==== TABLE OF CONTENTS ====
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//accepted-inputs.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-classical.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-entropy.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-initialization.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-multiscale.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-static.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//dynamics.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//introduction.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//model.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//paper.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//preamble.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-amplification.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-entropy.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-local.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-matrices.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-orbits.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-positive.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-roots.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-spectral.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//proof-overview.tex
Critical-mixing-in-the-Sherrington-Kirkpatrick-model//references.bib
===========================
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//accepted-inputs.tex ====
\section{Companion inputs and the critical endpoint}\label{sec:accepted-inputs}
The critical companion supplies exactly the two statements collected
in \Cref{thm:accepted-critical}: its Theorems~1.2 and~1.3
\cite{AcceptedCritical}. The spectral, noncentral spherical, and
spectral-cap estimates required by the upper bounds are proved below.
The high-temperature gap companion supplies observation identities
and deterministic field and residual tools, together with fixed-subcritical
consequences under their stated hypotheses \cite{AcceptedGap}.
We describe these hypotheses before using the tools at a critical
endpoint.
The field equations are of Thouless--Anderson--Palmer type
\cite{TAP1977}. We use the following normalization, including the
rank-one derivative of the self-consistent overlap.
For an interaction of variance $j/n$, set
\begin{align*}
m&=\tanh y,&q&=n^{-1}\norm m^2,&V_y&=\diag(1-m_i^2),\\
F_{J,h}(y)&=y-h+[j(1-q)I-J]m,&&&\\
H_J(y,A)&=I+A^{1/2}
[j(1-q)I-J-2jmm^{\mathsf T}/n]A^{1/2}.&&&
\end{align*}
The stable-field hypothesis is uniform over a region: every
$y$ with $\norm{F_{J,h}(y)}\le\rho_0\sqrt n$ and every diagonal
$0\le A\le(1+\eta)I$ sufficiently close to $V_y$ in normalized
Frobenius norm must satisfy $H_J(y,A)\succ cI$.
The constants $\rho_0,\eta,c$ are fixed positive margins.
The whole-region assertion permits root continuation and spin-flip
buffers; positivity at one selected root would not supply these uses.
The matrix diagnostics used below concern finite \emph{words}: ordered
products of bounded diagonal matrices and copies of the interaction
matrix, with at most one controlled inverse factor. They bound the
operator norm, the entrywise $\ell^4$ norm off the diagonal, and the
Euclidean norm of the diagonal error from its Gaussian contraction
prediction. The permitted diagonal classes and the actual inverse
margins are stated in \Cref{prof:endpoint-words,cap:restricted-words};
the distinction between adaptive diagonals and prescribed vector tests
is recorded below.
\paragraph{Deterministic consequences.}
Under the specified finite-word diagnostics and stable-field hypotheses,
the root $r$ is unique. The gap paper's Proposition~7.2 gives
\begin{equation}\label{eq:accepted-directional}
\norm{\Cov_{\mu_h}(G,X)}
\le C\bigl(\Var_{\mu_h}(G)+\D_{\mu_h}(G)\bigr)^{1/2}.
\end{equation}
Its proof also establishes
$|\mu_h[G(X-\tanh r)^{\mathsf T}e]|
\le C(\mu_h G^2+\D_{\mu_h}(G))^{1/2}$ for $\norm e\le1$.
In particular,
\begin{equation}\label{eq:accepted-ordinary-mean}
\norm{\mu_hX-\tanh r}\le C.
\end{equation}
This is an unnormalized Euclidean mean estimate; it is not an
approximation for the derivative of the mean map.
For $N=\mu_h f^2>0$ and $\nu=f^2\mu_h/N$, Proposition~7.3 gives, for
each sufficiently small fixed $\delta>0$,
\begin{equation}\label{eq:accepted-weighted-mean}
\frac{\norm{\nu X-\tanh r}}{\sqrt n}
\le C\delta+\frac{C_\delta}{\sqrt n}
\left(1+\frac{\D_{\mu_h}(f)}N\right).
\end{equation}
The leading constant $C$ is fixed by the stability margins. The
constant $C_\delta$ and the finite diagnostic length may then depend on
$\delta$ and on the resulting residual-recipe depth. This distinction
will matter when we choose an integrated entropy-drift budget before
choosing the residual tolerance that realizes it. The displayed
statements and the uncentered proof consequence come from
\cite[Section~6, Lemma~7.1, Proposition~7.2 and its proof,
Proposition~7.3]{AcceptedGap}.
\paragraph{Provider hypotheses.}
In the provider's random word estimates, $j<1$ and
$j(1+\eta)^2<1$ are fixed. When the interaction is $\beta$ times a
standard SK matrix, this variance parameter is $j=\beta^2$.
Words have fixed length and at most one inverse factor; that inverse
requires a separate positive margin for the actual matrix being
inverted. Bilinear estimates apply to prescribed finite families of
deterministic vectors. The random stable-field theorem is along ordinary
Gaussian observations of a zero-field Gibbs sample. It is not a theorem for every field.
The terminal observation-gap theorem also fixes $j<1$,
although its input spin law may be arbitrary
\cite[Proposition~5.1, Corollary~5.2, Proposition~8.3]{AcceptedGap}.
Some deterministic ingredients have broader domains. The root/inverse
lemma permits every fixed $j>0$ provided $H_J(y,V_y)\succeq cI$
throughout its entire small-residual region. That root lemma alone
needs only $A=V_y$; the nearby-diagonal condition is an additional
hypothesis for the implicit residual recipes. The local-volume lemma and
the signed two-spin curvature inequality similarly have their explicit
deterministic hypotheses
\cite[Lemmas~5.6, 5.7 and~8.1]{AcceptedGap}.
We apply these deterministic tools at $j=1$ only after verifying their
hypotheses. The two routes prove the needed restricted norm--trace
random diagnostics, actual adaptive-inverse margins, whole-region
stability, and terminal extensions locally. Recipe depth is fixed before
the number of allowed spin flips, buffer widths, and eventual large-$n$
threshold are used. No critical random estimate is obtained by taking
$\beta\uparrow1$ in the fixed-temperature spectral-gap theorem.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//accepted-inputs.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-classical.tex ====
\section{Classical boundary fields}\label{cap:classical}
The cap construction starts at a fixed positive scale. At this boundary we
need an approximation to the gradient in unnormalized Euclidean norm, a
bounded covariance, and an estimate that remains useful after changing the
spin law by a square density. We first specify the companion high-temperature
input and its restricted endpoint extension. We then analyze small positive
fields at the endpoint; it is this second analysis that supplies the curvature
and value of the initial auxiliary potential. All scales and all error
tolerances in this section are fixed before letting the dimension tend to
infinity.
For a scale-one GOE matrix $W$, let $J_*=\beta W_{\mathrm{off}}$ and
$j=\beta^2$. The diagonal of $W$ can be restored when convenient, because it
is constant in the spin Hamiltonian and tends to zero in operator norm. For
$y,h\in\mathbb R^n$ set
\begin{equation}\label{cap:classical-map}
m=\tanh y,\qquad q=\frac{\|m\|^2}{n},\qquad r=1-q,\qquad
V=\operatorname{diag}(1-m_i^2),\qquad
F_h(y)=y+(jrI-J_*)m-h.
\end{equation}
Write \(V_0(z)=\operatorname{sech}^2z\), so \(V=\operatorname{diag}V_0(y_i)\).
We write $\langle u v\rangle=n^{-1}u^{\mathsf T}v$ and
$|u|_n=n^{-1/2}\|u\|$. A field passes the stability test with parameters
\(b,\eta,\epsilon_A,\rho>0\) if the following implication holds for every
\(y\in\mathbb R^n\) and every diagonal \(A\):
\begin{equation}\label{cap:stability-test}
\left.
\begin{gathered}
|F_h(y)|_n\le 2\rho,\qquad
0\le A\le(1+\eta)I,\\
n^{-1/2}\|A-V\|_{\mathrm F}\le\epsilon_A
\end{gathered}
\right\}
\quad\Longrightarrow\quad
I+A^{1/2}\left(jrI-J_*-\frac{2j}{n}mm^{\mathsf T}\right)A^{1/2}
\succeq bI.
\end{equation}
The small enlargement of the residual region will allow later changes in
the external field and the finite spin-flip neighborhoods needed in the
residual identities.
\subsection{The posterior input and its hypotheses}
\begin{proposition}[Classical posterior input]\label{cap:classical-input}
Fix $0<\beta<1$, an operator-norm bound for $J_*$, and fixed positive
parameters in \eqref{cap:stability-test}, with
$j(1+\eta)^2<1$. On the fixed-length matrix-word events described below,
every field satisfying \eqref{cap:stability-test} has a unique root $y_*$
of $F_h$. If $m_* =\tanh y_*$ and
\[
\mu_h(x)\propto\exp\{x^{\mathsf T}J_*x/2+h^{\mathsf T}x\},
\qquad
\mathcal D_h(f)=\sum_i\mathbb E_{\mu_h}
\operatorname{Var}(f\mid x_{\ne i}),
\]
then
\begin{equation}\label{cap:classical-mean-covariance}
\|\mathbb E_{\mu_h}x-m_*\|\le C,
\qquad \|\operatorname{Cov}_{\mu_h}(x)\|\le C.
\end{equation}
For every sufficiently small fixed $\epsilon>0$, every real $f$ with
$N=\mathbb E_{\mu_h}f^2>0$, and $\nu=f^2\mu_h/N$,
\begin{equation}\label{cap:square-reweighting}
|\mathbb E_\nu x-m_*|_n
\le C\epsilon+\frac{C_\epsilon}{\sqrt n}
\left(1+\frac{\mathcal D_h(f)}{N}\right).
\end{equation}
The event and constants are independent of $h$ and $f$ under the stated
diagnostics and stability parameters. The required word length is finite
and may depend on $\epsilon$.
The same conclusions hold at $j=1$ if, in addition, every approximate root
in the residual region has $q\ge q_{\min}>0$, and the inverse-word event
is restricted to the diagonal class specified in
Lemma~\ref{cap:restricted-words}. Constants may depend on $q_{\min}$.
This endpoint assertion is a consequence of the proof below, rather than
an application of a theorem with $\beta<1$ at $\beta=1$.
\end{proposition}
Here and below a word means an ordered product of bounded real diagonal
matrices and copies of $J_*$, with at most one factor
\[
K_A(z)=\left(I-zAJ_*+z^2j\langle A\rangle A\right)^{-1},
\qquad \langle A\rangle=n^{-1}\operatorname{Tr}A,
\quad 0\le z\le1.
\]
Its prediction is obtained by the noncrossing contractions $J_*DJ_*\mapsto
j\langle D\rangle I$, after formally expanding the one inverse factor. For
each fixed word length the diagnostic bounds are bounded operator norm,
bounded off-diagonal entrywise $\ell^4$ norm, and bounded Euclidean norm of
the diagonal minus its prediction. Inverse words are only used when
\[
I+z^2j\langle A\rangle A-zA^{1/2}J_*A^{1/2}\succeq cI
\]
for a fixed $c>0$. For $j<1$ and $j(1+\eta)^2<1$, these are precisely
the simultaneous diagnostics of \cite[Lemma~3.1]{AcceptedGap}.
\begin{lemma}[Restricted endpoint matrix diagnostics]
\label{cap:restricted-words}
Fix $a_+<\infty$ and $\kappa\in(0,1)$. At $j=1$, restrict inverse
diagonals to
\begin{equation}\label{cap:restricted-diagonals}
\mathcal A_\kappa=
\{A:\ A\text{ diagonal},\ 0\le A\le a_+I,
\ \|A\|\langle A\rangle\le1-\kappa\}.
\end{equation}
For each fixed word length and bounded diagonal coefficients, the preceding
word diagnostics hold simultaneously over this class and all stable
inverse parameters, with probability $1-Ce^{-cn}$. The fixed-parameter
bilinear equivalents of \cite[Lemma~3.4]{AcceptedGap} also hold, uniformly
for deterministic $A\in\mathcal A_\kappa$ and finitely many deterministic
bounded-norm test vectors. They are not asserted simultaneously over all
test vectors. The regularized determinant bound of
\cite[Lemma~3.5]{AcceptedGap} holds simultaneously for diagonal \(0\le V\le I\)
with $\langle V\rangle\le1-\kappa$, including its bounded-rank,
bounded-norm perturbations.
\end{lemma}
\begin{proof}
We isolate the two places in the companion matrix proof
\cite[Section~3]{AcceptedGap} where the
temperature gap is used. The first is a lower bound on the symmetric
inverse along its scalar parameter path; the second is a coefficient below
one in the loop equation. The restricted diagonal condition supplies both
margins at the endpoint. Once these margins are available, truncation and
chaining give the simultaneous assertion by the same finite recursions.
For a deterministic $A$, the Gaussian process comparison in its
Lemma~3.2 bounds the relevant supremum by $2d-d^2$, where now
\[
0\le d\le \sqrt{\langle A\rangle\|A\|}
\le\sqrt{1-\kappa}.
\]
It therefore gives the fixed margin $c_0=(1-\sqrt{1-\kappa})^2/4$ along the
whole path $0\le z\le1$, after concentration and a fixed mesh of that path.
The constants are uniform over deterministic $A$ in
\eqref{cap:restricted-diagonals}. Next, the loop equation for
$u(z)=n^{-1}\mathbb E\operatorname{Tr}(K_A(z)A)-\langle A\rangle$ has linear
coefficient
\[
z^2 n^{-1}\sum_i A_{ii}^2\mathbb E K_A(z)_{ii}.
\]
When $u$ is small, the diagonal expectations are $1+O(u)+O(n^{-1})$. Since
$\langle A^2\rangle\le\|A\|\langle A\rangle\le1-\kappa$, this coefficient
stays below $1-\kappa/2$. Continuity from $z=0$ then gives $u(z)=O(n^{-1})$
throughout the path. All subsequent noncrossing loop recursions are finite and
unchanged.
For clarity, the simultaneous assertion does not condition on a randomly
chosen diagonal. Use the globally truncated inverse from
\cite[Section~3.2]{AcceptedGap}, with a smooth cutoff equal to $x^{-1}$ on a
neighborhood of $[\min(c,c_0)/2,C]$. Its word maps and their increments have
Frobenius Lipschitz constants $C$ and $C\|\theta-\theta'\|_\infty^{1/2}$,
respectively. Cover the restricted parameter set by meshes with
representatives in that set, or in $\mathcal A_{\kappa/2}$ when a buffer is
needed. At resolution $2^{-l}$ there are at most $\exp(C(l+1)n)$
representatives. Gaussian concentration and chaining bound the centered word
increments by $C\sqrt{l+1}\,2^{-l/2}$, with summable failure $Ce^{-c(l+1)n}$.
The deterministic diagonal expectation error is $O(n^{-1})$ coordinatewise by
the preceding loop calculation. This proves the word assertion even for
diagonals selected from the observed matrix. The bilinear assertion follows by
applying concentration to each fixed truncated bilinear form, exactly as in
the cited lemma.
For the determinant take $A=V$. The fixed-diagonal margin and loop
calculation give
\[
\mathbb E\frac1n\log\det(L^{\mathsf T}L+\gamma I)
\le\langle V\rangle^2+C_\kappa\gamma+O(n^{-1}),
\qquad L=I+(\langle V\rangle I-W)V.
\]
The regularized normalized log determinant is $(n\gamma)^{-1/2}$-Lipschitz in
Frobenius norm, so its concentration has rate $n^2$ at fixed tolerance. A
fixed mesh of the restricted diagonal class costs only $\exp(Cn)$; use its
slight enlargement for continuity. The bounded-rank perturbation changes the
answer by at most $\|E\|\sqrt{\operatorname{rank}E/(n\gamma)}$. This proves
the stated uniform determinant bound. Removing the GOE diagonal changes the
truncated words by a bounded Frobenius amount and the stability matrices by
$o(1)$ in operator norm, as in the companion proof. \end{proof}
\begin{proof}[Proof of Proposition~\ref{cap:classical-input}]
For fixed $j<1$ the root assertion is
\cite[Lemma~5.7]{AcceptedGap}; its proof is deterministic and allows any
fixed \(j>0\). It gives a bounded inverse for \(F'_h\) and
\(\|y-y_*\|\le C\|F_h(y)\|\) whenever \(|F_h(y)|_n<2\rho\);
the later uses lie in a strictly smaller residual region. The last estimate
in the proof of \cite[Proposition~7.2]{AcceptedGap}, before centering its
test function, is
\begin{equation}\label{cap:uncentered-directional}
\big|\mathbb E_{\mu_h}[G(x-m_*)^{\mathsf T}e]\big|
\le C\left(\mathbb E_{\mu_h}G^2+\mathcal D_h(G)\right)^{1/2},
\qquad \|e\|\le1.
\end{equation}
Taking $G=1$ gives the first assertion in
\eqref{cap:classical-mean-covariance}. For covariance, apply the centered
statement of that proposition to $G=e^{\mathsf T}x$, with $e$ a unit
eigenvector for the largest covariance eigenvalue $\lambda$. Since $\mathcal
D_h(e^{\mathsf T}x)\le\sum_i e_i^2=1$, it yields $\lambda\le
C\sqrt{\lambda+1}$, hence $\lambda\le C'$. The square-density estimate is
exactly \cite[Proposition~7.3]{AcceptedGap}, in the present rate-one-per-site
normalization.
The endpoint word event is useful only if the diagonals actually produced by
the posterior proof belong to its restricted class. We therefore inspect the
implicit recipe, rather than substituting the endpoint into the companion
statement. The root comparison places its primary vectors near the same root
throughout a buffered residual region. This controls both the largest diagonal
entry and the average diagonal entry, giving the strict product margin
required by the endpoint inverse.
Its primary iterates are
\[
m^0=x,\quad h^1=h+J_*x,\quad m^l=\tanh h^l,\quad
b_l=1-|m^l|_n^2,\quad
h^{l+1}=h+J_*m^l-jb_lm^{l-1}.
\]
Write $R_l=m^{l-1}-m^l$ and
$\Omega_{2\rho}=\{\max(\|R_k\|,\|R_{k-1}\|)\le2\rho\sqrt n\}$.
The root comparison gives, independently of $k$,
\[
|h^k-y_*|_n+|m^{k-1}-m_*|_n\le C\rho,
\qquad |1-b_{k-1}-q_*|\le C\rho.
\]
The only inverse diagonal in the implicit recipe is
\[
A_{ii}=\Phi(h_i^k;(y_*)_i,a),\qquad
a=j(1-b_{k-1}-q_*),\qquad
\Phi(z;v,a)=\int_0^1 e^{a(1-u^2)/2}
\frac{\cosh(u(z-v))}{\cosh z\cosh v}\,du.
\]
Its secant-slope representation at $a=0$ and its bounded derivatives
give, on the entire buffer,
\[
\|A\|\le e^{C\rho},\qquad
\langle A\rangle\le1-q_*+C\rho,
\qquad n^{-1/2}\|A-V_{h^k}\|_{\mathrm F}\le C\rho.
\]
Choose $\rho$ after $q_{\min}$ and the stability margins so that
$e^{C\rho}(1-q_{\min}+C\rho)\le1-q_{\min}/2$. Thus all these diagonals lie in
the interior of \(\mathcal A_{q_{\min}/4}\). At \(h^k\),
\(\langle V_{h^k}\rangle=b_k\), so the same Frobenius bound gives
\[
|\langle A\rangle-b_k|
\le n^{-1/2}\|A-V_{h^k}\|_{\rm F}\le C\rho.
\]
Removing the negative rank from \eqref{cap:stability-test} and then changing
the scalar coefficient from \(jb_k\) to \(j\langle A\rangle\) therefore gives
\[
I+j\langle A\rangle A-A^{1/2}J_*A^{1/2}
\succeq (b-jC\rho\|A\|)I .
\]
Choose \(\rho\) to retain a fixed positive margin. This is the symmetric
matrix for the actual recipe inverse \(K_A(1)\); the path \(0\le z\le1\)
is used to establish its formal word prediction.
Every later auxiliary recipe is ordinary; the closed differentiated formulas
contain at most this one inverse.
These bounds must also hold at the finitely many neighboring spin
configurations used by the residual identities. The recipe depth is fixed
before the dimension grows. Consequently the total Euclidean change across its
required flips is bounded independently of the dimension, while the residual
buffer has radius proportional to its square root. Choosing the dimension last
keeps every required coefficient inside the same restricted diagonal class.
The finite-flip buffers require no uniform bound over all depths. After fixing
a recipe and its depth, let $Q$ be the largest number of successive spin flips
appearing in its residual induction, and let $C_Q$ bound the one-flip
Euclidean changes of all involved primary vectors. Both are finite by
\cite[Lemma~6.1]{AcceptedGap}. Taking $n>(2QC_Q/\rho)^2$ places every required
neighborhood of $\Omega_{3\rho/2}$ inside $\Omega_{2\rho}$. All inverse
coefficients there retain the preceding trace margin. This is the buffer
argument in the released version of \cite[Section~6.3]{AcceptedGap}.
The order of choices is consequently: the root and diagonal stability margins;
the restricted diagonal margin; a sufficiently small $\rho$; the finite
residual-selection depth; the resulting finite recipes and word length; and
finally $n$. The residual-selection depth may increase when the accuracy in
\eqref{cap:square-reweighting} decreases. None of these choices depends on $f$
or on a directional seed. Ordinary words have no restricted inverse parameter
at all. The proof of \cite[Section~6 and Propositions~7.2--7.3]{AcceptedGap}
now applies term by term using Lemma~\ref{cap:restricted-words}, proving the
endpoint version and its claimed uniformity.
For a compact coefficient interval strictly inside $(0,1)$, choose the same
strict diagonal margin throughout and include $\beta$ as one more bounded
scalar in the parameter meshes. The additional mesh cardinality does not
change their exponential-in-$n$ entropy, and all Lipschitz and loop constants
have the same margin. The word and posterior conclusions are therefore
simultaneous on that interval. In the restricted endpoint case the same
argument applies to a sufficiently small coefficient neighborhood that
preserves \eqref{cap:restricted-diagonals}. \end{proof}
\subsection{The positive-time endpoint stability test}
\begin{lemma}\label{cap:endpoint-stability}
For a critical GOE interaction the following assertions hold with
probability tending to one over the disorder.
For every fixed $q_0>0$ there is $\rho_0>0$ such that
$|F_0(y)|_n\le\rho_0$ implies $q(y)<q_0$.
For every fixed $0<t_b\le T<\infty$, there are positive constants in
\eqref{cap:stability-test} and $a>0$ such that, conditionally on typical
disorder, that test holds simultaneously along
\[
Y(t)=tX+B(t),\qquad t_b\le t\le T,
\qquad X\sim\mu_0,
\]
outside conditional probability $e^{-an}$. It also holds throughout a
fixed normalized neighborhood of these fields. The conclusions persist
for sufficiently small fixed changes of the interaction coefficient and
sufficiently small normalized field displacements.
\end{lemma}
\begin{proof}
We adapt the conditional Gaussian, scalar-rate, and local-volume proof of
\cite[Section~5]{AcceptedGap}. The adaptation discards a small-$q$
region by an observation estimate, rather than using a temperature gap.
Here are the details needed at the endpoint.
First the scalar entropy inequality in \cite[Lemma~4.1]{AcceptedGap} extends
to $j=1$ when $q>0$. Its proof uses $k=jq/s\le1$ and $B=j(1-q)<1$, which
remain valid. In its large-$C$ case the quantity $C^2/(jk)$ is still strictly
greater than one, since $C>1$. The strictness and equality statements are
therefore unchanged. The consistent Gaussian laws at $j=1$ are $N(s,s)$ with
\[
s=t+\mathbb E_{N(s,s)}\tanh Y.
\]
Indeed both scalar one-Lipschitz comparisons in the proof of
\cite[Lemma~4.2]{AcceptedGap} become strictly contracting on every
nondegenerate compact interval: their Gaussian derivatives are strictly less
than one. More explicitly, if $f(s)=\mathbb E_{N(s,s)}\tanh Y$, then $f(0)=0$,
$0\le f'(s)<1$ for $s>0$, and $f(s)\le1$. Thus $s-f(s)$ is strictly increasing
from zero to infinity. The solution is unique, equals zero at $t=0$, and is
continuous by this monotonicity and compactness. The second scalar comparison
forces equality of the consistent mean and variance because its Lipschitz
inequality is strict between distinct nonnegative arguments. These facts
replace the global contraction constant $j<1$; no bound involving $(1-j)^{-1}$
is used.
At $t=0$, restrict the scalar-rate integral to $q\ge q_0/2$. Its variance is
then bounded below, and no consistent equality law lies in this set. The
compactness and Gaussian-tail argument of \cite[Lemma~5.4]{AcceptedGap} gives
a strictly negative exponential rate, including small residual smoothing. The
determinant diagnostic is only required on this retained set, where $\langle
V\rangle=1-q\le1-q_0/2$, so Lemma~\ref{cap:restricted-words} supplies it. A
small normalized ball around a witness with $q\ge q_0$ stays in the retained
set. The local-volume lower bound of \cite[Lemma~5.6]{AcceptedGap} is
deterministic and has no subcritical restriction. It turns the negative
integral rate into exclusion of approximate zeros.
On $[t_b,T]$, very small $q$ is excluded first. A sufficiently accurate
approximate root satisfies $|y-Y(t)|_n\le C\sqrt q+\rho$. Since $\tanh$ is
one-Lipschitz, this would force $|\tanh Y(t)|_n$ to be small. For each fixed
spin the coordinates of $tX+B(t)$ are independent Gaussians with means $\pm t$
and variance $t$, so their bounded squared hyperbolic tangents have an
empirical average bounded below, with exponential probability uniformly on the
compact time interval. A fixed time mesh and Brownian maximal bounds give the
same conclusion along the path. We may therefore retain $q\ge q_{\min}>0$ with
slack.
On this retained set, the scalar equality curve and all scalar-rate arguments
are compact and nondegenerate. We make the conditional matrix law explicit.
During the Gaussian reduction use the temporary planted interaction
\(\widetilde J=W+\mathbf1\mathbf1^\top/n\) with full GOE \(W\).
For fixed \(y\), let
\[
\begin{gathered}
b=1-q,\quad M=\langle m\rangle,\quad d=t+M,\quad
s=t+\sigma^2+q,\quad S=s+q,\\
a_y=\langle(y-d\mathbf1)m\rangle,\qquad
z_{\rm cond}=y-d\mathbf1+bm .
