\le N_\tau$
and the stopped density identity to obtain
\[
 \E[\one_{\mathrm{bad}}V_\tau]
 \le\E[\one_{\mathrm{bad}}N_\tau]
 =\Prob^f(\mathrm{bad})\le v/4.
\]
On a good path $\tau=T$ and the terminal gap applies, so
\[
 \E[\one_{\mathrm{good}}V_\tau]
 \le2\E\D_{Y(T)}(f)\le2d_0.
\]
Combining these estimates with \eqref{eq:variance-lower} gives
\[
 v-(1-v)d_0\le2d_0+v/4,
 \qquad\text{hence}\qquad
 d_0\ge\frac{3v}{4(3-v)}\ge\frac v4.
\]
The case $d_0>1$ already satisfies this bound. Center and rescale any
nonconstant function to obtain Theorem~\ref{thm:main} once the three
inputs have been proved, with the finite constant
$C_\beta=4e^{C_{\mathrm d}T}$. We do not optimize this constant.
For the clock conversion, the self-adjoint operator
$H=\sum_i(I-P_i^0)$ has form $\D_0$, and one discrete attempt is
$I-H/n$. Its gap is therefore the asserted unscaled gap divided by $n$.
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//observation.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//references.tex ====
\phantomsection
\addcontentsline{toc}{section}{References}
\begin{thebibliography}{99}

\bibitem{ABXY2024}
A.~Adhikari, C.~Brennecke, C.~Xu, and H.-T.~Yau,
\emph{Spectral gap estimates for mixed $p$-spin models at high temperature},
Probability Theory and Related Fields \textbf{189} (2024), nos.~3--4,
879--907.
\href{https://doi.org/10.1007/s00440-024-01261-9}{doi:10.1007/s00440-024-01261-9}.

\bibitem{ALR1987}
M.~Aizenman, J.~L.~Lebowitz, and D.~Ruelle,
\emph{Some rigorous results on the Sherrington--Kirkpatrick spin glass model},
Communications in Mathematical Physics \textbf{112} (1987), 3--20.
\href{https://doi.org/10.1007/BF01217677}{doi:10.1007/BF01217677}.

\bibitem{AJKPV2022}
N.~Anari, V.~Jain, F.~Koehler, H.~T.~Pham, and T.-D.~Vuong,
\emph{Entropic independence: optimal mixing of down-up random walks},
Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
(STOC 2022), 1418--1430.
\href{https://doi.org/10.1145/3519935.3520048}{doi:10.1145/3519935.3520048}.
Full Ising proof: \href{https://arxiv.org/abs/2106.04105v2}{arXiv:2106.04105v2}.

\bibitem{AKV2024}
N.~Anari, F.~Koehler, and T.-D.~Vuong,
\emph{Trickle-down in localization schemes and applications},
Proceedings of the 56th Annual ACM Symposium on Theory of Computing
(STOC 2024), 1094--1105.
\href{https://doi.org/10.1145/3618260.3649622}{doi:10.1145/3618260.3649622}.

\bibitem{BER2026}
A.~S.~Bandeira, A.~El Alaoui, and A.~R\"odder,
\emph{Mixing of Glauber Dynamics on High Overlap Gibbs Measures},
\href{https://arxiv.org/abs/2607.06813v1}{arXiv:2607.06813v1},
July 7, 2026.

\bibitem{BB2019}
R.~Bauerschmidt and T.~Bodineau,
\emph{A very simple proof of the LSI for high temperature spin systems},
Journal of Functional Analysis \textbf{276} (2019), no.~8, 2582--2588.
\href{https://doi.org/10.1016/j.jfa.2019.01.007}{doi:10.1016/j.jfa.2019.01.007}.

\bibitem{BLO2026}
M.~Boban, A.~Li, and S.~Oveis Gharan,
\emph{Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics
for the Sherrington--Kirkpatrick model up to $\beta\leq\frac12+\varepsilon$},
\href{https://arxiv.org/abs/2609.13138v1}{arXiv:2609.13138v1},
September 11, 2026.

\bibitem{Bolthausen2014}
E.~Bolthausen,
\emph{An iterative construction of solutions of the TAP equations for the
Sherrington--Kirkpatrick model}, Communications in Mathematical Physics
\textbf{325} (2014), no.~1, 333--366.
\href{https://doi.org/10.1007/s00220-013-1862-3}{doi:10.1007/s00220-013-1862-3}.

\bibitem{BSXY2024}
C.~Brennecke, A.~Schertzer, C.~Xu, and H.-T.~Yau,
\emph{The two point function of the SK model without external field at
high temperature}, Probability and Mathematical Physics
\textbf{5} (2024), 131--175.
\href{https://doi.org/10.2140/pmp.2024.5.131}{doi:10.2140/pmp.2024.5.131}.

\bibitem{Celentano2024}
M.~Celentano,
\emph{Sudakov--Fernique post-AMP, and a new proof of the local convexity of
the TAP free energy}, Annals of Probability \textbf{52} (2024), no.~3,
923--954.
\href{https://doi.org/10.1214/23-AOP1675}{doi:10.1214/23-AOP1675}.

\bibitem{CE2025}
Y.~Chen and R.~Eldan,
\emph{Localization schemes: A framework for proving mixing bounds for
Markov chains}, Duke Mathematical Journal \textbf{174} (2025), no.~8,
1431--1510.
\href{https://doi.org/10.1215/00127094-2024-0063}{doi:\nolinkurl{10.1215/00127094-2024-0063}}.
Preprint: \href{https://arxiv.org/abs/2203.04163v2}{arXiv:2203.04163v2}.

\bibitem{EAG2024}
A.~El Alaoui and J.~Gaitonde,
\emph{Bounds on the covariance matrix of the Sherrington--Kirkpatrick model},
\href{https://arxiv.org/abs/2212.02445v3}{arXiv:2212.02445v3},
November 12, 2024.

\bibitem{ElAlaouiMontanari2021}
A.~El Alaoui and A.~Montanari,
\emph{An information-theoretic view of stochastic localization},
\href{https://arxiv.org/abs/2109.00709v2}{arXiv:2109.00709v2}, September 9, 2021.

\bibitem{EAMS2022}
A.~El Alaoui, A.~Montanari, and M.~Sellke,
\emph{Sampling from the Sherrington--Kirkpatrick Gibbs measure via
algorithmic stochastic localization},
Proceedings of the 63rd IEEE Annual Symposium on Foundations of
Computer Science (FOCS 2022), 323--334.
Full version: \href{https://arxiv.org/abs/2203.05093v2}{arXiv:2203.05093v2}.

\bibitem{Eldan2013}
R.~Eldan,
\emph{Thin shell implies spectral gap up to polylog via a stochastic
localization scheme}, Geometric and Functional Analysis \textbf{23}
(2013), no.~2, 532--569.
\href{https://doi.org/10.1007/s00039-013-0214-y}{doi:10.1007/s00039-013-0214-y}.

\bibitem{EKZ2022}
R.~Eldan, F.~Koehler, and O.~Zeitouni,
\emph{A spectral condition for spectral gap: fast mixing in high-temperature
Ising models}, Probability Theory and Related Fields \textbf{182}
(2022), nos.~3--4, 1035--1051.
\href{https://doi.org/10.1007/s00440-021-01085-x}{doi:10.1007/s00440-021-01085-x}.

\bibitem{Glauber1963}
R.~J.~Glauber, \emph{Time-dependent statistics of the Ising model},
Journal of Mathematical Physics \textbf{4} (1963), no.~2, 294--307.
\href{https://doi.org/10.1063/1.1703954}{doi:10.1063/1.1703954}.

\bibitem{Lehec2013}
J.~Lehec,
\emph{Representation formula for the entropy and functional inequalities},
Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques
\textbf{49} (2013), no.~3, 885--899.
\href{https://doi.org/10.1214/11-AIHP464}{doi:10.1214/11-AIHP464}.

\bibitem{SK1975}
D.~Sherrington and S.~Kirkpatrick, \emph{Solvable model of a spin-glass},
Physical Review Letters \textbf{35} (1975), no.~26, 1792--1796.
\href{https://doi.org/10.1103/PhysRevLett.35.1792}{doi:10.1103/PhysRevLett.35.1792}.

\bibitem{Speicher1998}
R.~Speicher, \emph{Combinatorial theory of the free product with
amalgamation and operator-valued free probability theory},
Memoirs of the American Mathematical Society \textbf{132} (1998), no.~627.
\href{https://doi.org/10.1090/memo/0627}{doi:10.1090/memo/0627}.

\bibitem{TAP1977}
D.~J.~Thouless, P.~W.~Anderson, and R.~G.~Palmer,
\emph{Solution of `Solvable model of a spin glass'},
Philosophical Magazine \textbf{35} (1977), no.~3, 593--601.
\href{https://doi.org/10.1080/14786437708235992}{doi:10.1080/14786437708235992}.

\bibitem{Vitale2000}
R.~A.~Vitale, \emph{Some comparisons for Gaussian processes},
Proceedings of the American Mathematical Society \textbf{128} (2000),
no.~10, 3043--3046.
\href{https://doi.org/10.1090/S0002-9939-00-05367-3}{doi:10.1090/S0002-9939-00-05367-3}.

\bibitem{Wang2026}
S. Wang, \emph{Optimal Mixing of Glauber Dynamics for the
Sherrington--Kirkpatrick Model at $\beta<1/2$},
\href{https://arxiv.org/abs/2608.22159v2}{arXiv:2608.22159v2},
August 28, 2026.

\bibitem{Wu2006}
L.~Wu, \emph{Poincar\'e and transportation inequalities for Gibbs measures
under the Dobrushin uniqueness condition}, Annals of Probability
\textbf{34} (2006), no.~5, 1960--1989.
\href{https://doi.org/10.1214/009117906000000368}{doi:10.1214/009117906000000368}.

\end{thebibliography}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//references.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//residuals.tex ====
\section{Weak residual identities and discrete differentiation}
\label{sec:residuals}

We derive the residual identities needed for the two posterior estimates.
Moments of a fixed finite sequence of primary iterates will locate two
consecutive small increments. Ordinary test-vector recipes then control
means under square reweighting; one additional implicit recipe supplies
the linear-response test for the directional covariance estimate. The
proof proceeds from discrete derivative bounds to a weak residual rule,
and ends with the terminal identity for that implicit test.

This section is deterministic.  We fix $J$ satisfying the word estimates of
Lemma~\ref{lem:words}, fix an external field $h_0\in\R^n$, and take all
expectations with respect to $\mu_{h_0}$. We fix the depth, the finite
construction, and
its coefficient bounds independently of $n$. These choices determine a
finite required word length, which can increase when the choices change.
Constants
in derivative estimates can depend on these choices and on the corresponding
word bounds; they are uniform in $n$, $h_0$, the spin configuration, and the
permitted seeds.  The size and inverse bounds for the initial implicit
construction below have the stronger, depth-independent uniformity explicitly
stated there.

We use the discrete derivatives, product rule
\eqref{eq:discrete-product}, and norm $\norm G_*$ introduced above,
with $dg=(d_i g)_{i=1}^n$ for scalar $g$.  All vector functions such as
$\tanh h^1$ are understood coordinatewise.

\subsection{Primary fields}

The following recursion has the Onsager-corrected form of iterative TAP
constructions, as in Bolthausen~\cite{Bolthausen2014}. Here it starts from
a Gibbs spin sample and is used only through a fixed finite depth.
Define
\begin{equation}
 \begin{gathered}
 m^0=x,\quad b_0=0,\qquad
 m^l=\tanh h^l,\quad b_l=\av{1-(m^l)^2},\quad R_l=m^{l-1}-m^l,\\
 h^{l+1}=h_0+Jm^l-jb_lm^{l-1}\quad(l\geq1).
 \end{gathered}
 \label{eq:amp}
\end{equation}
The finite-depth objective is visible in the scalar pairings
$S_l=R_l^{\mathsf T}m^l/\sqrt n$. Expanding the squared increment gives
\begin{equation}\label{eq:residual-telescoping}
 \frac{\norm{R_l}^2}{n}=b_l-b_{l-1}-\frac{2S_l}{\sqrt n}.
\end{equation}
Since $0\le b_l\le1$, control of $S_l/\sqrt n$ through a sufficiently
large fixed depth forces two consecutive residuals to be small. We will
prove the moment bounds
\[
 \E S_l^2\le C_l,\qquad
 \E_\nu\left(\frac{S_l}{\sqrt n}\right)^2
 \le\frac{C_l}{\sqrt n}\left(1+\frac{\D_{h_0}(f)}{N}\right),
 \quad N=\E f^2>0,\quad \nu=\frac{f^2\mu_{h_0}}{N}.
\]
These bounds will supply the ordinary and square-reweighted selection
estimates in Section~\ref{sec:consequences}; convergence of the primary
recursion is not required.

The first residual explains what is needed from a test vector $U$.
Conditional covariance in one spin gives the exact identity
\[
 \E G R_1^{\mathsf T}U=\E\sum_i v_i^1d_i(GU_i).
\]
The product rule bounds its right side by $C\norm G_*$ whenever
$\norm U$ and $\sum_i|d_iU_i|$ are bounded pointwise. Thus the essential
property of our tests is a bound on the sum of their diagonal derivatives,
in addition to a bound on their Euclidean norm. The Onsager corrections
will preserve this property through the finite constructions below.
We begin with the primary fields, whose derivatives enter those tests.
We use
$\norm{Q}_{\ell^4}=(\sum_{i,j}|Q_{ij}|^4)^{1/4}$ for the unnormalized
entrywise norm.

\begin{lemma}[Primary differentiation]\label{lem:primary-derivatives}
Let
\begin{equation}
 H_1=J,\qquad M_0=I,\qquad
 M_l=D_{1-(m^l)^2}H_l,\qquad
 H_{l+1}=JM_l-jb_lM_{l-1}.
 \label{eq:primary-formal}
\end{equation}
For each fixed $l\geq1$,
\begin{align}
 &\norm{\nabla h^l-H_l}_{\Frob}
   +\norm{\nabla m^l-M_l}_{\Frob}\leq C,\label{eq:primary-error}\\
 &\norm{\nabla h^l}+\norm{\nabla m^l}
   +\norm{\nabla h^l}_{\ell^4}
   +\norm{\diag\nabla h^l}_2\leq C,
   \qquad \norm{db_l}\leq Cn^{-1/2}.
 \label{eq:primary-bounds}
\end{align}
More generally, let $g_i$ be a smooth function of a fixed list of coordinates
$h_i^l$ and averages $b_l$, with uniformly bounded derivatives through the
orders being used.  Deterministic site labels are permitted, provided the
same bounds hold uniformly in $i$.  Its derivative matrix has bounded
operator norm, bounded diagonal $\ell^2$ norm, and bounded entrywise
$\ell^4$ norm.
\end{lemma}

\begin{proof}
We first bound the formal matrices in \eqref{eq:primary-formal}, then
compare them with the actual discrete derivatives. Expanding $H_{l+1}$
produces the chain
\[
 J D_{1-(m^l)^2}J\cdots D_{1-(m^1)^2}J
\]
and the terms obtained by contracting disjoint pairs of originally adjacent
$J$'s.  Contracting $J D_{1-(m^s)^2}J$ contributes
$-j b_s I$.  This description follows inductively from the two terms defining
$H_{l+1}$: either keep its first link, or contract that link with the next
one.  Multiplicities are retained when different contractions happen to give
the same matrix.

The contraction terms cancel the diagonal predictions. To see this, fix
a complete noncrossing pairing of the original chain.  Reinserting each
contracted adjacent pair identifies its contribution with the prediction of
this full pairing, multiplied by the signs of the contractions.  The
originally adjacent pairs belonging to a pairing are mutually disjoint, and
every nonempty complete noncrossing pairing contains at least one such pair.
Summing over their subsets gives $(1-1)^a=0$, where $a\geq1$ is their
number.  An odd chain has no complete pairing.  Lemma~\ref{lem:words}
therefore bounds the diagonal $\ell^2$ norm of $H_l$, and bounds its operator
and off-diagonal $\ell^4$ norms term by term.  Its full $\ell^4$ norm is
bounded as well, since $\norm{\diag H_l}_4\leq\norm{\diag H_l}_2$.

We now compare these formal bounds with the actual derivatives. For a
scalar function $\psi$ with bounded second derivative, Taylor's theorem gives
\[
 \abs{d_j\psi(h_i^l)-\psi'(h_i^l)d_jh_i^l}
 \leq C|d_jh_i^l|^2.
\]
The Frobenius norm of the remainder is at most
$C\norm{\nabla h^l}_{\ell^4}^2$.  This proves the magnetization part of
\eqref{eq:primary-error} once the field part is known.  The same argument
for $\psi(z)=\sech^2z$ gives an operator bound for its site derivative
matrix $Q_l$.  Averaging its rows yields
\[
 db_l=n^{-1}Q_l^{\mathsf T}\one,\qquad
 \norm{db_l}\leq n^{-1/2}\norm{Q_l}.
\]
In differentiating $b_lm^{l-1}$, the term from $db_l$ is a rank-one
matrix of Frobenius norm at most
$\norm{m^{l-1}}\norm{db_l}\leq C$.  The discrete product error is
$\nabla m^{l-1}$ times a diagonal whose diagonal has norm at most
$2\norm{db_l}$.  Its Frobenius norm is bounded, since
$\norm{\nabla m^{l-1}}\leq C$.  Starting at
$\nabla h^1=J$, these estimates close the induction. The resulting bounded
Frobenius perturbation preserves the operator, diagonal $\ell^2$, and
entrywise $\ell^4$ bounds of the formal matrices.

For the final assertion, apply the same comparison to a smooth function
of several arguments. Expand along the segment between the arguments at
$x$ and $x^{(j)}$. The preceding bounds control the primary linear terms.
A linear term involving $db_s$ is a
rank-one matrix with a bounded-entry left vector, hence has bounded
Frobenius norm.  The squared-average remainder is controlled by
\[
 n\sum_j |d_jb_s|^4\leq n\norm{db_s}^4\leq C.
\]
Mixed remainders are bounded by sums of pure squares, and primary quadratic
remainders are controlled by the entrywise $\ell^4$ bounds.  This proves
the assertion.
\end{proof}

We have in particular bounded the Euclidean change of every primary
vector under a single spin flip, at each fixed depth. This is the estimate
that will keep the local constructions inside their buffers.

\subsection{Finite recipes for small vectors}

The residuals will be tested against vectors built by finitely many
operations. We call these vectors ``small'' when their Euclidean norm is
bounded; the word does not refer to coordinatewise magnitude. The following
definition specifies the operations and the meaning of their site partials.

\begin{definition}[Ordinary admissible recipe]\label{def:small-recipe}
Choose finitely many deterministic seeds $s\in\R^n$ of bounded Euclidean
norm.  Seeds may depend on $J$ and $h_0$, but are fixed as functions of $x$.
Choose a finite tuple of scalar functions $\theta$ satisfying
\[
 \sup_x\bigl(|\theta_\alpha(x)|+\norm{d\theta_\alpha(x)}\bigr)\leq C
\]
for each component.  Such scalar functions will be called parameters; they
may include the averages $b_l$.