\end{gathered}
\]
The matrix factor in \cite[Eq.~(5.8)]{AcceptedGap} is the Gaussian
conditional law given
\[
Wm+\sqrt{t+\sigma^2}\,g'=z_{\rm cond}.
\]
Here \(g'\) is independent standard Gaussian and \(\sigma>0\) is the
residual smoothing. Put
\[
u=\frac{m}{\sqrt{nq}},\qquad
Z_1=\frac{y-d\mathbf1}{\sqrt{ns}},\qquad P_u=I-uu^\top .
\]
Writing \(W=P_uWP_u+uw^\top+wu^\top+\alpha_c uu^\top\) with \(w\perp u\), the elementary
conditional Gaussian formulas give a representation using a fresh GOE \(W_0\):
\[
\begin{split}
P_uWP_u&=P_uW_0P_u,\\
w&=\sqrt{q/s}\,Z_1-(a_y/s)u+
\sqrt{(t+\sigma^2)/s}\,P_uW_0u,\\
\alpha_c&=\frac{2(a_y+bq)}S+
\sqrt{(t+\sigma^2)/S}\,u^\top W_0u .
\end{split}
\]
This is \cite[Eq.~(5.12)]{AcceptedGap} at \(j=1\).
For the moment fix an admissible diagonal \(A\), deterministic after \(y\)
has been fixed. Set
\[
\chi=\langle A\rangle,\qquad
S_A=I+\chi A-A^{1/2}W_0A^{1/2},\qquad
Y_A(v)=S_A^{-1/2}A^{1/2}v .
\]
Lemma~\ref{cap:restricted-words} supplies a positive lower bound for \(S_A\).
Its fixed-diagonal bilinear estimates apply to the three deterministic
vectors \(\mathbf1/\sqrt n,u,Z_1\), whose norms are bounded by the retained
second-moment bound. Their transformed Gram entries are close to
\(v^\top Av'\) for the corresponding pair \(v,v'\); the extra vector \(Y_A(W_0u)\) has mixed entries close to
\(\chi v^\top Au\) and squared norm close to
\(\chi+\chi^2u^\top Au\). These are Gram comparisons under the fresh
\(W_0\) law. At the limiting scalar equality law with \(A=V\) and
\(\sigma=0\), the whitened finite-rank test has
determinant
\[
(1-b+2g_1)(1-b+g_1)^2>0,
\qquad
g_1=\mathbb E_{N(s,s)}[\tanh^2Y\,\operatorname{sech}^2Y].
\]
The rank perturbation has at most one possibly nonpositive direction; the
positive determinant excludes that direction, exactly as in
\cite[Lemma~5.5]{AcceptedGap}. Compactness gives a uniform margin. For $A$
close to $V$, choose $\eta$ and its normalized Frobenius tolerance so that
$\|A\|\langle A\rangle\le1-\kappa$ on the retained range. To include
data-dependent \(A\), keep \(y\) fixed and round the entries exceeding a fixed sup-norm tolerance on
their exceptional coordinate set. That set has fraction at most
$\epsilon_A^2/\zeta^2$, and the logarithm of the number of choices is at most
\[
n\{H(\epsilon_A^2/\zeta^2)
+(\epsilon_A^2/\zeta^2)\log(C/\zeta)+o(1)\}.
\]
Choose \(\zeta\) after fixing the vector-norm bounds, Gram tolerance, and
fixed-diagonal exponent, then \(\epsilon_A\) so that this counting rate is
below that exponent. For this fixed \(y\), the approximants are deterministic,
so a union bound makes the conditional matrix comparison simultaneous in
\(A\). The omitted scalar mass has norm at most
\((1+\eta)|\chi-b|\le(1+\eta)n^{-1/2}\|A-V\|_{\rm F}\) and is included in
the reserved margin. The probability bound is uniform in the retained
\(y\)-data, as required inside the scalar integral. Enlarging the retained ranges slightly before the
local-volume step preserves all these margins.
The preceding calculations are under the usual size-biased planted law. The
critical zero-field partition comparison in
Proposition~\ref{cap:static-comparison} gives $\log Z-\log\mathbb EZ=o(n)$ in
probability. Thus negative exponential rates transfer to typical original
disorder: restrict to $Z/\mathbb EZ\ge e^{-\delta n}$, choose $\delta$ below
the rate, and use Markov's inequality. A finite time mesh and the Brownian
maximal bound supply the whole path. Finally, $F_h$ is one-Lipschitz in its
field and its stability matrix does not use $h$; coefficient and diagonal
changes are uniformly continuous under the norm bounds. Reserving slack in all
thresholds proves the neighborhood and perturbation assertions. \end{proof}
\subsection{Noise diagnostics at a fixed small scale}
We next work at $j=1$ with the full GOE matrix $W$. A spectral Gaussian means
$g=O\Sigma^{1/2}\xi$, where $W=O\Lambda O^{\mathsf T}$, $\xi$ is standard
Gaussian independent of $O$, and $\Sigma$ is positive semidefinite and fixed
after conditioning on the eigenvalues. We allow a fixed smooth compact family
of such covariances with $\|\Sigma\|\le Ct$. Write the field as
\begin{equation}\label{cap:boundary-field}
h=g+h_1,\qquad |h_1|_n\le Ct.
\end{equation}
The shift may be selected after seeing the frame and noise in every frame
estimate below. A lower bound for a shifted Gaussian projection has a
different quantifier, specified separately in Lemma~\ref{cap:boundary-noise}.
Let $\Pi_d$ denote the projections onto dyadic bands for the gap $2-W$, with rank at most
$Cnd^{3/2}$. At a bottom label $a$, combine all smaller gaps into one band
with that label. Separately, write $P_t^W=\mathbf1_{\{2-W\le t\}}$ for
the cumulative projection onto gaps at most $t$; thus its cap-width parameter
is $\sqrt t$. If a vector class obeys $|\Pi_d v|_n\le b_d$, put
\begin{equation}\label{cap:band-complexity}
\mathscr L(v)=\sum_{d\ge a}b_d d^{3/4}.
\end{equation}
Bounds for the individual band sizes can be chosen on arbitrarily fine fixed
grids. A finite collection of known spectral-coordinate vectors, including the
filtered noise, is included in the Gaussian comparison but is not charged this
complexity. Its number is fixed independently of $n$; smooth scalar spectral
filters can be netted before $n$ increases.
We record the two frame estimates used repeatedly. Up to logarithmic
factors in the band count, for a vector of normalized size at most $B$,
\begin{equation}\label{cap:sparse-restriction}
\sup_{|E|\le fn}|\mathbf1_Ev|_n
\le C\{B\sqrt{f\log(2/f)}+\mathscr L(v)\}.
\end{equation}
Empirical averages of smooth clipped functions of finitely many vectors,
normalized to have size at most one, differ from the corresponding centered
Gaussian averages with the same Gram matrix by
\begin{equation}\label{cap:frame-comparison}
C_\psi\sum_v\mathscr L(v).
\end{equation}
Here the complexities are those of the normalized vectors and $C_\psi$ is
controlled by the first two derivatives of the clipped function. Both
estimates permit an additional fixed slack and a failure allowance
$C\sqrt{\mathfrak h/n}$, multiplied by the indicated size and derivative
bounds, at failure rate $e^{-c\mathfrak h}$. We use $\mathfrak h$ of order
$nt^{3/2}$.
For completeness, \eqref{cap:sparse-restriction} follows by testing each band
against a sparse coordinate unit vector. Nets of its coefficient sphere and of
the chosen rows have logarithmic size $Cnd^{3/2}+Cnf\log(2/f)$, respectively.
The Gaussian representation of a Haar unit vector bounds its scalar products
at the square root of this entropy divided by $n$. Sum the band bounds and use
Cauchy--Schwarz for the total-size term. To prove
\eqref{cap:frame-comparison}, first fix \(k\) spectral coefficient columns
\(C=(c^1,\ldots,c^k)\), with \(k\) fixed and \(|c^j|_n\le1\). Put
\[
Q=C^\top C/n,\qquad
F_\psi(O,C)=n^{-1}\sum_i\psi((OC)_i).
\]
For \(\omega\) uniform on \(S^{n-1}\), rotation invariance gives the exact
one-row expectation
\[
\E_OF_\psi(O,C)
=\E_\omega\psi\bigl(\sqrt n\,Q^{1/2}(\omega_1,\ldots,\omega_k)^\top\bigr)
=\E\psi(G_Q)+o_n(1),\qquad G_Q\sim N(0,Q).
\]
The formula is interpreted on the support of a possibly singular \(Q\).
Thus the Gaussian uses the actual Gram matrix of the chosen columns.
For the usual Frobenius metric on the orthogonal group,
\[
\operatorname{Lip}_O F_\psi
\le C_\psi\|C\|_{\rm op}/\sqrt n=O(C_\psi),\qquad
\operatorname{Lip}_O(F_\psi(\cdot,C)-F_\psi(\cdot,C'))
\le C_\psi\|C-C'\|_{\rm F}/\sqrt n.
\]
The first estimate follows by differentiating the row average and using
\(\|\nabla\psi(OC)\|_{\rm F}\le C_\psi\sqrt n\); the second uses the
bounded second derivative. Haar concentration has exponent proportional to
\(n\) at this Lipschitz scale. Chaining over the band balls therefore costs
\(C_\psi\sum b_d\sqrt{\operatorname{rank}\Pi_d/n}\), and a failure budget
\(\mathfrak h\) costs \(C_\psi\sqrt{\mathfrak h/n}\).
This proves \eqref{cap:frame-comparison}. The event is uniform over the
coefficient balls, so vectors may be selected from those balls after the
frame is observed.
We intersect the full-frame event with the following residual-rotation
event. For this calculation fix the eigenvalues \(\Lambda\), the low
physical frame \(O_0\), and a high spectral multiplier \(D\) at a fixed
scalar net point. Choose an isometry \(H\) onto the complement of the low
frame and write the remaining columns as \(O_>=HU\), where \(U\) is Haar
in dimension \(m\). Put \(M=HUDU^\top H^\top\) and
\(P_\perp=HH^\top\). Its exact conditional mean is
\[
\E[M\mid\Lambda,O_0,D]=\mu P_\perp,\qquad \mu=\Tr D/m.
\]
For a finite net of vectors \(b,t\) fixed under this conditioning, let
\(h_{\rm net}\) be its logarithmic cardinality and \(h_{\rm tail}>0\).
The Haar quadratic-form bound and polarization give, with
\(\mathfrak H=C(h_{\rm net}+h_{\rm tail})\), outside conditional probability
\(2e^{-h_{\rm tail}}\),
\begin{equation}\label{cap:conditional-high-test}
\begin{split}
|\langle b,Mt\rangle-\mu\langle P_\perp b,P_\perp t\rangle|
\le C|b|_n|t|_n\left(
\frac{\|D\|_{\rm F}}{\sqrt m}\sqrt{\frac{\mathfrak H}{m}}
+\|D\|\frac{\mathfrak H}{m}\right).
\end{split}
\end{equation}
The probability is over the remaining \(U\); coefficient and sparse nets
are included in \(h_{\rm net}\). This is the conditional estimate used for
the high multipliers below. The low coordinate profiles remain fixed.
We specify the finite families included in the noise event. Scalar meshes
cover the covariance parameters, \(r\), the small values \(s=q+\sqrt t\),
and the derivative shift \(0\le\ell\le Cs\), with cutoffs
\(a=s^{7/5}\) and \(u=s^{1/10}\).
The uncharged spectral columns are the noise and the finitely many scalar
resolvent filters used in the root and derivative bases, including
\(r^{-1}\mathbf1_{\{d>a\}}(r+r^{-1}-\Lambda)^{-1}\Sigma^{1/2}\xi\).
The normalized root, derivative, base, correction, and band-test vectors
range over their charged coefficient balls. Sparse tests cover the root and
derivative tails, their union with a band-test tail, and the exceptional
\(A-V\) coordinates. The clipped recipe types are the root and derivative
sources against one band test, the scalar functions \(m^2,m^4,m^6,V'_0(y)w\),
their low-input versions, and the differentiated functions in the single
second-order Taylor expansion below. The number of these types and the
number of columns in each average are fixed before \(n\); the possible
directions within a type are paid for by the coefficient and sparse nets.
For each type the event is uniform over every band ball whose radii lie on
the chosen fixed grids; the later localization bounds select those radii.
\begin{lemma}[Uniform shifts and conditional noise]\label{cap:boundary-noise}
At every fixed sufficiently small $t>0$, the spectral band, sparse-row,
and clipped-average diagnostics just described can be imposed on an
event depending only on $(W,g)$ and the fixed parameter family.
They hold there uniformly over all shifts in \eqref{cap:boundary-field}
and the coefficient balls, sparse supports, and recipe and filter types
specified above. Their conditional failure
probability tends to zero for typical $W$; with fixed buffered tolerances
it is at most $e^{-c_tn}$ after decreasing $c_t>0$. The event is independent
of any square-density test function.
Suppose additionally that the covariance of $g$ is at least $ct$ on a
spectral band contained in $\{2-W\le t\}$ of rank at least $cnt^{3/2}$.
Then, for every fixed center $v\in\mathbb R^n$,
\begin{equation}\label{cap:small-ball}
\mathbb P_g\bigl\{|P_t^W(g+v)|_n<c't^{5/4}\mid W\bigr\}
\le e^{-c''nt^{3/2}}.
\end{equation}
The constants are uniform in the center. This is a probability bound
for each center, not a simultaneous event over all centers.
In particular, both types of estimate apply after averaging over any
spin law that is fixed before the fresh Gaussian noise is drawn.
\end{lemma}
\begin{proof}
There are two different uniformity statements here. The frame estimates
hold on one event for all coefficient vectors in the specified balls, so
they admit shifts selected after the frame and noise are seen. The
projection lower bound instead holds in probability for each fixed center,
with a constant independent of that center. A spin sampled before the
fresh noise can therefore be averaged into either statement. There is no
simultaneous lower bound over every possible center.
The frame estimates are simultaneous coefficient-ball estimates. The shift
contributes at most $Ct$ to each unfiltered band norm; after a high-band
inverse it contributes at most $Ct/d$. Thus every shift in the stated ball is
already included in their deterministic domains. The Gaussian band norms and
the norms of their finite smooth filtered families have exponential tails at
the specified fixed scales. If \(p(W)\) is the conditional noise failure
probability and \(\E_Wp(W)\le e^{-2c_tn}\), Markov gives
\(\Pr_W\{p(W)>e^{-c_tn}\}\le e^{-c_tn}\). This gives the asserted disorder
event and conditional bounds, without any reference to a test function.
The same argument integrated over a fixed time interval gives
the integrated conditional statement, when that is all that is needed.
For \eqref{cap:small-ball}, restrict to the nondegenerate band of rank $r_t\ge
cnt^{3/2}$. The Gaussian integral of a Euclidean ball is maximized when its
center is the mean; alternatively, exponential Markov applied to
$\exp(-\lambda\|g+v\|^2/t)$ gives a product bounded by
$(1+c\lambda)^{-r_t/2}$, since the noncentral exponential factors are at most
one. Taking the ball radius squared to be a sufficiently small multiple of
$tr_t$ proves the claim. Finally, condition on each spin and then average this
bound over its law. No union bound over spin configurations is involved, and
no initial density bound is used. \end{proof}
\subsection{The small-field TAP expansion}
\begin{proposition}[Boundary expansion]\label{cap:tap-expansion}
Assume the preceding spectral and noise diagnostics, the zero-field
approximate-root exclusion in Lemma~\ref{cap:endpoint-stability}, the field
representation \eqref{cap:boundary-field}, and
$|P_t^Wh|_n\ge ct^{5/4}$. The test \eqref{cap:stability-test} holds with
positive fixed slack; in particular $F_h$ has a unique root $y_*(h)$.
At all roots of $F_h$,
and with arbitrarily small additional errors at sufficiently accurate
approximate roots, set
\[
z=r+r^{-1},\qquad R=(z-W)^{-1}.
\]
For sufficiently small fixed $t$ and then sufficiently large $n$,
\begin{equation}\label{cap:tap-leading}
\begin{gathered}
q\ge c\sqrt t,\qquad |y-r^{-1}Rh|_n=o(\sqrt q),\\
K_{\mathrm{TAP}}
=R-\frac{4q\,Ryy^{\mathsf T}R/n}
{1+4q\langle yRy\rangle}+o(q^{-2}).
\end{gathered}
\end{equation}
Here $K_{\mathrm{TAP}}=\partial_h\tanh y_*(h)$, and the last error
is in operator norm. The errors tend to zero as the boundary scales
decrease, after which all dimensional errors and perturbation tolerances
are chosen.
If $q$ is comparable to $\sqrt t$, define the TAP potential per site by
\[
\mathcal T(h)=\langle hm\rangle+\tfrac12\langle mWm\rangle
-\langle I(m)\rangle+\tfrac14r^2,
\quad
I(v)=\tfrac12(1+v)\log(1+v)+\tfrac12(1-v)\log(1-v).
\]
With $\kappa_0=1/3$, the value expansion is
\begin{equation}\label{cap:tap-value}
\mathcal T(h)-\tfrac12\langle hRh\rangle
=\frac{r^2}{4}+\frac{zq}{2}-\langle I(m)\rangle
-\frac{\langle eRe\rangle}{2r^2}
=\frac14+\kappa_0q^3+o(q^3),
\qquad e=\tanh y-ry.
\end{equation}
All assertions persist to any prescribed small macroscopic error under
sufficiently small changes of the interaction coefficient, operator-norm
perturbations, and normalized field displacements, with the TAP potential
using the actual interaction and $jr^2/4$ at its actual root.
\end{proposition}
\begin{proof}
We first control the nonlinear source on the coordinates where a root
is large, uniformly over every approximate root in the specified region.
This gives initial localization bounds for the root and its derivative
tests. A second gain is then necessary: an error of order $q^2$ would still
be as large as the spectral separation. The later high-rotation calculation
provides that gain before we invert on the low bands.
Lemma~\ref{cap:endpoint-stability} at zero, applied with the small field
as an additional residual, first makes $q$ as small as required.
The field equation gives $|y|_n\le C(\sqrt q+\sqrt t)$.
Put
\begin{equation}\label{cap:clipping-scales}
s=q+\sqrt t,\qquad a=s^{7/5},\qquad
S=\{2-W\le a\},\qquad u=s^{1/10},\qquad
P_0=\mathbf1_{\{2-W\le u\}}.
\end{equation}
We only invert on $S^\perp$ until the lower bound for $q$ proves separation.
Dyadic bands there have gaps between $a$ and a constant. In all estimates
below $o_n(1)$ errors at fixed scales are chosen after the smallest scale and
may be below any specified power of $s$.
The derivative estimates below test vectors $w$ normalized by $|w|_n\le1$.
For a diagonal $A$ allowed by the stability test, put $v=Aw$ and
$\dot r=-2\langle mv\rangle$. Equation \eqref{cap:derivative-equation}
will specify the equation these tests satisfy; the present notation lets
us account for their clipping errors before that calculation.
The scalar variance profile satisfies
$V'_0(z)=-2\tanh z\,\operatorname{sech}^2z$. The cutoffs below are applied
to recipes built from this profile, including $V'_0(y)w$.
\paragraph{Clipping convention and its power budget.}
Write $T_s=s^{-\kappa}$, where $\kappa>0$ is chosen after the finite
clip-loss constant specified below. Use a smooth cutoff that is one on
$[-1,1]$, zero off $[-2,2]$, and has fixed bounded derivatives. Its
arguments are $y/(\sqrt sT_s)$, $w/T_s$, and a normalized linear test
divided by $T_s$. The normalized root source
$e/s^{3/2}$ and derivative source
$[(q-m^2)w-\dot r\,y]/s$ have degree-three envelopes on this region;
with a linear test they have degree-four envelopes. The normalized
scalar recipes for $m^2,m^4,m^6$, and $V'_0(y)w$ have envelopes of
degrees $2,4,6$, and $2$, respectively. In these statements ``degree''
means that every derivative through the order used is bounded by a
constant times that power of $T_s$. The frame comparison uses at most
two derivatives, and its differentiated Taylor tests use at most three.
Cutoff differentiation contributes a nonpositive power of $T_s$.
Thus all the needed envelopes are polynomial in $T_s$.
The proof uses a fixed finite sequence of operations: band projection and
dyadic summation, sparse inverses of norm at most two, two complexity
absorptions, replacement of high inputs by \(P_0\) inputs, and one
second-order Taylor expansion. The scalar moment and value estimates use the
same recipes and one further product or contraction. The polynomial
derivative bounds above and the logarithmic sizes of the Gaussian and Haar
nets therefore give each displayed error the form
\(Cs^\alpha T_s^{C_i}\), with a finite \(C_i\) independent of
\(\kappa,s,n\). The powers from \(a^{-1}\) and \(u^{-1}\) are displayed
as powers of \(s\); they are not included in \(C_i\).
There is a corresponding finite bound on the clipping exponents in the
absorption coefficients. In an inequality
\(X\le Cs^\alpha T_s^c+Cs^\theta T_s^dX\) with \(\theta>0\),
the second term is absorbed once \(\kappa d<\theta\) and the upper scale is
small. Let \(C_{\rm clip}\) dominate all the finitely many exponents
produced by these operations, including those \(d\). Choose
\begin{equation}\label{cap:clip-choice}
0<\kappa<\frac1{1000(1+C_{\mathrm{clip}})}.
\end{equation}
This choice precedes the upper boundary scale. Below, \(T_s^C\) denotes one
of these fixed losses. The smallest raw gain in the displayed estimates is
\(1/20\), so \eqref{cap:clip-choice} leaves a positive power in each
absorption and in the final errors.
The natural-size term in every sparse restriction is kept as
$CB\sqrt{f\log(2/f)}$, with no $T_s^C$ multiplier. Only its complexity term
incurs clipping losses. Thus threshold absorption uses
$\sqrt{\log(2/f)}/T_s\to0$ and is not circular. First separate fractions below
an arbitrarily high power of $s$, whose contributions already lie below the
target errors; otherwise $\log(2/f)=O(|\log s|)$. The finite remaining
logarithms fit into a further fixed power of $T_s$. The failure allowance
$\sqrt{\mathfrak h/n}=O(t^{3/4})\le O(s^{3/2})$ also lies below the errors for
which it is used.
\paragraph{The first localization of the root.}
We first localize the root strongly enough to make the nonlinear source
small on high spectral bands. This first estimate is intentionally weaker
than the final low-band cancellation: it controls the tail set and makes a
second approximation possible. The low-band root component and the filtered
shift form the charged part of the base vector; the known Gaussian noise is
included in the frame comparison without paying a vector-class complexity.
The exact high-band equation is
\begin{equation}\label{cap:root-band-equation}
y=b_y+R'_{\mathrm{hi}}e,\qquad
b_y=P_Sy+r^{-1}R_{\mathrm{hi}}h,\qquad
R'=r^{-2}R-r^{-1}I.
\end{equation}
Both $y$ and $b_y$ have normalized size $O(\sqrt s)$. The uncharged
noise term has band norm $O(s d^{-1/4})$. The low-band vector and the
filtered shift give
\begin{equation}\label{cap:base-root-complexity}
\mathscr L(b_y)
\le C\{\sqrt s\,a^{3/4}+s^2a^{-1/4}\}
=C\{s^{31/20}+s^{33/20}\},
\end{equation}
up to logarithms. Let $L_y$ be this complexity plus the complexities of
the correction bands in \eqref{cap:root-band-equation}. Initially
$L_y=O(\sqrt s)$ is sufficient.
Choose a smooth scalar cutoff $0\le\chi\le1$ equal to one on $[-1,1]$ and
zero outside $[-2,2]$, and put
$e_0(y)=\chi(y/(\sqrt sT_s))(\tanh y-ry)$ coordinatewise.
Then $|e_0|\le Cs^{3/2}T_s^3$ coordinatewise and $|e'_0|\le CsT_s^2$.
On the transition and tail coordinates, $\tanh y/y\le r$ for sufficiently
small $s$, so
\[
e-e_0=-D_Ey,\qquad 0\le D_E\le rI,
\qquad E\subseteq\{|y|>\sqrt sT_s\}.
\]
The sparse restriction estimate, with its first term absorbed against $\sqrt
sT_s\sqrt f$, gives $\sqrt f\le Cs^{-1/2}T_s^{-1}L_y$, apart from the already
reserved tiny errors. Testing the clipped source on an arbitrary normalized
unit vector in a band gives
\begin{equation}\label{cap:clipped-root-source}
|\Pi_de_0|_n\le CT_s^C
\{s^{3/2}d^{3/4}+sL_y\}.
\end{equation}
Here is the cancellation in this estimate. Gaussian integration by parts
predicts its linear projection with coefficient $\mathbb E e'_0(Y)$, where
$\operatorname{Var}Y=|y|_n^2$. The identity $\langle q-m^2\rangle=0$ and the
frame comparison bound the corresponding unclipped coefficient by
\(CT_s^C\sqrt sL_y\). For the cutoff error, the threshold estimate above
and sparse restriction give
\[
\sqrt f\le Cs^{-1/2}T_s^{-1}L_y,\qquad
|\mathbf1_Ey|_n\le CT_s^CL_y .
\]
Since the coarse input is \(L_y=O(\sqrt s)\),
\[
|\mathbf1_Ey|_n^2+sf
\le CT_s^CL_y^2\le C\sqrt s\,T_s^CL_y .
\]
This bounds the change in \(\mathbb E e'_0(Y)\). Multiplication by
\(|\mathbb E[YU]|\le C\sqrt s\) for the normalized Gaussian band test \(U\)
gives the \(sL_y\) term in
\eqref{cap:clipped-root-source}. The centered part has band-complexity cost
\(Cs^{3/2}T_s^Cd^{3/4}\). Clipping a linear test
first and then removing its clip costs its complexity times the size of the
bounded source, again by \eqref{cap:sparse-restriction}.
Summing \eqref{cap:clipped-root-source} with weight $d^{-1/2}$ yields
\[
|R_{\mathrm{hi}}^{1/2}e_0|_n+|e_0|_n
\le CT_s^C(s^{3/2}+sa^{-1/2}L_y).
\]
The tail equation is coercive: on the high bands,
$I+D_E^{1/2}R'_{\mathrm{hi}}D_E^{1/2}$ dominates a fixed positive multiple of
$I+D_E^{1/2}R_{\mathrm{hi}}D_E^{1/2}$. Indeed, the initial bound $|y|_n\le
C\sqrt s$ already gives $f=|E|/n\le CT_s^{-2}$. The uniform sparse restriction
bound on $S$ therefore implies
\[
\|P_S\mathbf1_E\|
\le C\{a^{3/4}+\sqrt{f\log(2/f)}\}+o_n(1)=o(1),
\]
before any improved tail estimate. Put $X_E=D_E^{1/2}R_{\mathrm{hi}}D_E^{1/2}$
and $Q_E=D_E^{1/2}P_SD_E^{1/2}$. Since $D_E$ is supported on $E$, $\|Q_E\|\le
r\|P_S\mathbf1_E\|^2=o(1)$, and $R_{\mathrm{hi}}\succeq cP_{\mathrm{hi}}$
gives $X_E\succeq cD_E-o(1)I$. Thus, for a sufficiently small fixed
$\theta>0$,
\[
\begin{aligned}
&I+D_E^{1/2}R'_{\mathrm{hi}}D_E^{1/2}-\theta X_E\mathstrut\\
&\quad=I-r^{-1}D_E+r^{-1}Q_E+(r^{-2}-\theta)X_E\\
&\quad\succeq I+\{c(r^{-2}-\theta)-r^{-1}\}D_E-o(1)I\\
&\quad\succeq c'I,
\end{aligned}
\]
where \(c'>0\) is fixed because \(D_E\le rI\) and \(r\to1\). This
domination uses only the coarse tail fraction.
To apply it, put \(u_E=D_E^{1/2}y\). Substituting \(e=e_0-D_Ey\) into
\eqref{cap:root-band-equation} gives the exact tail equation
\[
(I+D_E^{1/2}R'_{\rm hi}D_E^{1/2})u_E
=D_E^{1/2}b_y+D_E^{1/2}R'_{\rm hi}e_0 .
\]
Test against \(u_E\) and use the lower bound by a fixed multiple of
\(I+X_E\). Decomposing \(R'_{\rm hi}=r^{-2}R_{\rm hi}-r^{-1}P_{\rm hi}\)
and applying Cauchy--Schwarz in that energy norm bounds the right side by
the dual sizes \(|D_E^{1/2}b_y|_n\), \(|e_0|_n\), and
\(|R_{\rm hi}^{1/2}e_0|_n\). Since \(D_E\le rI\), the resulting estimate is
\begin{equation}\label{cap:root-tail-energy}
|D_Ey|_n+|R_{\mathrm{hi}}^{1/2}D_Ey|_n
\le C\{|D_E^{1/2}b_y|_n+|e_0|_n
+|R_{\mathrm{hi}}^{1/2}e_0|_n\}.
\end{equation}
The first term is $o(1)L_y+C\mathscr L(b_y)$ by sparse restriction. The high
correction complexity is bounded by the right side of
\eqref{cap:root-tail-energy}; the coefficient $sa^{-1/2}=s^{3/10}$ absorbs
after clipping losses. We conclude
\begin{equation}\label{cap:first-root-localization}
\begin{gathered}
L_y\le Cs^{3/2}T_s^C,\qquad f\le Cs^2T_s^C,\\
|y-b_y|_n\le CT_s^C
\{s^{3/2}a^{-1/4}+s^{5/2}a^{-1}\}
\le Cs^{11/10}T_s^C.
\end{gathered}
\end{equation}
For the last bound, the sparse part is tested bandwise using
$\|\Pi_d\mathbf1_E\|\le CT_s^C(d^{3/4}+s)$; the nonsparse part uses
\eqref{cap:clipped-root-source}. This also shows $|e|_n\le Cs^{3/2}T_s^C$.
\paragraph{Localization of derivative tests.}
Normalize $|w|_n\le1$, put $v=Aw$ and
$\dot r=-2\langle mv\rangle$, and consider
\begin{equation}\label{cap:derivative-equation}
w+(rI-W)v+\dot r\,m=h_v-\ell v,
\qquad |h_v|_n\le Cs^2,\quad 0\le\ell\le Cs.
\end{equation}
This includes possible nonpositive stability eigenvalues, after a scalar
shift, and the response to bounded sources. Set $R_\ell=(z+\ell-W)^{-1}$,
$R'_\ell=r^{-2}R_\ell-r^{-1}I$, and $d_v=(A-rI)w-\dot r\,y$. Exact elimination
gives
\begin{equation}\label{cap:derivative-band-equation}
\begin{split}
w&=b_w+R'_{\ell,\mathrm{hi}}d_v,\\
b_w&=P_Sw+r^{-1}R_{\ell,\mathrm{hi}}h_v\\
&\quad+\dot r\{R_{\ell,\mathrm{hi}}((r^{-2}-1)y-e/r)
-r^{-1}P_{\mathrm{hi}}y\}.
\end{split}
\end{equation}
Using the noise band sizes, \eqref{cap:first-root-localization}, and
$|\dot r|\le C\sqrt s$ gives
\[
\mathscr L(b_w)\le CT_s^C
(a^{3/4}+s^{3/2}+s^2a^{-1/4}),
\qquad |P_{\mathrm{hi}}b_w|_n
\le CT_s^C(s+s^2/a).
\]
The first bound starts at $s^{21/20}$; this small gain will be retained.
Let $L_w$ include the complexity of the correction bands.
Split $d_v=d_0+D'_Ew+e_A$, where $d_0$ uses the bounded clipped coefficient
$q-m^2$ and retains $-\dot r\,y$. Here $D'_E$ is bounded and supported on the
tail set; $e_A$ is an arbitrarily small normalized error. To obtain this
decomposition for all allowed $A$, fix a diagonal sup-norm tolerance
$\delta_A$, split $A-V$ outside that tolerance, and use
\[
\frac{|\{i:|A_{ii}-V_{ii}|>\delta_A\}|}{n}
\le\epsilon_A^2/\delta_A^2.
\]
At fixed bottom scale choose $\delta_A$ and then $\epsilon_A$ so that the
first part is below every requested error and the added fraction is below
every requested power of $s$. This choice is made before $n$, and the frame
tests remain simultaneous over the added sparse set.
Uniformly on the high bands,
\begin{equation}\label{cap:clipped-derivative-source}
|\Pi_dd_0|_n\le CT_s^C
\{sd^{3/4}+sL_w+\sqrt sL_y\}.
\end{equation}
For jointly centered Gaussian surrogates $Y,G,U$, Gaussian integration
by parts gives the exact identity
\[
\mathbb E[\operatorname{sech}^2(Y)GU]
=\mathbb E\operatorname{sech}^2(Y)\,\mathbb E[GU]
+\mathbb E[(\operatorname{sech}^2)'(Y)G]\,\mathbb E[YU].
\]
Centering by $r$ removes the first term; $\dot r$ removes the second. The
frame comparison for the latter scalar coefficient has natural size $\sqrt s$,
and its unclipping cost is bounded by $C|\mathbf1_Ey|_n|\mathbf1_Ew|_n$. This
proves \eqref{cap:clipped-derivative-source}; the diagonal error is handled by
the preceding split. Removing clips on its linear factors uses sparse
Cauchy--Schwarz, so it charges the sum of their complexities, not their full
norms.
The compression of \(M=R'_{\ell,\mathrm{hi}}\) to the enlarged sparse set
has operator norm \(o(1)\). Its spectral mean is \(O(\sqrt a+\sqrt s)+o_n(1)\), its
Frobenius norm divided by $\sqrt n$ is at most $Ca^{-1/4}$, and its operator
norm is at most \(C/a\). Put
\(r_f=f\log(2/f)+\mathfrak h/n\). The Haar quadratic-form bound and the
sparse nets give
\[
\|\mathbf1_E M\mathbf1_E\|
\le \frac{|\Tr M|}{n}
+C a^{-1/4}\sqrt{r_f}+C a^{-1}r_f+o_n(1).
\]
Since \(f\le Cs^2T_s^C\), \(\mathfrak h/n=O(t^{3/2})=O(s^3)\), and
\(a=s^{7/5}\), the two deviation terms have raw powers at least
\(s^{13/20}\) and \(s^{3/5}\). The clip choice makes them tend to zero.
Restricting \eqref{cap:derivative-band-equation} to the sparse coordinates
gives
\[
(I-\mathbf1_E M\mathbf1_E D'_E)\mathbf1_Ew
=\mathbf1_Eb_w+\mathbf1_E M(d_0+e_A).
\]
Because \(D'_E\) is bounded, the inverse on the left has norm at most two
after the upper scale is decreased. The \(e_A\) term is below the reserved
error. Bandwise testing of the other term gives
\begin{equation}\label{cap:derivative-tail-energy}
\begin{split}
|\mathbf1_Ew|_n\le C\bigg[|\mathbf1_Eb_w|_n+
T_s^C\sum_{d\ge a}d^{-1}(d^{3/4}+\sqrt f)
(sd^{3/4}+sL_w+\sqrt sL_y)\bigg].
\end{split}
\end{equation}
The factor multiplying $sL_w$ sums to $O(a^{-1/4}+sT_s^Ca^{-1})$, before the
displayed clipping loss. Multiplication by $s$ thus gives a positive power;
the sparse-source complexity costs only $C(1+sa^{-1/4})$, apart from clips and
logarithms. Together with the preceding bound for $\mathscr L(b_w)$ this
absorbs the $L_w$ terms and proves
\begin{equation}\label{cap:first-derivative-localization}
L_w\le CsT_s^C,\qquad
|P_{\mathrm{hi}}w|_n
\le CT_s^C(sa^{-1/4}+s^2/a)
\le Cs^{3/5}T_s^C.
\end{equation}
\paragraph{A second gain from the independent high rotation.}
The first root and derivative estimates have not yet beaten the spectral
separation $s^2$. We now separate each vector into a part with smaller complexity
and a structured high-frequency correction. The middle bands provide a
power gain by summation. On the remaining high bands, replacing each source
by a recipe of its low-frequency inputs leaves the high frame available for
a signed estimate. Full operator-norm bounds would lose this gain, so the
contractions are estimated before taking absolute operator norms.
We now prove a stronger decomposition,
with some fixed $\zeta>0$; one may take $\zeta=1/100$ after
\eqref{cap:clip-choice}:
\begin{equation}\label{cap:refined-decomposition}
\begin{array}{ll}
y=B_y+R'_{>u}t_y,&
|t_y|_n\le Cs^{3/2}T_s^C,\quad
\mathscr L(B_y)\le Cs^{3/2+\zeta},\\
w=B_w+R'_{\ell,>u}t_w,&
|t_w|_n\le CsT_s^C,\quad
\mathscr L(B_w)\le Cs^{1+\zeta}.
\end{array}
\end{equation}
As before, the filtered noise is included but not charged in $\mathscr
L(B_y)$. To specify the recipes, put $Y_0=P_0y$, $W_0=P_0w$ and let $\chi$ be
the preceding scalar cutoff. We may take
\begin{equation}\label{cap:low-input-recipes}
\begin{split}
t_y&=\chi\left(\frac{Y_0}{\sqrt sT_s}\right)
(\tanh Y_0-rY_0),\\
t_w&=\chi\left(\frac{Y_0}{\sqrt sT_s}\right)
\chi\left(\frac{W_0}{T_s}\right)
\{(\operatorname{sech}^2Y_0-r)W_0-\dot r\,Y_0\}.
\end{split}
\end{equation}
All products here are coordinatewise. Small operator and sparse $A-V$ errors
are assigned to $B_w$, using the tail equations below. Net the scalar
parameters on their bounded ranges and let the low inputs vary over all
vectors of the stated sizes in $P_0$. Once these coefficients and the low
columns have been fixed, the recipes are determined. The high columns are
still free to rotate, which is the independence used in the next signed
estimate.
First sum the source bounds on $a\le d\le u$. Both the weighted
complexity and the norm restricted to a fraction
$f\le s^2T_s^C$ gain, relative to the source size, the factor
\begin{equation}\label{cap:middle-band-gain}
T_s^C(u^{1/2}+sa^{-1/4}+s^2a^{-1})
=T_s^C(s^{1/20}+s^{13/20}+s^{3/5}).
\end{equation}
This follows by multiplying \eqref{cap:clipped-root-source} and
\eqref{cap:clipped-derivative-source} by $d^{-1}(d^{3/4}+\sqrt f)$ and summing
dyadically. The same bounds with only the complexity weight prove the claimed
gain in complexity. Above $u$ we replace the bounded sources by
\eqref{cap:low-input-recipes}. The necessary input estimates are
\begin{equation}\label{cap:high-input-bounds}
\frac{|P_{>u}y|_n}{\sqrt s}
\le CT_s^C(s^{19/40}+s^{3/5}),
\qquad |P_{>u}w|_n\le CT_s^Cs^{3/5}.
\end{equation}
For the first, the filtered noise contributes $s u^{-1/4}$ before division by
$\sqrt s$, and \eqref{cap:first-root-localization} contributes $s^{11/10}$;
the second follows from \eqref{cap:first-derivative-localization}. The source
derivatives have natural sizes $s$ for the root source, $s$ in the $w$
variable for the derivative source, and $\sqrt s$ in its $y$ variable after
clipping $w$. Including the high inverse norm $C/u$, the replacement errors
are at most
\[
CT_s^Cs^{15/8}\quad\hbox{for }y,\qquad
CT_s^C(s^{11/8}+s^{3/2})\quad\hbox{for }w.
\]
These are smaller than the gained sizes in \eqref{cap:refined-decomposition}.
Clipping the linear $w$ input at this stage uses its first sparse
localization: the bounded coefficient has size $sT_s^C$, and the resulting
error after division by $u$ is $O(s^{19/10}T_s^C)$. Thus the refined joint
tail estimate is not being assumed in constructing these recipes.
We justify the signed estimate for the structured high terms. Conditional on
$P_0$ and its frame, the remaining rotation is Haar. For either high
multiplier $M=R'_{>u}$ or $R'_{\ell,>u}$,
\begin{equation}\label{cap:high-multiplier-sizes}
\left|\frac1n\operatorname{Tr}M\right|
\le C(\sqrt u+\sqrt s)+o_n(1),\qquad
\frac{\|M\|_{\mathrm F}}{\sqrt n}\le Cu^{-1/4},
\qquad \|M\|\le C/u.
\end{equation}
Net the low coefficient balls and the bounded scalar parameters before
substituting the actual, possibly frame-dependent coefficients. At polynomial
resolution in \(s\), their logarithmic cardinality is
\(O(nu^{3/2}|\log s|)\). Here the fixed complement has dimension
\(m=n-O(nu^{3/2})\asymp n\). Equation~\eqref{cap:conditional-high-test}
bounds the deviation from the exact conditional mean
\(\mu\langle P_\perp b,P_\perp t\rangle\), with \(\mu=\Tr D/m\), by
\[
C\left(u^{-1/4}\sqrt{u^{3/2}|\log s|}
+u^{-1}u^{3/2}|\log s|\right)
\le Cu^{1/2}T_s^C
\]
times the two vector sizes. The mean itself is bounded by
\(C(\sqrt u+\sqrt s)+o_n(1)\) from \eqref{cap:high-multiplier-sizes}.
The estimate is uniform over the nets and then over the full
classes by their derivative bounds. Adding sparse coordinate unit tests costs
$O(nf\log(2/f))$, which is smaller than this entropy. Applying this test to a
sparse output gives a factor $s^{1/20}T_s^C$ relative to the source norm.
Applying it to another low-input recipe gives the same factor relative to the
product of the two norms. These are the two forms of the gain used below. The
operator norm $u^{-1}$ is not charged a second time to a signed contraction:
it already appears in the second term of the quadratic-form estimate above.
The known high-noise column causes no conditioning problem here. It was
included in the finite-column frame comparison, where its spectral
coefficients are independent of the frame. For the present signed tests, first
replace each differentiated input by its $P_0$ version. Estimate the discarded
part in full norm, including the high-noise contribution $s^{19/40}$ in
\eqref{cap:high-input-bounds}. The power calculation below puts this error at
higher order. The remaining recipe no longer uses the high-noise image. The
retained recipes therefore depend only on the low frame. This distinction is
essential: we do not condition on the image of the high noise and then call
the remaining rotation unrestricted Haar.
It remains to improve the unclipped sparse sources. First separate the
exceptional \(A-V\) coordinates using the previously chosen diagonal
tolerances; their contribution is below the errors in this paragraph.
On the root tail \(E\), the exact restriction is
\[
(I+\mathbf1_E R'_{\mathrm{hi}}\mathbf1_E D_E)\mathbf1_Ey
=\mathbf1_Eb_y+\mathbf1_E R'_{\mathrm{hi}}e_0.
\]
The preceding sparse compression estimate includes \(\ell=0\), so the
inverse on the left has norm at most two. This is the unweighted sparse
inverse; the earlier energy estimate on \(D_Ey\) alone would not control
\(\mathbf1_Ey\). The middle-band gain and the structured sparse-output
estimate then give
\[
|\mathbf1_Ey|_n
\le C\sqrt s\sqrt{f\log(2/f)}+Cs^{3/2+1/20}T_s^C.
\]
The threshold \(\sqrt sT_s\sqrt f\le|\mathbf1_Ey|_n\) absorbs the first
term, with its natural-size coefficient unchanged. Hence
\[
\sqrt f\le Cs^{1+1/20}T_s^C,\qquad
|\mathbf1_Ey|_n\le Cs^{3/2+1/20}T_s^C.
\]
The derivative sparse equation uses this same root-coefficient tail,
together with the separately controlled \(A-V\) set. Its base restriction
is now bounded by
\(C\sqrt{f\log(2/f)}+Cs^{1+1/20}T_s^C\). The inverse of norm at most two
and the middle and structured output gains therefore give
\[
|\mathbf1_Ew|_n
\le C\sqrt{f\log(2/f)}+Cs^{1+1/20}T_s^C
\le Cs^{1+1/20}T_s^C.
\]
For a sparse source of row norm \(B_E\), its contribution to the weighted
high-band complexity is at most \(B_E T_s^C\) times
\[
\sum_{d\ge a}d^{-1}d^{3/4}(d^{3/4}+\sqrt f)
\le C(1+\sqrt f\,a^{-1/4}),
\]
by the sparse band test and dyadic summation. Thus the improved row norms
give the raw bounds
\begin{equation}\label{cap:refined-raw-complexity}
\mathscr L(B_y)\le Cs^{3/2+1/20}T_s^C,\qquad
\mathscr L(B_w)\le Cs^{1+1/20}T_s^C.
\end{equation}
The middle-band and replacement errors have at least these powers.
The clip choice weakens \eqref{cap:refined-raw-complexity} to
\eqref{cap:refined-decomposition} with \(\zeta=1/100\).
Separate tail estimates are insufficient for the products used in a derivative
test. We therefore clip the root, derivative vector, and low-band test on the
union of their tail sets. The refined decomposition bounds all three vectors
on this same set. Their natural-size terms can be absorbed against the
clipping thresholds, leaving a positive power gain for every product discarded
when the clips are removed.
If $z_S$ is any normalized unit vector in $S$, then $\mathscr L(z_S)\le
Ca^{3/4}=Cs^{21/20}$. Let $E$ be the union of the tails of $y/\sqrt s$, $w$,
and $z_S$, each at threshold $T_s$, together with the sparse $A-V$ set. The
first bounds put its fraction below \(s^2T_s^C\). The raw bounds
\eqref{cap:refined-raw-complexity}, the structured sparse-output estimate,
and sparse restriction give
\begin{equation}\label{cap:joint-tail-inequalities}
\begin{split}
|\mathbf1_Ey|_n&\le
C\sqrt s\sqrt{f\log(2/f)}+Cs^{3/2+1/20}T_s^C,\\
|\mathbf1_Ew|_n+|\mathbf1_Ez_S|_n&\le
C\sqrt{f\log(2/f)}+Cs^{1+1/20}T_s^C.
\end{split}
\end{equation}
The threshold inequality, after removing the arbitrarily smaller
$A-V$ fraction, is
\[
T_s\sqrt f\le
|\mathbf1_Ey|_n/\sqrt s
+|\mathbf1_Ew|_n+|\mathbf1_Ez_S|_n.
\]
The unmultiplied natural-size terms in
\eqref{cap:joint-tail-inequalities} absorb. Therefore
\begin{equation}\label{cap:joint-tail-gain}
\sqrt f\le Cs^{1+1/20}T_s^C,\qquad
|\mathbf1_Ey|_n\le Cs^{3/2+1/20}T_s^C,\qquad
|\mathbf1_Ew|_n+|\mathbf1_Ez_S|_n
\le Cs^{1+1/20}T_s^C.
\end{equation}
For an explicit permitted diagonal tolerance, split $A-V$ at $s^{20}$ and take
its normalized Frobenius tolerance at most $s^{40}$. The exceptional fraction
is then at most $s^{40}$; its contribution is smaller than all the displayed
errors. Across a fixed interval of boundary scales, use its smallest scale in
these choices.
\paragraph{The low-band cancellations.}
All parts needed for the low-band estimate are now available. The base
vectors have small enough complexity for comparison with Gaussian
averages; the linear structured corrections use the signed high-rotation
bound; and the quadratic Taylor remainder is smaller still. We list the
powers separately before removing the joint clips. After these errors have
been bounded, the Gaussian predictions cancel because the overlap and its
derivative were defined from the same root.
\begin{equation}\label{cap:low-source-cancellation}
|P_Se|_n=o(s^{5/2}),\qquad |P_Sd_v|_n=o(s^2).
\end{equation}
We give the power ledger explicitly. Normalize the variables as $y/\sqrt
s,w,z_S$. The base complexities in \eqref{cap:refined-decomposition} are
$s^{1+\zeta}$, $s^{1+\zeta}$, and $s^{21/20}$. Applying the frame comparison
to the clipped averages therefore gains more than one power of $s$ relative to
the source sizes $s^{3/2}$ and $s$. The structured normalized perturbations
have source norm $sT_s^C$ and full norm at most $s/u$ times a clipping loss.
Their second-order Taylor remainder consequently has relative size
$s^{9/5}T_s^C$. For a linear term use the signed high contraction, giving
relative size $su^{1/2}T_s^C=s^{21/20}T_s^C$. Replacing its differentiated
recipe by a $P_0$-input recipe has relative error at most
$su^{-1}(s^{19/40}+s^{3/5})T_s^C$. The resulting exponents, before clip
losses, are
\begin{equation}\label{cap:boundary-power-ledger}
\begin{array}{c|c}
\text{operation}&\text{relative power of }s\\ \hline
\text{base frame comparison}&1+\min(\zeta,1/20)\\
\text{structured linear contraction}&21/20\\
\text{second-order Taylor remainder}&9/5\\
\text{high-input replacement}&11/8\quad\text{and}\quad3/2.
\end{array}
\end{equation}
The same calculation controls changes of Gram entries in the Gaussian
predictions; the discarded high parts are included in the replacement error.
The differentiated recipes are exactly derivatives, through order two, of $e
z_S$ and $d_v z_S$, and the scalar recipes $m^2,V'_0(y)w,m^4$. Their
normalized polynomial envelopes have degree at most six. The sixth moment is
needed only to leading order, for which the first localization and a crude
perturbation bound suffice; no differentiated degree-eight recipe is required.
Removing the joint clips costs, by
\eqref{cap:joint-tail-gain},
\[
|\langle\mathbf1_Ee z_S\rangle|
\le Cs^{13/5}T_s^C,\qquad
|\langle\mathbf1_Ed_v z_S\rangle|
\le Cs^{21/10}T_s^C.
\]
We used $|e|\le C|y|$, boundedness of $m,V,A$, and $|\dot r|\le C\sqrt s$. The
sparse $A-V$ term obeys the second bound by the same joint row-energy product.
Scalar unclipping uses $m^2\le y^2$ and $|V'_0(y)|\le2|y|$; its products obey
the identical ledger. Gaussian unclipping has superpolynomially small tails
because the normalized surrogate variances are bounded.
We have now bounded the comparison, Taylor, and unclipping errors. The
Gaussian variables \(Y,G,U\) in the remaining calculation are centered and
have the current empirical Gram matrix of \(y,w,z_S\); they are the
full-frame surrogates, rather than the noncentral scalar equality laws used
earlier. The scalar comparisons and the justified replacement of \(A\) by
\(V\) give the precise target errors
\[
|q-\E\tanh^2Y|=o(s^2),\qquad
|\dot r-\E[V'_0(Y)G]|=o(s^{3/2}).
\]
Indeed the natural scalar sizes are \(s\) and \(\sqrt s\), and their
comparison and unclipping errors gain more than one power of \(s\).
Gaussian integration by parts now gives
\[
\begin{split}
\E[(\tanh Y-rY)U]
&=(q-\E\tanh^2Y)\E[YU],\\
\E[((V_0(Y)-r)G-\dot rY)U]
&=(q-\E\tanh^2Y)\E[GU]
+(\E[V'_0(Y)G]-\dot r)\E[YU].
\end{split}
\]
The Gram bounds are \(|\E[GU]|\le C\) and \(|\E[YU]|\le C\sqrt s\).
The two displays therefore give \(o(s^{5/2})\) and \(o(s^2)\),
respectively. Every entry of
\eqref{cap:boundary-power-ledger} is strictly greater than one, and each
unclipping exponent is strictly above its target. By \eqref{cap:clip-choice}
the finite clipping losses leave positive margins. The supremum over unit
$S$-tests proves \eqref{cap:low-source-cancellation}. Quantitatively, after
the fixed-scale dimensional errors are decreased, its two right-hand sides are
bounded by $Cs^{5/2+1/200}$ and $Cs^{2+1/200}$, respectively; the smallest
remaining margin is $\zeta-1/1000>1/200$. This also proves the scalar
comparison for $m^4$ with error $o(s^3)$ and for $m^6$ to leading order.
\paragraph{Separation, stability, and response.}
Suppose \(q\le\varepsilon\sqrt t\), where \(\varepsilon>0\) will be fixed
below the field lower-bound constant.
The exact field equation is
\[
h=r(z-W)y+(r-W)e.