Given auxiliary fields $y^1,\ldots,y^{a-1}$ already constructed, a source
has the form
\begin{equation}
 w_i^a=\sum_{s}\phi_{a,s,i}\bigl((h_i^l)_{l\in I_a},\theta\bigr)s_i
       +\sum_{b<a}\phi_{a,b,i}\bigl((h_i^l)_{l\in I_a},\theta\bigr)y_i^b.
 \label{eq:source-form}
\end{equation}
All sums and index sets are finite.  Coefficients are smooth and bounded,
with bounded derivatives of every fixed order needed in the construction,
uniformly on their argument ranges and the connecting line segments.
Define the new auxiliary field by
\begin{equation}
 y^a=Jw^a
       -j\sum_l\av{\partial_lw^a}\,m^{l-1}
       -j\sum_{b<a}\av{\partial_{y^b}w^a}\,w^b.
 \label{eq:small-field}
\end{equation}
The partial $\partial_l$ differentiates the explicitly displayed local
argument $h_i^l$, holding every parameter and every auxiliary argument fixed.
In particular it holds $b_l$ fixed even when that average is among the
parameters.  The partial $\partial_{y^b}$ differentiates the indicated
linear auxiliary argument, with its coefficient held fixed.  A partial in
an absent argument is zero.  Thus $\partial_{y^b}w^a$ is simply the
coefficient vector $\phi_{a,b}$.

A terminal vector is another source of the form \eqref{eq:source-form},
using any already constructed auxiliary fields.  A source $w^b$ used again
as a terminal vector or in a new source is replaced by its expression
\eqref{eq:source-form}; auxiliary fields themselves are not expanded.
The recipe is \emph{admissible from level $p$} if every explicit primary
argument appearing in its source formulas or terminal formula has index
at least $p$.  This condition does not restrict the parameters, or the
vectors $m^{l-1}$ in the correction terms of \eqref{eq:small-field}.
\end{definition}

The definition permits two operations that we will use repeatedly.
Multiplication by a bounded smooth coefficient of the same allowed
arguments gives another source. At any finite stage we may also add
parameters satisfying the stated size and difference bounds.  We will use
the deterministic seed $\one/\sqrt n$ freely; thus $m^l/\sqrt n$ is a
source with the single primary argument $h^l$.  Linearity in the
auxiliary arguments is preserved in every source. It will be used in both
the derivative estimates and the trace cancellation.

For a terminal vector $U$ admissible from level $p$, the estimate we
seek is
\[
 \left|\E G R_p^{\mathsf T}U\right|\le C\norm G_*.
\]
Its square-density counterpart bounds
$|\E f^2H_0R_p^{\mathsf T}U|$ by
$C(\E f^2+\D_{h_0}(f))$ when $H_0$ has bounded size and difference norm.
With $U=m^p/\sqrt n$, the choices $G=S_p$ and $H_0=S_p/\sqrt n$
will give the moment estimates above. Lemma~\ref{lem:residual} proves
these rules after we establish the required diagonal derivative bounds.

\subsection{A scalar identity for comparing fields}

To pass from $R_{p+1}=\tanh h^p-\tanh h^{p+1}$ to earlier residuals,
we use a scalar identity valid for arbitrary real fields. For
$z,r_0,a\in\R$, define
\begin{equation}
 \Phi(z;r_0,a)=\int_0^1 e^{a(1-u^2)/2}
       \frac{\cosh(u(z-r_0))}{\cosh z\cosh r_0}\,\dd u.
 \label{eq:phi}
\end{equation}
Differentiating $e^{a(1-u^2)/2}\sinh(u(z-r_0))$ in $u$ gives
\begin{equation}
 \bigl[(z-r_0-a\tanh z)-a\partial_z\bigr]\Phi(z;r_0,a)
       =\tanh z-\tanh r_0.
 \label{eq:phi-stein}
\end{equation}
The boundary at $u=0$ is zero; the boundary at $u=1$, after division
by $\cosh z\cosh r_0$, is the right side. The identity holds for either
sign of $a$. At $a=0$, $\Phi$ is the secant slope of $\tanh$, with
continuous value $\sech^2r_0$ at $z=r_0$.

These formulas also give the uniform coefficient bounds needed for the
recipes. Positivity of the integrand and the secant-slope formula imply
$0<\Phi\le e^{|a|/2}$. For $a$ in a fixed bounded interval, every
fixed-order derivative in $z,r_0,a$ is bounded by a constant times $\Phi$:
derivatives of the hyperbolic factors introduce bounded ratios, and those
of the exponential introduce bounded powers of $(1-u^2)/2$.
The quotient rule then bounds every fixed-order derivative of each ratio
$(\partial^\alpha\Phi)/\Phi$. These assertions are uniform in the two
real field arguments and require no stability assumption.

In the ordinary residual induction we will substitute
$(z,r_0,a)=(h_i^p,h_i^{p+1},j(b_p-b_{p-1}))$. The same identity also
provides the two tests comparing an approximate field with the stable
root, which we construct next.

\subsection{An initial implicit auxiliary field}

The directional estimate needs one auxiliary field defined by a linear
implicit equation. We construct it near the stable root, establish its
inverse margin, and then control its discrete derivatives on a smaller
region.

Assume the stable-field implication \eqref{eq:stable-field} holds at $h_0$
through the fixed residual threshold required in Lemma~\ref{lem:root}.
Let $r$ be its unique root, and put
$t_*=\tanh r$ and $q_*=\av{t_*^2}$.  Fix $k\geq2$ and consider
\begin{equation}
 \Omega_\rho=
 \{x:\max(\norm{R_k(x)},\norm{R_{k-1}(x)})\leq\rho\sqrt n\}.
 \label{eq:small-region}
\end{equation}
All choices of $\rho$ below depend only on the stable-field constants and
the operator bound on $J$, until derivative estimates are considered.
For sufficiently small $\rho$, the primary recursion and
Lemma~\ref{lem:root} give on $\Omega_{2\rho}$
\begin{equation}
 \norm{h^k-r}+\norm{m^{k-1}-t_*}\leq C\rho\sqrt n,
 \qquad |1-b_{k-1}-q_*|\leq C\rho.
 \label{eq:implicit-near-root}
\end{equation}
The comparison is independent of $k$: the primary recursion gives
\[
 F(h^k)=JR_k-jb_{k-1}(R_{k-1}+R_k)
                     +j(b_k-b_{k-1})m^k,
 \qquad
 |b_k-b_{k-1}|\leq2\norm{R_k}/\sqrt n,
\]
so the residual is at most $C\rho\sqrt n$. Applying the root estimate
therefore gives the asserted comparison with the same depth independence.

Use the scalar kernel with
\begin{equation}
 a=j(1-b_{k-1}-q_*),\qquad A_i=\Phi(h_i^k;r_i,a).
 \label{eq:phi-data}
\end{equation}
Write $A=D_{(A_i)}$ and $\chi=\av{A_i}$.  The initial source $w_0$
is one of
\[
 w_{0,i}=A_i e_i\quad(\norm e\leq1),\qquad
 w_{0,i}=n^{-1/2}\partial_{r_0}\Phi(h_i^k;r_i,a).
\]
The partial in the second formula holds $a$ fixed. The two choices
have complementary roles. Holding $r_i,a$ fixed in the site partial
$\partial_k$, the scalar identity and its derivative in $r_0$ give
\begin{align}
 \sum_i[(h_i^k-r_i-am_i^k)-a\partial_k](A_i e_i)
   &=(m^k-t_*)^{\mathsf T}e,\notag\\
 \sum_i[(h_i^k-r_i-am_i^k)-a\partial_k]
   \frac{\partial_{r_0}\Phi(h_i^k;r_i,a)}{\sqrt n}
   &=\sqrt n\{\chi-(1-q_*)\}.
 \label{eq:implicit-source-actions}
\end{align}
Indeed differentiating \eqref{eq:phi-stein} in $r_0$ gives
$[(z-r_0-a\tanh z)-a\partial_z]\partial_{r_0}\Phi
=\Phi-\sech^2r_0$. Thus the first source tests a direction of the
magnetization error, while the second tests the discrepancy of the
averaged coefficient $\chi$. Both quantities enter the posterior
comparison. To put these scalar identities into a form controlled by the
weak residual rule, introduce the coupled source and field
\begin{equation}
 \begin{gathered}
 w=w_0+A(y^{\rm imp}-jc_1t_*),\qquad
 y^{\rm imp}=Jw-j\chi w-jc_1m^{k-1},\\
 c_1=\av{\partial_k w}.
 \end{gathered}
 \label{eq:implicit}
\end{equation}
In this partial, $y^{\rm imp}$, $c_1$, and $a$ are independent arguments
held fixed.  Consequently $\partial_{y^{\rm imp}}w=A$, and the field
equation in \eqref{eq:implicit} is exactly \eqref{eq:small-field} with
one allowed self-dependence: $\chi$ is its auxiliary-partial average,
and $c_1$ its primary-partial average. The correction $jc_1m^{k-1}$
will supply the scalar cancellation in Lemma~\ref{lem:terminal-residual}.
Its only primary local argument is $h^k$.
Once constructed, $y^{\rm imp}$ is available as the first auxiliary field
in any ordinary finite recipe.  The scalar $\sqrt n c_1$ is then an
additional parameter, and $t_*/\sqrt n$ is an additional seed.

\begin{lemma}[Implicit construction]\label{lem:implicit-construction}
For sufficiently small $\rho>0$, \eqref{eq:implicit} has a unique solution
on $\Omega_{2\rho}$.  There are constants independent of the primary
depth $k$ such that, throughout this region,
\begin{equation}
 \norm w+\norm{y^{\rm imp}}+\sqrt n|c_1|\leq C,
 \qquad
 I+j\chi A-A^{1/2}JA^{1/2}\succeq cI.
 \label{eq:implicit-size}
\end{equation}
On $\Omega_{3\rho/2}$, for all sufficiently large $n$ depending on the
fixed depth, the solution also satisfies
\begin{equation}
 \norm{\nabla w}_{\Frob}+\norm{\nabla y^{\rm imp}}_{\Frob}
       +\sqrt n\norm{dc_1}\leq C_k.
 \label{eq:implicit-frobenius}
\end{equation}
\end{lemma}

\begin{proof}
We first control the coefficients and solve the implicit linear system.
Only then do we differentiate it, using the larger region to cover spin
flips from the smaller one. The uniform kernel and quotient bounds proved
above will control the coefficients even when some $A_i$ are small.

Put $\ell_i=(\partial_z\Phi/\Phi)(h_i^k;r_i,a)$.
At $z=r_0$ this ratio equals $-\tanh r_0$, for every $a$.  Comparison
with $z=r_0,a=0$, followed by \eqref{eq:implicit-near-root}, gives
\begin{equation}
 \norm{A-D_{1-(m^k)^2}}_{\Frob}\leq C\rho\sqrt n,
 \qquad \norm{\ell+t_*}\leq C\rho\sqrt n,
 \qquad |\chi-b_k|\leq C\rho.
 \label{eq:implicit-coefficients}
\end{equation}
In particular, the comparisons put $A$ among the diagonals allowed by
the stable-field implication, once $\rho$ is decreased.

Let $u_0=w_0/A$ coordinatewise and $p_0=A\partial_k u_0$.  The quotient
bounds just proved give $\norm{u_0}+\norm{p_0}\leq C$; their local
derivatives have coordinatewise bounds proportional to $|e_i|$ in the
first case and to $n^{-1/2}$ in the second.  Direct differentiation with
the specified partial convention gives
\[
 \partial_k w=D_\ell w+p_0,\qquad
 c_1=\av{\ell w+p_0}.
\]
We can now eliminate $y^{\rm imp}$ and $c_1$ and solve for the source
alone. The resulting linear equation is
\begin{equation}
 \left[I-A(J-j\chi I)
       +\frac jn A(m^{k-1}+t_*)\ell^{\mathsf T}\right]w
 =A\left[u_0-j(m^{k-1}+t_*)\av{p_0}\right].
 \label{eq:implicit-linear-system}
\end{equation}
Write the matrix on the left as $I+AL_2$.  Its symmetrically normalized
version $I+A^{1/2}L_2A^{1/2}$ differs in operator norm by $O(\rho)$ from
\[
 I+jb_k A-A^{1/2}JA^{1/2}
       -\frac{2j}{n}A^{1/2}m^k(m^k)^{\mathsf T}A^{1/2}.
\]
For the rank term, use \eqref{eq:implicit-near-root} and
\eqref{eq:implicit-coefficients} in the elementary inequality
$\norm{uv^{\mathsf T}-u'v'^{\mathsf T}}
\leq\norm{u-u'}\norm v+\norm{u'}\norm{v-v'}$.
The displayed symmetric matrix has the fixed positive margin supplied
by \eqref{eq:stable-field}. Decrease $\rho$ until the perturbation is less
than half that margin. The normalized matrix then has bounded inverse,
even though it need not itself be symmetric.  The identity
\[
 (I+AL_2)^{-1}
 =I-A^{1/2}(I+A^{1/2}L_2A^{1/2})^{-1}A^{1/2}L_2
\]
gives the same bound for the unnormalized inverse, since $\norm{L_2}$ is
bounded.  The right side of \eqref{eq:implicit-linear-system} is bounded
in norm, because $|\av{p_0}|\leq\norm{p_0}/\sqrt n$.
This establishes the size bounds. To obtain the last assertion of
\eqref{eq:implicit-size}, remove the negative rank term from the stable-field
matrix and change $b_k$ to $\chi$. The constants used to solve the system
and obtain these bounds depend only on the fixed stability and operator
margins.

It remains to bound the discrete derivatives. Compare the equations at
$x$ and $x^{(i)}$. On the stated inner region both configurations belong
to $\Omega_{2\rho}$ for large $n$, by
Lemma~\ref{lem:primary-derivatives}. In every coefficient-difference term,
retain the solution vector at the common point $x$.  In particular,
\begin{align*}
 d_i c_1
 &=\av{(d_i\ell)w(x)+d_i p_0}
             +n^{-1}\ell(x^{(i)})^{\mathsf T}d_iw,\\
 d_i\{c_1(m^{k-1}+t_*)\}
 &=c_1(x)d_i m^{k-1}
               +(m^{k-1}(x^{(i)})+t_*)d_ic_1.
\end{align*}
Use the same product arrangement for $A$ and $\chi$ in
\[
 w=A\{u_0+(J-j\chi I)w-jc_1(m^{k-1}+t_*)\}.
\]
For clarity, put $v=m^{k-1}+t_*$ and
$\mathcal B=u_0+(J-j\chi I)w-jc_1v$, and use primes for evaluation at
$x^{(i)}$ in the next display.  With
$\zeta_i=n^{-1}\{(d_i\ell)^{\mathsf T}w+\one^{\mathsf T}d_ip_0\}$,
the exact equation is
\begin{align}
 &\left[I-A'(J-j\chi'I)+\frac jn A'v'(\ell')^{\mathsf T}\right]d_iw
 \notag\\
 &\qquad=(d_iA)\mathcal B+A'd_iu_0
       -jA'w\,d_i\chi-jc_1A'd_im^{k-1}-jA'v'\zeta_i.
 \label{eq:implicit-flip-system}
\end{align}
Thus the left matrix is precisely the matrix in
\eqref{eq:implicit-linear-system} evaluated at $x^{(i)}$. The bracket
$\mathcal B$ on the right is consequently one common-point
vector for all columns. This allows the following estimates to control
the Frobenius norm of the entire right side.

We bound its terms in turn.  A site coefficient difference
times a common vector $s$ of bounded norm has bounded Frobenius norm,
since
\[
 \sum_{i,j}s_i^2|d_j\psi_i|^2
 \leq \norm s^2\max_i\sum_j|d_j\psi_i|^2\leq C.
\]
This applies to $A,\ell,u_0,p_0$, using their coefficient bounds and
Lemma~\ref{lem:primary-derivatives}; the seed factors are retained for
$u_0,p_0$.  The bracket multiplying $d_iA$ is bounded in norm by the
already established size estimates, so no bound on $1/A_i$ is needed.
Also $\norm{d\chi}\leq C/\sqrt n$, because $A_i$ is a primary site
function with an average parameter of that derivative scale.  If
$\zeta_i=\av{(d_i\ell)w+d_i p_0}$, then
\[
 \norm\zeta\leq n^{-1/2}
   \bigl(\norm{D_w\nabla\ell}_{\Frob}
                      +\norm{\nabla p_0}_{\Frob}\bigr)
 \leq C/\sqrt n.
\]
Its multipliers $m^{k-1}(x^{(i)})+t_*$ have norm at most $2\sqrt n$,
which still gives a bounded Frobenius norm after summing the squared
column norms.  Finally
$|c_1|\norm{\nabla m^{k-1}}_{\Frob}\leq C$, by its operator bound and
$|c_1|\leq C/\sqrt n$.  Applying the uniformly bounded flipped inverses
column by column proves $\norm{\nabla w}_{\Frob}\leq C_k$.
Returning to the two displayed difference identities gives
$\norm{dc_1}\leq C_k/\sqrt n$ and the required bound for
$\nabla y^{\rm imp}$.
The operator in \eqref{eq:implicit-linear-system}, the regions
$\Omega_\rho$, and the required buffer inclusions do not depend on the
directional seed $e$.  All bounds for its right side and its derivatives
use only $\norm e\leq1$; indeed $u_0=e$ and $p_0=0$ for that source.
The inverse margins, the threshold on $n$, and all these estimates are
therefore uniform over the full Euclidean unit ball of seeds.
\end{proof}

\paragraph{Local recipes and their buffers.}
A local recipe starts with this implicit field and adds only ordinary
auxiliaries. Such
recipes are evaluated on the inner
region $\Omega_\rho$, with all estimates also available on its fixed
finite flip-neighborhoods.  To define them as functions on the whole cube,
keep the actual solution through $\Omega_{2\rho}$ and assign finite values
to $y^{\rm imp},c_1$ outside that region, defining sources by their stated
expressions. The implicit equation is required only inside the buffer.
Every test
in a local statement is supported on $\Omega_\rho$. For a fixed recipe,
once $n$ is sufficiently large, every difference used in the proof stays
strictly inside the buffer. The arbitrary extension therefore does not
enter the estimates.