\]
On the band \(2-W\le t\), its first multiplier is at most \(C(t+q^2)\)
in norm. Also \(s\asymp\sqrt t\) under the supposition, and
\(t\le s^2\le s^{7/5}=a\) at small scale, so
\(P_t^W\preceq P_S\). The norm bound for \(y\) and
\eqref{cap:low-source-cancellation} therefore give
\[
|P_t^Wh|_n
\le\bigl(C\sqrt\varepsilon+o_t(1)+o_n(1)\bigr)t^{5/4}.
\]
Choose \(\varepsilon\) so that \(C\sqrt\varepsilon\) is below half the
prescribed field lower-bound constant, then decrease the upper boundary
scale, and finally the dimensional error. This contradicts the field lower
bound, proving \(q\ge c\sqrt t\). In particular \(s\) is now comparable to
\(q\), and the identity
$z-2=q^2/(1-q)$ gives the positive separation needed to use the full
resolvent: $R>0$ and $\|R\|\le Cq^{-2}$. Inverting on all bands now gives
$|y-r^{-1}Rh|_n=o(\sqrt q)$ and $|m-Rh|_n=o(\sqrt q)$. Replacing $z$ by
$2+q^2$ incurs only a relative $o(1)$ error; consequently $q$ approximately
solves the scalar equation
\[
q=|(2+q^2-W)^{-1}h|_n^2.
\]
Its right-hand side is decreasing in \(q>0\); equivalently,
\(q-|(2+q^2-W)^{-1}h|_n^2\) has derivative at least one.
For derivative tests, use \eqref{cap:derivative-band-equation},
\eqref{cap:first-derivative-localization}, and
\eqref{cap:low-source-cancellation} to obtain, uniformly under
\eqref{cap:derivative-equation},
\begin{equation}\label{cap:rank-response}
w=r^{-1}R_\ell h_v
-4qR_\ell y\langle yw\rangle+o(1)
\quad\text{in normalized norm}.
\end{equation}
To see the coefficient, substitute $v=rw+\dot r\,y+d_v$ into
\eqref{cap:derivative-equation} and use $r-r^{-1}=-2q+O(q^2)$, $\dot
r=-2\langle yw\rangle+o(\sqrt q)$. The unenhanced term $\dot r\,y$ is $O(q)$,
whereas the resolvent term has coefficient $2q\dot r=-4q\langle
yw\rangle+o(q^{3/2})$. The low source errors are multiplied by at most
$Cq^{-2}$; their strict power gain therefore leaves an $o(1)$ error. The
high-band errors are controlled by the preceding localization estimates. This
justifies the remainder in the rank-response formula uniformly over the tests.
The rank-one operator $I+4qR_\ell yy^{\mathsf T}/n$ has uniformly bounded
inverse: its denominator is at least one, and its numerator correction has
norm at most $4q\|R_\ell\||y|_n^2\le C$. Thus \eqref{cap:rank-response} with
zero source excludes a normalized unit solution. If the stability matrix had a
nonpositive eigenvalue, increase $r-W$ by $\ell I$ until its minimum
eigenvalue reaches zero. The norm bound, $A\le(1+\eta)I$, and $q=O(s)$ ensure
$0\le\ell\le Cs$, choosing $\eta$ sufficiently small at the fixed bottom
scale. For invertible $A$, writing its eigenvector as $A^{1/2}w$ produces
exactly the zero-source test above, a contradiction. Likewise a positive
eigenvalue $\varepsilon\ll q^2$ at $\ell=0$ produces the source
$h_v=\varepsilon w$ and contradicts the same inverse estimate. Slightly
smaller tolerances and continuity include noninvertible \(A\). Hence the
stability test has positive fixed slack. The deterministic root lemma now
gives a unique root, and the implicit function theorem defines its response
on this stable field family. For \(A=V\) and arbitrary sources of
size $O(q^2)$, \eqref{cap:rank-response} first bounds the response and then,
by the rank-one inverse formula, gives \eqref{cap:tap-leading};
$|(I-V)w|_n=o(1)$ by the localization just proved.
\paragraph{The value of the potential.}
The root and response estimates determine the local curvature. To build a
single extension at initialization we also need to compare values at
separated fields. We therefore expand the stationary TAP potential through
cubic order in the overlap. The scalar fourth and sixth moments come from
the same Gaussian comparison, while the quadratic source correction uses
the low-band cancellation and the signed high-rotation estimate already
proved. Their coefficients must be combined before discarding the lower
powers.
The stationary TAP expression has gradient $m$. Since $h=(z-W)m-e/r$,
completing the square gives the first equality in \eqref{cap:tap-value}
exactly. Joint tail control permits the entropy expansion
\[
\langle I(m)\rangle
=q/2+\langle m^4\rangle/12+\langle m^6\rangle/30+o(q^3).
\]
The scalar comparisons just established express these moments by their
Gaussian Taylor expansions. If $\tau=|y|_n^2$ denotes the matching
variance, then
\[
\tau=q+2q^2+o(q^2),\quad
\langle m^4\rangle=3q^2-8q^3+o(q^3),\quad
\langle m^6\rangle=15q^3+o(q^3).
\]
For the quadratic correction, the immediate goal is
\(\langle eR'e\rangle=o(q^3)\). The low spectral region contributes at
most \(Cq^{-2}|P_Se|_n^2=o(q^3)\) by
\eqref{cap:low-source-cancellation}. On \(a<d\le u\), the clipped source
\(e_0\) contributes \(O(q^3u^{1/2}+q^5/a)T_s^C=o(q^3)\) by
\eqref{cap:clipped-root-source}.
Keep the coordinate-sparse source \(e_{\rm sp}=e-e_0\) separate on all
bands \(d>a\). Its improved row norm is
\(|e_{\rm sp}|_n\le Cq^{31/20}T_s^C\). The bounded sparse compression of
\(R_{\rm hi}\), together with \(R'=r^{-2}R-r^{-1}I\), bounds its
quadratic form by \(O(q^{31/10}T_s^C)=o(q^3)\). The bandwise sparse estimate
following \eqref{cap:first-root-localization} and the improved row bound
make its cross terms on the middle bands \(o(q^3)\). Here the spectral projection
belongs to the operator: for example,
\(\langle P_{>u}e_{\rm sp},R'P_{>u}e_{\rm sp}\rangle
=\langle e_{\rm sp},R'_{>u}e_{\rm sp}\rangle\). The vector tested by the
sparse estimate is still \(e_{\rm sp}\).
Above \(u\), write \(e_0=t_y+e_{\rm rep}\), where
\(e_{\rm rep}=e_0(y)-e_0(Y_0)\); the definition of \(t_y\) in
\eqref{cap:low-input-recipes} uses the same cutoff \(\chi\).
For the structured term,
\(|t_y|_n\le Cq^{3/2}T_s^C\), and the signed high test gives
\[
|\langle t_yR'_{>u}t_y\rangle|
\le Cq^{1/20}T_s^C|t_y|_n^2=o(q^3).
\]
The high-input replacement estimates give
\(|R'_{>u}e_{\rm rep}|_n\le Cq^{15/8}T_s^C\) and
\(|e_{\rm rep}|_n\le Cq^{79/40}T_s^C\). Its cross term with \(t_y\)
is therefore \(O(q^{27/8}T_s^C)\), and its self term is of still higher
order. For the cross term with \(e_{\rm sp}\), the structured sparse-output
bound gives
\(|\mathbf1_E R'_{>u}t_y|_n\le Cq^{31/20}T_s^C\), while the full
replacement bound above handles \(R'_{>u}e_{\rm rep}\). Multiplication by
\(|e_{\rm sp}|_n\) gives powers \(31/10\) and \(137/40\), respectively.
All exceed three after the clipping losses. This proves
\(\langle eR'e\rangle=o(q^3)\). It follows from \(R=r^2R'+rI\) that
\[
\langle eRe\rangle=r\langle e^2\rangle+o(q^3)
=\tfrac23q^3+o(q^3).
\]
The last scalar coefficient follows by expanding $e=qY-Y^3/3+O(Y^5)$ under the
matched Gaussian law. Substitution cancels the quadratic terms and proves
\eqref{cap:tap-value}, with $\kappa_0=1/3$. This coefficient is used in
the own-saddle support comparison at initialization.
Residuals at approximate roots and changes of coefficient or matrix add small
normalized sources to the two exact equations used above. At a fixed bottom
scale all inverses, clip derivatives, and tests have fixed bounds. Choose
these perturbations after the displayed scales and error targets, and repeat
the estimates with their reserved slack. The TAP expression then uses the
actual $J_*$ and $jr^2/4$. This proves the final perturbation assertion and
completes the proof. \end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-classical.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-entropy.tex ====
\section{Entropy transfer along the observation path}\label{cap:entropy}
The curvature estimates of Theorem~\ref{cap:true-curvature} control the
information revealed by the edge observations. We now use that control to
retain enough relative entropy until the posterior becomes a product measure.
The product inequality then gives a lower bound on the original heat-bath
Dirichlet form. All constants in this section may depend on fixed scale
cutoffs and fixed tolerances, but not on the test function or on $n$.
For a probability measure $\eta$ on the cube, write
\[
\operatorname{Ent}_{\eta}(u)
=\eta[u\log u]-\eta[u]\log\eta[u],\qquad
\mathcal D_{\eta}(f)=\sum_{i=1}^n
\eta\big[\operatorname{Var}_{\eta}(f\mid X_{-i})\big].
\]
Here $0\log0=0$, $\log_+u=\max\{\log u,0\}$, and $\mathcal D_0=\mathcal
D_\mu$. The interaction matrix is the GOE matrix $J$, with off-diagonal
variance $1/n$; its diagonal does not change $\mu(x)\propto\exp(x^{\mathsf
T}Jx/2)$.
\subsection{A reduction to bounded tests}\label{cap:capacity-reduction}
It is enough to prove the following estimate for every fixed $r>0$:
\begin{equation}\label{cap:restricted-energy}
\mathcal D_0(f)\ge n^{-2/3-r}N L,
\qquad N=\mu[f^2]=e^{-L},\qquad
\nu=\frac{f^2\mu}{N},
\end{equation}
whenever $0\le f\le1$, $0<N$, $\mu(f>0)\le1/2$, and
\begin{equation}\label{cap:restricted-entropy}
D(\nu\Vert\mu)\ge L/2,\qquad L\le Cn.
\end{equation}
The probability event on which this holds must be independent of $f$. The
observation argument is most effective for a bounded function whose
squared-density tilt has entropy comparable to its logarithmic inverse mass.
We first show that proving the energy estimate only for those tests is enough.
Equal height layers turn it into a capacity estimate for sets; dyadic layers
then control arbitrary half-supported functions. A median decomposition and an
entropy shift recover every real-valued test.
For this class, $\log2\le L$ and
$L/2\le D(\nu\Vert\mu)\le L$, while $d\nu/d\mu=f^2/N\le e^L$.
\begin{lemma}[Layers and capacity]\label{cap:layer-lemma}
Let $\eta$ be a strictly positive probability measure on a finite state
space, let $\mathcal E$ be a reversible Markov Dirichlet form, and suppose
$\min_x\eta(x)\ge e^{-M}$, with $M\ge\log2$. Suppose that, for some $A>0$,
\[
\mathcal E(g)\ge A\eta[g^2]\log\frac1{\eta[g^2]}
\]
for every $0\le g\le1$ supported on a set of measure at most $1/2$ and
satisfying
\[
D\!\left(\frac{g^2\eta}{\eta[g^2]}\middle\Vert\eta\right)
\ge\frac12\log\frac1{\eta[g^2]},\qquad
0<\log\frac1{\eta[g^2]}\le M.
\]
Then, with an absolute constant $C$,
\begin{equation}\label{cap:layer-lsi}
\operatorname{Ent}_{\eta}(f^2)
\le \frac{C(1+\log_+M)^2}{A}\,\mathcal E(f)
\qquad\text{for every real-valued }f.
\end{equation}
\end{lemma}
\begin{proof}
Fix a nonempty set $E$ of measure $a\le1/2$ and a test $h$ equal to one
on $E$ and zero outside a set of measure at most $1/2$. Clipping $h$ to
$[0,1]$ decreases its form. Set
\[
K=\left\lceil\log_2\!\left(1+
\frac{\log(1/a)}{\log2}\right)\right\rceil,
\qquad
g_i=K\big((h-i/K)_+\wedge K^{-1}\big),\quad 0\le i<K.
\]
Let $S_i=\eta(h>i/K)$ and $T_i=\eta(h\ge i/K)$. Some $i$ satisfies $T_{i+1}\ge
S_i^2$. Otherwise $T_1<S_0^2\le1/4$ and $T_{i+1}<T_i^2$, which would give
$a\le T_K<2^{-2^K}<a$. For this layer put $N_i=\eta[g_i^2]$. Then $a\le
T_{i+1}\le N_i\le S_i\le1/2$ and
\[
D\!\left(\frac{g_i^2\eta}{N_i}\middle\Vert\eta\right)
\ge\log(1/S_i)\ge\tfrac12\log(1/N_i).
\]
The first inequality follows by conditioning on its support. Also
$\log(1/N_i)\le\log(1/a)\le M$, and $\mathcal E(g_i)\le K^2\mathcal E(h)$.
Since $s\log(1/s)\ge c a\log(1/a)$ for $a\le s\le1/2$, we obtain
\begin{equation}\label{cap:set-capacity}
\mathcal E(h)\ge
\frac{cA}{(1+\log_+M)^2}\,a\log(1/a).
\end{equation}
This is the required capacity bound, where capacity is the infimum of
$\mathcal E(h)$ over these admissible tests.
For $v\ge0$ supported on a set of measure at most $1/2$, apply
\eqref{cap:set-capacity} to the layers $\phi_j=(v-2^j)_+\wedge2^j$,
$j\in\mathbb Z$, normalized by $2^j$. Writing $a_j=\eta(v\ge2^j)$ and omitting
empty upper level sets gives
\[
\mathcal E(v)\ge\sum_j\mathcal E(\phi_j)
\ge\frac{cA}{(1+\log_+M)^2}
\sum_j4^j a_{j+1}\log(1/a_{j+1}).
\]
For the first inequality, fix an edge and orient its endpoints so that the
increment of $v$ is nonnegative. The increments of all clipped layers are then
nonnegative and sum to that increment. The sum of their squares is at most its
square; summing over edges gives the inequality. Denote the last sum by $S$.
The pointwise dyadic sum $\sum_j4^j\mathbf1_{\{v\ge2^{j+1}\}}$ lies between
$v^2/12$ and $v^2/3$, so $S\ge(\log2)\eta[v^2]/12$. On $2^{j+1}\le v<2^{j+2}$,
the inequalities $\eta[v^2]\ge4^{j+1}a_{j+1}$ and $v^2\le16\cdot4^j$ give
\[
\log_+\frac{v^2}{\eta[v^2]}
\le\log\frac4{a_{j+1}}\le3\log\frac1{a_{j+1}}.
\]
Consequently $\operatorname{Ent}_{\eta}(v^2)\le48S$.
Choose a median $b$ of $f$ and apply these estimates to $v_+=(f-b)_+$ and
$v_-=(f-b)_-$. Their supports have measure at most $1/2$, their forms sum to
at most $\mathcal E(f)$, and
\[
\operatorname{Ent}_{\eta}((f-b)^2)
\le\operatorname{Ent}_{\eta}(v_+^2)
+\operatorname{Ent}_{\eta}(v_-^2).
\]
Finally we use an entropy shift inequality of Rothaus type; compare
\cite[Lemma~4]{BR2003}. In the form needed here it is
\begin{equation}\label{cap:entropy-shift}
\operatorname{Ent}_{\eta}((u+b)^2)
\le C\operatorname{Ent}_{\eta}(u^2)+C\eta[u^2]
\end{equation}
for every $u$ and every constant $b$. For completeness, normalize
$\eta[u^2]=1$. If $|b|\le2$, use $(u+b)^2\le2u^2+8$ and the positive part of
$s\log s$; its negative part costs at most an absolute constant. If $|b|>2$,
use
\[
\operatorname{Ent}_{\eta}((u+b)^2)
\le b^2\eta\!\left[
\Phi\!\left((1+u/b)^2\right)\right],
\qquad \Phi(s)=s\log s-s+1.
\]
On $|u|\le|b|/2$ the integrand is at most $Cu^2$ by Taylor's theorem; on its
complement it is at most $Cu^2(1+\log_+u^2)$. This proves
\eqref{cap:entropy-shift}, and hence \eqref{cap:layer-lsi}. The case
$\eta[u^2]=0$ is immediate. \end{proof}
On $\|J\|\le3$, all Gibbs atoms are at least $\exp(-Cn)$.
Lemma~\ref{cap:layer-lemma} therefore turns \eqref{cap:restricted-energy} into
the desired log-Sobolev inequality, after absorbing $(\log n)^2$ by choosing
$r$ smaller than the final exponent allowance. It remains to prove the
restricted estimate.
\subsection{Information and energy under observations}
This is the Gaussian observation form of stochastic localization
\cite{Eldan2013StochasticLocalization,ChenEldan2025Localization}. The
conditional-variance contraction below is also the mechanism of \cite[Lemma~9,
arXiv version~2]{EldanKoehlerZeitouni2022}. We give the identities directly
for the present matrix precisions and keep the stopped-path estimates separate
from any all-field assertion.
Fix a bounded test as in \eqref{cap:restricted-entropy}. For a deterministic
increasing, piecewise smooth matrix precision $B_s$, let $Q_\mu$ and $Q_\nu$
be the observation-path laws obtained from the initial spin laws $\mu$ and
$\nu$, and let $\mathcal F_s$ be the observation filtration. The field has increments
\[
dh_s=G_sX\,ds+G_s^{1/2}\,dW_s,\qquad G_s=\dot B_s\succeq0.
\]
Let $\mu_s,\nu_s$ be the respective spin posteriors,
$N_s=\mu_s[f^2]$, and write
\[
l_s=D(\nu_s\Vert\mu_s),\qquad
H(s)=\mathbb E_{Q_\nu}l_s,\qquad
m_{\mu,s}=\mu_s[X],\quad m_{\nu,s}=\nu_s[X].
\]
The posterior interaction is $J-B_s$, and
$\nu_s=f^2\mu_s/N_s$. The posterior need not retain the original
half-support or lower entropy-to-log-mass condition. The likelihood and
averaged identities below replace those pointwise restrictions.
These definitions also apply to precision jumps,
by using the corresponding Gaussian channel.
\begin{lemma}[Observation identities]\label{cap:observation-identities}
Along the continuous portions of this path,
\begin{align}
-\dot H(s)
&=\tfrac12\mathbb E_{Q_\nu}
\|m_{\nu,s}-m_{\mu,s}\|_{G_s}^2,
\label{cap:entropy-identity}\\
\mathbb E_{Q_\nu}
\frac{\mathcal D_{\mu_s}(f)}{N_s}
&\le\frac{\mathcal D_0(f)}N.
\label{cap:energy-conditioning}
\end{align}
The continuous entropy-loss integral up to a stopping time, together with
the information from preceding precision jumps, gives the relative entropy
of the stopped observation laws. Moreover, for $v\ge0$,
\begin{equation}\label{cap:conditional-entropy-tail}
Q_\nu(l_s>L+v)\le e^{-v},\qquad
Q_\nu(E)\le e^L Q_\mu(E)
\end{equation}
for every observation event $E$, including stopped-path events.
\end{lemma}
\begin{proof}
The likelihood ratio of the observation laws through time $s$ is
\[
\frac{dQ_\nu}{dQ_\mu}\bigg|_{\mathcal F_s}=\frac{N_s}{N}\le e^L.
\]
The
conditional drifts are $G_sm_{\mu,s}$ and $G_sm_{\nu,s}$, with common
innovation covariance $G_s$. The Gaussian likelihood formula and the
chain rule for relative entropy give
\[
D(\nu\Vert\mu)
=D(Q_\nu|_{[0,s]}\Vert Q_\mu|_{[0,s]})+H(s),
\]
and then \eqref{cap:entropy-identity}. The same calculation with the stopped
likelihood proves the stopping assertion; zero-rate directions are omitted.
The finite spin space and bounded finite precision remove any integrability
difficulty.
To prove \eqref{cap:energy-conditioning}, first change the path expectation
from the tilted law to the ordinary law:
\[
\mathbb E_{Q_\nu}\frac{\mathcal D_{\mu_s}(f)}{N_s}
=\frac1N\mathbb E_{Q_\mu}\mathcal D_{\mu_s}(f).
\]
The remaining assertion is the conditional-variance
decomposition, applied site by site under the joint ordinary law of the spin
and observations. Conditioning additionally on the observations can only
decrease the expected variance given $X_{-i}$. Finally $f^2\le1$ implies
$N_s/N\le e^L$ and $l_s\le-\log N_s$. Hence
\[
Q_\nu(l_s>L+v)
\le\mathbb E_{Q_\mu}
\left[\frac{N_s}{N}\mathbf1_{\{N_s/N<e^{-v}\}}\right]
\le e^{-v}.
\]
\end{proof}
We record the ordinary-path estimates used below. Under the original
zero-field Gibbs law observed to precision $B_p$, the quantities
$\|P_pX\|/M_p$ and $\|h_p\|/(p^2M_p)$ have bounded-range tails with
arbitrarily large fixed exponential rates per $k_p$, with the same control for
polynomial moments. The parameter $\alpha$ inferred from the cap field
satisfies
\begin{equation}\label{cap:ordinary-saddle-test}
Q_\mu\big(|\alpha(h_p)|>\zeta p^2\big)
\le e^{-a_\zeta k_p}
\end{equation}
for every fixed $\zeta>0$, after reducing the upper cutoff on $p$. These are
quenched estimates on events of probability tending to one. For integration
over logarithmic scale intervals it suffices that the same bounds hold in the
integrated form with a smaller positive exponent.
Here is the additional verification of \eqref{cap:ordinary-saddle-test}. Put
\[
R_\xi=(2+\xi p^2-J)^{-1},\qquad
R_{p,\xi}=(2+\xi p^2-J_p)^{-1},
\]
where $\xi>0$ is a sufficiently small fixed constant. Under the
unconditioned Gaussian reference, sample $X^{\rm G}\sim N(0,R_\xi)$ and
$h_p=B_pX^{\rm G}+\eta$, with $\eta\sim N(0,B_p)$ independently. Then
\[
a_{p,\xi}:=R_{p,\xi}h_p\sim N(0,R_\xi-R_{p,\xi}),\qquad
\mathbb E\|a_{p,\xi}\|^2=\operatorname{Tr}(R_\xi-R_{p,\xi}).
\]
This expected squared norm differs from
$n-\operatorname{Tr}R_{p,0}$ by $O(\sqrt\xi\,np)+o(np)$.
The covariance has operator norm $O((\xi p^2)^{-1})$ and trace of its square
$O(n/(\sqrt\xi p))$, so a fixed-tolerance deviation has exponent at most
$-c_\zeta\xi k_p$. The one-coordinate lower norm-density estimate for
$X^{\rm G}$ costs $C\xi^{3/2}k_p+O(\log k_p)$ in the logarithm, besides
the density scale $\sqrt{p/n}$. The quadratic tilt retains a spectral band
whose upper norm-density bound has the same scale. Choose $\xi$ small
enough that the $O(\sqrt\xi)$ target error fits the chosen saddle tolerance
and $C\xi^{3/2}<c_\zeta\xi/2$.
To connect this norm test to the saddle, use $h_p\in H_p$ and
$R_{p,0}h_p=(1+\xi)R_{p,\xi}h_p$. The monotone saddle equation and its
scaled inverse bound then place every field with
$|\alpha_p(h_p)|>\zeta p^2$ inside the relaxed quadratic-deviation event.
The preceding conditioning calculation gives that event a positive
spherical rate per $k_p$. The first-moment transfer and zero-field concentration in
Proposition~\ref{cap:static-comparison} prove the claim. The projection and
field tails follow there in the same way, including polynomial moments.
Markov's inequality and Fubini's theorem, summed over geometric scales, give
the stated integrated versions. At a fixed finite number of constant-scale
intervals, time nets and Gaussian maximal inequalities also give whole-path
versions.
The two paths below use at most two fixed endpoint constants $T'>3$,
chosen from their path parameters before $f$. At each endpoint precision
$T'I+J$, the posterior interaction is diagonal and its spin law is product.
On the same kind of disorder event, imposed for this finite collection and
simultaneously for $0\le L\le Cn$,
\begin{equation}\label{cap:product-field-tail}
Q_\mu\left(\|h_{\rm end}\|_\infty>
C\sqrt{\log n+L}\right)
\le e^{-2L}n^{-10},
\end{equation}
and the omitted tails admit polynomial moment bounds. To check this, condition
on the eigenvalues in the Haar first moment and then on the spin and its
spherical image. The component of $JX$ parallel to $X$ has bounded
coefficient, and its perpendicular component is a uniformly rotated vector of
norm at most $3\sqrt n$. Each coordinate thus has a Gaussian-type tail. Add
the Gaussian observation noise, whose covariance has bounded norm, and take a
coordinate union bound. Increase the rate before the first-moment transfer and
use zero-field partition concentration. A scalar grid in $L$ gives
simultaneity, with stronger tails between grid points. The same calculation
with tail integrals gives the polynomial moment assertion.
\subsection{Retention at small and moderate entropy}\label{cap:entropy-retention}
Fix a small constant $\lambda>0$, to be chosen after the curvature and
ordinary-path tolerances, and first suppose that $L\le\lambda n$. Choose $p_1$
so that, up to fixed endpoint factors,
\begin{equation}\label{cap:first-entropy-scale}
k_1=np_1^3=\max\{k_{\min},c_0L\},\qquad
k_{\min}\asymp\sqrt{\log n},
\end{equation}
where $c_0>0$ is a sufficiently small absolute constant. The coarse
endpoint factors and $c_0$ are chosen before the sharp curvature
tolerances.
The initial observation jump leaves the following amount of entropy:
\begin{equation}\label{cap:jump-retention}
H(p_1)\ge
\begin{cases}
cL,& k_1=c_0L,\\
e^{-C_0k_1},& k_1=k_{\min}.
\end{cases}
\end{equation}
To prove the first line, let $Q_0$ be the pure-noise output of the
precision-$B_{p_1}$ channel. Convexity of relative entropy gives $D(Q_\nu\Vert
Q_0)\le\frac12\nu[X^{\mathsf T}B_{p_1}X]$. By Jensen's inequality and
$\mu[X]=0$,
\[
\log\frac{dQ_\mu}{dQ_0}(h)
=\log\mu\!\left[e^{h^{\mathsf T}X-X^{\mathsf T}B_{p_1}X/2}\right]
\ge-\tfrac12\mu[X^{\mathsf T}B_{p_1}X].