The buffer sizes are chosen before the construction depth; the threshold
on dimension is chosen afterward. To verify this order, choose a finite
recipe and let $q$ be the largest number of successive spin flips used by
any difference in its proof. The induction below adds only finitely many
operations at each smaller residual index, so $q$ is finite. The primary
bounds give, for every involved level $l$,
\[
 \|R_l(x^{(i)})-R_l(x)\|
 \le 2\|\nabla R_l(x)e_i\|\le C_l.
\]
With $C=\max_l C_l$, a path of at most $q$ flips increases either
residual defining $\Omega_\rho$ by at most $qC$. Thus
$n>(2qC/\rho)^2$ puts every required neighborhood of
$\Omega_{3\rho/2}$ strictly inside $\Omega_{2\rho}$. Here $q,C$, and the
threshold on $n$ may depend on the already fixed recipe and depth;
$\rho$ still depends only on the earlier stability and size margins.

\subsection{Trace control for small derivatives}

The primary derivative bounds do not yet give the diagonal trace control
needed for integration by parts. For small vectors, the source structure
and its correction terms supply this additional cancellation. We prove
it together with the size and derivative bounds used to construct later
auxiliary fields.

\begin{lemma}[Small-vector differentiation]\label{lem:small-derivatives}
Every source, auxiliary field, and terminal vector $U$ in a fixed ordinary
admissible recipe satisfies
\begin{equation}
 \norm U+\norm{\nabla U}_{\Frob}
                   +\norm{\diag\nabla U}_1\leq C.
 \label{eq:small-derivative-bound}
\end{equation}
The same holds locally for recipes with the single initial implicit field
above.  In these recipes the initial partial $\partial_k w$ is also a
small source.  For every primary coefficient and auxiliary coefficient
in \eqref{eq:small-field}, respectively,
\begin{equation}
 \begin{split}
 \alpha_l=\av{\partial_lw^a}:&\quad
       |\alpha_l|+\norm{d\alpha_l}\leq C/\sqrt n,\\
 \gamma_b=\av{\partial_{y^b}w^a}:&\quad
       |\gamma_b|+\norm{d\gamma_b}\leq C.
 \end{split}
 \label{eq:small-average-bounds}
\end{equation}
All local statements include the required fixed flip buffers.  Constants
can depend on the entire finite recipe and its primary depth.
\end{lemma}

\begin{proof}
The proof has two parts. We bound the errors introduced by discrete
differentiation, then cancel the trace predictions of the retained formal
terms. Write $\norm E_{\rm nuc}$ for the sum of the singular values of
$E$. The error estimates repeatedly use
\begin{equation}
 |\Tr(PE)|\leq\norm P\norm E_{\rm nuc},\qquad
 \norm{BC}_{\rm nuc}\leq\norm B_{\Frob}\norm C_{\Frob},\qquad
 \norm E_{\rm nuc}\leq\sum_j\norm{E_{\cdot j}}.
 \label{eq:nuclear-tools}
\end{equation}
Every bounded-length diagnostic word $P$ has bounded operator norm and
bounded off-diagonal entrywise $\ell^4$ norm. For the errors arising before
we solve the differentiated implicit
equations, $P$ may contain one inverse $K$ from \eqref{eq:K}. Its required
margin is supplied by the last assertion of \eqref{eq:implicit-size}.

\paragraph{Source errors.}
Consider one source term $\phi_i s_i$.  Retain the derivative
\begin{equation}
 D_\phi\nabla s+\sum_lD_{s\partial_l\phi}\nabla h^l,
 \label{eq:retained-source}
\end{equation}
and put parameter derivatives into the error rather than into the formal
expression. Set
\[
 Q_{ij}=\sum_l|d_jh_i^l|,\qquad
 \Theta_j=\sum_\alpha|d_j\theta_\alpha|.
\]
The finite sums have the bounds
\[
 \max_i\sum_jQ_{ij}^2+\norm Q_{\ell^4}^2
       +\norm{\diag Q}_2^2\leq C,
 \qquad \norm\Theta\leq C.
\]
Taylor expansion of $d_j\phi_i$ has linear parameter terms and a remainder
bounded entrywise by
\begin{equation}
 C\{Q_{ij}^2+Q_{ij}\Theta_j+\Theta_j^2\}.
 \label{eq:source-taylor-error}
\end{equation}
After multiplication by $s_i$, the linear parameter terms have bounded
nuclear norm: each is a rank-one matrix with a bounded-norm left vector
and a bounded-norm right vector.  For the $Q^2$ part, the matrix before
multiplication by $D_s$ has bounded Frobenius norm.  Its product with
$D_s$ has bounded nuclear norm by \eqref{eq:nuclear-tools} and
$\norm{D_s}_{\Frob}=\norm s\leq C$.
For the mixed part the sum of column norms is at most
\[
 C\sum_j\Theta_j\Bigl(\sum_i s_i^2Q_{ij}^2\Bigr)^{1/2}
 \leq C\norm\Theta
       \Bigl(\sum_i s_i^2\sum_jQ_{ij}^2\Bigr)^{1/2}\leq C.
\]
For the last part it is at most $C\norm s\sum_j\Theta_j^2\leq C$.
These separate estimates also apply to the combined remainder in
\eqref{eq:source-taylor-error}: entry by entry, assign each summand in
the bound its proportional share of the actual remainder.

The product rule also has a remainder with entries bounded by
\[
 C|d_js_i|\{Q_{ij}+\Theta_j\}.
\]
Assume $\norm{\nabla s}_{\Frob}\leq C$, which is zero for a seed,
holds for the initial implicit field by
Lemma~\ref{lem:implicit-construction}, and is inductive for later fields.
The $\Theta$ part has bounded sum of column norms by Cauchy--Schwarz.
For the $Q$ part, test against $P$ and split the trace into diagonal and
off-diagonal terms.  H\"older's inequality gives
\begin{align*}
 \sum_{i\ne j}|P_{ji}|\,|d_js_i|Q_{ij}
 &\leq \norm{P_{\rm off}}_{\ell^4}
       \norm{\nabla s}_{\Frob}\norm Q_{\ell^4}\leq C,\\
 \sum_i|P_{ii}|\,|d_is_i|Q_{ii}
 &\leq\norm P\norm{\diag\nabla s}_2\norm{\diag Q}_2\leq C.
\end{align*}
The first line uses exponents $4,2,4$. In particular, the argument does
not assume the diagonal $\ell^1$ bound for $\nabla s$ that we are still
proving. Bounded entries of $Q$ also give a bounded Frobenius norm for
this error. We have therefore controlled each source error both in
Frobenius norm and in trace against the indicated words.

\paragraph{Average corrections and size bounds.}
Next we control the averaged coefficients in the correction terms.
Differentiating a coefficient function in a primary argument gives another
source with the same inputs and bounds. Apply the size and Frobenius
parts of the preceding estimates to that source, then average:
\[
 |\av u|\leq\norm u/\sqrt n,\qquad
 \norm{d\av u}\leq\norm{\nabla u}_{\Frob}/\sqrt n,
\]
which proves the first line of \eqref{eq:small-average-bounds}.
The implicit coefficient $c_1$ already has these bounds from its direct
construction. For an auxiliary partial the scale is different: it is a
bounded coefficient vector, rather than a small vector. Averaging its
difference estimate
$|d_j\phi_i|\leq C(Q_{ij}+\Theta_j)$ and using the row bounds above
gives $|\av\phi|+\norm{d\av\phi}\leq C$.  This proves the second
line, including the self-coefficient $\chi$ when it occurs.

For a primary correction $\alpha_lm^{l-1}$ retain
$\alpha_l\nabla m^{l-1}$.  The dropped rank-one matrix has nuclear norm
at most $\norm{m^{l-1}}\norm{d\alpha_l}\leq C$.  The dropped product
error is $\nabla m^{l-1}$ times a diagonal of signed entries
$2d_i\alpha_l$.  Its nuclear norm is bounded by
\[
 2\norm{\nabla m^{l-1}}_{\Frob}\norm{d\alpha_l}
 \leq C\sqrt n\,n^{-1/2}\leq C.
\]
This also holds for $m^0=x$.  For an auxiliary correction $\gamma_bw^b$,
the analogous rank-one and product errors have bounded nuclear norm using
$\norm{w^b}+\norm{\nabla w^b}_{\Frob}\leq C$ and
$|\gamma_b|+\norm{d\gamma_b}\leq C$.
When primary derivatives in retained terms are replaced by
$H_l,M_l$, their Frobenius errors from
Lemma~\ref{lem:primary-derivatives} are multiplied either by a
small-vector diagonal or by an $O(n^{-1/2})$ scalar.  In the former case
\eqref{eq:nuclear-tools} applies; in the latter use
$\norm E_{\rm nuc}\leq\sqrt n\norm E_{\Frob}$.  Both give bounded
nuclear norm.

We can now close the induction for the size and Frobenius parts of
\eqref{eq:small-derivative-bound}. In \eqref{eq:small-field}, the terms are a
bounded-norm leading term, primary corrections bounded by
$C\sqrt n\,n^{-1/2}$, and bounded-norm auxiliary corrections.
The retained source derivative \eqref{eq:retained-source} is
Frobenius-bounded: use the preceding auxiliary derivative bounds and
$\norm{D_{s\partial_l\phi}\nabla h^l}_{\Frob}
\leq\norm{s\partial_l\phi}\norm{\nabla h^l}$.
The retained primary correction is bounded in Frobenius norm by
$|\alpha_l|\norm{\nabla m^{l-1}}_{\Frob}\leq C$.
Thus the size and Frobenius induction closes before we use or prove
the diagonal $\ell^1$ assertion.

\paragraph{Solving the differentiated self-dependence.}
The initial implicit source needs one additional step before the trace
cancellation. Retain its site partials, freeze the parameters and
averages, and replace the primary derivatives by their formal matrices.
This gives
\begin{equation}
 \begin{split}
 \nabla w&=A\nabla y^{\rm imp}+D_{\partial_k w}H_k+E_1,\\
 \nabla y^{\rm imp}&=(J-j\chi I)\nabla w-jc_1M_{k-1}+E_2.
 \end{split}
 \label{eq:implicit-formal-differentiation}
\end{equation}
The errors have bounded Frobenius norm and bounded trace against words
containing at most one $K$.  The parameter $c_1$ causes no exception:
write the corresponding source term as
$-j(\sqrt n c_1)A(t_*/\sqrt n)$, so that it is exactly of the source
form above.  Its difference bound follows from
\eqref{eq:implicit-frobenius}.
Solving \eqref{eq:implicit-formal-differentiation} yields
\begin{equation}
 \nabla w
 =K\{D_{\partial_k w}H_k-jc_1AM_{k-1}+E_1+AE_2\},
 \qquad K=(I-AJ+j\chi A)^{-1}.
 \label{eq:implicit-derivative-solved}
\end{equation}
Testing a propagated error against an ordinary word now puts at most
one $K$ into the test word. The preceding error estimates therefore give
both bounded trace and bounded Frobenius norm. Substitution gives the same
conclusion for
$\nabla y^{\rm imp}$. Subsequent fields are ordinary: propagating these
errors multiplies
them only by ordinary words. Induction therefore bounds the trace of
every final error against an ordinary word. Since the recipe is fixed
and finite, the required increases in word length are finite as well.

\paragraph{Cancellation of the formal traces.}
All differentiated errors are now controlled. To finish, we must bound
the diagonal $\ell^1$ norm of the retained formal derivative. A single
ordinary field already illustrates the cancellation. From
$w_i=\phi(h_i^1)s_i$ we obtain
$y=Jw-j\av{\phi'(h^1)s}x$. If $P=D_{\phi'(h^1)s}$, its retained
derivative is $JPJ-j\av{\phi'(h^1)s}I$, whose two diagonal predictions
cancel. We now show that the correction at each node gives this same
cancellation along every dependency path, including paths through the
implicit field.

At the point $x$, freeze the local partial vectors and all their averages. Index
each primary or auxiliary field by a node $a$, and attach its source to
the same node: for example, $h^l$ has source $m^{l-1}$, and $y^a$ has
source $w^a$. Let $p_{ad}$ be the partial of the source at node $a$
with respect to the field at node $d$. Introduce a formal variable $z$,
and denote the formal source and field derivatives by
$\mathfrak S_a$ and $\mathfrak H_a$, respectively. They obey
\begin{equation}
 \mathfrak S_a=\sum_dD_{p_{ad}}\mathfrak H_d,
 \qquad
 \mathfrak H_a=zJ\mathfrak S_a
             -z^2j\sum_d\av{p_{ad}}\mathfrak S_d.
 \label{eq:formal-graph}
\end{equation}
At the spin leaf, the source derivative for $h^1$ is $I$ and its field
derivative is $zJ$.  A terminal source uses just the first equation.
At $z=1$ these are precisely the retained formal derivatives.  In the
implicit case the only cyclic node is $y^{\rm imp}$ and its source $w$;
closing it gives
\begin{equation}
 \mathfrak S_{\rm imp}
 =K(z)\{D_{\partial_k w}\mathfrak H_{h^k}
                 -z^2jc_1A\mathfrak S_{h^k}\},
 \quad K(z)=(I-zAJ+z^2j\chi A)^{-1}.
 \label{eq:formal-self-loop}
\end{equation}
Every other dependency is acyclic and linear. Solving the single cyclic
node therefore leaves at most one inverse $K(z)$ in each product of the
closed formulas.

The averaged term in \eqref{eq:formal-self-loop} is essential even for
$k=2$. Write $D=D_{1-(m^1)^2}$ and $P=D_{\partial_2w}$, so that
$b_1=n^{-1}\Tr D$ and $c_1=n^{-1}\Tr P$. Then
\[
 \mathfrak H_{h^2}=z^2(JDJ-jb_1I),\qquad
 \mathfrak S_{h^2}=zDJ,
\]
and the implicit derivative is
$K(z)\{z^2P(JDJ-jb_1I)-z^3jc_1ADJ\}$.
At degree four, the two noncrossing pairings of $AJAJPJDJ$ have
predictions
\[
 j^2\chi b_1 AP,
 \qquad j^2c_1\frac{\Tr(AD)}n A.
\]
The first is canceled by the terms from $-j\chi A$ in the
quadratic coefficient of $K(z)$ and $-jb_1I$ in
$\mathfrak H_{h^2}$. The second is canceled by the degree-four
term $-jc_1AJADJ$ from the averaged correction. This calculation exhibits
the first cancellation involving both the
self-dependent node and its primary input. We now account for every
degree by following dependency paths.

We work coefficient by coefficient.  An uncontracted
dependency path, followed backwards from a field to the spin leaf, has
successive links $JD_{p_{ad}}$ and ends with the $J$ link of $h^1$.
A terminal source contributes an initial partial diagonal and then such
a path.  The choices of input nodes index these paths separately, preserving
all multiplicities.  Expanding \eqref{eq:formal-graph} is
inclusion--exclusion over disjoint originally adjacent link pairs: the
first term keeps the first link, while the second contracts it with the
next link, with factor $-j\av{p_{ad}}$, and continues from the input
node's source.  That continuation skips exactly the link just contracted.
Every contraction has degree two in $z$, equal to the degree of the
replaced links.  Consequently a coefficient of degree $L$ involves only
uncontracted paths of length $L$, of which there are finitely many even
with the self-dependence.

To compute the trace prediction, test the formal derivative by a bounded
diagonal $D$. Fix a nonempty path and a full noncrossing pairing of its
links. Reinserting contracted adjacent pairs restores that same
full-pairing prediction.
Each contraction contributes its minus sign.  Conversely, all subsets of
the originally adjacent pairs of that full pairing occur, with precisely
these signs.  Such a full pairing has at least one originally adjacent
pair, so its total coefficient is zero.  Paths of odd length have no full
pairing.  There is no empty path contributing a derivative of a small
seed, because all seeds are fixed with respect to $x$; parameter
derivatives were already put into the error. Thus the trace prediction
vanishes as a formal power series. The
polynomial-prediction assertion for single-inverse words in
Lemma~\ref{lem:words} then makes it zero at $z=1$ in the closed formulas.

To pass from this zero prediction to a bounded trace, we use the
small-vector structure. Every product in a closed formal derivative
of a small vector contains
either a small-vector diagonal or a scalar of size $O(n^{-1/2})$:
a path giving a nonzero derivative must enter the primary nodes before
reaching the spin leaf.  Entry from a small source uses
either its primary partial, a small vector, or the average of that partial
in a correction term.  The latter has the indicated scalar size.
For the implicit starting node these are exactly
$\partial_k w$ and $c_1$ in \eqref{eq:formal-self-loop}.

For a term containing a small diagonal $D_s$, rotate the tested trace
to put that factor first.  The word estimate for the remaining factors
and Cauchy--Schwarz give
\[
 \left|\Tr(D_sQ)-\sum_i s_i\,[\mathsf P(Q)]_{ii}\right|
 \leq\norm s\norm{\diag(Q-\mathsf P(Q))}_2\leq C.
\]
The normalized-trace prediction is cyclically invariant by
Lemma~\ref{lem:words}, so the rotation does not change the prediction
being cancelled.  For a term with a scalar of size $O(n^{-1/2})$, its
unscaled trace error is at most $C\sqrt n$ by the diagonal $\ell^2$
word estimate, and multiplication gives the same $O(1)$ bound.
The closed formula has finitely many terms. Combining their errors
with the zero prediction and the already controlled differentiated errors
gives
\[
 |\Tr(D\nabla U)|\leq C\qquad(\norm D\leq1,\ D\text{ diagonal}).
\]
Taking $D_{ii}$ to be the sign of $d_iU_i$ proves the remaining assertion
$\norm{\diag\nabla U}_1\leq C$.  All arguments were pointwise inside
the prescribed buffers, which proves the local version as well.
\end{proof}

\subsection{The weak residual rule}

We now apply the diagonal $\ell^1$ estimate to the exact conditional
integration-by-parts identity at the first primary field. The scalar
integral identity then reduces each higher residual to earlier ones.
We carry out this induction for ordinary tests and square-density tests
together.

\begin{lemma}[Weak residual rule]\label{lem:residual}
Let $U$ be a terminal vector in a fixed recipe admissible from level
$p\geq1$.  For an ordinary recipe and every scalar $G$,
\begin{equation}
 \left|\E G R_p^{\mathsf T}U\right|\leq C\norm G_*.
 \label{eq:weak-residual}
\end{equation}
For a local recipe with implicit starting index $k$, the same conclusion
holds when $p\leq k$ and $G$ is supported on $\Omega_\rho$.