\]
Subtract the logarithmic likelihood bound from the pure-noise relative entropy
estimate. The information revealed by the jump is therefore at most half the
sum of these two quadratic moments. The projection tails, transferred also on
an auxiliary dyadic grid, bound this sum by
\[
Ck_1\left(1+(L/k_1)^{1/3}\right).
\]
For example the $v^6$ tail and $d\nu/d\mu\le e^L$ give
$\nu[\|P_{p_1}X\|^2/M_{p_1}^2]\le C(1+(L/k_1)^{1/3})$. When $k_1=c_0L$, the
displayed loss is at most $C(c_0+c_0^{2/3})L$, which is smaller than $L/4$
after choosing $c_0$; the initial entropy is at least $L/2$.
This argument gives proportional entropy retention when the initial cap
dimension is chosen proportional to L. At the smallest permitted cap that
choice may be unavailable. We then use the support condition instead: the
original law assigns a fixed positive mass to a bounded-projection set that
the tilted law never visits. After a bounded field observation its posterior
mass may be exponentially small in $k_1$, but it remains positive and yields
the weaker entropy bound required in the bottom branch.
For the second line, a set of $\mu$-mass at least $1/4$ lies outside the
support of $f$ and has bounded scaled projection. On a sufficiently large
fixed field ball its posterior mass is at least $e^{-Ck_1}$: the numerator is
bounded below on these projections, and the denominator is bounded above by
integrating the projection tails. The field ball has $Q_\nu$-probability
bounded below because $L\le k_1/c_0$ and its ordinary complement can have
arbitrarily large fixed rate per $k_1$. On this ball $\nu_h$ assigns zero mass
to the indicated complement of the support. Data processing gives
$D(\nu_h\Vert\mu_h)\ge-\log(1-e^{-Ck_1})\ge e^{-C_0k_1}$, after adjusting
$C_0$. These are all coarse constants.
Parameterize the cap observations by $s=\log p$. Their precision rate is
$G=2p^2P_p$. Write $\delta m_h=m_{\nu,h}-m_{\mu,h}$. Suppose that, for
every shift $u\in H_p$ in the neighborhood under consideration,
\[
P_p\nabla^2\phi_p(h+u)P_p\preceq \frac{C_{\rm ball}}{p^2}P_p.
\]
The required neighborhood contains all such shifts whose radius, in
units $p^2M_p$, is at most
\begin{equation}\label{cap:entropy-shift-radius}
\left(\frac{2l_h}{k_p C_{\rm ball}}\right)^{1/2}.
\end{equation}
Then the variational formula for relative entropy, tested against linear
functions, gives
\begin{equation}\label{cap:entropy-mean-bound}
p^2\|P_p\delta m_h\|^2\le2C_{\rm ball}l_h.
\end{equation}
Indeed integrate the Hessian along a shift $u$ to bound its centered log
moment generating function by $C_{\rm ball}\|u\|^2/(2p^2)$, and test in the
direction of $P_p\delta m_h$. The trial shift of length $p\sqrt{2l_h/C_{\rm ball}}$
is available precisely under \eqref{cap:entropy-shift-radius}; the resulting
inequality also bounds the length of the unconstrained optimizer.
We next trade covariance control for a differential inequality on the
remaining entropy. The variational argument requires an entire ball of linear
field shifts, with radius depending on the conditional entropy. A coarse first
interval makes the cap dimension large compared with that entropy. On the
later interval the required shifts fit inside the region of sharp curvature,
apart from errors whose weighted drift contribution is exponentially small.
Keeping those contributions inside the differential inequality is necessary
for the bottom branch, where the retained entropy is itself small.
Here is the bound used for every discarded event. Put
$Y_p(X)=\|P_pX\|/M_p$. Conditional Jensen gives
\[
p^2\|P_p\delta m_h\|^2
\le2k_p\bigl(\nu_h[Y_p^2]+\mu_h[Y_p^2]\bigr).
\]
For any observation event $E$, including one depending on $f$, and any
fixed $V>0$, disintegration and the likelihood bound yield
\begin{equation}\label{cap:discarded-drift}
\begin{split}
\mathbb E_{Q_\nu}\!\left[
p^2\|P_p\delta m_h\|^2\mathbf1_E\right]
\le{}&4k_pV^2Q_\nu(E)\\
&+4k_pe^L\mu\!\left[Y_p^2\mathbf1_{\{Y_p>V\}}\right].
\end{split}
\end{equation}
For example, the tilted posterior term averages to a moment under $\nu$;
the ordinary posterior term uses the path density at most $e^L$.
The projection tail pays the weighted term, so a probability estimate for
$E$ is used together with a moment estimate.
Let $k_2=R_*k_1$ with $R_*$ a sufficiently large fixed constant. On
$[p_1,p_2]$, use the bounded-ball curvature bound of
Theorem~\ref{cap:true-curvature}. Choose $A_\ell\ge A>C_0$, where $C_0$ is
from \eqref{cap:jump-retention}. Discard fields outside a fixed ball and
the event $l_h>L+A_\ell k_p$. Since $L/k_p\le1/c_0$, the latter event has
probability at most $e^{-A_\ell k_p}$ by
\eqref{cap:conditional-entropy-tail}. Enlarge the eventual curvature
constant to satisfy $C_{\rm ball}\ge1$. Before obtaining that constant,
choose the ball to contain shifts of radius
$\sqrt{2(c_0^{-1}+A_\ell)}$ in scaled units from every retained field;
this contains the radius in \eqref{cap:entropy-shift-radius}.
Choose its ordinary field-tail rate and the projection threshold in
\eqref{cap:discarded-drift} larger than $A+1/c_0$.
After paying $e^L$, the discarded drift is at most $Ck_pe^{-Ak_p}$.
The field range and these coarse tail choices precede the sharp tolerances
and $R_*$. Its curvature constant, even if large, causes only a fixed loss
on this interval. Equations \eqref{cap:entropy-identity} and
\eqref{cap:entropy-mean-bound} give
\[
\frac{dH}{d\log p}\ge-2C_{\rm ball}H-Ck_pe^{-Ak_p}.
\]
The interval has fixed logarithmic length. Integrating this inequality
therefore multiplies the initial lower bound in \eqref{cap:jump-retention} by
a fixed positive factor. The additive error is negligible relative to either
branch of that lower bound.
On $[p_2,p_{\rm top}]$, use instead the curvature $(1+\epsilon)/p^2$ on the
near-zero-parameter region. Choose an ordinary saddle tolerance $\zeta$
strictly inside that region, leaving a field buffer of radius
$r_{\rm buf}$ in units $p^2M_p$. Put
$b_{\rm buf}=(1+\epsilon)r_{\rm buf}^2/2$. The shift radius fits this
buffer whenever $l_h\le b_{\rm buf}k_p$. At these scales
$L/k_p\le(R_*c_0)^{-1}$. Choose $R_*$ so this ratio is below fixed
fractions of both $b_{\rm buf}$ and the rate $a_\zeta$ in
\eqref{cap:ordinary-saddle-test}. Then
\[
Q_\nu(l_h>b_{\rm buf}k_p)
\le e^{-(b_{\rm buf}-L/k_p)k_p},\qquad
Q_\nu(|\alpha_p(h_p)|>\zeta p^2)
\le e^{-(a_\zeta-L/k_p)k_p}.
\]
Equation~\eqref{cap:discarded-drift}, with another fixed projection
threshold, leaves $Ck_pe^{-ak_p}$ for some fixed $a>0$. Thus
\[
\frac{dH}{d\log p}\ge-2(1+\epsilon)H-Ck_pe^{-ak_p}.
\]
Increase $R_*$ again so that $aR_*>C_0$ with slack.
For either differential inequality, an exponent $\gamma_H$ and error
$Ck_pe^{-bk_p}$ give
\[
H(p)\ge\left(\frac{p_a}{p}\right)^{\gamma_H}
\left[H(p_a)-C\int_{p_a}^{p}
\left(\frac{u}{p_a}\right)^{\gamma_H}
k_u e^{-bk_u}\,d\log u\right].
\]
The coarse error starts at $k_1$ with $A>C_0$; the sharp error starts at
$R_*k_1$ with $aR_*>C_0$. Their weighted integrals are therefore negligible
relative to either branch of \eqref{cap:jump-retention}. Solving yields
\begin{equation}\label{cap:retained-entropy}
H(p_{\rm top})\ge
c\left(\frac{p_1}{p_{\rm top}}\right)^{2+2\epsilon}
\begin{cases}
L,& k_1=c_0L,\\
e^{-C_0k_1},& k_1=k_{\min}.
\end{cases}
\end{equation}
Pointwise control of the discarded drift is not needed here. With the
integrated bounds, the integrating factor charges each geometric interval only
a fixed power of $p/p_1$. Its error decreases exponentially in $k_p$, and
$k_1\to\infty$, so their weighted sum is still negligible. Finally choose
$\lambda$ small enough that $R_*c_0\lambda<p_{\rm top}^3$, up to the fixed
endpoint factors. Then $p_2<p_{\rm top}$ for all $L\le\lambda n$.
\subsection{The remaining fixed scales}\label{cap:constant-scales}
We next transfer \eqref{cap:retained-entropy} to a product endpoint, allowing
fixed multiplicative losses. The initialized scale is $p_0$ from
Section~\ref{cap:initialization}, with $t=C_*p_0^2$. Starting from
$p_{\rm top}$, first traverse the finite remaining cap intervals to $p_0$.
Use the actual finer frame for the terminal bridge there. Its increment is
\[
\Delta_{\rm base}=tI+(1-\beta)J-B_{p_0},
\]
so the total precision is exactly $tI+(1-\beta)J$. The posterior interaction
is $\beta J-tI$, with the same spin law as $\beta J_{\rm off}$.
Now increase scalar precision along
\[
B_u=uI+(1-\beta)J,\qquad t\le u\le T,
\]
for a sufficiently large fixed $T$. The choice of the single coefficient
$\beta<1$ in Section~\ref{cap:initialization} includes the persistence
requirement for this interval. Namely, fix $T$ and the normalized buffers
for the ordinary critical test after $p_0$ is fixed and before choosing
$\beta$. Adding an independent increment of precision
$(1-\beta)(4I+J)$ to scalar precision $(u-4(1-\beta))I$ gives $B_u$.
The associated common-spin field displacement has normalized size
$O((1-\beta)+\sqrt{1-\beta})$ outside an exponentially rare event.
Time nets and Gaussian maximal bounds preserve the buffered tests along the
interval. Lemma~\ref{cap:endpoint-stability} then supplies a fixed
whole-path rate $e^{-an}$ and a fixed field neighborhood.
The auxiliary removal is performed with this already fixed $\beta$;
it determines $p_{\rm top}$, and $\lambda$ is chosen afterward.
Next use
\[
B_a=(T+4a)I+(1-\beta+a)J,\qquad 0\le a\le\beta.
\]
Its rate is $4I+J\succ0$, and it ends at
$T'_\beta I+J$ with $T'_\beta=T+4\beta>3$. For $T$ large, uniformly in the underlying spin, the field has large
absolute coordinates except on a small site fraction: the main shift is $TX$,
the other deterministic shifts have bounded normalized norm, and the noise has
squared normalized norm $O(T)$ outside probability $e^{-an}$. At an
approximate root the diagonal $A$ in the stability test is therefore small
outside a small site set. Gaussian nets give a small operator norm for every
sufficiently sparse principal submatrix of a typical GOE matrix. Together with
the full norm bound and the small normalized norm of $A^{1/2}m$, this verifies
the stability test, again on a field neighborhood. While the coefficient of
$J$ belongs to a fixed compact subinterval of $(0,1)$, the classical
covariance bound and the entropy shift argument consequently give
\begin{equation}\label{cap:constant-entropy-ode}
\dot H\ge-CH-e^{-an},
\end{equation}
after reducing $a$ and $\lambda$ to absorb the path tilt $e^L$.
For sufficiently small interaction coefficients, the covariance is bounded
globally. Indeed add a scalar diagonal to the interaction to obtain a matrix
$Q$ with $0\prec Q\prec I/2$, and introduce a Gaussian latent field with
covariance $Q$. Its posterior log density has Hessian at most
$-Q^{-1}+I\preceq-I$. Brascamp--Lieb bounds the covariance of the
coordinatewise $\tanh$ by $I$, while the conditional spin variances contribute
another $I$. Thus the spin covariance is at most $2I$. The variational
argument now gives $\dot H\ge-CH$ for every field and every $L$, without any
discarded set. Word and residual diagnostics along varying coefficients are
consequently needed only on closed subintervals of $(0,1)$.
The sharp cap estimate stops at $p_{\rm top}$, below the initialization
scales. There are only finitely many intervals left before the classical
field, so bounded amplification and fixed losses suffice there. Starting from
the terminal covariance bound, subtract the two mean-transport identities of
Lemma~\ref{cap:error-transport}. One term transports the difference of
terminal means; the other compares expectations of the bounded terminal
approximation error under two path laws. Pinsker's inequality controls the
latter by the square root of conditional entropy. This propagates the required
quadratic mean estimate backward through the finite list of intervals.
At the terminal classical field, the covariance bound holds on a fixed
normalized neighborhood outside an ordinary event of probability $e^{-an}$.
The linear entropy test fits this neighborhood whenever $l_h\le bn$ for a
fixed $b>0$. Choose $\lambda<\min(a,b)/2$ after these constants. The ordinary
exception costs at most $e^{-(a-\lambda)n}$ under $Q_\nu$, and
\eqref{cap:conditional-entropy-tail} gives
$Q_\nu(l_h>bn)\le e^{-(b-\lambda)n}$. Since both spin means have norm at
most $\sqrt n$, these exceptions have exponentially small quadratic cost.
The neighborhood covariance estimate therefore gives, after reducing the
positive exponent,
\[
\mathbb E_{Q_\nu}\|m_\nu-m_\mu\|^2\le CH+e^{-an}.
\]
Fix any current time $s$ in one preceding interval and a current field $h$.
Let $Q_{\nu,h}^{\rm cont}$ and $Q_{\mu,h}^{\rm cont}$ be the two laws of the
future observations through the end of that interval, starting from the
posteriors $\nu_h$ and $\mu_h$. They live on the same path space, and data
processing gives
$D(Q_{\nu,h}^{\rm cont}\Vert Q_{\mu,h}^{\rm cont})\le l_h$.
Let $g$ be the gradient of one extended auxiliary potential evolved by heat
convolution, and let $U$ be its linear transport from time $s$ to the interval
end. The same path function $U$ is evaluated under both continuation laws,
and its amplification is bounded by a fixed constant. Subtracting the two
transport identities gives, in every direction $v$ of the current rate subspace,
\begin{align*}
\langle v,m_\nu-m_\mu\rangle
={}&\mathbb E_{Q_{\nu,h}^{\rm cont}}
\langle Uv,m_{\nu,\rm end}-m_{\mu,\rm end}\rangle\\
&+(\mathbb E_{Q_{\nu,h}^{\rm cont}}-\mathbb E_{Q_{\mu,h}^{\rm cont}})
\langle Uv,m_{\mu,\rm end}-g_{\rm end}\rangle.
\end{align*}
On agreement events the second observable is bounded by a constant times
$\|v\|$, by the raw gradient comparison, including at the classical terminal
field. Truncate it there and use the displayed continuation entropy bound and
Pinsker's inequality to bound its expectation difference by
$C\sqrt{l_h}\|v\|$. To bound the complements after averaging, let $W\ge0$
be a terminal cost, such as the squared raw error outside agreement. The
amplification bound lets us dominate the transported exceptional terms by
such costs. Disintegration and the observation likelihood give
\[
\begin{split}
\mathbb E_{Q_\nu}\mathbb E_{Q_{\nu,h}^{\rm cont}}W
&=\mathbb E_{Q_\nu}W
\le e^L\mathbb E_{Q_\mu}W,\\
\mathbb E_{Q_\nu}\mathbb E_{Q_{\mu,h}^{\rm cont}}W
&=\mathbb E_{Q_\mu}\!\left[
\frac{N_s}{N}\mathbb E_{Q_{\mu,h}^{\rm cont}}W\right]
\le e^L\mathbb E_{Q_\mu}W.
\end{split}
\]
The ordinary terminal agreement estimates and their polynomially weighted
versions therefore give exponentially small averaged costs after choosing
$\lambda$ below the finitely many fixed rates. This uses full-path and
current-prefix likelihood bounds, separately from conditional Pinsker.
Apply Cauchy--Schwarz first conditionally on
the current field and then over the tilted observation path. The terminal
entropy satisfies $H_{\rm end}\le H(s)$. If $P_{\rm rate}$ projects onto the
current rate subspace, these bounds give
\[
\mathbb E_{Q_\nu}\|P_{\rm rate}(m_{\nu,s}-m_{\mu,s})\|^2
\le C H(s)+e^{-an}.
\]
The transported directions end in the appropriate terminal rate subspace on
cap intervals; for a full-rank bridge $P_{\rm rate}=I$. Since $s$ was
arbitrary, this estimate supplies the differential inequality throughout
the interval.
For the bridge use the actual-refinement TAP extension; for cap intervals use
the auxiliary or true-cap extension supplied by the induction. At each cap
endpoint, \eqref{cap:ordinary-saddle-test} gives agreement outside an
exponentially rare event, and the extension has polynomial gradient moments.
For the bridge, apply Lemma~\ref{cap:marked-terminal-tests}
on the agreement annulus at $p_0$; its ordinary complement is paid by the same
saddle test and polynomial moments. The amplification bound holds from every
intermediate time, while these terminal estimates enter through the averaged
comparison above. No agreement condition at the current field is needed in
the subtraction, since its common approximate gradient cancels.
Iterating through this finite list, even when $p_{\rm
top}$ is interior to one interval, proves \eqref{cap:constant-entropy-ode}.
Its constants may depend on the fixed initialization length. Solving it proves
\eqref{cap:retained-entropy}, up to constants, at the product endpoint.
\subsection{The product estimate}\label{cap:product-estimate}
At precision $T'I+J$ the posterior has independent coordinates with fields
$h_i$. A two-point measure at field $h_i$ has inverse log-Sobolev constant at
most $C(1+|h_i|)$ in conditional-variance normalization. One direct proof
centers the test, uses \eqref{cap:entropy-shift}, and bounds the entropy of
the centered square by its second moment times the logarithm of the inverse
smaller atom; that logarithm is at most $C(1+|h_i|)$. Tensorization and
the posterior entropy identity first give
\[
l_h\le C(1+\|h\|_\infty)\,
\frac{\mathcal D_{\mu_h}(f)}{N_h}.
\]
Averaging this bound with \eqref{cap:energy-conditioning} gives
\begin{equation}\label{cap:product-entropy-bound}
H_{\rm end}\le
C\sqrt{\log n+L}\,\frac{\mathcal D_0(f)}N+O(n^{-3}).
\end{equation}
To justify the truncation implicit here, apply tensorization on the
event in \eqref{cap:product-field-tail}. On its complement use
\[
l_h\le\log\frac1{\min_x\mu_h(x)}
\le Cn(1+\|h\|_\infty).
\]
The moment version of \eqref{cap:product-field-tail}, followed by the
tilt bound $e^L$, makes its contribution $O(n^{-3})$, uniformly for
$L\le Cn$.
Combining this with \eqref{cap:retained-entropy} proves
\eqref{cap:restricted-energy} for $L\le\lambda n$. Here is the exponent
bookkeeping. In the first case, $p_1^3\asymp L/n$ and $L\gtrsim\sqrt{\log n}$,
so
\[
\frac{H_{\rm end}}{\sqrt{\log n+L}}
\ge c n^{-(2+2\epsilon)/3}L
\frac{L^{(2+2\epsilon)/3}}{\sqrt{\log n+L}}.
\]
The last quotient is bounded below by a negative fixed power of $\log n$ when
$L\le\log n$, and by a constant when $L\ge\log n$. In the second case
$k_1\asymp\sqrt{\log n}$, $p_1^3=k_1/n$, and $L\le C\sqrt{\log n}$; hence
\[
\frac{H_{\rm end}}{\sqrt{\log n+L}}
\ge n^{-(2+2\epsilon)/3-o(1)}.
\]
Both lower bounds dominate $n^{-2/3-r}L$ by choosing $\epsilon$ small
relative to $r$. They also dominate the additive $O(n^{-3})$ error.
\subsection{Large entropy and stopping}\label{cap:large-entropy}
The remaining entropy levels are proportional to the dimension. At these
levels an ordinary rare-event bound cannot simply be multiplied by the initial
density bound. We instead argue under the proposed small energy ratio. The
posterior mean estimate then makes the revealed information small. Relative
entropy of the stopped observation laws transfers the ordinary stopping
probability to the tilted law, and bounded spin means control the loss after
stopping. This retains order-n entropy and conflicts with the product
estimate.
Now suppose $\lambda n\le L\le Cn$. We prove the stronger estimate
$\mathcal D_0(f)/N>n^{-2/3}L$ by contradiction. Thus assume
\begin{equation}\label{cap:large-entropy-assumption}
\frac{\mathcal D_0(f)}N\le n^{-2/3}L\le Cn^{1/3}.
\end{equation}
The objective is to retain a fixed positive multiple of $\lambda n$ until the
product endpoint of this path. Formula \eqref{cap:product-entropy-bound} would
then bound that retained entropy by $O(n^{5/6})+O(n^{-3})$, a contradiction.
We specify the uniformity required at the start. The small-positive-time
statement in Proposition~\ref{cap:tap-expansion}, with the noise interface of
Lemma~\ref{cap:boundary-noise}, uses diagnostics of the centered Gaussian
noise and admits every additional shift of normalized norm at most $Ct$. Its
separate low-band condition is a Gaussian small-ball bound uniform in the
center, not a simultaneous event for all centers. More precisely, on a
disorder event independent of $f$, the noise-only diagnostics fail with
conditional probability $o(1)$, integrated over each fixed interval
$[t_*,t_b]$, and
\begin{equation}\label{cap:small-ball-centers}
\sup_{z\in\mathbb R^n}
\mathbb P_g\!\left(
\|P_t^J(g_t+z)\|<c\sqrt n\,t^{5/4}\right)
\le e^{-c'nt^{3/2}}.
\end{equation}
Here $P_t^J=\mathbf1_{\{2I-J\le tI\}}=P_{\sqrt t}$ is the cumulative
projection used in Lemma~\ref{cap:boundary-noise}, with $W=J$.
The covariance on a band of rank comparable to $nt^{3/2}$ is comparable to
$tI$; projection onto that band and the Gaussian ball bound prove
\eqref{cap:small-ball-centers}. Translation cannot increase the mass of a
centered Gaussian Euclidean ball. For fixed $J$ the spin is independent of the
observation noise. Since its drift shift has normalized norm $O(t)$ for every
$X\in\{-1,1\}^n$, the deterministic shift allowance and the uniform-in-center
probability give the same $o(1)$ integrated failure bound under every initial
spin law, including $f^2\mu/N$. One first uses Fubini and Markov for the
noise-only diagnostic event to select the disorder event; no union over the
$2^n$ spin centers and no factor $e^L$ occurs in this step. The
exclusion-at-zero and matrix-word events are also selected independently of
$f$.
First fix the large scalar endpoint $T$ and the interaction coefficient below
which covariance is globally bounded. Next choose $t_b>0$ sufficiently small
for Proposition~\ref{cap:tap-expansion}, and obtain the buffered ordinary test
and its failure rate $a>0$ on $[t_b,T]$ and the subsequent weakening interval.
These margins are uniform for all sufficiently small changes of the
interaction coefficient from one. Fix a target integrated drift budget
$\eta_1$ sufficiently small relative to $a\lambda$. Choose $0<t_*\ll t_b$ so
that $Ct_*$ fits that budget, and finally choose $1-\beta'>0$ sufficiently
small relative to these fixed scales and $t_*$.
After fixing $\beta'$, reduce the permitted diagonal enlargement in the
stability test to a positive $\eta_{\rm stab}<(\beta')^{-1}-1$. The
stability implication persists on this smaller diagonal class, and
$(\beta')^2(1+\eta_{\rm stab})^2<1$ supplies the strict margin required by
Proposition~\ref{cap:classical-input}.
Once the resulting stability margins are known, choose the residual tolerance
and finite diagnostic depth in the square-reweighted mean estimate of
Proposition~\ref{cap:classical-input} to realize the target budget. The target
$\eta_1$ is distinct from that proposition's accuracy parameter: the leading
constant multiplying its accuracy may depend on the fixed stability margins.
Its remaining constants may be arbitrarily large but are all fixed before
$n\to\infty$. The factor $n^{-1/2}$ in the posterior estimate will therefore
still make the energy-dependent error vanish under
\eqref{cap:large-entropy-assumption}.
Start directly with precision $t_*I+(1-\beta')J$, increase scalar precision to
$T$, and then weaken the interaction at rate $4I+J$ for duration $\beta'$.
The endpoint is $T'_{\beta'}I+J$ with $T'_{\beta'}=T+4\beta'$, the second
fixed endpoint included in \eqref{cap:product-field-tail}. Positivity of the initial
precision follows from choosing $1-\beta'$ sufficiently small after $t_*$. For comparing this path with
the ordinary critical scalar observation, one may add an independent increment
of precision $(1-\beta')(4I+J)\succ0$ and increase scalar noise time by
$O(1-\beta')$. This produces a small normalized field displacement outside an
exponentially rare event, so the buffered ordinary tests remain valid with
rate $a$ after a fixed decrease.
Let $s_g$ be the onset of the globally bounded-covariance portion. On the
interval $[t_*,t_b]$, the uniform noise and small-ball argument just given
verifies the stability hypotheses with integrated failure tending to zero
under $Q_\nu$. On their validity set the tilted mean estimate from
Proposition~\ref{cap:classical-input}, together with
\eqref{cap:energy-conditioning}, yields an arbitrarily small integrated bound
on $\mathbb E_{Q_\nu}|m_\nu-m_\mu|_n^2$. Indeed the relevant error is bounded
by
\[
C\eta+\frac{C_\eta}{\sqrt n}
\left(1+\frac{\mathcal D_{\mu_h}(f)}{N_h}\right),
\]
and its average is $C\eta+O_\eta(n^{-1/6})+O_\eta(n^{-1/2})$ by
\eqref{cap:large-entropy-assumption}. This estimate initially controls a
first moment. Since $|m_\nu-m_\mu|_n\le2$, its square is at most twice its
first power, giving the needed quadratic bound. The integrated failure probability contributes
only $o(1)$ by the same uniform bound.