There is also an ordinary square-density version.  If
$\sup_x(|H_0(x)|+\norm{dH_0(x)})\leq C_0$, then for every $f$,
\begin{equation}
 \left|\E f^2H_0 R_p^{\mathsf T}U\right|
       \leq C\{\E f^2+\D_{h_0}(f)\}.
 \label{eq:square-residual}
\end{equation}
Constants are uniform in $f,G,H_0$ subject to the stated bound, and may
depend on $p$, the recipe, its coefficient bounds, and the finite set of
word diagnostics needed in the proof.
\end{lemma}

\begin{proof}
We induct strongly on $p$ over all finite admissible recipes, proving
the two assertions together. Each step may enlarge the original recipe,
but it adds only finitely many auxiliary operations and invokes only
smaller residual indices. Consequently every particular instance of the
induction uses finitely many operations and a finite word-length bound.

At the base level $p=1$, condition on all spins except the one being
differentiated. Its conditional mean is $m_i^1$ and its conditional
variance is $v_i^1$. A function of this single spin is affine, with linear
coefficient $d_i$ of the function. Taking its conditional covariance
with the spin gives the exact identity
\begin{equation}
 \E\sum_i(x_i-m_i^1)GU_i
       =\E\sum_i v_i^1d_i(GU_i).
 \label{eq:conditional-ibp}
\end{equation}
By \eqref{eq:discrete-product}, its right side is
\[
 \E\sum_i v_i^1Gd_iU_i
   +\E\sum_i v_i^1(d_iG)\{U_i-2x_id_iU_i\}.
\]
The diagonal derivative bound makes the first term at most $C\E|G|$.
For the second term, the coefficient vector has pointwise norm at most
$\norm U+2\norm{\diag\nabla U}_2\leq C$.
Weighted Cauchy--Schwarz and $0<v_i^1\leq1$ therefore bound that term
by $C\D_{h_0}(G)^{1/2}$.  This proves \eqref{eq:weak-residual}.
For a supported local test, its derivatives vanish outside the one-flip
neighborhood of its support; all bounds used here are available there.

For \eqref{eq:square-residual}, absorb the bounded multiplier by replacing
$U$ in the base identity with $\wt U=H_0U$. This vector still has bounded
norm and bounded diagonal derivative
$\ell^1$ norm: the additional term obeys
$\sum_i|U_i d_iH_0|\leq\norm U\norm{dH_0}$, and the product-error
term is bounded by the same argument using
$\norm{\diag\nabla U}_2$.  Apply \eqref{eq:conditional-ibp} with
$G=f^2$ and use
\[
 d_i(f^2)=2f\,d_i f-2x_i(d_i f)^2.
\]
The diagonal-derivative term is bounded by $C\E f^2$; the remaining
terms are bounded by
$C\sqrt{\E f^2\,\D_{h_0}(f)}+C\D_{h_0}(f)$.
The base case is now proved in both forms. Before proceeding to higher
residuals, we record how bounded scalar multipliers affect the tests. If
$|\vartheta|+\norm{d\vartheta}\leq C$, then
\begin{equation}
 \norm{\vartheta G}_*\leq C'\norm G_*.
 \label{eq:star-multiplier}
\end{equation}
Indeed arrange the product rule as
$d_i(\vartheta G)=\vartheta(x^{(i)})d_iG+G(x)d_i\vartheta$,
square, and sum with weights $v_i^1\leq1$.  In the square-density
case such a multiplier can be absorbed into $H_0$; the product retains
its bounded size and difference norm.  For local tests only the buffer
bounds are needed, since multiplication does not enlarge their support.

For the induction step, suppose both assertions hold through level $p$
and take $U$ admissible from level $p+1$. The scalar integral identity
suggests the following new source. Set
\[
 a_p=j(b_p-b_{p-1}),\qquad
 U'_i=\Phi(h_i^p;h_i^{p+1},a_p)U_i.
\]
This is a source admissible from level $p$.  Its coefficient and all
needed derivatives are bounded because $|a_p|\leq j$ and the bounds for
$\Phi$ proved with \eqref{eq:phi-stein} hold uniformly in its two real
field arguments.
The level condition on $U$ makes its site partial at level $p$ vanish:
$\partial_pU$ differentiates only explicit primary arguments. Parameters
and auxiliary arguments remain fixed, including when their values depend
on earlier primary fields.  Applying \eqref{eq:phi-stein} coordinatewise
therefore yields
\[
 R_{p+1}^{\mathsf T}U
 =(h^p-h^{p+1}-a_pm^p)^{\mathsf T}U'
                      -a_p n\av{\partial_pU'}.
\]
The primary recursion gives
\[
 h^p-h^{p+1}-a_pm^p
       =JR_p+a_pR_p-jb_{p-1}R_{p-1},
\]
where the last term is absent for $p=1$.  Add a last auxiliary field
$y'$ by \eqref{eq:small-field} with source $U'$, and write
$\alpha_l=\av{\partial_lU'}$ and
$\gamma_b=\av{\partial_{y^b}U'}$.  Symmetry of $J$ now gives
\begin{align}
 R_{p+1}^{\mathsf T}U
 ={}&R_p^{\mathsf T}y'+a_pR_p^{\mathsf T}U'
              -jb_{p-1}R_{p-1}^{\mathsf T}U'
              +j\sum_b\gamma_bR_p^{\mathsf T}w^b\notag\\
    &+j\sum_{l\geq p}\alpha_lR_p^{\mathsf T}m^{l-1}
                   -a_pn\alpha_p.
 \label{eq:residual-induction-expansion}
\end{align}
The first line now consists of earlier residuals with admissible sources
and bounded multipliers. After testing, \eqref{eq:star-multiplier},
\eqref{eq:small-average-bounds}, and the level constraints allow us to
apply the induction hypothesis to each term. In particular the enlarged
recipe is admissible from level $p$, which is
also sufficient for its $R_{p-1}$ term.

On the second line, the terms with $l>p$ use the terminal source
$m^{l-1}/\sqrt n$, admissible from level $p$, and multiplier
$\sqrt n\alpha_l$. The multiplier bounds follow from
Lemma~\ref{lem:small-derivatives}. The remaining term is handled by the
identity
\begin{equation}
 R_p^{\mathsf T}m^{p-1}
       =n(b_p-b_{p-1})-R_p^{\mathsf T}m^p.
 \label{eq:residual-inner-product}
\end{equation}
This follows from
$(m^{p-1}-m^p)^{\mathsf T}(m^{p-1}+m^p)
=\norm{m^{p-1}}^2-\norm{m^p}^2$.
The first term on the right of \eqref{eq:residual-inner-product}
cancels $a_pn\alpha_p$ in
\eqref{eq:residual-induction-expansion}; the remaining term is handled
with the admissible source $m^p/\sqrt n$ and multiplier
$\sqrt n\alpha_p$. This completes the induction for both
\eqref{eq:weak-residual} and
\eqref{eq:square-residual}. In the local version it stops at level $k$;
each added source respects the support and buffer requirements of the
initial implicit construction.
\end{proof}

The last step puts the implicit source into the form needed for the
posterior comparison. The following identity is a direct application of
the weak residual rule, with the averaged correction retained.

\begin{lemma}[Terminal residual identity]\label{lem:terminal-residual}
For the implicit construction at index $k$ and every $G$ supported on
$\Omega_\rho$,
\begin{equation}
 \left|\E G\left[
 \{h^k-h_0-(J-jb_{k-1}I)m^k\}^{\mathsf T}w
                   -j(b_k-b_{k-1})n c_1\right]\right|
       \leq C\norm G_*.
 \label{eq:terminal-residual}
\end{equation}
\end{lemma}

\begin{proof}
The vector in braces is $JR_k-jb_{k-1}(R_{k-1}+R_k)$.
To express its pairing with the implicit source in terms of earlier
residuals, use symmetry of $J$, the equation
$Jw=y^{\rm imp}+j\chi w+jc_1m^{k-1}$, and
\eqref{eq:residual-inner-product}. The bracketed expression becomes
\[
 R_k^{\mathsf T}y^{\rm imp}
   +j(\chi-b_{k-1})R_k^{\mathsf T}w
   -jb_{k-1}R_{k-1}^{\mathsf T}w
   -jc_1R_k^{\mathsf T}m^k.
\]
Apply Lemma~\ref{lem:residual} to the first three terms, absorbing their
bounded parameters into the test. For the last term, use source
$m^k/\sqrt n$ and multiplier $\sqrt n c_1$. The bounds needed for this
multiplier are precisely \eqref{eq:implicit-size} and
\eqref{eq:implicit-frobenius}.
\end{proof}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//residuals.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//stability.tex ====
\section{Stable fields along Gaussian observations}
\label{sec:stability}

The observation argument requires control of every approximate solution
of the field equation at the fields visited by the observation process.
We first prove this stability statement, including perturbations of the
diagonal variance matrix, and then derive a deterministic existence,
uniqueness, and inverse estimate for the field equation.

For a symmetric matrix $J$, a field $h_0\in\R^n$, and $y\in\R^n$, define
\begin{equation}
 \begin{split}
 m(y)&=\tanh y,\qquad q(y)=\av{m(y)^2},\qquad
 b(y)=1-q(y),\\
 V_y&=D_{1-m(y)^2},\qquad B(y)=jb(y),\\
 F_{J,h_0}(y)&=y-h_0+(B(y)I-J)m(y).
 \end{split}
 \label{eq:field-map}
\end{equation}
For fixed $J$, abbreviate $F_h=F_{J,h}$; we also suppress $h$ when
there is no ambiguity. Hyperbolic functions act coordinatewise. To measure
stability while allowing the diagonal factor to vary, define, for each
nonnegative diagonal matrix $A$,
\begin{equation}
 \mathsf H_J(y,A)
 =I+A^{1/2}\left(B(y)I-J-\frac{2j}{n}m(y)m(y)^{\mathsf T}\right)A^{1/2}.
 \label{eq:field-hessian}
\end{equation}
The matrix $\mathsf H_J(y,V_y)$ is the symmetric version of the
possibly nonsymmetric derivative $F'_{J,h_0}(y)$. The external field
$h_0$ enters the residual $F_{J,h_0}(y)$ but not
$\mathsf H_J(y,A)$. This separation will let us use the same matrix
witness when the observation field moves.

\begin{proposition}[Stability along observations]
\label{prop:stable-fields}
Fix $j\in(0,1)$ and $0<T<\infty$.  There are constants
$\eta,c,\epsilon_A,\rho_0,a>0$, depending only on $j,T$, and events
$\mathcal J_n$ for the zero-diagonal Gaussian matrix $J$, such that
$\Prob(J\in\mathcal J_n)\longrightarrow1$ and the following assertion
holds for every $J\in\mathcal J_n$.  Let $X\sim\mu_0$, let $B_0$ be an
independent standard $n$-dimensional Brownian motion, and put
$Y(t)=tX+B_0(t)$.  With conditional probability at least $1-e^{-an}$,
simultaneously for $0\le t\le T$, $y\in\R^n$, and all diagonal $A$,
\begin{equation}
 \left.
 \begin{gathered}
 \norm{F_{J,Y(t)}(y)}\le2\rho_0\sqrt n,\\
 0\le A\le(1+\eta)I,\qquad
 \norm{A-V_y}_{\Frob}\le\epsilon_A\sqrt n
 \end{gathered}
 \right\}
 \quad\Longrightarrow\quad
 \mathsf H_J(y,A)\succ cI.
 \label{eq:stable-field}
\end{equation}
We may require $j(1+\eta)^2<1$ and
$\norm J\le2\sqrt j+\theta$ on $\mathcal J_n$, for any sufficiently
small fixed $\theta>0$ chosen in the proof.
\end{proposition}

\begin{corollary}[Stopping formulation]\label{cor:stopping}
For every $J\in\mathcal J_n$ there is an observation stopping time $\tau_0$
such that
\[
 \Prob(\tau_0\le T\mid J)\le e^{-an},
\]
and, for every $t\in[0,T]$ with $t<\tau_0$, the implication \eqref{eq:stable-field} holds
at $Y(t)$ with residual threshold $\rho_0\sqrt n$ in place of
$2\rho_0\sqrt n$. The stopping rule depends on $J$ and the observations,
and is independent of any test function.
\end{corollary}

\begin{proof}
We express failure of stability through a witness set that is fixed
before observing the path. Let $\mathcal W_J$ consist of pairs $(y,A)$ satisfying
$0\le A\le(1+\eta)I$ and $\norm{A-V_y}_{\Frob}\le\epsilon_A\sqrt n$
for which $\mathsf H_J(y,A)\not\succ cI$. Define the least
normalized residual among these witnesses by
\[
 d_J(h)=\inf_{(y,A)\in\mathcal W_J}
             \frac{\norm{F_{J,h}(y)}}{\sqrt n},
 \qquad \inf\varnothing=+\infty.
\]
If $\mathcal W_J$ is nonempty, $d_J$ is finite and $n^{-1/2}$-Lipschitz,
because $F_{J,h}(y)=F_{J,0}(y)-h$. Let
\[
 \tau_0=\inf\{t\in[0,T]:d_J(Y(t))\le\rho_0\},
\]
with value $+\infty$ when the set is empty. Continuity makes the first
hitting time an observation stopping time.
Before the hit, every violating witness has residual greater than
$\rho_0\sqrt n$. At a hit, the definition of the infimum supplies a
violating witness with residual strictly below $2\rho_0\sqrt n$;
attainment of the infimum is unnecessary.
Proposition~\ref{prop:stable-fields} therefore bounds the hitting probability.
\end{proof}

The proof of Proposition~\ref{prop:stable-fields} has three parts. A
Gaussian integral bounds the weight of unstable configurations; its
scalar rate is controlled by the entropy inequality, and its conditional
matrix factor is controlled near the equality laws. A local-volume
estimate then shows that any unstable approximate zero would contribute
too much to that integral. Finally we transfer the resulting estimate to
the original observation law, uniformly over the time interval.

\subsection{A Gaussian integral and its scalar rate}

We carry out the integral calculation first under a planted Gaussian
law, whose relation to the original law will be proved when we assemble
the path estimate. Under this law,
\begin{equation}
 \wt J=W+\frac jn\one\one^{\mathsf T},\qquad
 h_0=t\one+\sqrt t\,g,
 \label{eq:planted-field}
\end{equation}
where $W$ is centered GOE of scale $j$ and $g$ is independent standard
Gaussian. We retain the diagonal of $\wt J$ during this calculation.
For a Borel set $\mathcal E\subseteq\R^n\times\mathrm{Sym}_n$, define
\begin{equation}
 I_{t,\sigma}(\mathcal E)
 =(2\pi\sigma^2)^{-n/2}\int
 \one_{\{(y,\wt J)\in\mathcal E\}}
 \exp\left\{-\frac{\norm{F_{\wt J,h_0}(y)}^2}{2\sigma^2}
              +\frac{njb(y)^2}{2}\right\}\,\dd y,
 \qquad \sigma>0.
 \label{eq:field-integral}
\end{equation}
The weight $e^{njb(y)^2/2}$ compensates for the exponential scale of
the local determinant: Lemma~\ref{lem:local-field-volume} will use the
regularized determinant bound to turn an approximate zero into a lower
bound for this integral. The sets to which we apply the integral depend
on $y,\wt J$ and are independent of $h_0$. Thus the observation can be
averaged while retaining the restriction to $\mathcal E$.

For fixed $y,t,\sigma$, introduce the scalar quantities
\begin{equation}
 \begin{gathered}
 m=\tanh y,\quad q=\av{m^2},\quad b=1-q,\quad B=jb,
 \quad M=\av m,\\
 d=t+jM,\qquad s=t+\sigma^2+jq,\qquad S=s+jq,\\
 a_y=\av{(y-d\one)m},\qquad z=y-d\one+Bm,\qquad
 \psi=\frac{j(a_y-sb)^2}{2sS}.
 \label{eq:scalar-parameters}
 \end{gathered}
\end{equation}
Write $\varphi_{d,s}$ for the density of $N(d,s)$, with $s$ denoting
the variance, and define
\begin{equation}
 G_{t,\sigma,n}(y)
 =\prod_{i=1}^n\varphi_{d,s}(y_i)\,e^{n\psi}.
 \label{eq:scalar-density}
\end{equation}
The Gaussian parameters in this expression depend on $y$ through the
preceding averages; the product is therefore not a fixed product density.

\begin{lemma}[Conditional Gaussian reduction]
\label{lem:gaussian-reduction}
With the preceding definitions, and with $g'$ an independent standard
Gaussian vector,
\begin{equation}
 \E I_{t,\sigma}(\mathcal E)
 =\int \sqrt{\frac sS}\,G_{t,\sigma,n}(y)
 \Prob\left((y,\wt J)\in\mathcal E\,\middle|\,
       Wm+\sqrt{t+\sigma^2}\,g'=z\right)\,\dd y.
 \label{eq:conditional-integral}
\end{equation}
The conditional distribution in this identity is the usual Gaussian
conditional distribution and is defined for every $y$, since
$\sigma>0$.
\end{lemma}

\begin{proof}
We first evaluate the integral without its indicator. For fixed $m$,
the GOE covariance is
\[
 \Cov(Wm)=jqI+\frac jnmm^{\mathsf T}.
\]
Convolution with the heat kernel of variance $\sigma^2$ and with
$\sqrt t\,g$ therefore produces the Gaussian vector
$Wm+\sqrt{t+\sigma^2}\,g'$, of covariance
$sI+jmm^{\mathsf T}/n$.  Its determinant is $s^nS/s$, and its inverse
is
\[
 \frac1sI-\frac{j}{nsS}mm^{\mathsf T}.
\]
Moreover,
$F_{\wt J,h_0}(y)=z-Wm-\sqrt t\,g$.  The Gaussian density at $z$,
multiplied by $e^{njb^2/2}$, differs from
$\prod_i\varphi_{d,s}(y_i)$ by the prefactor $\sqrt{s/S}$ and the
exponential whose logarithm divided by $n$ is
\[
 -\frac{Ba_y}{s}-\frac{B^2q}{2s}
 +\frac{j(a_y+Bq)^2}{2sS}+\frac{jb^2}{2}
 =\frac{j(a_y-sb)^2}{2sS}.
\]
This calculation gives the identity for the unrestricted integral.
To retain the indicator, condition the joint Gaussian pair formed by
$W$ and the convolved observation on the value $z$ of the latter.
The resulting conditional probability is exactly the factor in
\eqref{eq:conditional-integral}. All integrands are nonnegative, so
Tonelli's theorem justifies the interchange of integrations.
\end{proof}

The scalar entropy inequality controls the remaining integral in
\eqref{eq:conditional-integral}. To apply it, replace empirical averages
in \eqref{eq:scalar-parameters} by integration against a probability
measure $P$ with finite second moment, keeping the notation
$M,q,b,d,s,S,a_y,\psi$. Where needed, write $a(P)$ instead of $a_y$.
Lemma~\ref{lem:scalar-entropy} then gives
\begin{equation}
 \psi(P,t,\sigma)-D(P\Vert N(d,s))\le0
 \qquad (s>0).
 \label{eq:rate-nonpositive}
\end{equation}
At $\sigma=0$, Lemma~\ref{lem:gaussian-equality} identifies all equality
cases: they form the curve
\[
 P_t=N(s_*(t),s_*(t)),\qquad
 s_*(t)=t+j\E_{P_t}\tanh y,
\]
with the point-mass convention $P_0=\delta_0$.  In particular,
$s_*(t)\le t/(1-j)$ and
$q(P_t)=(s_*(t)-t)/j\le t/(1-j)$.