For times at least $t_b$, use the fixed buffered ordinary stability test,
selected independently of $f$ before observing the path. Let $\tau\le s_g$
be its first violation, taking $\tau=t_b$ if it already fails there
and $\tau=s_g$ if no violation occurs. Up to $\tau$ the same integrated
tilted-mean bound applies deterministically on the diagnostic event. Write
$Q_\nu^\tau,Q_\mu^\tau$ for the laws of the observations stopped at $\tau$.
The initial jump leaks at most $Ct_*n$, by the pure-noise comparison above
without the projection improvement. The stopped version of
\eqref{cap:entropy-identity} then gives
\begin{equation}\label{cap:stopped-path-entropy}
D(Q_\nu^\tau\Vert Q_\mu^\tau)
\le (Ct_*+\eta_1+o(1))n,
\end{equation}
where $\eta_1>0$ can be made arbitrarily small by the preceding fixed
accuracy choices. This includes the possibility of immediate stopping
at $t_b$.
Let $E$ be the event that a violation occurs before the globally
controlled portion. Its ordinary probability is at most $e^{-an}$.
Data processing for the indicator of $E$ gives
\[
D(Q_\nu^\tau\Vert Q_\mu^\tau)
\ge Q_\nu(E)\log\frac1{Q_\mu(E)}-\log2.
\]
Consequently
\begin{equation}\label{cap:stopping-probability}
Q_\nu(E)\le\frac{Ct_*+\eta_1+o(1)}a.
\end{equation}
On a path that stops, use the spin norm bound for the remaining portion of the
observation. Every mean has norm at most $\sqrt n$, and the precision rate has
bounded norm over a bounded duration. Its entropy-loss integral is
consequently at most $Cn$. Multiplying by the stopping probability bounds the
averaged cost of these paths. By the constant choices,
\eqref{cap:stopped-path-entropy} and \eqref{cap:stopping-probability} make the
total loss before $s_g$ smaller than $\lambda n/4$. Initially $H(0)\ge
L/2\ge\lambda n/2$, so $H(s_g)\ge c\lambda n$. The subsequent global
covariance estimate propagates a fixed fraction to the product endpoint. This
is the claimed contradiction with \eqref{cap:product-entropy-bound}.
All disorder, word, and stability events in this argument have fixed
thresholds selected independently of $f$. The only function-dependent objects
are the tilted law and the entropy level $L$. The proof uses the exponential
ordinary failure rate through stopped-path relative entropy, and therefore
remains valid when $L$ is of order $n$.
\subsection{The log-Sobolev bound}\label{cap:dynamics}
We have proved \eqref{cap:restricted-energy} in both entropy ranges.
Lemma~\ref{cap:layer-lemma} now gives the promised functional inequality.
\begin{theorem}[Critical log-Sobolev upper bound]\label{cap:thm-lsi}
For every fixed $\rho>0$, with probability tending to one over the
Gaussian couplings, the critical zero-field Gibbs measure satisfies
\begin{equation}\label{cap:final-lsi}
\operatorname{Ent}_\mu(f^2)
\le n^{2/3+\rho}\mathcal D_0(f)
\qquad\text{for every }f:\{-1,1\}^n\longrightarrow\mathbb R.
\end{equation}
The form is normalized for the generator
$\mathcal L=\sum_{i=1}^n(Q_i-I)$, where $Q_i$ is conditional expectation
given $X_{-i}$.
\end{theorem}
\begin{proof}
Apply the preceding argument with an exponent allowance strictly
smaller than $\rho$. On $\|J\|\le3$ the value $M$ in
Lemma~\ref{cap:layer-lemma} is $Cn$. Its fixed constant and
$(1+\log M)^2$ factor are absorbed in the remaining power of $n$.
\end{proof}
Section~\ref{sec:dynamics} derives the mixing and relaxation bounds from
this inequality and combines them with the companion's lower bounds.
The cap comparison also gives the following direct linear test. It explains
the same worst-start lower exponent with weaker quantifiers.
\begin{remark}[An independent linear lower test]\label{cap:lower-test}
The static comparison also gives a direct, weaker-in-quantifiers route
to the matching worst-start lower exponent. Fix $0<\epsilon<1/3$ and take
$p=n^{-1/3+\epsilon}$. In the uncapped zero-field spherical model,
$\|P_pX\|\ge cM_p$ with probability tending to one. To see this,
compare with the Gaussian of precision $2+\xi p^2-J$. A spectral band
with gaps comparable to $p^2$ already has this lower bound outside
$e^{-ck_p}$. The lower norm-density estimate and the bounded density
of the remaining coordinates transfer it through norm conditioning,
when $\xi>0$ is sufficiently small. Haar first moments and the
zero-field second-moment comparison in
Proposition~\ref{cap:static-comparison} transfer the conclusion to the cube
with high Gibbs probability for typical disorder. Spin symmetry and
$\operatorname{rank}P_p\le Cnp^3$ imply that some Euclidean unit vector
$a$ has
\[
\operatorname{Var}_\mu(a^{\mathsf T}X)\ge c p^{-2},
\qquad
\mathcal D_0(a^{\mathsf T}X)
=\sum_i a_i^2\mu[\operatorname{Var}(X_i\mid X_{-i})]\le1.
\]
Thus the rate-one spectral gap is at most $Cp^2$. Testing a gap
eigenfunction from a state where its absolute value is maximal gives
$t_{\rm mix}(1/4)\ge c p^{-2}$; the discrete gap is smaller by $n$.
Letting the fixed $\epsilon$ decrease proves the same lower exponents.
The realized-start statement and the bounds for every arbitrarily slowly
diverging $M_n$ remain the separate conclusions of
Theorem~\ref{thm:accepted-critical}.
\end{remark}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-entropy.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-initialization.tex ====
\section[Initializing the cap induction and removing the auxiliary potential]{Initializing the cap induction and\\removing the auxiliary potential}
\label{cap:initialization}
The induction of Proposition~\ref{cap:curvature-induction} needs a negative
radial correction at its first scale. We obtain this correction from the
small-field TAP expansion, initially for an auxiliary potential. The auxiliary
potential differs from the true cap potential by a bounded Euclidean gradient
error. A finite number of cap steps contracts this error, after which a
covariance comparison permits us to use the true potential at every remaining
scale.
We retain the notation $P_p,k_p,M_p,R_p,a_p$ of the cap construction and
write
\[
\mathcal A_p(s)=\{h\in P_p:|\alpha_p(h)|\le s p^2\},
\qquad \phi_p(h)=\log Z_p(h).
\]
Here $\alpha_p(h)$ is determined by $n=\operatorname{Tr}R_p+\|R_ph\|^2$.
Whenever an auxiliary potential $\widetilde\phi_p$ is used, its derivatives
and its discrepancy from $\phi_p$ are restricted to $P_p$. An auxiliary object
is called \emph{intrinsic to the parent data} if it is measurable with respect
to the capped matrix, the eigenvalues, and auxiliary randomness independent of
the as yet unused finer eigenframe. This requirement is essential for the
contraction argument below.
All parameters in the initialization and the finite comparison phase are fixed
before $n$ tends to infinity. Fix first a finite field radius $L_0$ for the
eventual bounded-ball estimate. The construction tolerances may depend on
$L_0$; the coarse constant supplied by
Lemma~\ref{cap:frozen-evaluation-support} depends only on $L_0$ and is
fixed before the sharp curvature tolerances. Choose a sufficiently large absolute $C_*$
and the small excess $\eta_*$ used below, and then the constant $K_0$ in
\eqref{cap:invariant}. Choose the scale ratio
$\gamma$ sufficiently small, the annulus tolerance $\sigma$ a sufficiently
high power of $\gamma$, and $\delta$ a still higher power. Increasing the
exponents is allowed, and the choice of $\delta$ is made last among these
curvature tolerances. The finitely many error tolerances in the static
comparison are chosen after these choices. Finally choose the starting scale
$p_0>0$ sufficiently small. The coefficient $1-\beta>0$ introduced below is
chosen still smaller, including the fixed-scale persistence requirement in
Section~\ref{cap:constant-scales}; its scalar endpoint and buffers are fixed
from $p_0$ before this choice. The coefficient is thereafter fixed. Constants in the companion
fixed-temperature theorem may depend on this $\beta$ and on $p_0$.
\subsection{A terminal TAP potential and its extension}
Given the data at the starting cap $p=p_0$, restore the eigenvalues above the
cap using an independent conditional Haar frame in $H_p$. Conditional on
the capped matrix and eigenvalues, this auxiliary refinement and the
original refinement are independent. Denote the resulting matrix by $W$
and its cap subtraction by $B_p^{\rm aux}$, so that $W-B_p^{\rm aux}=J_p$.
Its unconditional law is GOE. The sigma-field
$\mathscr F_p^{\rm aux}$ generated by the capped data, eigenvalues, and
auxiliary Haar seed excludes the original finer frame. Set $t=C_*p^2$ and observe with
precision
\begin{equation}\label{cap:initial-precision}
\Delta=tI+(1-\beta)W-B_p^{\rm aux}.
\end{equation}
For $C_*$ large and $1-\beta$ sufficiently small this is positive definite.
The terminal interaction is $\beta W-tI$, which gives the same spin law as
$\beta W_{\rm off}$. Scalar and Gaussian diagonal terms change only the
partition-function normalization.
On the stable terminal region constructed below, let $\Psi(z)$ be the TAP
potential at its unique root, with interaction $J_*=\beta W_{\rm off}$ and
$j=\beta^2$. Thus $\nabla\Psi(z)=m(z)$, the TAP magnetization.
Propositions~\ref{cap:classical-input} and~\ref{cap:tap-expansion} give the
following construction on an event of probability tending to one. We describe
its domain carefully because it need not be convex.
For a starting field $h\in\mathcal A_p(\sigma)$, freeze its parameter
$\alpha=\alpha_p(h)$. Write $\mathsf G_h$ for the canonical Gaussian
terminal law
\[
Z=h+\Delta a_p+\zeta,\qquad
\zeta\sim N(0,\Delta+\Delta R_p\Delta).
\]
Its filtered field satisfies
\begin{equation}\label{cap:initial-frozen}
R_f=(2-W+t+\alpha+(1-\beta)W)^{-1},\qquad a_f=R_fz,
\qquad
a_f\sim N(a_p,R_p-R_f)\quad\text{under }\mathsf G_h.
\end{equation}
In particular,
\[
\mathbb E\|a_f\|^2=n-\operatorname{Tr}R_f
=(1+o(1))n\sqrt{t+\alpha}.
\]
Under $\mathsf G_h$, the centered noise has covariance norm $O_{C_*}(t)$
and depends on the scalar $\alpha$ in a compact smooth family. Its shift
$h+\Delta a_p$ has normalized norm $O_{C_*}(t^{5/4})$. The scalar and
recipe nets in Proposition~\ref{cap:tap-expansion} cover this family and
all input directions. The exact mean above, the spectral trace bounds,
and a Gaussian quadratic deviation estimate give positive rates per $k_p$
for the fixed filtered-length tolerances.
For comparison, let $\mathsf T_h$ be the true terminal law conditional on
$\mathscr F_p^{\rm aux}$ and $h$:
\[
X\sim\mu_{p,h},\qquad Z=h+\Delta X+\sqrt\Delta\,\xi,\qquad
\xi\sim N(0,I)\ \text{independently of }X.
\]
The drift $h+\Delta X$ has normalized norm $O_{C_*}(t)$ for every spin.
The noise diagnostics hold on one event for all permitted shifts. The
low-band lower bound has a different quantifier: on a band with
$2-W\asymp t$ below $t$, condition first on $X$ and apply the Gaussian
small-ball bound to that fixed center, then average over $X$. This gives
failure $e^{-c nt^{3/2}}=e^{-cC_*^{3/2}k_p}$ with no union over centers.
The exact filtered Gaussian law in \eqref{cap:initial-frozen} is not the
law under $\mathsf T_h$. For those tests the spherical reference is the
canonical bridge conditioned on the prior spin norm. The tested norm-density
calculation gives a positive rate for the named relaxed quadratic tests.
The positive-moment transfer then attaches independent Gaussian noise to
each replica and uses the specified Gram-cell enlargement with the noise
marks fixed. Its field-uniform conclusion and parameter order are recorded
after the terminal regions are defined. The same interface is needed for
the actual finer frame at the starting cap.
The initial convolution uses one terminal potential for all input fields.
We construct it on three nested regions $D_0\subset D_1\subset D_2$,
determined by the parent data and the auxiliary frame. The region $D_0$
contains the predicted terminal fields, $D_1$ supplies the supporting
quadratics, and $D_2$ is the larger region on which the TAP Hessian is
controlled.
Fix $\eta_*>0$ at the choice of $C_*$, sufficiently small for the spectral
contraction and the Gaussian distance-tail comparison below. Choose a fixed
geometric tolerance $\theta$ sufficiently small compared with $\eta_*$.
The earlier finite choice of the annulus tolerance is made so that
$\sigma\ll\theta$. To define membership from a terminal field, let
$\Lambda=\{\mathrm{obs}\}\sqcup[-\sigma p^2,\sigma p^2]$ label the noise
covariances
\[
\Sigma_{\mathrm{obs}}=\Delta,\qquad
\Sigma_\alpha=\Delta+\Delta R_{p,\alpha}\Delta .
\]
Let $\mathcal G_W$ be the set of labelled noise vectors $(\lambda,g)$
passing the buffered diagnostics of Proposition~\ref{cap:tap-expansion}
for every allowed shift. Use closed relaxed versions of these tests and a
fixed Gaussian norm cutoff; their stricter versions retain the stated
probability bounds. The diagnostic families are continuous on their compact
coefficient sets, so $\mathcal G_W$ may be taken with a Borel graph and compact
sections. The low-band lower bound is a test of the resulting field, not a
condition imposed simultaneously on all shifts in $\mathcal G_W$.
Put $R_t=(2-W+t)^{-1}$ and $P_t^W=\mathbf1_{\{2-W\le t\}}$.
Choose shift radii $L_0^{\rm shift}<L_1^{\rm shift}<L_2^{\rm shift}$
inside the allowed range, with the first containing every drift
$h+\Delta X$ above and with fixed margins between the radii. Choose lower
thresholds $\ell_0>\ell_1>\ell_2>0$ below the small-ball threshold, and
$(\theta_0,\theta_1,\theta_2)=(\theta/4,\theta/2,\theta)$. Define
\begin{equation}\label{cap:terminal-regions}
\begin{split}
D_j=\bigl\{z=g+v:\;&(\lambda,g)\in\mathcal G_W,\quad
|v|_n\le L_j^{\rm shift}t,\\
&|P_t^Wz|_n\ge\ell_jt^{5/4},\quad
\bigl||R_tz|_n^2/\sqrt t-1\bigr|\le\theta_j\bigr\},
\qquad j=0,1,2.
\end{split}
\end{equation}
Membership means that at least one labelled noise and shift pair satisfies
these conditions. All labels are included before $h$ is chosen. These are
nested parent-measurable closed sets: the defining graph has compact
sections, and its projection under $(g,v)\mapsto g+v$ is compact.
The geometric tolerances remain fixed throughout the argument.
The next lemma supplies the terminal tests at the fixed initialization
scale. Its filtered-length estimate has a rate available before the replica
order and upper scale are chosen. The noise diagnostics are then imposed at
that chosen scale; their rate need only be positive there.
\begin{lemma}[Terminal tests at the initialization scale]
\label{cap:marked-terminal-tests}
Fix the preceding geometric tolerances and a filtered-length tolerance
$\varepsilon_{\rm len}>0$. For a sufficiently small fixed $p=p_0$ and
then sufficiently small fixed $1-\beta>0$, with probability tending to one
in the capped data and auxiliary refinement, uniformly for
$h\in\mathcal A_p(\sigma)$,
\begin{equation}\label{cap:terminal-test-input}
\mathsf T_h(D_0^c)\le e^{-a k_p},
\end{equation}
and
\begin{equation}\label{cap:terminal-filtered-test}
\mathsf T_h\!\left\{
\left|\|R_fZ\|^2-(n-\operatorname{Tr}R_f)\right|
>\varepsilon_{\rm len}n\sqrt t\right\}\le e^{-a k_p}.
\end{equation}
Here $a>0$ may depend on $p_0$ and the fixed parameters. For either
exceptional event, the same conclusion holds, with constants $C_j,a_j>0$,
after inserting any fixed power of
$1+\|Z\|/\sqrt n+\|R_fZ\|/M_p$ in the expectation. A finite collection
of stricter filtered-length tolerances may be imposed simultaneously.
The same assertions hold for the actual refinement at this starting cap.
\end{lemma}
\begin{proof}
We first prove the filtered tests, keeping their rate uniform as the upper
scale decreases. Under the canonical joint law, sample
$X\sim N(a_p,R_p)$ and independent standard Gaussian $\xi$, and set
$Z=h+\Delta X+\sqrt\Delta\,\xi$. With $A=R_fZ$ and $T=R_p-R_f$,
Gaussian conditioning gives the joint representation
\begin{equation}\label{cap:terminal-joint-gaussian}
A\sim N(a_p,T),\qquad X=A+\eta_f,\qquad
\eta_f\sim N(0,R_f)\ \text{independently of }A.
\end{equation}
In particular $\mathbb E\|A\|^2=n-\operatorname{Tr}R_f$. The spectral
estimates, uniformly for $|\alpha|\le\sigma p^2$ and
$(1-\beta)/t$ sufficiently small, give
\[
\|T\|\le Cp^{-2},\qquad
\operatorname{Tr}T^2+a_p^{\mathsf T}Ta_p\le Cn/p.
\]
The Gaussian quadratic moment formula therefore implies, for each fixed
$\varepsilon>0$,
\begin{equation}\label{cap:canonical-terminal-length}
\Pr\!\left\{\left|\|A\|^2-\mathbb E\|A\|^2\right|
>\varepsilon n\sqrt t\right\}
\le 2e^{-c_{\varepsilon,C_*}k_p}.
\end{equation}
Indeed its centered logarithmic moment is at most
$C\lambda^2 n/p$ for $|\lambda|\le cp^2$; choose a sufficiently small
fixed multiple of $p^2$ in each sign. These constants are independent of
sufficiently small $p$.
We must retain this rate after conditioning $\|X\|^2=n$. For a spectral
quadratic tilt $e^{\lambda A^{\mathsf T}HA+v^{\mathsf T}A}$, the normalized
tilted law of $X$ is Gaussian with covariance
\[
R_f+T^{1/2}
(I-2\lambda T^{1/2}HT^{1/2})^{-1}T^{1/2}.
\]
Choose the tilt with its bracket between fixed positive multiples of $I$.
This covariance is then comparable to $R_f+T=R_p$, regardless of the
tilted mean. A band of order $k_p$ eigenvalues remains comparable to
$p^{-2}$. The characteristic-function estimate
\eqref{cap:norm-characteristic} bounds its squared-norm density by
$Cp^2/\sqrt{k_p}=C\sqrt{p/n}$. The original saddle density is at least
$c\sqrt{p/n}$. Thus the density ratio in
\eqref{cap:conditioned-tail} is bounded by a constant, and
\eqref{cap:canonical-terminal-length} holds with a decreased positive
rate under the spherical terminal law. Independent marked replicas have
the corresponding product bound; equivalently one may use
\eqref{cap:joint-norm-density}.
This argument also covers the $R_tZ$ test defining $D_0$. Write
$R_tZ=(R_tR_f^{-1})A$. The spectral multiplier is bounded and differs
from $I$ by $O(|\alpha|/t+(1-\beta)/t)$, so its quadratic tilt has the same
precision margin. Its canonical mean square divided by $n\sqrt t$ is
$1+o(1)+O(\sigma/C_*)+O((1-\beta)/t)$. The trace estimate and the
resolvent identity give this comparison. Choose these errors below a
fixed fraction of $\theta_0$ and reserve fixed gaps between the tested
and relaxed thresholds. The spherical failure rate for both filtered
tests is consequently a fixed positive multiple of $k_p$ before $p_0$
is chosen.
We now transfer these concrete tests to cube posteriors by
\eqref{cap:tested-transfer}. Give every replica its own independent
standard Gaussian mark in spectral coordinates, and keep the marks fixed
in the Gram-cell comparison. A cell of the size used there changes a
spin representative by $O_l(n^{-1/2})$ in Euclidean norm. Hence it changes
$Z$ by $O_l(tn^{-1/2})$ and $R_fZ$ or $R_tZ$ by
$O_{l,C_*}(n^{-1/2})$. The corresponding changes in the normalized
filtered lengths tend to zero at the fixed scale. Norm inequalities in
both directions show that a failure with tolerance $\varepsilon$ implies
failure with tolerance $\varepsilon/2$ throughout the cell, even for unbounded
noise marks. This verifies the marked inclusion required after
\eqref{cap:tested-transfer}. Integrating the independent marks leaves
the same cube--sphere Gram factor. The small-overlap moment bound thus
has exponent $-lc k_p/2+\eta_l k_p+o_n(k_p)$; the discarded large-overlap
term is added after normalization as in that equation.
For uniformity in $h$, take a mesh of fixed resolution $\zeta$ in
$p^2M_p$ units on a slightly enlarged annulus. Its logarithmic cardinality
is at most $C_\zeta k_p$. Couple neighboring fields by using the same
spin and mark. The scaled saddle is Lipschitz, and the resolvent identity
gives
\[
\|R_{f,h}Z_h-R_{f,h'}Z_{h'}\|
\le C\zeta\bigl(M_p+\|R_{f,h'}Z_{h'}\|\bigr).
\]
Also $|\operatorname{Tr}R_{f,h}-\operatorname{Tr}R_{f,h'}|
\le C\zeta np$, so the centers of the squared-length tests respect the
same fixed gaps. The $R_t$ comparison is simpler, since its resolvent is fixed. Thus a
sufficiently fine fixed mesh respects the threshold gaps. On
$\|P_pX\|\le VM_p$, changing the spin posterior between these fields
costs at most $e^{C\zeta V k_p}$, including the partition normalizer;
use the static projected-mean bound on a prescribed ball containing the
annulus and the joining segment. The
complement has the uniform projected $V^6$ tail from
Proposition~\ref{cap:static-comparison}. Choose $V$ first to give the
required tail rate and then $\zeta$ to fit both the threshold and
likelihood budgets. In \eqref{cap:tested-transfer}, choose a posterior
exponent below $c/2$, the value error below the remaining margin, and
then a fixed replica order $l$ whose Haar exponent exceeds $C_\zeta$.
Only afterward decrease $p_0$ so that the Gram error $C_lnp^4$ and all
comparison errors fit this budget. The field net then proves both
filtered tests uniformly with a positive exponent per $k_p$.
It remains to impose the other conditions of $D_0$. The deterministic
shift $h+\Delta X$ is within the chosen shift radius for every spin.
Lemma~\ref{cap:boundary-noise} gives one conditional noise event for all
these shifts, with failure at most $e^{-c_tn}$ for typical $W$. For the
low-band lower bound, condition on each spin and use that lemma's
uniform-in-center small-ball estimate, then average. This bound is
uniform in the spin law and in $h$, without a field net or a union over
spins. At the chosen $p_0$, the minimum of these positive rates and the
filtered-test rate is a positive rate per $k_{p_0}$, proving
\eqref{cap:terminal-test-input}.
At this fixed scale, the conditional Gaussian observation and
$\|X\|=\sqrt n$ bound every fixed moment of
$1+\|Z\|/\sqrt n+\|R_fZ\|/M_p$, uniformly in $h$. Cauchy--Schwarz
with a doubled moment gives the weighted exceptional bounds with a
smaller exponent. The auxiliary refinement has marginal GOE law and is
independent of the true finer frame conditional on the capped data.
The event just proved is therefore measurable before that true frame is
revealed. Repeating the same argument gives the actual-refinement event;
it is imposed separately, without including it in
$\mathscr F_p^{\rm aux}$. A finite intersection proves the final assertion.
\end{proof}
The true terminal field now lies in $D_0$ with the weighted error bounds
needed for gradient transport. We next extend the TAP potential while
preserving its values and derivatives on a neighborhood of these fields.
A sufficiently small fixed $b_*>0$ gives a ball of radius
\begin{equation}\label{cap:agreement-buffer}
b_*\sqrt n\,t^{5/4}
\end{equation}
inside $D_1$ around every point of $D_0$, and inside $D_2$ around every point
of $D_1$. Indeed, retain a witnessing $(\lambda,g)$ and assign the
displacement to $v$. A displacement of normalized length $b_*t^{5/4}$
changes the low-band norm by at most that amount and changes
$|R_tz|_n^2/\sqrt t$ by at most $Cb_*+Cb_*^2$. The fixed gaps between the
shift, lower-bound, and filtered-length thresholds give both buffers.
We also reserve an outer shift and low-band margin beyond $D_2$. Choose one
residual radius $\rho_{\mathrm{res}}$ inside these margins, with
$\rho_{\mathrm{res}}\ll t^{5/4}$. If
$e=F_{J_*,z}(y)$ and $|e|_n\le\rho_{\mathrm{res}}$, then
$F_{J_*,z+e}(y)=0$. The same noise witness remains admissible at $z+e$,
and the realized low-band bound is preserved by the triangle inequality.
The pre-stability part of Proposition~\ref{cap:tap-expansion} therefore
applies to every such approximate root. Its zero-field exclusion first
bounds $q$ above, the low-band estimate bounds it below by $c\sqrt t$,
and its scalar constraint, together with the filtered-length condition and
monotonicity, places it in the common band used below.
The derivative estimates then give the stability implication on this
whole residual region, with common parameters. Only at this point do we
apply the deterministic root lemma in Proposition~\ref{cap:classical-input}:
it gives a globally unique stable root for every $z\in D_2$.
The root is measurable in the data and field, for example as the limit of
measurably selected rational approximate roots; locally it is smooth by
the inverse function theorem.
The local curvature and the comparison of distant support points require
different arguments. To make the latter comparison without changing the
geometric regions, use the fixed interval
$I_t=[(1-2\theta)\sqrt t,(1+2\theta)\sqrt t]$ and define
\begin{equation}\label{cap:own-saddle-potential}
R_q=(2+q^2-W)^{-1},\qquad
F_q(z)=\tfrac12 z^{\mathsf T}R_qz+\frac n3q^3,
\qquad
\partial_qF_q(z)=q\bigl(nq-\|R_qz\|^2\bigr).
\end{equation}
The resolvents are positive on $I_t$ for the fixed boundary scale and large
$n$. The function $q-\|R_qz\|^2/n$ is strictly increasing. Its value at
$\sqrt t$ has absolute value at most $\theta\sqrt t$ on $D_2$, so it has a
unique zero $\widehat q(z)$ in the interior of $I_t$. Thus $\widehat q(z)$ is
the minimizer of $F_q(z)$ on this same interval for every $z\in D_2$.
The value calculation in Proposition~\ref{cap:tap-expansion} gives the exact
coefficient $\kappa_0=1/3$. Together with its mean expansion it implies,
uniformly on $D_2$, for a field-independent constant $c$,
\begin{equation}\label{cap:own-saddle-accuracy}
\left|\Psi(z)-c-\min_{q\in I_t}F_q(z)\right|
\le\epsilon nt^{3/2},\qquad
\left\|\nabla\Psi(z)-R_{\widehat q(z)}z\right\|
\le\epsilon\sqrt n\,t^{1/4}.
\end{equation}
Here $\epsilon$ is an approximation tolerance, separate from $\theta$ and
the two buffers. It can be decreased after those geometric quantities are
fixed. For clarity, the TAP resolvent initially has parameter
$r+r^{-1}=2+q^2/(1-q)$. Replacing it by $2+q^2$ changes its quadratic value
by $O(nq^4)$ and its mean by $O(\sqrt n\,q^{3/2})$, uniformly on the
coarse band $q\asymp\sqrt t$. The mean estimate and the definition of the
TAP overlap give $q-\|R_qz\|^2/n=o(q)$. Strict monotonicity then gives
$q-\widehat q(z)=o(q)$. On this fixed band the first two scalar derivatives
of $F_q/n$ have orders $q^2$ and $q$, so this replacement changes the value
by $o(nq^3)$ and the mean by $o(\sqrt n\,q^{1/2})$. These errors, and the
perturbations from the interaction coefficient and diagonal completion, give
\eqref{cap:own-saddle-accuracy}: choose the diagnostic accuracies, then the
boundary scale sufficiently small, then the coefficient perturbation
sufficiently small, and finally $n$ large. The noise diagnostic accuracy may improve, but assigning displacements to
the allowed shift still preserves those diagnostics. These choices do not
reduce the geometric tolerances or either fixed buffer.
By the resolvent identity and the fixed choice $\theta\ll\eta_*$,
$R_q\preceq R_t+\eta_*I/(4t)$ on $I_t$. The local TAP Hessian, whose leading
correction is negative semidefinite, therefore obeys the required upper
bound on $D_2$. Put
\begin{equation}\label{cap:initial-support-curvature}
U=(2-W+t)^{-1}+\frac{\eta_*}{t}I,
\qquad \|\Delta^{1/2}U\Delta^{1/2}\|\le1-b_{**}
\end{equation}
for a fixed $b_{**}>0$. The last inequality follows in the common spectral
basis: before the $\eta_*$ and $1-\beta$ perturbations,
$1-\Delta_i/(d_i+t)\ge c/C_*$ on both sides of the cap. The approximation
tolerance is also chosen small enough that $\nabla^2\Psi\preceq U$ on $D_2$.
For $z,z'\in D_1$ at distance less than
\eqref{cap:agreement-buffer}, their segment lies in $D_2$; integration of
this Hessian bound gives the upper-support inequality. For more distant
points put $d=z'-z$ and choose the minimizing parameter at $z$.
Since $\min_{q\in I_t}F_q(z')\le F_{\widehat q(z)}(z')$,
\eqref{cap:own-saddle-accuracy} gives
\[
\Psi(z')-\Psi(z)-\nabla\Psi(z)^{\mathsf T}d
\le\tfrac12d^{\mathsf T}R_{\widehat q(z)}d
+2\epsilon nt^{3/2}
+\epsilon\sqrt n\,t^{1/4}\|d\|.
\]
At these distances the two error terms are at most
$\epsilon(2b_*^{-2}+b_*^{-1})\|d\|^2/t$. Choose
$\epsilon\ll\eta_*\min(b_*^2,b_*)$ after the fixed buffers. The curvature
slack then bounds the right side by $d^{\mathsf T}Ud/2$. This is a
one-sided comparison using the saddle at the support point; it does not
require an arbitrarily accurate absolute comparison with a single frozen
quadratic on the whole geometric region.
Take the infimum of the upper supporting quadratics with curvature $U$
over $D_1$ and call the result $\overline\Psi$. The preceding inequalities
hold for every pair of support points. They show that the extension equals
$\Psi$ on $D_1$, hence on a full fixed neighborhood of $D_0$. The extension
has global upper Hessian $U$ in the distributional sense and at most
quadratic growth; its support gradients have at most linear growth. One may
mollify and pass to the limit at fixed $n$.
For later estimates we also need to transfer canonical tests to the heat
convolution of this extension. At each input field, restrict the supports
further to a stricter predicted subset $H_h$ of $D_0$ where $q/\sqrt{t+\alpha}$
and the filtered norm at $R_f$ have arbitrarily small fixed errors. This subset has canonical probability
tending uniformly to one. On it, $\Psi$ and its gradient agree with the frozen
canonical quadratic up to an arbitrarily small error per $k_p$ and per $M_p$,
respectively. The distance to this subset, in $\sqrt n\,t^{5/4}$ units, has
Gaussian upper tails with a positive rate per $nt^{3/2}$, uniformly as $p$
decreases. Indeed, Gaussian enlargement uses only
$\mathsf G_h(H_h)\ge1/2$ and the covariance bound $C_{C_*}t$;
the large-$n$ threshold for this mass bound may depend on the fixed $p$,
while the distance-tail rate does not. The support comparison bounds the additional logarithmic weight by
a small constant times $k_p$ plus
\[
C(\eta_*+\sigma)nt^{3/2}
\operatorname{dist}(z,\text{the subset})^2,
\]
where distance is in those scaled units. Choose $\eta_*$ below the
distance-tail rate. Integrating the distance tails, or applying H\"older's
inequality, transfers the required tests with a smaller positive exponent; the
same subset gives the normalization lower bound. For tests excluding $D_1$,
the tolerances and exponents can be fixed
independently of small $p$. Only then decrease $p$ and $1-\beta$ to control
the approximation errors.
For the lower Fisher estimate, localize before differentiating the extension.
Choose a cutoff equal to one on the stricter predicted subset in $D_0$ and
supported strictly inside $D_1$, where $\overline\Psi=\Psi$. This ensures
that the calculation uses only the original smooth potential, even though
$D_1$ need not be convex. The $D_0$-to-$D_1$ buffer
\eqref{cap:agreement-buffer} gives polynomial derivative bounds for a
distance cutoff. The larger domain $D_2$ is used for the local Hessian
segments in the support comparison. Integration by parts with the cutoff and the
score uses only the classical Hessian where the extension agrees locally.
Negative distributional Hessian mass elsewhere is never integrated. The
omitted terms have exponential probability and polynomial moments for the
fixed parameters.
\subsection{The initial radial deficit}
\begin{proposition}\label{cap:auxiliary-initialization}
For the preceding choices, with probability tending to one there is
an intrinsic auxiliary cap potential
\[
\widetilde\phi_p(h)=
\log\mathbb E_g\exp\overline\Psi(h+\sqrt\Delta\,g),
\qquad h\in P_p,
\]
whose Hessian satisfies \eqref{cap:invariant} on the used annulus,
with slack. Moreover, for a finite constant $C$ independent of $n$,
\begin{equation}\label{cap:initial-gradient-error}
\sup_{h\in\mathcal A_p(\sigma)}
\|\nabla\widetilde\phi_p(h)-P_p\nabla\phi_p(h)\|\le C.
\end{equation}
The constant may depend on all the fixed initialization parameters.
\end{proposition}
\begin{proof}
We first verify curvature. Write $\widetilde{\mathsf T}_h$ for the terminal
field law with density proportional to $e^{\overline\Psi(z)}$ relative to
$N(h,\Delta)$. All expectations in the curvature calculation below use this
law. An indicator $\mathbf1_{H_h}$ means that the integrand is integrated only
over $H_h$, including when it is defined only there. The canonical identities
in \eqref{cap:initial-frozen} enter through
the transferred tests, rather than as identities under
$\widetilde{\mathsf T}_h$.
On $H_h$, put $\nu=n/(4q)$; no value of $q$ or $\nu$ outside $H_h$ is used
in these retained integrals. The ideal form of the TAP response there is
\[
K_f^0=R_f-\frac{R_fa_fa_f^{\mathsf T}R_f}
{\nu+a_f^{\mathsf T}R_fa_f}.
\]
Proposition~\ref{cap:tap-expansion} has the same two resolvent factors with
the root vector $y$. Replacing its resolvent by $R_f$ and
$y=r^{-1}a_f+o(M_p)$ by $a_f$ changes the response by an arbitrarily small
amount in $t^{-1}$ units; the scalar matching is detailed below. In the
ideal calculation, $R_f^{-1}=R_p^{-1}+\Delta$ and rank-one inversion give
the exact identity
\begin{equation}\label{cap:initial-rank-transport}
\Delta^{-1}(\Delta^{-1}-K_f^0)^{-1}\Delta^{-1}-\Delta^{-1}
=R_p-\frac{R_pa_fa_f^{\mathsf T}R_p}
{\nu+a_f^{\mathsf T}R_pa_f}.
\end{equation}
The left side is the Brascamp--Lieb covariance bound after the
increment-covariance transformation. Thus the transported correction has
two $R_p$ factors, and its denominator contains $R_p$.
The transferred canonical tests give
\[
\mathbb E[\mathbf1_{H_h}a_f]=a_p+o(M_p),\qquad
\|a_p\|^2=(b+o(1))np,
\]
and
\[
\begin{split}
\mathbb E[\mathbf1_{H_h}a_f^{\mathsf T}R_pa_f]
&\le a_p^{\mathsf T}R_pa_p+
\operatorname{Tr}\bigl(R_p(R_p-R_f)\bigr)+o(n/p)\\
&\le (3b+o(1))n/p.
\end{split}
\]
Here $b=4/(3\pi)$, while
$\mathbb E[\mathbf1_{H_h}\nu]=O(n/(\sqrt{C_*}p))$.
On $H_p$, $R_p=(p^2+\alpha)^{-1}I$. For
$e_p=a_p/\|a_p\|$, Cauchy--Schwarz therefore bounds the retained expected
radial subtraction in $p^2$ Hessian units below by
\begin{equation}\label{cap:initial-radial-deficit}
\frac{p^2}{(p^2+\alpha)^2}
\frac{\bigl(\mathbb E[\mathbf1_{H_h}e_p^{\mathsf T}a_f]\bigr)^2}
{\mathbb E[\mathbf1_{H_h}(\nu+a_f^{\mathsf T}R_pa_f)]}
\ge \frac{1}{3+o(1)+O(C_*^{-1/2})}.
\end{equation}
The errors also tend to zero with $\sigma$. Taking $C_*$ large leaves a
strict margin above the radial constant $0.30$.
For each fixed unit transverse direction $v\perp a_p$, the retained second
moment $\mathbb E[\mathbf1_{H_h}(v^{\mathsf T}(a_f-a_p))^2]$ is $o(np)$,
with bounds uniform in that direction. The Fisher lower bound and the
Brascamp--Lieb upper bound therefore give arbitrarily small transverse error
and a bounded radial lower error. The contribution from $H_h$ in the localized
Fisher calculation is linear in the terminal correction and retains its denominator
$\nu+a_f^{\mathsf T}R_fa_f$; the upper bound uses the denominator in
\eqref{cap:initial-rank-transport}. The directional second-moment test
controls both on transverse directions.
The reference resolvent must be matched at the actual frozen parameter.
Put $s=t+\alpha$ and $q_0=\sqrt s$. On $H_h$, the
$R_f$ filtered-length test is within $\epsilon_{\rm len}q_0$ of its exact
mean $1-\operatorname{Tr}R_f/n$, and the latter is within
$\epsilon_{\rm tr}q_0$ of $q_0$. The TAP mean expansion gives
$|q-|R_qz|_n^2|\le\epsilon_{\rm TAP}q$. Since
\[
\frac{d}{dv}\bigl(v-|R_vz|_n^2\bigr)
=1+\frac{4v}{n}z^{\mathsf T}R_v^3z\ge1,
\qquad
R_f^{-1}=R_{q_0}^{-1}+(1-\beta)W,
\]
these tests imply
\[
\frac{|q-q_0|}{q_0}
\le C\left(\epsilon_{\rm len}+\epsilon_{\rm tr}
+\epsilon_{\rm TAP}+\frac{1-\beta}{t}\right).
\]
Consequently the TAP parameter $q^2/(1-q)$ differs from $t+\alpha$ by
an arbitrarily small fraction of $p^2$ after the prescribed accuracy,
scale, and coefficient choices. This uses the exact $R_f$ test, so no fixed
$O(\sigma)$ error remains. The exceptional set is already covered by the
terminal transition bounds. The transverse error can therefore be made as
small as required at fixed $C_*$ and $\sigma$.
Positive-matrix Cauchy--Schwarz applied to the two matrix bounds gives the
mixed-block estimate in \eqref{cap:invariant}. The cutoffs described above
justify the lower bound, and \eqref{cap:initial-support-curvature} controls
the upper bound on the exceptional set.
The curvature invariant is now established for the auxiliary potential. We
next compare its gradient with the true cap mean in unnormalized Euclidean
norm. At the terminal field this is the uncentered consequence of the
companion's directional estimate. We transport that estimate backward through
the fixed precision interval. This gives a bounded gradient error; it does not
yet give a Hessian comparison.
At every terminal field in the agreement region the companion
fixed-$\beta<1$ posterior theorem gives
\begin{equation}\label{cap:accepted-mean-error}
\|m_{\rm Gibbs}(z)-m(z)\|\le C.
\end{equation}
Precisely, use \cite[Section~7.2, proof of Proposition~7.2]{AcceptedGap}: its
estimate $|\mathbb E[G(X-t_*)^{\mathsf T}v]| \le C\|G\|_*$, before centering
$G$, with $G=1$ and $\|v\|\le1$ gives \eqref{cap:accepted-mean-error}. Its
assumptions are the fixed-length word diagnostics and the stable-field
implication through a fixed residual threshold. After fixing $\beta<1$,
restrict the permitted diagonal enlargement to a positive
$\eta<\min(\eta_{\rm TAP},\beta^{-1}-1)$, where $\eta_{\rm TAP}$ is the
enlargement supplied by the boundary test. The stability implication remains
valid on this smaller class, and $\beta^2(1+\eta)^2<1$ supplies the
companion's strict diagonal margin. Impose its fixed-length word diagnostics
with this choice. All constants are uniform in the terminal field and $n$
under these diagnostics. No assertion about the constant as $\beta\uparrow1$
is needed.
Transport \eqref{cap:accepted-mean-error} backward through the observation,
using Lemma~\ref{cap:error-transport} below. The strict upper curvature bound
\eqref{cap:initial-support-curvature} gives bounded amplification on this
fixed precision interval. Outside agreement, true observation probabilities
and their polynomially weighted moments are exponentially small by
Lemma~\ref{cap:marked-terminal-tests}. The extension gradients have at
most linear growth, so their contribution there tends to zero. This proves
\eqref{cap:initial-gradient-error}. Finally the conditional auxiliary
completion uses only the parent data and an independent Haar seed. Its
high-probability properties may be imposed before revealing the true finer
frame, which proves the asserted intrinsic measurability. \end{proof}
\subsection{Transporting and contracting gradient errors}
\begin{lemma}\label{cap:error-transport}\label{cap:gradient-transport}
Consider a deterministic increasing matrix precision $B_s$, with
derivative $G_s\succeq0$, and a terminal potential extended with a
strict semiconcavity margin relative to the total remaining
precision. Let $f_s$ be its backward log heat convolution and
$K_s=\nabla^2 f_s$. Along the true observation process, write
$m_s$ for the posterior mean and $e_s=m_s-\nabla f_s$.
For a starting direction $v$, let $v_s$ solve
\begin{equation}\label{cap:linear-flow}
dv_s=dB_s\,K_s v_s=G_sK_sv_s\,ds.
\end{equation}
Then
\[
e_{s_0}^{\mathsf T}v
=\mathbb E[e_{s_1}^{\mathsf T}v_{s_1}\mid h_{s_0}],
\]
whenever the indicated polynomial moments are finite. The same
identity holds by approximation for the extensions used here.
\end{lemma}
\begin{proof}
The true posterior mean has zero drift. Apply It\^o's formula to
the approximate gradient along the same observation, and use the backward
heat equation for $f_s$. Subtracting the two evolutions gives
$de_s=-K_sG_se_s\,ds+dM_s$.
The finite-variation equation \eqref{cap:linear-flow} cancels this
drift in $e_s^{\mathsf T}v_s$. Localization and then the moment
bounds give the identity. A global upper Hessian bound for the
terminal potential gives a global upper bound for the convolved
Hessian by Brascamp--Lieb. For a constant precision rate $G$
this bounds amplification in the $G^{-1}$ metric. On the cap
path the Euclidean estimate applies instead, because the
variation stays in the support of $P_u$, on which the rate is
scalar. Mollification,
the same bound, and dominated convergence justify the limiting
extension. In \eqref{cap:linear-flow}, $dB_s$ is a precision
increment, not a Brownian increment.
\end{proof}
\begin{proposition}\label{cap:gradient-contraction}
Suppose the parent auxiliary potential is intrinsic, satisfies the
curvature invariant with slack, and its true gradient discrepancy
is bounded by $D/p$ on its agreement annulus, where $D<\infty$
is fixed. Put $c=\gamma p$. Extend the parent potential as in
Proposition~\ref{cap:curvature-induction} and take its log heat
convolution to $c$. For the parameter choices above, with
probability tending to one over the unused refinement,
\begin{equation}\label{cap:gradient-contraction-bound}
\sup_{h\in\mathcal A_c(\sigma)}
\|P_c\nabla\phi_c(h)-\nabla\widetilde\phi_c(h)\|
\le\frac1c\bigl(\tfrac12D+o_n(1)\bigr).
\end{equation}
This conclusion is uniform in the child field. For a fixed finite
number of comparison steps, the additive errors tend to zero
simultaneously.
\end{proposition}
\begin{proof}
Parameterize observation from $c$ to $p$ by $\log u$, so its rate
is $2u^2P_u$. A variation initially in $P_c$ remains in $P_u$.
Freeze the starting parameter $\alpha$ and write
$a_u=R_uh_u$ along the canonical bridge. These vectors form a
Gaussian martingale with independent increments and
\[
\operatorname{Cov}(a_u-a_s)=R_s-R_u\qquad(c\le s\le u\le p).
\]
Norms and mixed products concentrate at their predictions, in particular
$a_s^{\mathsf T}a_u=\|a_s\|^2+o(M_p^2)$, with arbitrarily small fixed scaled
errors. The inferred parameter is within an arbitrarily small fraction of
$u^2$ of its frozen value. These assertions are uniform in $u$: use finite
time grids and Gaussian maximal bounds. In parent units the covariance of an
increment has operator norm at most $O_\gamma((\operatorname{Tr}P_p)^{-1})$
times its logarithmic time length and trace at most $O_\gamma(1)$ times that
length. Transfer of path tests to the true law uses its terminal density. The
exceptional-tail estimates are uniform conditionally on the starting field.
Outside the retained paths the global semiconcavity estimate still bounds flow
amplification.
If $h_u^{\rm lin}$ denotes the variation in \eqref{cap:linear-flow}, rescale
it to $v_u=(c^2+\alpha)h_u^{\rm lin}/(u^2+\alpha)$. The scalar reference
curvature cancels. By the intermediate estimates of
Proposition~\ref{cap:curvature-induction}, the remaining equation, up to a
small operator perturbation, is
\begin{equation}\label{cap:rank-flow}
\frac{dv_u}{d\log u}
=-2d_u\frac{a_ua_u^{\mathsf T}}{\|a_u\|^2}v_u,
\qquad 0.26\le d_u\le C.
\end{equation}
The upper bound is absolute: use the variance-formula lower Hessian bound when
$u/p\le0.9$, and the Fisher bound closer to the parent. The perturbation is
bounded by $O_\gamma(\delta^{b'})$ for an absolute $b'>0$, plus errors chosen
arbitrarily small by the approximation tolerances. Variation of constants
absorbs it after these choices. The starting parameter tolerance contributes
no additional fixed scalar error, because the rescaling uses the actual frozen
parameter. All intermediate support conditions have slack; near $p$ use the
parent agreement domain itself.
Let $V_u:P_c\longrightarrow P_u$ be the pure flow
\eqref{cap:rank-flow}, with the identity inclusion at $c$.
It is a contraction. Its terminal radial component satisfies
\begin{equation}\label{cap:radial-flow}
\left\|\frac{a_p^{\mathsf T}V_p}{\|a_p\|}\right\|
\le C\sqrt H\,\gamma^{1.02}+\tau,
\qquad H=C_1\log(1/\gamma),
\end{equation}
where $\tau>0$ can be chosen arbitrarily small before $n$ is large. Removing
the scalar reference growth leaves a flow that contracts only in the current
radial direction. That direction changes along the bridge. To measure the
cumulative contraction, test the flow against the terminal radial vector and
use the bridge's mixed scalar products. The resulting scalar integral equation
has an integrating factor bounded by a fixed power of the scale ratio. We use
pathwise integration by parts, so no adaptedness claim is needed for that
factor.
To see the exponent, put $G_s=a_s^{\mathsf T}V_s$.
The mixed-product estimates in the integral equation give
\[
G_s=a_s^{\mathsf T}V_c-
\int_c^s2d_uG_u\,\frac{du}{u}
+\text{an arbitrarily small scaled error}.
\]
Write $Q(p,u)=\exp(-\int_u^p2d_s\,ds/s)\le(u/p)^{0.52}$.
The integrating-factor formula is
\[
G_p=Q(p,c)a_c^{\mathsf T}V_c+
\int_c^pQ(p,u)\,d(a_u^{\mathsf T}V_c)
+\text{an arbitrarily small scaled error}.
\]
The integral is interpreted by pathwise integration by parts; its integrand
can depend on future values of $d_s$, so an adapted stochastic-integral
interpretation is unnecessary. No differentiability of $d_s$ is needed. On a
logarithmic dyadic interval starting at $u$, increments and partial increments
of $P_ca_s$, divided by $\|a_p\|$, are at most $C\sqrt H\,\gamma^{3/2}p/u$
with exponential probability. This follows from the covariance formula and
$\operatorname{Tr}P_c\asymp nc^3$. Integration by parts against the
integrating factor gives a summable series with largest term $C\sqrt
H\,\gamma^{3/2}\gamma^{-0.48}$. The initial term is at most
$C\gamma^{1/2}\gamma^{0.52}$. This proves \eqref{cap:radial-flow}. Time and
mixed-product errors have at most polynomial amplification in $\gamma^{-1}$.
We next prove the transverse gain, including its conditional law. Let
$\mathcal F_p$ contain the capped matrix, the eigenvalues, and the entire
intrinsic auxiliary construction. Write $Q$ for the unused Haar map into
$P_p$. For a fixed child-coordinate field, sample the canonical path in
spectral coordinates independently of $Q$. Condition on these path coordinates
and on the terminal physical image $Q\xi_p=h_p$, where $\xi_p$ is the terminal
field in spectral coordinates. The terminal discrepancy, as a function of
$(\mathcal F_p,h_p)$, is now fixed. Its transverse part $e_p^\perp$ is
perpendicular to $h_p$, hence to $a_p$. The residual map from $\xi_p^\perp$ to
$h_p^\perp$ is exactly Haar. We do not condition on the intermediate physical
field images.
The elementary Haar projection bound, applied conditionally
as just specified, therefore gives
\[
\|P_cQ^{\mathsf T}e_p^\perp\|
\le C\sqrt H\,\gamma^{3/2}\|e_p^\perp\|,
\qquad
|(e_p^\perp)^{\mathsf T}a_u|/\|a_u\|
\le C\sqrt H\,\gamma^{3/2}\|e_p^\perp\|
\]
on a fixed polynomially fine time grid, with failure at most $\exp(-A Hk_c)$
for a prescribed fixed $A$ after increasing the constant. Here $P_c$ in the
first expression denotes the child coordinate subspace before applying $Q$.
The bound uses $\operatorname{Tr}P_c/\operatorname{Tr}P_p \asymp\gamma^3$;
adding finitely many grid directions does not affect it once $n$ is large.
Gaussian modulus tests fill the gaps with any prescribed fixed error. Since
$\|V_u\|\le1$ and $d_u\le C$, the integral equation for $V_p$ yields
\begin{equation}\label{cap:transverse-flow}
\|V_p^{\mathsf T}e_p^\perp\|
\le\bigl[C\sqrt H\,\gamma^{3/2}
(1+|\log\gamma|)+\tau\bigr]\|e_p^\perp\|.
\end{equation}
This step also permits the coefficients $d_u$ to depend on $Q$. The projection
event controls the vectors appearing in the integral equation, while their
coefficients use only the deterministic bounds already established. No
additional independence of those coefficients is required.
The residual Haar statement is used only under canonical
sampling. To pass to the actual heat-tilted path, its
Radon--Nikodym derivative for each fixed $Q$ is the terminal
weight
\[
\frac{\exp\{\phi_p(Q\xi_p)-q_p^{\rm fr}(\xi_p)\}}
{\mathbb E_{\rm can}
\exp\{\phi_p(Q\xi_p)-q_p^{\rm fr}(\xi_p)\}},
\]
where $q_p^{\rm fr}$ is the frozen quadratic. An additive calibration of the
numerator is immaterial. On an arbitrarily narrow fixed terminal shell, the
static comparison and the auxiliary gradient calibration bound this derivative
by $\exp(\eta k_p)$ for any prescribed $\eta>0$, and the same shell gives its
normalization lower bound. This normalization may depend on $Q$. We transfer
the canonical Haar failure estimate through the density ratio, rather than
asserting that the tilted law is Haar. If a finite grid or replica transfer
has $r$ terminal factors, choose $r\eta\ll H\gamma^3$. More explicitly, if
$\mathcal E_H$ is the high-rate Haar failure event on the retained terminal
shell, the canonical bound and the $r$ density factors give
\[
\mathbb E_Q\,\mathsf T_{h,Q}^{(r)}(\mathcal E_H)
\le \exp\{-[AH-r\eta\gamma^{-3}]k_c\}.