\begin{lemma}[Uniform scalar rate bounds]
\label{lem:scalar-rates}
Fix $j\in(0,1)$ and $T<\infty$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For every $s_{\min}>0$ and $L<\infty$, one can choose $R<\infty$
such that, for all sufficiently large $n$, uniformly over
$0\le t\le T$ and $0<\sigma\le1$,
\[
 \int_{\{s\ge s_{\min},\,\av{y^2}>R^2\}}
 G_{t,\sigma,n}(y)\,\dd y\le e^{-Ln}.
\]
For every $\varepsilon>0$, the unrestricted integral over
$\{s\ge s_{\min}\}$ is at most $e^{\varepsilon n}$ for all
sufficiently large $n$.
\item Fix $t_0>0$.  Outside any prescribed open neighborhood of the
equality curve $\{(P_t,t):t_0\le t\le T\}$, there are
$a,\sigma_*>0$ such that the integral of $G_{t,\sigma,n}$ is at most
$e^{-an}$, uniformly over $t_0\le t\le T$ and
$0<\sigma\le\sigma_*$, for all sufficiently large $n$.
The neighborhood can impose weak closeness of the empirical law and
closeness of any finite list of continuous, at most linearly growing
averages, including all parameters in
\eqref{eq:scalar-parameters}.
\item For every $q_0>0$, there are $t_0,\sigma_*,a>0$ such that
\[
 \int_{\{q\ge q_0\}}G_{t,\sigma,n}(y)\,\dd y\le e^{-an}
\]
uniformly over $0\le t\le t_0$ and $0<\sigma\le\sigma_*$, for all
sufficiently large $n$.
\item Fix $t_0>0$ and $\nu>0$.  For all sufficiently large $R_1$ there
is $a>0$ such that
\[
 \int_{\{\av{y^2\one_{\{|y|>R_1\}}}>\nu\}}
 G_{t,\sigma,n}(y)\,\dd y\le e^{-an},
\]
uniformly over $t_0\le t\le T$ and $0<\sigma\le1$, for all
sufficiently large $n$.
\end{enumerate}
All constants are independent of $n,t,\sigma$ in the indicated ranges.
In \textup{(ii)}, neighborhoods are interpreted on each fixed
second-moment ball; the complement of a sufficiently large such ball
is covered by \textup{(i)}.
\end{lemma}

\begin{proof}
We first obtain a Gaussian tail envelope, then prove a compact-set
bound by a finite cover. Strict negativity away from the equality laws
will give the second and third assertions; an exponential tilt will give
the fourth.

Throughout, $|d|\le T+j$ and $s\le T+1+j$. Also
$a(P)^2\le q\int(y-d)^2\,\dd P$ and $jq/S\le1/2$.  The elementary
inequality $(r-u)^2\le\frac32r^2+3u^2$ consequently gives
\[
 n\psi\le\frac{3jq}{4sS}\sum_i(y_i-d)^2+\frac{3nj}{2}
 \le\frac{3}{8s}\sum_i(y_i-d)^2+\frac{3nj}{2}.
\]
On $s\ge s_{\min}$, it follows that
\begin{equation}
 G_{t,\sigma,n}(y)\le
 \exp\left(Cn-c\sum_i y_i^2\right),
 \label{eq:scalar-tail-envelope}
\end{equation}
where $c>0$ and $C<\infty$ depend only on $j,T,s_{\min}$.
On $\sum_i y_i^2>nR^2$, use half of the remaining Gaussian square
for integration and the other half for the required exponential decay.
This proves the first assertion of \textup{(i)}.

The estimates on bounded second moments follow from one compactness
argument, which we now give in full. Put
\[
 \mathcal P_R=\left\{P:\int y^2\,\dd P\le R^2\right\}.
\]
Tightness and preservation of the second-moment bound under weak limits
make this set weakly compact. Integration of a continuous function of at
most linear growth is continuous on it: the second-moment bound makes
the contribution of $|y|>K$ uniformly $O(K^{-1})$, while bounded
truncations are weakly continuous. In particular, $d,s,a(P),\psi$ are
continuous on
$\mathcal P_R\times[0,T]\times[0,1]$ where $s\ge s_{\min}$.

For every compact subset $\mathcal C$ of that region,
\begin{equation}
 \limsup_{n\to\infty}\frac1n\log
 \sup_{t,\sigma}\int_{\{(P_y,t,\sigma)\in\mathcal C\}}
 G_{t,\sigma,n}(y)\,\dd y
 \le\sup_{\mathcal C}\bigl\{\psi-D(P\Vert N(d,s))\bigr\},
 \label{eq:compact-rate-bound}
\end{equation}
where $P_y=n^{-1}\sum_i\delta_{y_i}$; empty integrals are zero.
To establish this bound, fix a strict finite upper bound $r$ for its
right-hand side. At each $\xi=(P,t,\sigma)\in\mathcal C$, choose a
bounded continuous test function $\varphi_\xi$ from the variational
formula for relative entropy so that
\[
 \psi(\xi)-\int\varphi_\xi\,\dd P
       +\log\int e^{\varphi_\xi}\,\dd N(d,s)<r.
\]
Such a test also exists when the entropy is infinite. Retain positive
slack in the inequality and then choose a neighborhood of $\xi$ on
which continuity controls $\psi$, the empirical average of
$\varphi_\xi$, and $d,s$.  Moreover, on
$\sum_i y_i^2\le nR^2$, the logarithmic density ratio between
$N(d',s')^{\otimes n}$ and the fixed law $N(d,s)^{\otimes n}$ is at
most $n$ times a quantity tending uniformly to zero as
$(d',s')\to(d,s)$; this follows directly by expanding their Gaussian
log densities.  For $y$ in that neighborhood, insert
$e^{\sum_i\varphi_\xi(y_i)}$ and bound its expectation under the
fixed product Gaussian.  The integral on the neighborhood is then at
most $e^{nr}$, after using less than the reserved slack for the three
continuity errors. Cover $\mathcal C$ by finitely many such
neighborhoods. An arbitrarily
small enlargement of $r$ absorbs their number and proves
\eqref{eq:compact-rate-bound}. Combining this bound with
\eqref{eq:rate-nonpositive} on the ball and
\eqref{eq:scalar-tail-envelope} outside it proves the second assertion
of \textup{(i)}.

To obtain a strictly negative rate, observe that the same variational
formula expresses relative entropy as a supremum of continuous
functions. Thus it is jointly lower semicontinuous in $P,d,s$ when
$s$ is bounded away from zero, and $\psi-D$ is upper semicontinuous.
At $\sigma=0$, its zero set is precisely the equality curve identified
above.  On a compact set disjoint from any open neighborhood of that
curve its supremum is strictly negative.  The same remains true for
all sufficiently small $\sigma$: otherwise a convergent sequence with
$\sigma\downarrow0$ and rates approaching zero would give an equality
point outside that neighborhood.  Apply
\eqref{eq:compact-rate-bound}, with $s_{\min}=t_0$, and then choose the
second-moment tail rate in \textup{(i)} strictly negative.  This proves
\textup{(ii)}.

Near time zero, the restriction on $q$ supplies the positive variance
needed for the same argument. For \textup{(iii)}, choose $t_0$ so small that
$t_0/(1-j)<q_0/2$.  The set $q\ge q_0$ contains no equality point at
$\sigma=0$ and $0\le t\le t_0$.  On this set $s\ge jq_0>0$, so the
preceding compactness argument and the tail envelope apply without
change, including at $t=0$.

It remains to control the contribution of large coordinates in
\textup{(iv)}. Fix $t_0,\nu>0$, and take
$0<\lambda<1/[8(T+1+j)]$.  Let
$h_{R_1}(y)=y^2\one_{\{|y|>R_1\}}$.  On a fixed second-moment ball,
repeat the proof of \eqref{eq:compact-rate-bound}, this time inserting
$e^{\lambda\sum_i h_{R_1}(y_i)}$.  Begin with any small positive rate
bound $\varepsilon$ in that proof and its finite collection of tests
$\varphi_\xi$.  Uniformly for $|d|\le T+j$ and
$t_0\le s\le T+1+j$,
\[
 \int\bigl(e^{\lambda h_{R_1}(y)}-1\bigr)\,\dd N(d,s)(y)
 \longrightarrow0\qquad(R_1\to\infty),
\]
by Gaussian tails; the choice of $\lambda$ leaves a uniformly negative
quadratic exponent.  The same holds with the integrand multiplied by
any of the finitely many bounded $e^{\varphi_\xi}$. The additional
logarithmic moment can therefore be made arbitrarily
small in every member of the finite cover. For sufficiently large $R_1$,
the tilted integral on the ball is at most $e^{2\varepsilon n}$.  On
$\av{h_{R_1}(y)}>\nu$, remove the tilt at cost
$e^{-\lambda\nu n}$.  Choosing $2\varepsilon<\lambda\nu/2$ gives a
strictly negative rate there.  The integral off the ball is estimated
without the tilt, using \textup{(i)} with a larger negative rate.
This proves \textup{(iv)}.
\end{proof}

\subsection{The conditional Hessian}

The scalar bounds concentrate the integral near the equality curve.
We now show that, there, the conditional Hessian is positive
with an exponential probability bound. The conclusion is simultaneous
over all admissible nearby diagonal matrices, which will be needed to
exclude every unstable approximate zero.

\begin{lemma}[Conditional stability near the scalar equality curve]
\label{lem:conditional-field-stability}
Fix $0<t_0\le T$, $R<\infty$, and $\eta>0$ with
$j(1+\eta)^2<1$.  There are $c_*,\epsilon_*,a_*,\sigma_*>0$,
a neighborhood $\mathcal U$ of
$\{(P_t,t):t_0\le t\le T\}$, a tail tolerance $\nu>0$, and a finite
$R_1$ such that the following holds.  Suppose
\[
 \av{y^2}\le R^2,\qquad (P_y,t)\in\mathcal U,\qquad
 \av{y^2\one_{\{|y|>R_1\}}}\le\nu,
 \qquad t_0\le t\le T,\quad0<\sigma\le\sigma_*.
\]
Under the Gaussian conditioning in
\eqref{eq:conditional-integral}, with probability at least
$1-e^{-a_*n}$, simultaneously for all diagonal $A$ satisfying
\[
 0\le A\le(1+\eta)I,\qquad
 \norm{A-V_y}_{\Frob}\le\epsilon_*\sqrt n,
\]
one has $\mathsf H_{\wt J}(y,A)\succeq c_*I$.
All constants are uniform over the displayed ranges and sufficiently
large $n$.  The tail cutoff $R_1$ may be required to be arbitrarily
large after choosing $\nu$; $\epsilon_*$ is then decreased if
necessary.
\end{lemma}

\begin{proof}
We first compute the Gaussian regression and the finite-rank correction
at an equality law. We then show that the empirical-law and diagonal
hypotheses preserve its positive margin, uniformly over all admissible
diagonals.

\paragraph{Gaussian regression and whitening.}
In a sufficiently small neighborhood of the equality curve, $s$ and
$q$ have positive lower bounds depending on $j,t_0,T$.
Indeed $s_*(t)\ge t_0$, and
$q(P_t)=\E_{P_t}\tanh^2 y$ is positive and continuous there.  Set
\[
 u=\frac{m}{\sqrt{nq}},\qquad
 Z_1=\frac{y-d\one}{\sqrt{ns}},\qquad P_u=I-uu^{\mathsf T}.
\]
The norms of $\one/\sqrt n,u,Z_1$ are bounded uniformly by the assumed
second-moment bound.  Write
\[
 W=P_uWP_u+uw^{\mathsf T}+wu^{\mathsf T}+\alpha uu^{\mathsf T}.
\]
The three blocks in this decomposition are independent before
conditioning, with $\Cov(w)=(j/n)P_u$ and $\Var(\alpha)=2j/n$.
The projection of the
observation in \eqref{eq:conditional-integral} perpendicular to $u$
observes $\sqrt{nq}\,w$ with independent noise of covariance
$(t+\sigma^2)P_u$; projecting along $u$ observes
$\sqrt{nq}\,\alpha$ with noise variance $t+\sigma^2$.  The elementary
Gaussian conditional-mean and conditional-variance formulas therefore
give the following representation, using a fresh centered GOE $W_0$:
\begin{equation}
 \begin{split}
 P_uWP_u&=P_uW_0P_u,\\
 w&=\bar w+\kappa P_uW_0u,\qquad
 \bar w=j\sqrt{q/s}\,Z_1-\frac{ja_y}{s}u,\qquad
 \kappa=\sqrt{\frac{t+\sigma^2}{s}},\\
 \alpha&=\frac{2j(a_y+Bq)}S
       +\sqrt{\frac{t+\sigma^2}S}\,u^{\mathsf T}W_0u.
 \end{split}
 \label{eq:conditional-W}
\end{equation}
For example, the perpendicular conditional mean is
$j\sqrt{q/s}\,P_uZ_1$, which equals the displayed $\bar w$ because
$u^{\mathsf T}Z_1=a_y/\sqrt{qs}$.  The parallel conditional mean follows
from $u^{\mathsf T}z=\sqrt{n/q}(a_y+Bq)$.  The parallel residual
variance is $(2j/n)(t+\sigma^2)/S$. These conditional means and
residual variances verify all three blocks
of \eqref{eq:conditional-W}.

For the moment, fix a deterministic admissible $A$. We compare the
conditional Hessian with the positive matrix built from the fresh GOE
by setting
\[
 \chi=\frac1n\Tr A,\qquad B_A=j\chi,\qquad
 S_A=I+B_AA-A^{1/2}W_0A^{1/2},\qquad
 \mathcal Y(p)=S_A^{-1/2}A^{1/2}p.
\]
Lemma~\ref{lem:matrix-path-stability} gives $S_A\succeq c_0I$ except
with exponentially small probability, where $c_0>0$ is uniform in
$A$.  Let $G_A=A^{1/2}S_A^{-1}A^{1/2}$.  Lemma~\ref{lem:bilinear},
applied to the three deterministic vectors
$\one/\sqrt n,u,Z_1$, says that their Gram entries after $\mathcal Y$
are as close as desired to their $A$-weighted Gram entries, and that
the extra vector $\mathcal Y(W_0u)$ has Gram entries as close as desired
to those of
\begin{equation}
 B_A\mathcal Y(u)+\sqrt{B_A}\,N,
 \label{eq:extra-whitened-vector}
\end{equation}
where $N$ is a unit vector orthogonal to the three deterministic
images in this comparison.  Specifically, its mixed entries are
$B_Ap^{\mathsf T}Au$ and its squared norm is
$B_A+B_A^2u^{\mathsf T}Au$. Here $N$ describes only Gram entries;
this notation does not assert a coupling with an additional random or
deterministic vector. The comparison concerns finite Gram matrices; the
associated symmetric finite-rank eigenvalues depend continuously on their
entries, as we verify when returning to the empirical parameters. Each fixed
Gram tolerance has failure probability at most $Ce^{-a n}$ uniformly
over the bounded deterministic vectors and over $A$.  Also
$u^{\mathsf T}W_0u$ is smaller than any fixed tolerance with
exponentially high probability, since its variance is $2j/n$.

\paragraph{The correction at an equality law.}
We now compute the finite-rank perturbation in the ideal Gram comparison.
Set $\sigma=0$, take $s=s_*(t)$, and replace each weighted average
$n^{-1}\sum_i A_{ii}f(y_i)$ by
$\E_{N(s,s)}[\sech^2 y\,f(y)]$. Evaluate all other empirical
averages under the same law. Thus $a_y=sb$, $B_A=B=jb$, and
$\kappa^2=1-jq/s$. Write
\[
 X_1=\mathcal Y(\one/\sqrt n),\qquad
 U=\mathcal Y(u),\qquad Z_2=\mathcal Y(Z_1),
\]
using the Gram interpretation just described. Subtracting
$W_0$ from \eqref{eq:conditional-W} gives exactly
\begin{equation}
 \begin{split}
 W-W_0={}&u\bar w^{\mathsf T}+\bar w u^{\mathsf T}
 -(1-\kappa)\{u(W_0u)^{\mathsf T}+(W_0u)u^{\mathsf T}\}\\
 &+\{\alpha+(1-2\kappa)u^{\mathsf T}W_0u\}uu^{\mathsf T}.
 \end{split}
 \label{eq:conditional-rank-difference}
\end{equation}
At equality, $\bar w=j\sqrt{q/s}\,Z_1-Bu$ and $\alpha=2B$ after
discarding the vanishing Gaussian scalar.  Whiten by $S_A$, and use
\eqref{eq:extra-whitened-vector} with $B_A=B$.  The perturbation to
subtract from the identity consists of the planted term
$jX_1X_1^{\mathsf T}$, the term $2jqUU^{\mathsf T}$, and the whitened
version of \eqref{eq:conditional-rank-difference}.  The terms
$-2BUU^{\mathsf T}$ and $+2BUU^{\mathsf T}$ cancel.  Eliminating the
orthogonal direction $N$ by Schur complement adds
$(1-\kappa)^2BUU^{\mathsf T}$ to the perturbation.  Its total
$UU^{\mathsf T}$ coefficient becomes
\[
 2jq-2(1-\kappa)B+(1-\kappa)^2B
 =jq(2-B/s).
\]
Consequently the Schur complement is the identity minus
\begin{equation}
 j\{X_1X_1^{\mathsf T}+(2-B/s)M_1M_1^{\mathsf T}
                 +M_1G_1^{\mathsf T}+G_1M_1^{\mathsf T}\},
 \qquad M_1=\sqrt q\,U,\quad G_1=Z_2/\sqrt s.
 \label{eq:limiting-rank-perturbation}
\end{equation}

To prove positivity of this Schur complement, it suffices to examine
its three-vector Gram matrix. Put
$e=\E_{N(s,s)}[m^2(1-m^2)]$.  The Gram matrix of
$(X_1,M_1,G_1)$ equals
\begin{equation}
 H=\begin{pmatrix}
 b&e&-2e\\
 e&e&b-3e\\
 -2e&b-3e&b/s-2b+6e
 \end{pmatrix}.
 \label{eq:limiting-Gram}
\end{equation}
We derive the entries of this matrix explicitly. The density of
$N(s,s)$ satisfies
$p(y)/p(-y)=e^{2y}$.  Pairing $y$ and $-y$ shows that every integrable
odd function $f$ satisfies $\E f(y)=\E[f(y)\tanh y]$.  This gives
$\E[m(1-m^2)]=e$.  Gaussian integration by parts gives
\[
 \frac1s\E[(y-s)(1-m^2)]=-2e,\qquad
 \frac1s\E[(y-s)m(1-m^2)]=b-3e.
\]
Applying it twice, and using
$(1-m^2)''=4m^2(1-m^2)-2(1-m^2)^2$, gives the final diagonal entry
$b/s-2b+6e$.  The remaining entries are immediate, proving
\eqref{eq:limiting-Gram}.

The coefficient matrix in \eqref{eq:limiting-rank-perturbation} is
\[
 C=j\begin{pmatrix}1&0&0\\0&2-B/s&1\\0&1&0\end{pmatrix}.
\]
Set $d_0=1-jb$ and $g_1=je$.  The first and third columns of $I-HC$
are $(d_0,-g_1,2g_1)^{\mathsf T}$ and
$(-g_1,-g_1,d_0+3g_1)^{\mathsf T}$.  Adding $B/s$ times the third
column to the second makes the second column
$(0,d_0+g_1,0)^{\mathsf T}$.  Hence
\begin{equation}
 \det(I-HC)=(d_0+2g_1)(d_0+g_1)^2>0.
 \label{eq:positive-rank-determinant}
\end{equation}
The same determinant is obtained from the identity minus
\eqref{eq:limiting-rank-perturbation} on any ambient space containing
its three vectors. To determine its sign as a quadratic form, first
subtract only
$jX_1X_1^{\mathsf T}$; the result is positive definite because
$j\norm{X_1}^2=jb<1$.  The remaining two-vector perturbation has at
most one positive eigenvalue, since its coefficient block
$j\left(\begin{smallmatrix}2-B/s&1\\1&0\end{smallmatrix}\right)$
has one positive and one negative eigenvalue. The resulting matrix can
consequently have at most one nonpositive
direction. The positive determinant in
\eqref{eq:positive-rank-determinant} excludes both a zero eigenvalue
and a single negative eigenvalue. Hence the matrix is positive definite.
The Schur complement then also gives positivity before $N$ was eliminated.

\paragraph{Perturbing the scalar law and the diagonal.}
The equality calculation supplies the positive margin we need. We first
explain why the hypotheses put the actual weighted Gram entries near
these equality values.

Each deterministic vector has coordinates $n^{-1/2}$ times
one of $1,m/\sqrt q,(y-d)/\sqrt s$.  Products of two such coordinate
functions are bounded by $C(1+y^2)$.  For any such product $f$, the
tail and Frobenius bounds imply
\begin{equation}
 \left|\frac1n\sum_i(A_{ii}-(V_y)_{ii})f(y_i)\right|
 \le C_{R_1}\frac{\norm{A-V_y}_{\Frob}}{\sqrt n}+C\nu.
 \label{eq:A-moment-comparison}
\end{equation}
On $|y_i|\le R_1$, use Cauchy--Schwarz and boundedness of $f$; outside,
use $|A_{ii}-(V_y)_{ii}|\le2+\eta$ and the tail hypothesis, with
$R_1\ge1$.  The corresponding $V_y$-weighted functions are bounded
and continuous, even when multiplied by $y^2$, because
$\sech^2 y$ decays exponentially.  Their averages are therefore close
to the equality-law averages in a sufficiently small weak
neighborhood.  The parameters $d,s,q,a_y$ are continuous on the
second-moment ball, so their denominators and coefficients are close
as well.  Finally $|B_A-B|\le j\norm{A-V_y}_{\Frob}/\sqrt n$.

We retain the positivity uniformly for $t\in[t_0,T]$ and under
small Gram errors. If $Q$ has finitely many columns, the nonzero
eigenvalues of
$QCQ^{\mathsf T}$ agree with those of
$(Q^{\mathsf T}Q)^{1/2}C(Q^{\mathsf T}Q)^{1/2}$, including
multiplicity.  The largest eigenvalue, with zero adjoined, is therefore
a continuous function of the Gram matrix and the coefficients.
All equality parameters vary continuously on the compact interval
$[t_0,T]$, and $s,q$ stay positive there.  The exact calculation thus
gives a uniform positive whitened margin.  Choose a fixed tolerance
$\xi>0$ on coefficients and Gram entries small enough to preserve,
say, half of it.  Lemmas~\ref{lem:matrix-path-stability}
and~\ref{lem:bilinear}, the Gaussian scalar estimate, and
\eqref{eq:A-moment-comparison} show that this tolerance is achieved
for each fixed admissible $A$ with failure probability at most
$Ce^{-a n}$, after choosing the scalar neighborhood, tail tolerance,
$\sigma_*$, and Frobenius distance sufficiently small.  Returning
from whitening multiplies the margin by at least $c_0$.
The omitted mass $(B-B_A)A$ has norm at most
$j(1+\eta)\norm{A-V_y}_{\Frob}/\sqrt n$ and can be absorbed by
decreasing that distance again.

At this stage the margin is proved for each fixed $A$. To make it
simultaneous, we use that its failure exponent is uniform over the
bounded deterministic vector norms. Those bounds were fixed before
choosing the Frobenius distance.  Also \eqref{eq:conditional-W} implies
$\norm W\le K_1$ with failure $Ce^{-a_1n}$ for some uniform $K_1$:
the norms of the deterministic means are bounded, and the remaining
terms are controlled by $\norm{W_0}$.
On this event, $A\mapsto\mathsf H_{\wt J}(y,A)$ is uniformly
$1/2$-H\"older in diagonal sup norm.  Indeed
$\norm{A^{1/2}-\widehat A^{1/2}}\le
\norm{A-\widehat A}^{1/2}$, while the middle factor in
\eqref{eq:field-hessian} has bounded norm.

Choose a fixed mesh size $\zeta>0$ small enough that the H\"older
error is below one quarter of the margin.  Given $A$ with normalized
Frobenius distance at most $\epsilon$ from $V_y$, let
$\mathcal S_A=\{i:|A_{ii}-(V_y)_{ii}|>\zeta\}$.  It has at most
$\alpha n$ elements, where $\alpha=\epsilon^2/\zeta^2$.
Outside $\mathcal S_A$ replace $A_{ii}$ by $(V_y)_{ii}$; on $\mathcal S_A$, round it
within $\zeta$ to a grid of $[0,1+\eta]$, including the endpoints.
The resulting $\widehat A$ obeys
\[
 \norm{A-\widehat A}\le\zeta,\qquad
 \norm{\widehat A-V_y}_{\Frob}\le2\epsilon\sqrt n.
\]
For fixed $y$, these approximants are deterministic and number at
most
\[
 \sum_{k\le\alpha n}\binom nk N_\zeta^k,
 \qquad N_\zeta\le C(1+1/\zeta).
\]
Its logarithm divided by $n$ tends to zero as $\alpha\downarrow0$;
for instance it is at most
$-\alpha\log\alpha-(1-\alpha)\log(1-\alpha)
 +\alpha\log N_\zeta+o(1)$ when $\alpha<1/2$.
Choose $\epsilon_*$ so small that $2\epsilon_*$ meets all moment
tolerances and this counting rate is less than half the fixed-$A$
failure exponent.  A union bound, followed by the H\"older comparison,
proves the desired uniform margin.

For later use, record the order that preserves the concentration
exponents. First fix $R,t_0,T,\eta$,
the limiting margin, and the Gram tolerance $\xi$.  This fixes the
vector norm bounds, conditional concentration exponents, $K_1$, and
the mesh $\zeta$.  Next take the equality neighborhood and tail
tolerance $\nu$ small.  Choose any sufficiently large finite $R_1$,
then take $\epsilon_*$ small enough for
\eqref{eq:A-moment-comparison} and the union bound.  Finally decrease
$\sigma_*$ so all scalar coefficients lie within the required
tolerance.  No concentration exponent is lost when $\epsilon_*$ is
decreased. This also proves the last assertion of the lemma.
\end{proof}

\subsection{From a small integral to exclusion of approximate zeros}

The conditional estimate controls all nearby diagonals for each fixed
$y$ satisfying the scalar hypotheses. Combined with the scalar rate
bounds, it will make the expected integral over unstable configurations
exponentially small. To exclude approximate zeros at any $y$, the next
deterministic lemma gives the complementary lower bound: an approximate
zero satisfying a regularized determinant bound contributes a definite
exponential amount to the integral over a surrounding ball. The
regularization makes a sign assumption on the Jacobian unnecessary.

\begin{lemma}[Local volume at an approximate zero]
\label{lem:local-field-volume}
Fix $j,K<\infty$, $r>0$, and $\varepsilon>0$.  Suppose
$\norm{\wt J}\le K$, and a set $E\subseteq\R^n$ contains the ball
$\{y':\norm{y'-y}\le r\sqrt n\}$.  Put
\[
 L=F'_{\wt J,h_0}(y)
 =I+(BI-\wt J)V_y-\frac{2j}{n}mm^{\mathsf T}V_y,\qquad
 Q=L^{\mathsf T}L+\gamma I.
\]
If
\begin{equation}
 \frac1n\log\det Q\le jb(y)^2+\varepsilon_1,\qquad
 \norm{F_{\wt J,h_0}(y)}\le\rho\sqrt n,
 \label{eq:local-volume-hypotheses}
\end{equation}
then
\begin{equation}
 (2\pi\sigma^2)^{-n/2}\int_E
 e^{-\norm{F_{\wt J,h_0}(y')}^2/(2\sigma^2)+njb(y')^2/2}\,\dd y'
 \ge e^{-\varepsilon n}
 \label{eq:local-volume-conclusion}
\end{equation}
for all sufficiently large $n$, provided the constants are chosen in
this order: first $\varepsilon_1>0$ sufficiently small in terms of
$\varepsilon$; then any fixed $\gamma>0$; then $\sigma>0$
sufficiently small in terms of $j,K,r,\varepsilon,\gamma$; and finally
$\rho>0$ sufficiently small in terms of those constants and $\sigma$.
The bound is uniform in $y,h_0,n$ under the stated hypotheses.
\end{lemma}

\begin{proof}
We integrate over a Gaussian ellipsoid centered at the approximate
zero. Its covariance is chosen to control the linearized residual.
Let $g''$ be standard Gaussian and put
$\Delta=\sigma Q^{-1/2}g''$, and restrict to
$\mathcal B=\{\norm{g''}\le2\sqrt n\}$.  For
$2\sigma/\sqrt\gamma\le r$, all resulting points $y+\Delta$ belong
to $E$.  Changing variables in the integral gives the lower bound
\begin{equation}
 (\det Q)^{-1/2}\Prob(\mathcal B)
 \E\left[\exp\left\{
 \frac{\norm{g''}^2}{2}
 -\frac{\norm{F(y+\Delta)}^2}{2\sigma^2}
 +\frac{njb(y+\Delta)^2}{2}\right\}\,\middle|\,\mathcal B\right].
 \label{eq:volume-change-variables}
\end{equation}
Here and for the remainder of this proof $F=F_{\wt J,h_0}$.
The conditioning has probability tending to one, and all estimates
below remain valid under it with a uniform factor, since
$\Prob(\mathcal B)\ge1/2$ for large $n$.

We must control the nonlinear residual on this ellipsoid uniformly
in its center. Coordinatewise Taylor expansion gives
\[
 m(y+\Delta)=m+V_y\Delta+r_m,\qquad
 |(r_m)_i|\le C\Delta_i^2.
\]
Writing $b(y+\Delta)=b+\ell_b+r_b$, its linear part is
$\ell_b=-2\av{m(1-m^2)\Delta}$, and
$|r_b|\le C\norm\Delta^2/n$.  Thus, when the product $jb,m$ in
$F$ is expanded, its remainder is bounded in norm by
\[
 C\left(\norm{r_m}+\frac{\norm\Delta^2}{\sqrt n}\right).
\]
For the product of first differences use
$|b(y+\Delta)-b(y)|\le C\norm\Delta/\sqrt n$ and
$\norm{m(y+\Delta)-m(y)}\le\norm\Delta$.  The remaining term
$-\wt Jm$ has remainder bounded by $K\norm{r_m}$.
Since the covariance of $\Delta$ is bounded above by
$\sigma^2I/\gamma$, Gaussian fourth moments give
\[
 \E\sum_i\Delta_i^4\le Cn\sigma^4/\gamma^2,\qquad
 \E\norm\Delta^4\le Cn^2\sigma^4/\gamma^2.
\]
Combining these Taylor estimates with the Gaussian moments, and writing
$R_F=F(y+\Delta)-F(y)-L\Delta$, yields
\begin{equation}
 \E[\norm{R_F}^2\mid\mathcal B]
 \le Cn\sigma^4/\gamma^2,\qquad
 \E[|b(y+\Delta)^2-b(y)^2|\mid\mathcal B]
 \le C\sigma/\sqrt\gamma.
 \label{eq:volume-remainders}
\end{equation}
All constants depend only on $j,K$.

The choice of ellipsoid now controls the linear term: the singular
values of $LQ^{-1/2}$ are at most one. Hence
$\norm{L\Delta}\le\sigma\norm{g''}$ pointwise, including on
$\mathcal B$.  Expanding
$F(y+\Delta)=F(y)+L\Delta+R_F$ and applying Cauchy--Schwarz to its
cross terms yields
\begin{align*}
 &\frac1n\E\left[
 \frac{\norm{g''}^2}{2}
 -\frac{\norm{F(y+\Delta)}^2}{2\sigma^2}
 \,\middle|\,\mathcal B\right]\\
 &\hspace{1cm}\ge
 -C\left\{
 \frac\rho\sigma+\frac\sigma\gamma
 +\left(\frac\rho\sigma\right)^2
 +\left(\frac\sigma\gamma\right)^2
 +\frac\rho\gamma\right\}.
\end{align*}
The last mixed term may also be absorbed into the two squares.
Apply Jensen's inequality to the conditional expectation in
\eqref{eq:volume-change-variables}.  Its logarithm divided by $n$ is
at least
\[
 -\frac{1}{2n}\log\det Q+\frac{jb(y)^2}{2}
 -C\left\{\frac\rho\sigma+\frac\sigma\gamma
          +\left(\frac\rho\sigma\right)^2
          +\left(\frac\sigma\gamma\right)^2
          +\frac\sigma{\sqrt\gamma}\right\}+o(1).
\]
The determinant hypothesis cancels $jb(y)^2/2$, with a loss of at
most $\varepsilon_1/2$. Choose the constants in the order in the
statement; the other errors then have total size smaller than
$\varepsilon$. This proves the lower bound.
\end{proof}

\subsection{Proof of the observation-path statement}

\begin{proof}[Proof of Proposition~\ref{prop:stable-fields}]
We assemble the scalar, matrix, and local-volume estimates under the
planted law \eqref{eq:planted-field}. After excluding unstable
approximate zeros at a fixed time, we remove the auxiliary diagonal,
control the whole path by a fixed mesh, and undo the planting.

Let $J$ be $\wt J$ with its diagonal zeroed. Choose
$\eta,\theta,q_0>0$ sufficiently small that
$j(1+\eta)^2<1$ and, whenever $q\le2q_0$,
\begin{equation}
 I+(j-3jq-2\sqrt j-2\theta)A\succeq c_{\mathrm{sm}}I
 \qquad(0\le A\le(1+\eta)I)
 \label{eq:small-time-matrix-margin}
\end{equation}
for some $c_{\mathrm{sm}}>0$.  These choices are possible because at
$\eta=\theta=q=0$ the lower scalar bound is
$1+j-2\sqrt j=(1-\sqrt j)^2>0$.  On
\begin{equation}
 \norm J\le2\sqrt j+\theta,\qquad
 \max_i|\wt J_{ii}|\le\theta,
 \label{eq:planted-norm-restriction}
\end{equation}
we have $\norm{\wt J}\le2\sqrt j+2\theta$.  Since
$mm^{\mathsf T}/n\preceq qI$, \eqref{eq:small-time-matrix-margin}
implies $\mathsf H_{\wt J}(y,A)\succeq c_{\mathrm{sm}}I$ for all
$q\le2q_0$.  Choose $t_0\in(0,T]$ small enough for
Lemma~\ref{lem:scalar-rates}\textup{(iii)}.

For the remaining times $t\in[t_0,T]$, the conditional Hessian lemma
provides the margin near the equality curve. Choose a second-moment
ball whose complement has a fixed negative rate, using
Lemma~\ref{lem:scalar-rates}\textup{(i)} with $s_{\min}=t_0$.
Apply Lemma~\ref{lem:conditional-field-stability} on this ball.  Take
its equality neighborhood and tail tolerance, choose its tail cutoff
$R_1$ sufficiently large for
Lemma~\ref{lem:scalar-rates}\textup{(iv)}, and then choose its
diagonal-distance radius.  Decrease $\sigma_*>0$ so that
Lemma~\ref{lem:scalar-rates}\textup{(ii)} excludes the complement of
that equality neighborhood, and so that the small-time scalar bound
also holds. The scalar regions excluded by these choices all have fixed
strictly
negative rates, uniformly for $0<\sigma\le\sigma_*$. On the region
left over, the conditional matrix margin fails with probability at
most $e^{-a_*n}$.

Let $\epsilon_*>0$ denote the full diagonal-distance radius just chosen.
Choose $c_E>0$ smaller than both $c_*/2$ and $c_{\mathrm{sm}}/2$.
For $t\ge t_0$, define
\[
 E_t(W)=\left\{y:\exists A,\quad
 0\le A\le(1+\eta)I,\quad
 \norm{A-V_y}_{\Frob}\le\epsilon_*\sqrt n,\quad
 \lambda_{\min}(\mathsf H_{\wt J}(y,A))\le c_E\right\}.
\]
For $0\le t<t_0$, instead put $E_t(W)=\{y:q(y)\ge q_0\}$.
There is $a_0>0$ such that
\begin{equation}
 \E I_{t,\sigma}(E_t)\le e^{-a_0n}
 \quad(0\le t\le T, 0<\sigma\le\sigma_*)
 \label{eq:bad-integral-rate}
\end{equation}
for all sufficiently large $n$, uniformly in $t,\sigma$.
For small times, this follows from the scalar bound and
Lemma~\ref{lem:gaussian-reduction}. For larger times, split the
conditional integral into the excluded scalar regions and their complement.
The excluded regions have negative rates.  On the complement multiply
the conditional probability $e^{-a_*n}$ by the unrestricted scalar
integral, whose rate is at most any prescribed positive number, by
Lemma~\ref{lem:scalar-rates}\textup{(i)}.  Taking that number below
$a_*/2$ proves \eqref{eq:bad-integral-rate}.  Finitely many pieces can
be absorbed by decreasing $a_0$.

We now use local volume to exclude every unstable approximate zero.
An individual witness will force a ball of the excluded set to contribute
too much to the integral. Suppose
\begin{equation}
 \norm{F_{\wt J,h_0}(y)}\le\rho\sqrt n,\qquad
 \norm{A-V_y}_{\Frob}\le\tfrac12\epsilon_*\sqrt n,\qquad
 \lambda_{\min}(\mathsf H_{\wt J}(y,A))\le\tfrac12c_E
 \label{eq:bad-approximate-point}
\end{equation}
for some $0\le A\le(1+\eta)I$, and assume
\eqref{eq:planted-norm-restriction}.  There is a constant $r>0$,
independent of this witness and of $n,t$, such that
$\norm{y'-y}\le r\sqrt n$ implies $y'\in E_t(W)$.
For $t\ge t_0$, the same $A$ witnesses instability throughout a
sufficiently small ball. Indeed,
\[
 \norm{V_{y'}-V_y}_{\Frob}\le2\norm{y'-y},\qquad
 |b(y')-b(y)|\le2\norm{y'-y}/\sqrt n,
\]
and
\[
 \norm{m(y')m(y')^{\mathsf T}/n-m(y)m(y)^{\mathsf T}/n}
 \le2\norm{y'-y}/\sqrt n
\]
preserve the full diagonal-distance constraint and the bound
$\lambda_{\min}\le c_E$ if $r$ is small enough.  For $t<t_0$,
\eqref{eq:small-time-matrix-margin} and the chosen $c_E$ imply
$q(y)>2q_0$; the same Lipschitz bound for $q$ then gives
$q(y')\ge q_0$ for a smaller $r$.

Set $L=F'_{\wt J,h_0}(y)$.  It is a bounded-rank, bounded-norm
perturbation of
\[
 I+(BI-W)V_y:
 \quad L=I+(BI-W)V_y
       -\frac jn\one\one^{\mathsf T}V_y
       -\frac{2j}{n}mm^{\mathsf T}V_y.
\]
Here $n^{-1}\Tr V_y=b(y)$, so the determinant rate in
Lemma~\ref{lem:logdet} matches the weight in \eqref{eq:field-integral}.
That lemma says that, for any chosen $\varepsilon_1>0$, one may choose
a fixed $\gamma>0$ sufficiently
small so that, except on an event of probability $Ce^{-a_1n}$,
simultaneously at every $y$,
\[
 \frac1n\log\det(L^{\mathsf T}L+\gamma I)
 \le jb(y)^2+\varepsilon_1.
\]
The lemma's rank-perturbation allowance applies here with two rank-one
terms of norms at most $j$ and $2j$.
Choose $\varepsilon<a_0/2$, choose $\varepsilon_1$ for
Lemma~\ref{lem:local-field-volume}, then choose $\gamma$ by the
log-determinant lemma.  Next fix $\sigma\in(0,\sigma_*]$ sufficiently
small for the local-volume lemma, and finally fix $\rho>0$
sufficiently small for it.  A witness
\eqref{eq:bad-approximate-point} on the determinant event forces
$I_{t,\sigma}(E_t)\ge e^{-\varepsilon n}$.  Markov's inequality and
\eqref{eq:bad-integral-rate} show that the probability of such a
witness, under \eqref{eq:planted-norm-restriction}, is at most
$Ce^{-a_2n}$, uniformly in $t\in[0,T]$. Here the uniformity of the
negative rates permits the later choice of
$\sigma$ required by the local-volume estimate.

The integral argument used a full GOE matrix. We now pass to its
zero-diagonal version, retaining slack in both residual and Hessian bounds.
Choose a fixed
$\delta_D>0$ so small that
\[
 \delta_D\le\theta,\qquad
 \delta_D\le\rho/2,\qquad
 (1+\eta)\delta_D\le c_E/4.
\]
The diagonal entries of $\wt J$ are independent
$N(j/n,2j/n)$, so
$\Prob(\max_i|\wt J_{ii}|>\delta_D)\le Ce^{-a_Dn}$ for large $n$.
On its complement,
\[
 \frac{\norm{F_{\wt J,h_0}(y)-F_{J,h_0}(y)}}{\sqrt n}
 \le\delta_D,\qquad
 \norm{\mathsf H_{\wt J}(y,A)-\mathsf H_J(y,A)}
 \le(1+\eta)\delta_D.
\]
Thus a zero-diagonal witness with residual at most
$\rho\sqrt n/2$, radius $\epsilon_*\sqrt n/2$, and Hessian minimum
at most $c_E/4$ would give a witness
\eqref{eq:bad-approximate-point}.  We have proved an exponentially
small fixed-time failure probability for the zero-diagonal matrix,
restricted only to $\norm J\le2\sqrt j+\theta$.

To pass from a fixed time to the whole interval, we control observation
increments on a deterministic mesh. Take a mesh of size $\Delta>0$
that includes both endpoints.  For a scalar
Brownian motion, the reflection bound implies that
$\sup_{0\le u\le\Delta}|B(u)|^2/\Delta$ has an exponential moment
at a fixed positive argument.  The coordinate Brownian motions are
independent.  Exponential Markov applied to the sum of their squared
suprema consequently gives, for each desired $\kappa_0>0$, a mesh
$\Delta$ small enough that
\[
 \Prob\left(\max_k\sup_{t_k\le t\le t_{k+1}}
       \frac{\norm{B_0(t)-B_0(t_k)}}{\sqrt n}>\kappa_0\right)
 \le Ce^{-a_3n}.
\]
The deterministic drift contributes at most $\Delta\sqrt n$ to each
increment.  Choose the mesh so the full normalized increment is at
most $\rho/4$.  Set $\rho_0=\rho/8$,
$\epsilon_A=\epsilon_*/2$, and $c=c_E/4$.  A pathwise violation of
\eqref{eq:stable-field} then gives a fixed-time violation at a mesh
point: the Hessian and its witness do not use the field, and the
residual increases by at most $\rho\sqrt n/4$.  A union bound over
the fixed finite mesh proves an annealed path failure bound
$Ce^{-a_4n}$ under the planted law, restricted to
$\norm J\le2\sqrt j+\theta$.

We have obtained a path estimate under planting. To recover the
original disorder law, introduce the partition-function density
\[
 Z(J)=\sum_x e^{x^{\mathsf T}Jx/2},\qquad
 L(J)=Z(J)/\E Z(J).
\]
Size-bias the original joint law of $J,X$, with $X\sim\mu_0$, by
$L(J)$.  Its density in $J,X$ is proportional to
$e^{x^{\mathsf T}Jx/2}$ times the original Gaussian density.
Gaussian tilting makes $X$ uniform and, conditionally on $X$, makes
the off-diagonal entries of $J$ independent with means
$jX_iX_j/n$ and variance $j/n$.  Add independent diagonal entries with law $N(j/n,2j/n)$---a centered
GOE diagonal plus $jI/n$---and conjugate by $D_X$; the resulting matrix and observation have exactly
the law \eqref{eq:planted-field}.  All witness conditions are
equivariant under this conjugation, because $m(D_Xy)=D_Xm(y)$,
$V_{D_Xy}=V_y$, and diagonal $A$ commutes with $D_X$.  Conditional
observation paths given $J$ are unchanged by size bias.

Undoing this change of law requires a lower bound for $L(J)$ on
typical original matrices. For independent uniform signs $\varepsilon_i$,
Gaussian averaging gives
\[
 \frac{\E Z^2}{(\E Z)^2}
 =e^{-j/2}\E\exp\left\{\frac j{2n}
                       \left(\sum_i\varepsilon_i\right)^2\right\}
 \le(1-j)^{-1/2}.
\]
To obtain the inequality, introduce an independent standard Gaussian
$g$ and use
$\E_\varepsilon e^{\sqrt{j/n}g\sum_i\varepsilon_i}
 =\cosh(\sqrt{j/n}g)^n\le e^{jg^2/2}$.
Paley--Zygmund then yields a constant $p_j>0$ with
$\Prob(Z\ge\E Z/2)\ge p_j$.  As a function of the independent
standard off-diagonal Gaussians, $\log Z$ has gradient squared at
most $(j/n)\binom n2\le jn/2$.  Gaussian concentration about a median
therefore has scale $O(\sqrt n)$.  The preceding positive-probability
lower bound implies that this median is at least
$\log\E Z-O_j(\sqrt n)$: otherwise its upper concentration tail
would be smaller than $p_j$.  It follows that, for every fixed
$\delta>0$,
\begin{equation}
 \Prob\{L(J)<e^{-\delta n}\}\longrightarrow0.
 \label{eq:partition-lower-tail}
\end{equation}

We can now turn the annealed planted estimate into the required
conditional observation bound. Let $p_n(J)$ denote the conditional
observation-path failure probability for \eqref{eq:stable-field}.
The planted estimate is
\[
 \E\bigl[L(J)p_n(J)
       \one_{\{\norm J\le2\sqrt j+\theta\}}\bigr]
 \le Ce^{-a_4n}.
\]
Choose $\delta<a_4/2$.  On $L(J)\ge e^{-\delta n}$, this gives an
original expectation bound $Ce^{-(a_4-\delta)n}$ for the restricted
$p_n(J)$.  Markov's inequality makes $p_n(J)\le e^{-an}$ outside an
event of probability tending to zero, for a fixed sufficiently small
$a>0$.  Equation~\eqref{eq:partition-lower-tail} and the usual GOE norm
bound, including diagonal removal, show that the restrictions on
$L(J)$ and $\norm J$ hold with probability tending to one.  Intersect
these events to obtain $\mathcal J_n$ and finish the proof.

We finish by checking measurability of the witness events used in the
argument. At fixed $y$, the matrix-witness sets are projections over
the compact diagonal range.  For existence
of $y,A,t$ along a continuous path, additionally restrict
$\norm y$, $\norm J$, and $\sup_{t\le T}\norm{Y(t)}$ by integers.
The witness inequalities are closed and the witness parameter space
is compact, so these restricted projections are closed.  Taking a
countable union removes the bounds.  This also justifies the
conditional probabilities and the restricted integrals above.
\end{proof}

\subsection{A unique root and quantitative inversion}

The preceding proposition supplies stability along random observations
at fixed subcritical $j$. The following consequence is deterministic:
its hypothesis requires positivity throughout the region of small
residual. From that whole-region assumption we obtain a unique zero
and a bound on the distance to it. This separates the probabilistic
construction of stable fields from the inversion used later.

\begin{lemma}[Root and inverse bound at a stable field]
\label{lem:root}
Fix $j,K,c,r_0>0$.  Let $J$ be symmetric with $\norm J\le K$, and
let $h_0\in\R^n$.  Suppose
\begin{equation}
 \mathsf H_J(y,V_y)\succeq cI
 \quad\text{whenever}\quad
 \norm{F_{J,h_0}(y)}\le r_0\sqrt n.
 \label{eq:root-stability-assumption}
\end{equation}
Then $F_{J,h_0}$ has exactly one zero $r\in\R^n$, and
\begin{equation}
 \norm{y-r}\le C\norm{F_{J,h_0}(y)}
 \quad\text{if}\quad\norm{F_{J,h_0}(y)}<r_0\sqrt n,
 \qquad C=1+(K+3j)/c.
 \label{eq:root-distance}
\end{equation}
In particular this conclusion applies to every field satisfying
\eqref{eq:stable-field}, even if that implication is required only
through residual threshold $\rho_0\sqrt n$.
\end{lemma}

\begin{proof}
We first bound the derivative inverse on the region in the hypothesis.
We then prove existence and uniqueness of a zero, and integrate the
inverse bound along a path to that zero.

Put $L_1=BI-J-2jmm^{\mathsf T}/n$. Since
$F'(y)=I+L_1V_y$, the elementary inverse identity gives
\[
 F'(y)^{-1}
 =I-L_1V_y^{1/2}
       (I+V_y^{1/2}L_1V_y^{1/2})^{-1}V_y^{1/2}.
\]
Under \eqref{eq:root-stability-assumption}, this inverse exists and
has norm at most $1+(K+3j)/c$, since $0<V_y\le I$ and
$\norm{L_1}\le K+3j$.

For existence and uniqueness, pass to the coordinates
$m\in(-1,1)^n$ and define the potential
\[
 \Phi(m)=\sum_i\int_0^{m_i}\atanh z\,\dd z-h_0^{\mathsf T}m
 -\frac12m^{\mathsf T}Jm+\frac j2\norm m^2
 -\frac j{4n}\norm m^4.
\]
Its critical-point equation is
$F_{J,h_0}(\atanh m)=0$.  At any critical point,
\[
 V_y^{1/2}\Phi''(m)V_y^{1/2}
 =\mathsf H_J(y,V_y)\succeq cI,\qquad y=\atanh m,
\]
so the critical point has positive definite Hessian and is isolated.
Choose $a\in(0,1)$ close enough to one that
\[
 \atanh a>1+\max_i\left(|(h_0)_i|+\sum_j|J_{ij}|+j\right).
\]
Every critical point lies in the interior of $[-a,a]^n$, and on each
face of this box the negative gradient $-\nabla\Phi$ points strictly
inward.  The minimum of $\Phi$ on the box is therefore attained at an
interior critical point. This proves existence.

To prove uniqueness, consider the negative gradient flow from any
point of the box. The inward-pointing condition keeps the trajectory
in the box for all positive times.  Its potential decreases and is bounded
below, so
$\int_0^\infty\norm{\nabla\Phi(m(t))}^2\,\dd t<\infty$.
The squared gradient is uniformly continuous in time, because the
gradient and Hessian are bounded on the box.  It follows that the
gradient tends to zero: a nonzero limsup would, by uniform
continuity, force disjoint intervals of fixed positive integral.
Every limit point of a trajectory is therefore critical.
The critical set is finite, since it is a compact set of isolated
zeros.  The limit set of a trajectory is connected: it is the
intersection of the nested compact connected closures of its tails.
Each trajectory therefore converges to one critical point.

It remains to show that all trajectories converge to the same point.
The positive Hessian makes every critical point a strict local
attractor. For clarity, in a sufficiently small ball about it one has
$(m-m_*)^{\mathsf T}\nabla\Phi(m)\ge c_*\norm{m-m_*}^2$,
which gives exponential contraction of the distance along the flow
and invariance of the ball.  Continuous dependence of the flow on
its starting point, up to the time it enters that ball, shows that
each attraction basin is relatively open in $[-a,a]^n$.  These
disjoint nonempty basins cover the connected box.  Hence there is
only one basin and one critical point, proving uniqueness of $r$.

We now derive the distance estimate. Suppose
$\norm{F(y)}<r_0\sqrt n$. Starting at $y(0)=y$, solve on the
open region $\norm{F}<r_0\sqrt n$ the ordinary differential equation
\[
 \dot y(s)=-F'(y(s))^{-1}F(y).
\]
The chain rule gives $F(y(s))=(1-s)F(y)$ while the solution exists.
Its velocity has norm at most $C\norm{F(y)}$.  If its maximal
existence interval ended at or before time one, this velocity bound would
give a finite limiting point, whose residual is still strictly less
than $r_0\sqrt n$ and whose derivative is invertible.  Local existence
would extend the solution, a contradiction. The solution consequently
reaches time one, at which its value is the
unique zero $r$. Integrating the velocity bound along this path gives
\eqref{eq:root-distance}.
\end{proof}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//stability.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//style.tex ====
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\hypersetup{pdftitle={A spectral gap throughout the high-temperature Sherrington--Kirkpatrick phase},pdfauthor={OpenAI}}

==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//style.tex ====

==== BEGIN A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//terminal.tex ====
\section{The spectral gap at a large observation time}
\label{sec:terminal}

We complete the observation argument by proving a gap after a sufficiently
large fixed observation time. The deterministic two-spin curvature estimate
is uniform over external fields. The final probabilistic statement instead
concerns the specified observation law, at fixed $j<1$.
We adapt the signed two-spin method of Wang~\cite[Lemma~3.4]{Wang2026},
retaining the signed first-order interaction and estimating the remaining
terms by an entrywise-square matrix and a quartic row error. The complete
calculation below proves the precise variant used here.

For a symmetric matrix $J$ with zero diagonal, define the symmetric matrices
\[
 U_{ij}=\tanh J_{ij},\qquad Q_{ij}=U_{ij}^2,\qquad
 r_4(U)=\max_i\sum_j U_{ij}^4.
\]
In particular, $U_{ii}=Q_{ii}=0$.  The square defining $Q$ is entrywise.
At a field $h_0$, put
\[
 Hf=\sum_i(I-P_i)f
    =\sum_i(x_i-m_i^1)d_i f,\qquad
 g_i=d_i f,\qquad a_i=v_i^1g_i.
\]
Each $P_i$ is an orthogonal projection in $L^2(\mu_{h_0})$, and therefore
$H$ is self-adjoint and positive, with
\begin{equation}
 \E_{\mu_{h_0}}[fHf]=\D_{h_0}(f).
 \label{eq:terminal-generator-form}
\end{equation}

\begin{lemma}[Signed curvature inequality]
\label{lem:curvature}
There are universal constants $u_0>0$ and $C<\infty$ such that, for every
$n$, every symmetric zero-diagonal $J$ satisfying
$\max_{i,j}|U_{ij}|\le u_0$, every $h_0\in\R^n$, and every real function $f$
on the discrete cube,
\begin{equation}
 \E_{\mu_{h_0}}[(Hf)^2]
 \ge (1-Cr_4(U))\D_{h_0}(f)
       -\E_{\mu_{h_0}}\bigl[a^{\mathsf T}Ua
                         +C|a|^{\mathsf T}Q|a|\bigr].
 \label{eq:terminal-curvature}
\end{equation}
Here $|a|$ denotes coordinatewise absolute value.
\end{lemma}

\begin{proof}
The curvature bound is obtained by summing an exact two-spin calculation.
Fix $i\ne j$, condition on the other spins, and call the remaining spins
$x,y$. Their conditional density is proportional to
$\exp(\alpha x+\gamma y+J_{ij}xy)$. Introduce the scalar parameters
\[
 a=\tanh\alpha,\quad b=\tanh\gamma,\quad
 A=1-a^2,\quad B=1-b^2,\quad u=\tanh J_{ij}.
\]
The letters $a,b,A,B$ have these scalar meanings only during the pair
calculation. Use $\E_0$ for the product law of means $a,b$, and
$\E_{ij}$ for the interacting conditional law. The identity
$e^{J_{ij}xy}=\cosh(J_{ij})(1+uxy)$ gives their exact density ratio:
\begin{equation}
 \frac{\dd\mu_{ij}}{\dd\mu_{0,ij}}(x,y)
       =\frac{1+uxy}{1+uab}.
 \label{eq:pair-density}
\end{equation}
Expand $f$ in the four-dimensional space of functions of $x,y$. Its
two gradients then have the unique representation
\[
 g_i=p+r(y-b),\qquad g_j=s+r(x-a)
\]
with real $p,s,r$. The shared coefficient $r$ is the mixed-spin term
of this expansion. The conditional means and variances satisfy
\begin{align}
 x-m_i^1&=\frac{(x-a)(1-uxy)}{1+uay},&
 v_i^1&=\frac{A(1-u^2)}{(1+uay)^2},\notag\\
 y-m_j^1&=\frac{(y-b)(1-uxy)}{1+ubx},&
 v_j^1&=\frac{B(1-u^2)}{(1+ubx)^2}.
 \label{eq:pair-conditional-identities}
\end{align}

For a function $F$ of one spin, write $F(z)=\overline F+z\,dF$, and set
\[
 F_1(y)=\frac{g_i(y)}{1+uay},\qquad
 F_2(x)=\frac{g_j(x)}{1+ubx}.
\]
The identity $(1-uxy)^2(1+uxy)=(1-u^2)(1-uxy)$ and independence under
$\E_0$ give the following exact expression for the cross term:
\begin{equation}
 \frac{\E_{ij}[(x-m_i^1)g_i\,(y-m_j^1)g_j]}{AB}
 =\frac{1-u^2}{1+uab}
       \bigl(dF_1dF_2-u\overline F_1\overline F_2\bigr).
 \label{eq:pair-cross-exact}
\end{equation}
Both terms follow from the one-spin identities
$\E_0[(x-a)F_2(x)]=A\,dF_2$ and
$\E_0[(x-a)xF_2(x)]=A\overline F_2$, together with their $y$
counterparts. Thus the cross term has been reduced to four scalar
coefficients.

The four coefficients appearing here are
\begin{align*}
 dF_1&=\frac{r(1+uab)-uap}{1-u^2a^2},&
 \overline F_1&=\frac{p-(b+ua)r}{1-u^2a^2},\\
 dF_2&=\frac{r(1+uab)-ubs}{1-u^2b^2},&
 \overline F_2&=\frac{s-(a+ub)r}{1-u^2b^2}.
\end{align*}
Consequently the right-hand side of \eqref{eq:pair-cross-exact} is
\begin{equation}
 K\bigl[(-u+u^2ab)ps+u^2bA\,rp+u^2aB\,rs+Rr^2\bigr],
 \label{eq:pair-cross-coefficients}
\end{equation}
where
\begin{align*}
 K&=\frac{1-u^2}{(1+uab)(1-u^2a^2)(1-u^2b^2)},\\
 R&=(1+uab)^2-u(b+ua)(a+ub)\\
  &=1+uab+u^2(a^2b^2-a^2-b^2)-u^3ab.
\end{align*}
The mixed coefficients $rp$ and $rs$ begin at order $u^2$: their
first-order terms have canceled. Keeping this cancellation is what will
leave only a quartic row cost after Young's inequality.

For $|u|\le1/4$ and $|a|,|b|\le1$, all denominators above are bounded
away from zero.  There is thus a universal constant $C_1$ such that
\[
 |K-1|+|R-1|\le C_1|u|.
\]
Consequently the $ps$ coefficient in \eqref{eq:pair-cross-coefficients}
is $-u+O(u^2)$, both mixed coefficients are $O(u^2)$, and the $r^2$
coefficient is $1+O(|u|)$. These constants are uniform in the conditional
means $a,b$: all the displayed rational functions and their required
derivatives are bounded on the compact parameter range.

Return temporarily to the notation $a_i=v_i^1g_i$ and $a_j=v_j^1g_j$ for
the two gradient weights.  Equations \eqref{eq:pair-density} and
\eqref{eq:pair-conditional-identities} imply
\begin{equation}
 \frac{\E_{ij}[a_i a_j]}{AB}
       =\E_0\bigl[w_u(x,y)g_i(y)g_j(x)\bigr],\qquad
 w_u(x,y)=\frac{(1-u^2)^2(1+uxy)}
 {(1+uab)(1+uay)^2(1+ubx)^2}.
 \label{eq:pair-gradient-product}
\end{equation}
Uniformly in $a,b,x,y$, one has
\[
 c_1\le w_u(x,y)\le C_2,\qquad
 |w_u(x,y)-1|\le C_2|u|\qquad (|u|\le1/4)
\]
for universal positive constants $c_1,C_2$.  Expanding
$g_i(y)g_j(x)$ and using that the centered factors have zero product-law
mean gives
\begin{equation}
 \frac{\E_{ij}[a_i a_j]}{AB}
       =ps+e_0ps+e_1rp+e_2rs+e_3r^2,
       \qquad |e_\ell|\le C_3|u|.
 \label{eq:pair-gradient-coefficients}
\end{equation}
To obtain the four error bounds, expand the product into its $ps$,
$rp$, $rs$, and $r^2$ terms. Each centered spin is bounded by two, and
the uniform estimate for $w_u-1$ controls every resulting coefficient.

Adding $u$ times \eqref{eq:pair-gradient-coefficients} to
\eqref{eq:pair-cross-coefficients}, and decreasing the universal $u_0>0$
if necessary, yields
\begin{align*}
 &\frac{\E_{ij}[(x-m_i^1)g_i\,(y-m_j^1)g_j]
                    +u\E_{ij}[a_i a_j]}{AB}\\
 &\hspace{15mm}\ge
       \frac12r^2-C_4u^2|ps|
                       -C_4u^2|r|(|p|+|s|)\\
 &\hspace{15mm}\ge -C_5u^2|ps|-C_5u^4(p^2+s^2).
\end{align*}
The last inequality is Young's inequality, absorbing the mixed terms
into $r^2/2$.

We must now express the two scalar costs in the heat-bath weights that
can be summed over pairs. The lower bound for $w_u$ in
\eqref{eq:pair-gradient-product}, followed by product independence,
gives
\begin{align*}
 \E_{ij}|a_i a_j|
 &\ge c_1AB\,\E_0|g_i(y)g_j(x)|\\
 &=c_1AB\,\E_0|g_i(y)|\,\E_0|g_j(x)|
 \ge c_1AB|ps|.
\end{align*}
Similarly, the density in \eqref{eq:pair-density} and the ratios
$v_i^1/A$, $v_j^1/B$ are bounded above and below by universal positive
constants for $|u|\le u_0$.  Hence
\[
 \E_{ij}[v_i^1g_i^2]\ge c_2A\E_0g_i^2
                       \ge c_2Ap^2\ge c_2ABp^2,
 \qquad
 \E_{ij}[v_j^1g_j^2]\ge c_2ABs^2.
\]
The comparison constants do not deteriorate when a conditional variance
$A$ or $B$ is small. Substituting these bounds into the preceding scalar
inequality proves the pair estimate
\begin{align}
 \E_{ij}[(x-m_i^1)g_i\,(y-m_j^1)g_j]
 \ge{}&-u\E_{ij}[a_i a_j]-C_6u^2\E_{ij}|a_i a_j|\notag\\
 &-C_6u^4\E_{ij}[v_i^1g_i^2+v_j^1g_j^2].
 \label{eq:terminal-pair-bound}
\end{align}

To assemble the curvature form, first condition on all but spin $i$ in
the diagonal term of $\E(Hf)^2$. Their sum is
$\sum_i\E[v_i^1g_i^2]=\D_{h_0}(f)$. Next average
\eqref{eq:terminal-pair-bound} over the outside spins and sum over
ordered pairs $i\ne j$. The signed and absolute pair costs become the
two quadratic forms in \eqref{eq:terminal-curvature}. The quartic cost
is bounded by
\[
 2C_6\,r_4(U)\sum_i\E[v_i^1g_i^2].
\]
Increasing $C$ proves the lemma.
\end{proof}

The curvature inequality is useful when the few coordinates with appreciable
conditional variance occupy a small principal submatrix. Bandeira, El Alaoui,
and R\"odder~\cite[Theorem~1.1]{BER2026} combine uniform bounds on small
principal submatrices with a sufficiently large constant field to prove
polynomial mixing. Here the matrix bounds enter the signed curvature form
directly. We next obtain matrix control uniform over every such coordinate
set; the set will later depend on the spin configuration.

\begin{lemma}[Small principal submatrices]
\label{lem:terminal-matrices}
Fix $j\in(0,1)$.  There is a finite constant $M$, depending only on $j$,
with the following property.  For every $\varepsilon>0$ there is
$\alpha\in(0,1/2)$ such that, with probability tending to one over $J$,
\begin{gather}
 \max_{i,j}|J_{ij}|\le n^{-1/4},\qquad
 \|J\|,\|U\|,\|Q\|\le M,\qquad
 r_4(U)\le Mn^{-1/2},
 \label{eq:terminal-global-matrix}\displaybreak[1]\\
 \|U_{SS}\|\le\varepsilon,\qquad
 \|Q_{SS}\|\le\varepsilon
 \quad\text{for every }S\subseteq\{1,\ldots,n\}
                       \text{ with }|S|\le\alpha n.
 \label{eq:terminal-sparse-matrix}
\end{gather}
The same $M$ works for all $\varepsilon$.
\end{lemma}

\begin{proof}
For any deterministic unit vector $z$, the centered Gaussian variable
$z^{\mathsf T}Jz$ has variance
\[
 \frac{4j}{n}\sum_{i<k}z_i^2z_k^2\le\frac{2j}{n}.
\]
Thus
\begin{equation}
 \Prob\{|z^{\mathsf T}Jz|>t\}\le
                 2\exp\{-nt^2/(4j)\}.
 \label{eq:terminal-quadratic-tail}
\end{equation}
To pass from fixed quadratic forms to norms, choose a $1/4$-net of the
unit sphere in $\R^k$ with at most $9^k$ points. Every symmetric
$k\times k$ matrix $B$ then satisfies
\[
 \|B\|\le2\max_{z\text{ in the net}}|z^{\mathsf T}Bz|;
\]
to see this, approximate a unit vector maximizing the absolute quadratic
form and bound the change by $2\|B\|/4$.  Taking $k=n$ in
\eqref{eq:terminal-quadratic-tail} proves $\|J\|\le M_0$ with probability
at least $1-e^{-c n}$ for a sufficiently large constant $M_0=M_0(j)$.

For a fixed row $i$ and a fixed set $S$ of at most $k$ indices, Gaussian
independence within that row gives
\begin{equation}
 \Prob\left\{\sum_{\ell\in S}J_{i\ell}^2>t\right\}
 \le \exp\left\{-\frac{nt}{4j}+\frac{k}{2}\log2\right\}.
 \label{eq:terminal-row-tail}
\end{equation}
Use exponential Markov with parameter $n/(4j)$ to obtain this estimate:
each nonzero Gaussian square contributes the factor $\sqrt2$ to the
moment generating function. Choosing the full index set and taking a
union bound over rows bounds every row square sum by a fixed
$M_1=M_1(j)$, with failure at most $e^{-cn}$. Separately, the Gaussian
entry tail and a union bound give
\[
 \Prob\{\max_{i,\ell}|J_{i\ell}|>n^{-1/4}\}
 \le 2n^2\exp\{-\sqrt n/(2j)\}=o(1).
\]
Since $|\tanh z|\le|z|$, these bounds imply $\|Q\|\le M_1$ and
$r_4(U)\le M_1n^{-1/2}$, using the maximal absolute row-sum bound for
the operator norm of a symmetric matrix.  Moreover,
\[
 |\tanh z-z|\le |z|^3/3,
 \qquad
 \|U-J\|\le \frac13\max_i\sum_\ell|J_{i\ell}|^3
                \le\frac{M_1}{3}n^{-1/4}.
\]
The scalar inequality follows by integrating
$|1-\sech^2 z|=\tanh^2 z\le z^2$.  This proves
\eqref{eq:terminal-global-matrix}, with a fixed $M$.

The global bounds have fixed $M$; we now select the small-set fraction
$\alpha$. Put $k=\lfloor\alpha n\rfloor$. Bounding sets of size exactly
$k$ suffices, because a smaller set can be enlarged and restriction cannot
increase a principal-submatrix norm. The $\binom nk$ possible sets have
logarithmic count at most $n\mathfrak h(k/n)$, where
\[
 \mathfrak h(t)=-t\log t-(1-t)\log(1-t),
\]
and $\mathfrak h(t)\to0$ as $t\downarrow0$.  Applying the sphere-net
argument on each support and using \eqref{eq:terminal-quadratic-tail}
gives
\[
 \Prob\left\{\max_{|S|=k}\|J_{SS}\|>\varepsilon/2\right\}
 \le 2\binom nk9^k
                  \exp\{-n\varepsilon^2/(64j)\}.
\]
Choose $\alpha>0$ sufficiently small that this tends to zero
exponentially in $n$.  Then $\|U-J\|=o(1)$ proves the assertion for $U$.

Finally, \eqref{eq:terminal-row-tail}, with $t=\varepsilon$, and a union
bound over all rows and all $k$-element sets give
\[
 \Prob\left\{\max_i\max_{|S|=k}\sum_{\ell\in S}J_{i\ell}^2
                                     >\varepsilon\right\}
 \le n\binom nk
       \exp\left\{-\frac{n\varepsilon}{4j}+\frac{k}{2}\log2\right\}.
\]
Decreasing $\alpha$ further makes this probability exponentially small.
On its complement, the maximal row sum of every $Q_{SS}$ is at most
$\varepsilon$, which proves the remaining assertion.
\end{proof}

We apply these matrix bounds to a noisy observation. The following statement
allows any initial spin law, with the interaction still drawn at a fixed
$j\in(0,1)$; the exceptional probability concerns the observation at
that fixed matrix.

\begin{proposition}[Terminal observation gap]
\label{prop:terminal-gap}
Fix $j\in(0,1)$.  There are finite $T>0$, $c>0$, and events
$\mathcal E_n^{\mathrm{term}}$ depending only on $J$, with
$\Prob_J(\mathcal E_n^{\mathrm{term}})\to1$, such that the following holds
for every sufficiently large $n$ and every $J\in\mathcal E_n^{\mathrm{term}}$.
Let $X$ have any probability law on $\{-1,1\}^n$, possibly depending on
$J$, and let $Z$ be an independent standard Gaussian vector.  Set
$Y=TX+\sqrt T\,Z$.  Then, conditionally on $J$, except on an event of
probability at most $e^{-cn}$, the inequality
\begin{equation}
 \Var_{\mu_Y}(f)\le 2\D_Y(f)
 \label{eq:terminal-poincare}
\end{equation}
holds simultaneously for every real function $f$ on the cube.
All constants and matrix events are independent of the law of $X$.
\end{proposition}

\begin{proof}
Choose the tolerances in the order needed to absorb the curvature costs.
Let $C$ be the constant in Lemma~\ref{lem:curvature}, and choose
$\varepsilon>0$ with $(1+C)\varepsilon\le1/8$.
Lemma~\ref{lem:terminal-matrices} supplies $\alpha$ and the fixed global
bound $M$. Finally choose the fixed observation time $T$ large enough that
\begin{equation}
 \frac{16M^2}{T^2}\le\frac\alpha2,
 \qquad
 (1+C)M\bigl(2\sech(T/4)+\sech^2(T/4)\bigr)\le\frac18,
 \qquad
 e^{-T/8}\le\frac{\alpha}{4(e-1)}.
 \label{eq:terminal-T-choice}
\end{equation}
Let $\mathcal E_n^{\mathrm{term}}$ be the events supplied by that lemma.
For all sufficiently large $n$, their bounds also ensure
$\max|U_{ij}|\le u_0$ and $Cr_4(U)\le1/8$.

First count coordinates whose observation field remains small. Given
$X$, the indicators $\one_{\{|Y_i|<T/2\}}$ are independent; symmetry of
the Gaussian noise gives the same success probability $p_T$ for either
sign of $X_i$. The Gaussian tail bound yields
\[
 p_T=\Prob\{|T+\sqrt T Z_1|<T/2\}
       \le\Prob\{Z_1<-\sqrt T/2\}\le e^{-T/8}.
\]
For $B_Y=\#\{i:|Y_i|<T/2\}$, exponential Markov inequality yields
\[
 \Prob\{B_Y>\alpha n/2\mid X,J\}
 \le e^{-\alpha n/2}(1+p_T(e-1))^n
 \le e^{-\alpha n/4}.
\]
The same bound holds after averaging over $X$.

Suppose now that $B_Y\le\alpha n/2$.  For every spin configuration $x$,
$\|Jx\|^2\le M^2n$, and consequently
\[
 \#\{i:|(Jx)_i|>T/4\}\le\frac{16M^2n}{T^2}\le\frac\alpha2 n.
\]
It follows, simultaneously for all $x$, that
\begin{equation}
 S(x)=\{i:|Y_i+(Jx)_i|<T/4\}
 \quad\text{satisfies}\quad |S(x)|\le\alpha n.
 \label{eq:terminal-small-field-set}
\end{equation}
At this fixed field $Y$, set
$q_f(x)=\sum_i v_i^1(x)g_i(x)^2$ and
$\eta=\sech^2(T/4)$.  Since $0<v_i^1\le1$, one has pointwise
\[
 \|a\|^2\le q_f(x),\qquad
 \|a_{S(x)^c}\|^2\le\eta q_f(x).
\]
We may now use $S(x)$ even though it depends on the configuration,
because \eqref{eq:terminal-sparse-matrix} holds for every small set on
one matrix event. Split the quadratic form between $S(x)$ and its
complement. The small-set bound handles its first block, and the global
norm handles the cross and complementary terms, giving
\[
 a^{\mathsf T}Ua
 \le\bigl[\varepsilon+M(2\sqrt\eta+\eta)\bigr]q_f(x).
\]
The identical argument, with $|a|$ and $Q$, gives the same upper bound
for $|a|^{\mathsf T}Q|a|$.  Lemma~\ref{lem:curvature} and
\eqref{eq:terminal-T-choice} therefore imply
\begin{align*}
 \E_{\mu_Y}[(Hf)^2]
 &\ge\left[1-Cr_4(U)
       -(1+C)\{\varepsilon+M(2\sqrt\eta+\eta)\}\right]\D_Y(f)\\
 &\ge\frac12\D_Y(f).
\end{align*}

It remains to translate curvature into a gap. The kernel of the
self-adjoint operator $H$ contains exactly the constants: vanishing
$\E[fHf]$ forces every one-spin conditional variance to vanish, and the
strict positivity of the Gibbs law propagates equality across every
cube edge. For an eigenfunction with positive eigenvalue $\lambda$,
the curvature estimate and \eqref{eq:terminal-generator-form} imply
$\lambda^2\ge\lambda/2$, hence $\lambda\ge1/2$. Expand in an
orthonormal eigenbasis to obtain \eqref{eq:terminal-poincare}.
The observation failure bound was $e^{-\alpha n/4}$, uniformly in the
initial spin law, so we may set $c=\alpha/4$.
\end{proof}
==== END A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase//terminal.tex ====