\]
Here $\mathsf T_{h,Q}^{(r)}$ is the joint actual path law for those
$r$ terminal factors at the fixed grid field.
Choose $b_{\rm path},b_Q>0$ with
$b_{\rm path}+b_Q<AH-r\eta\gamma^{-3}$ and
$b_Q>C_{\rm net}\log(1/\gamma)$, where the child-field net has logarithmic
size at most $C_{\rm net}\log(1/\gamma)k_c$. Markov's inequality then gives
\[
\Pr_Q\!\left\{\mathsf T_{h,Q}^{(r)}(\mathcal E_H)
>e^{-b_{\rm path}k_c}\right\}
\le e^{-b_Qk_c}.
\]
The second exponent pays for the field net. The first remains as a
conditional path bound and absorbs the polynomial discrepancy and flow
moments at the fixed comparison scales. Here is the likelihood comparison
needed to pass from the net to nearby fields. Let
$\|h-h'\|\le\rho c^2M_c$ and keep the later physical grid fields fixed.
Only the first Gaussian increment and the initial normalizer then change.
Its inverse on $P_c$ is bounded by a fixed multiple of $c^{-2}$ once the
first positive grid time is fixed. On bounded retained grid fields, the
Gaussian exponent comparison therefore gives
\[
\left|\log\frac{d\mathsf T_h^{(r)}}{d\mathsf T_{h'}^{(r)}}\right|
\le rC_\gamma\rho k_c.
\]
The normalizer obeys the same bound: differentiate the exact full-interval
convolution to obtain
$\nabla_{P_c}\phi_c=\Delta_{P_c}^{-1}(\mathbb E Y_{P_c}-h)$,
and use the uniform parent-field moment bound before integrating along
the field segment. The constant $C_\gamma$ may also depend on the fixed
first grid time and buffers.
The intrinsic terminal discrepancy is the same function of the fixed
terminal field in this comparison, so no derivative bound on it is
required. Discarding an arbitrarily short initial grid interval costs an
arbitrarily small operator error in the rank flow. The remaining
projection and mixed-product tests, including their normalizations by
nonzero annular lengths, have polynomial continuity in the fixed
parameters. Choose a net of inverse-polynomial resolution $\rho$ so
that these changes fit the reserved test slack and
$rC_\gamma\rho<b_{\rm path}/2$. A relaxed bad event at a neighboring
field is then bounded by its net-point probability times
$e^{rC_\gamma\rho k_c}$, retaining exponent at least
$b_{\rm path}/2$. This comparison is made before taking conditional
expectations. Its net still has logarithmic size
$C_{\rm net}\log(1/\gamma)k_c$, paid by the previously reserved
orientation exponent. Modulus tests,
mixed-product tests, and shell complements are controlled by the separate
uniform conditional tail estimates. Their failure exponents need not equal the
high Haar rate, and no union bound over the child net is applied to those
tails.
Apply Lemma~\ref{cap:error-transport}. The prefactor from undoing the
rescaling is $O(\gamma^{-2})$. Multiplying the resulting bound by $c$, the
radial and transverse terms in \eqref{cap:radial-flow} and
\eqref{cap:transverse-flow} are bounded by
\[
C\sqrt H\left[\gamma^{0.02}
+\gamma^{1/2}(1+|\log\gamma|)\right]D.
\]
Choose $\gamma$ so this is below $1/4$, and choose the operator and
fixed-tolerance errors so their contribution is below $D/4$. The exceptional
sets contribute $o_n(1)$ in these units for fixed $D$ and fixed scales. This
proves \eqref{cap:gradient-contraction-bound}. Agreement at each extension
boundary is enough: true paths leave it only with the stated negligible
probability. Integrating the bounded gradient discrepancy along annular paths
and recalibrating an additive constant gives an $o(k_c)$ value oscillation
error, since $c$ is fixed during this finite phase. The next curvature step
therefore retains all its macroscopic comparison hypotheses. \end{proof}
\subsection{Replacing the auxiliary potential and evaluating all scales}
After finitely many contractions the gradient discrepancy is small in the
scale units needed for the next convolution. This is the point at which a
Hessian comparison becomes available. The logarithmic density discrepancy is
Lipschitz on the parent shell; strong log-concavity turns that Lipschitz bound
into small thermal fluctuations. Changing density therefore perturbs the
directional first and second moments of the terminal field by a small amount.
The exact increment covariance formula then compares the two child Hessians.
\begin{lemma}\label{cap:remove-auxiliary}
Fix $\gamma$ and the curvature tolerances. If the parent
gradient discrepancy is at most $D/p$ on the used annulus,
then one further convolution makes the true and auxiliary
output Hessians differ by $O_\gamma(D)+o_n(1)$ in child
$c^2$ units. In particular, for $D$ sufficiently small,
the true output satisfies \eqref{cap:invariant} whenever
the auxiliary output satisfies it with the established
slack.
\end{lemma}
\begin{proof}
On the parent shell the difference of logarithmic
potentials has Lipschitz constant at most $CD/p$.
Annular paths have a bounded ratio of path length to
chord length, so this is also a Euclidean Lipschitz
bound up to an absolute factor. Extend the difference
with the same bound, for example by the infimum formula
for Lipschitz extensions. Under the strongly log-concave
extended auxiliary bridge, its centered fluctuations
are subgaussian with scale $C_\gamma D$. Indeed the
bridge covariance scale is $O_\gamma(p^2)$ and its
strong convexity bound applies to every Lipschitz
function. Linear functions of the terminal field in
thermal units have a bounded subgaussian scale as well.
If $F$ is the centered logarithmic discrepancy and
$L$ such a normalized linear function, Cauchy--Schwarz
and the subgaussian exponential moments give
\[
\left|\frac{\mathbb E[e^F L^j]}{\mathbb E e^F}
-\mathbb E L^j\right|
\le C_\gamma D,\qquad j=1,2,
\]
for $D$ below a fixed constant. Center the linear observable in each tested
direction before applying these inequalities. Their constants are uniform over
unit directions, so the resulting variance estimates control the full
covariance operator. The true bridge and the tilted extension agree
conditionally on the shell. The true bridge has the preceding complementary
moment bounds. For the discrepancy-tilted extension, H\"older's inequality
and the centered subgaussian bound on $F$ transfer the auxiliary complementary
moment bounds with a decreased positive exponent, including the required
second moments. The increment covariance
formula, with inverse increment $O_\gamma(p^{-2})$ on $P_c$, now proves the
asserted Hessian comparison. \end{proof}
Starting with Proposition~\ref{cap:auxiliary-initialization}, write its finite
error bound as $D_0/p_0$. Apply Proposition~\ref{cap:gradient-contraction} a
fixed number of times until $D$ is as small as
Lemma~\ref{cap:remove-auxiliary} requires, and perform the final replacement
step. The number of steps may be chosen after $p_0$, $\beta$, and $D_0$ are
fixed. There is no restriction on its size as a function of $p_0$: it is
finite and independent of $n$. The tolerances in the curvature induction and
in the flow estimates depend only on the fixed ratios and curvature
tolerances, not on this number of steps or on $D_0$. Raw bounded gradient
errors give vanishing macroscopic calibration errors at every one of these
fixed scales. Thereafter all cap potentials in the induction are the true
potentials. The resulting curvature conclusion depends only on the original
disorder: if it fails for a given $J$, the joint construction fails for every
auxiliary seed. Its disorder failure probability is therefore bounded by the
joint failure probability.
\begin{theorem}[Uniform curvature of the true cap potentials]
\label{cap:true-curvature}
Fix a finite field radius $L_0$. There is $C=C(L_0)<\infty$ such that
for every $\epsilon>0$ there are constants $p_{\rm top}>0$ and $s_0>0$
for which, with probability tending to one,
the following assertions hold simultaneously for every
\[
p_{\min}(n)\le p\le p_{\rm top},\qquad
p_{\min}(n)=\left(\frac{\sqrt{\log n}}{n}\right)^{1/3}.
\]
For all $h\in P_p$ with $|\alpha_p(h)|\le s_0p^2$,
\begin{equation}\label{cap:true-sharp-curvature}
P_p\nabla^2\phi_p(h)P_p
\preceq \frac{1+\epsilon}{p^2}P_p.
\end{equation}
For all $h\in P_p$ with $\|h\|\le L_0p^2M_p$,
\begin{equation}\label{cap:true-ball-curvature}
P_p\nabla^2\phi_p(h)P_p\preceq\frac C{p^2}P_p.
\end{equation}
The constants are independent of $n,p,h$. The field
radius and all coarse constants may be fixed before
the sharp curvature tolerances. The lower endpoint
$p_{\min}(n)$ may be changed by a fixed factor.
\end{theorem}
\begin{proof}
Fix $L_0$ first and take the corresponding coarse constant from
Lemma~\ref{cap:frozen-evaluation-support}. Choose the construction
tolerances after this radius and the requested sharp error $\epsilon$;
the coarse constant is independent of $\epsilon$.
After the finite auxiliary phase, iterate
Proposition~\ref{cap:curvature-induction} on a geometric
list down to $np^3\asymp\sqrt{\log n}$. Its
positive-rate failure bounds are summable on this list.
The static comparison and spectral estimates hold on
the same event. This proves the invariant for all
retained true parent scales.
For \eqref{cap:true-sharp-curvature}, write the target scale as $p=xq$,
where $q$ is a true parent and $x\in[\gamma^2,\gamma]$. Choose $s_0$
small enough that the target frozen parameter lies strictly inside the
parent agreement range. Lemma~\ref{cap:deterministic-evaluation} applies
on this entire ratio interval. Its transverse and mixed errors can be
made arbitrarily small, while its radial correction is negative. Thus
$p^2(K_p-R_p^H)$ has arbitrarily small positive part. Choose the fixed
errors below the margin allowed by $\epsilon$, including the replacement
of $1/(p^2+\alpha)$ by $1/p^2$. This proves
\eqref{cap:true-sharp-curvature} without another random compression.
The sharp estimate concerns a thin parameter annulus. A prescribed bounded
ball may have a wider range of saddle parameters, so it requires a different
evaluation ratio. Choose the parent sufficiently coarser that this entire
range lies strictly inside the parent's agreement annulus. The child saddle
remains separated from its pole, and an upper support with smaller excess than
that separation gives the coarse ball bound. Each evaluation may use its own
support extension, since no induction is performed with its output.
For the ball bound again write $p=xq$, now with
$x\in[\gamma b_1,b_1]$, where
$b_1$ is supplied by Lemma~\ref{cap:frozen-evaluation-support} with
radius $L_0$. The lemma constructs supports around the frozen typical
length with excess below the child convolution margin, and proves the
complementary moment estimates for both the extended and true
transitions. Equation~\eqref{cap:ball-evaluation-bound} therefore gives
\eqref{cap:true-ball-curvature}, with a coarse constant depending only on
$L_0$. The construction is uniform in the indicated fixed ratio interval;
it uses no radial gain and need not agree with the extensions used for
other targets.
Every target scale below a sufficiently small fixed $p_{\rm top}$ has a parent
in each indicated ratio interval. The two evaluation lemmas are deterministic
and uniform under the parent hypotheses. Thus the assertions include all
intermediate scales, not just the geometric list. Fixed-factor enlargement of
the lowest list interval covers the stated bottom endpoint. \end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-initialization.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-multiscale.tex ====
\section{Transport of the radial covariance deficit}\label{cap:multiscale}
We now propagate a covariance estimate from one cap to the next. The
spherical saddle supplies a radial deficit, while a fresh Haar refinement
reduces the transverse error. Section~\ref{cap:initialization} constructs an
initial potential and replaces it, after finitely many levels, by the true
partition function. Work on successive levels \(p,c=\gamma p\) with
\(\gamma\) a very small fixed ratio. For \(s>0\), define
\begin{equation}
\mathcal A_p(s)=\{h\in H_p:|\alpha_p(h)|\le s p^2\};
\qquad\text{the induction uses }\mathcal A_p(\sigma)
\label{cap:annulus}
\end{equation}
with small fixed \(\sigma>0\). We will start with auxiliary smooth log
potentials close in \emph{gradient norm by a bounded amount} to the true ones
there. This auxiliary phase lasts only a fixed number of levels, which can be
arbitrarily large but is independent of \(n\). Thereafter the potentials are
the true ones. In either case write
\[
K_p=\nabla_{H_p}^2(\phi_p|_{H_p}),\qquad
R_p^H=(P_pR_pP_p)|_{H_p}=(p^2+\alpha_p(h))^{-1}I_{H_p}.
\]
For the true potential,
\(K_p=\operatorname{Cov}_{\mu_{p,h}}(P_pX)\) as an operator on \(H_p\).
Thus every cap curvature assertion below concerns \(H_p\). In
the auxiliary case we require smoothness only on the used annulus, and the
starting auxiliary function can itself be a Gaussian convolution. Potentials
at subsequent levels in the auxiliary phase are computed by heat convolution,
using a semiconcave extension at each step, defined below. Additive
calibrations of these potentials on \eqref{cap:annulus} are harmless. The
invariants are as follows; all uniformities on an annulus are pointwise
everywhere. The saddle equation gives
\[
\|h\|=(p^2+\alpha)\sqrt{n-\Tr R_{p,\alpha}}
=(\sqrt b+O(\sigma)+o(1))p^2M_p
\quad\text{when }|\alpha|\le\sigma p^2 .
\]
The annulus is therefore separated from zero. In the frame radial and
transverse to \(e=h/\|h\|\), put
\(D=p^2(K_p-R_p^H)\in\operatorname{Sym}(H_p)\). Then
\begin{equation}
\|D_{\perp\perp}\|\le\delta,\quad
\|D_{\perp e}\|\le K_0\sqrt\delta,\quad
-K_0 I_{H_p}\preceq D,\quad D_{ee}\le-d_0,\qquad d_0=0.30 .
\label{cap:invariant}
\end{equation}
Here \(K_0\) is a sufficiently large absolute constant. We can take \(\delta\)
a sufficiently high fixed power of \(\gamma\), and \(\sigma\) likewise.
The negative radial sign also controls the full positive part. If
\(z=D_{\perp e}\) and \(v=v_\perp+v_e e\), then
\[
v^\top Dv\le\delta\|v_\perp\|^2
+2|v_e|\|z\|\|v_\perp\|-d_0v_e^2
\le\left(\delta+\frac{\|z\|^2}{d_0}\right)\|v_\perp\|^2.
\]
Hence \(D\preceq C\delta I_{H_p}\) and
\(\|D\|\le\max(K_0,C\delta)\).
Throughout multiscale estimates constants polynomially bad in \(\gamma^{-1}\)
from inverses with fixed spectral formulas are allowed (the powers do not
depend on \(\delta,\sigma\)); the small powers used for tolerances can be
chosen with sufficiently large fixed exponents. Macroscopic value and gradient
errors relative to the spherical predictions can be taken as tiny as
subsequently required. This is by \eqref{cap:static-values}; in the auxiliary
phase we use the gradient comparison just mentioned with the true model, which
integrates to \(o(k_p)\) value oscillation error on \eqref{cap:annulus} since
the scales in that phase are fixed constants. The starting potential, its
auxiliary randomness if any, and the gradient error are intrinsic given the
current cap data and independent of the unused Haar refinement.
The extension used in an auxiliary step is one function selected from
the full parent annulus using only these parent data. It agrees with the
parent potential on an inner annulus with fixed slack, has at most
quadratic growth, and has a global upper Hessian bound
$(1+\gamma^4/100)I/p^2$ after the curvature tolerances are made sufficiently
small. This is the semiconcavity bound that controls its Gaussian
convolution. The supporting-quadratic construction below verifies these
properties without using the child eigenframe. For a true parent the
extension is used only to estimate the transition; the output remains
the true child potential.
\begin{proposition}[One cap step]\label{cap:curvature-induction}
Fix a bounded child field range for the one-off evaluation conclusion below.
There are a sufficiently large absolute $K_0$ and a sufficiently small
ratio $\gamma>0$ with the following property. Choose $\sigma$ as a
sufficiently high fixed power of $\gamma$, and then choose $\delta$ as a
still higher fixed power, allowing these tolerances to depend on the
prescribed field range. Let a parent potential at scale $p$ be intrinsic
to the capped data, satisfy the value and gradient comparisons above, and
satisfy \eqref{cap:invariant} at all fields in \eqref{cap:annulus}.
Use the semiconcave extension just described when forming an auxiliary
child potential. For a true parent, retain the true child potential.
Condition on the parent data and used auxiliary randomness
\(\mathscr F_p\) described below, on which these hypotheses and the
\(\mathscr F_p\)-measurable parent-level conclusions of the static
comparison hold, including the projected posterior tail and moment bounds
for this capped parent. The separate \(o(1)\) failure probability used to
obtain these parent conclusions remains outside the conditional estimate
below. Then,
uniformly over such parent data, the probability over the unused Haar
refinement \(O\) that \eqref{cap:invariant} fails at \(c=\gamma p\) anywhere
on the child annulus is at most
\[
C e^{-a k_p}+C e^{-a' k_c}
\]
for fixed positive \(a,a'\). The same inequalities hold with strict slack
on the complementary event. These failure probabilities are summable over
the geometric cap list.
At intermediate ratios $c/p\in[\gamma,1)$, if the frozen parameter stays
inside the agreement domain with fixed slack, the transverse error and
mixed block tend to zero with the parent errors, the total scaled Hessian
is bounded, and the radial deficit is at least $0.26$. The same deterministic
evaluation bounds hold for $c/p\in[\gamma^2,\gamma]$ under this
interior-parameter condition. For a true parent and the prescribed bounded
child field range there is also a sufficiently small fixed evaluation-ratio
interval on which the upper child curvature is at most $C/c^2$, with $C$ depending only on
that field range. This last evaluation uses supports centered at the frozen
parameter, constructed at the end of the proof.
All statements are uniform down to $np^3\asymp\sqrt{\log n}$.
\end{proposition}
The proof occupies the remainder of this section. Its probabilistic step is
used only at ratio $\gamma$; the intermediate-scale conclusions are
consequences of the deterministic covariance bounds.
\subsection{Semiconcave extensions and the observation bridge}
We make the measurability in a refinement explicit. Conditional on the
eigenvalues and the capped matrix $J_p$, let $\mathscr F_p$ include all
auxiliary randomness already used to construct the parent potential. The
construction is required to be measurable with respect to $\mathscr F_p$;
it does not reveal the orientation of the unequal eigenvalues within the
flat eigenspace $P_p$. Choose a fixed orthonormal identification of this space
with $\mathbb R^{d_p}$, where $d_p=\Tr P_p$. The remaining refinement is an
orthogonal matrix $O$ which is Haar and independent of $\mathscr F_p$.
All spectral projectors and increments below are obtained by conjugating
fixed diagonal matrices by $O$. This independence is the reason for the
auxiliary refinement used in Section~\ref{cap:initialization}.
After a step, the increment $\Delta=B_p-B_c$ is a function of $J_p$ and $J_c$
and is scalar on $P_c$. Thus the output auxiliary potential depends only on
the new capped data and previously introduced independent seeds; it does not
reveal the internal eigenframe of $P_c$. Conditional on these data the next
finer rotation is Haar.
For \(c=xp<p\), put
\[
\Delta=B_p-B_c,\qquad E=\operatorname{ran}\Delta\subset H_p,\qquad
U=H_c\subset E.
\]
Every inverse of \(\Delta\) is taken on \(E\), after omitting the zero
increment coordinates. An inverse is taken before its block is compressed
to \(U\). In particular \([M^{-1}]_{UU}\) generally differs from
\([M_{UU}]^{-1}\). The child increment is the scalar
\(\Delta_U=(1-x^2)p^2I_U\).
For a child field \(h_c\), the exact convolution and its transition
probability are
\begin{align}
\phi_c(h_c)&=\log\int \exp\{\phi_p(h_c+z)\}\,\mathsf N(0,\Delta)(\dd z),
\label{cap:heat-convolution}\\
\mathsf T_{c,p}^{h_c}(\dd h_p)
&=\frac{\exp\{\phi_p(h_p)\}\,
\mathsf N(h_c,\Delta)(\dd h_p)}{\exp\{\phi_c(h_c)\}}.
\label{cap:actual-transition}
\end{align}
For an auxiliary step, the function convolved in these equations is the
selected extension \(\bar\phi_p\) defined below; its logarithmic convolution
defines the child potential. For the true partition function the equations
follow by integrating the Gaussian linear tilt for each spin. Freeze the child
saddle parameter \(\alpha\) and put \(R_u=R_{u,\alpha}\) for \(u=c,p\).
Replacing the convolved potential by \(q_{p,\alpha}^{\rm fr}\) from
\eqref{cap:frozen-potential} gives the canonical probability
\(\mathsf G_{c,p}^{h_c}\). Under this law,
\begin{equation}\label{cap:canonical-bridge}
h_p=h_c+\Delta a_c+\zeta,\qquad
\zeta\sim\mathsf N(0,\Delta+\Delta R_c\Delta),\qquad
R_p h_p\sim\mathsf N(a_c,R_c-R_p).
\end{equation}
These identities follow from \(R_c^{-1}=R_p^{-1}-\Delta\), as in
\eqref{cap:static-canonical-bridge}. The covariance \(R_c-R_p\) is supported
on \(E\). The reference spin Gaussian is sampled without norm conditioning.
The parent Hessian estimate holds on an annulus, whereas the convolution
integrates over the full increment space. We extend the potential to make the
convolution globally controlled, while preserving it on the fields the bridge
visits with high probability. The extension must be determined by the parent
data alone: using the child refinement in its construction would consume the
randomness needed for transverse compression.
Here is one parent-measurable extension with the needed buffers. The radius
corresponding to a saddle parameter is
\[
\ell_p(\alpha)=(p^2+\alpha)\sqrt{n-\Tr R_{p,\alpha}},\qquad
\ell_p'(\alpha)=\sqrt{n-\Tr R_{p,\alpha}}+
\frac{(p^2+\alpha)\Tr R_{p,\alpha}^2}
{2\sqrt{n-\Tr R_{p,\alpha}}}\asymp M_p .
\]
Thus \(\mathcal A_p(s)\) is the radial annulus between
\(\ell_p(-sp^2)\) and \(\ell_p(sp^2)\). Put
\[
S_0=\mathcal A_p(\sigma/2),\qquad
S_1=\mathcal A_p(3\sigma/4),\qquad
S_2=\mathcal A_p(\sigma).
\]
Half the distance from \(S_0\) to \(H_p\setminus S_1\), and from \(S_1\)
to \(H_p\setminus S_2\), gives buffers \(\rho_{01},\rho_{12}\asymp
\sigma p^2M_p\). In particular \(S_0+B(0,\rho_{01})\subset S_1\),
and a segment of length at most \(\rho_{12}\) from a point of \(S_1\)
lies in \(S_2\).
Let \(f_p=\phi_p|_{H_p}\) be the parent before extension and set
\[
L_p=(1+\zeta)p^{-2}I_{H_p},\qquad \zeta=\gamma^4/100.
\]
We claim that every pair \(z,w\in S_1\) satisfies
\begin{equation}\label{cap:parent-upper-support}
f_p(w)\le f_p(z)+\nabla_{H_p}f_p(z)^\top(w-z)
+\tfrac12(w-z)^\top L_p(w-z).
\end{equation}
For \(\|w-z\|\le\rho_{12}\), the segment stays in \(S_2\). There
\eqref{cap:invariant} and its positive-part bound give
\[
K_p\preceq p^{-2}\bigl((1-\sigma)^{-1}+C\delta\bigr)I_{H_p}
\prec L_p
\]
after choosing \(\sigma,\delta\) below the excess \(\zeta\). Integrating the
Hessian on the segment proves \eqref{cap:parent-upper-support} in this case.
For \(\|w-z\|\ge\rho_{12}\), put \(d=w-z\) and freeze the own saddle at
\(z\). The value comparison, the gradient comparison, and
\eqref{cap:frozen-mgf} give
\[
f_p(w)-f_p(z)-\nabla_{H_p}f_p(z)^\top d
\le \frac{\|d\|^2}{2(p^2+\alpha_p(z))}
+2\varepsilon_v k_p+C_{\rm dens}
+\varepsilon_g M_p\|d\|.
\]
In an auxiliary step, \(\varepsilon_v,\varepsilon_g\) also include the
calibrated auxiliary errors. With
\(\beta_{\rm buf}=\rho_{12}/(p^2M_p)\), the last three terms are at most
\[
\frac{\|d\|^2}{p^2}
\left(\frac{2\varepsilon_v+C_{\rm dens}/k_p}{\beta_{\rm buf}^2}
+\frac{\varepsilon_g}{\beta_{\rm buf}}\right).
\]
Choose these errors after the buffers so that this quantity fits below the
remaining excess of \(L_p\). This proves
\eqref{cap:parent-upper-support} for every support pair.
Now define the single function
\begin{equation}\label{cap:parent-extension}
\bar\phi_p(y)=\inf_{z\in S_1}
\left\{f_p(z)+\nabla_{H_p}f_p(z)^\top(y-z)
+\tfrac12(y-z)^\top L_p(y-z)\right\}.
\end{equation}
At \(y\in S_1\), \eqref{cap:parent-upper-support} bounds every term below
by \(f_p(y)\), and the term \(z=y\) equals it. Thus \(\bar\phi_p=f_p\)
on \(S_1\), in particular on a neighborhood of \(S_0\). Subtracting
\(\tfrac12y^\top L_py\) leaves an infimum of affine functions, hence a
concave function. The support set is bounded and its values and slopes are
bounded, so this infimum is finite and has at most quadratic growth.
Consequently \(\nabla^2\bar\phi_p\preceq L_p\) in the distributional sense.
The construction uses only the parent annulus and parent function, so it is
\(\mathscr F_p\)-measurable and consumes no part of \(O\).
On \(E\) the convolution has the strict margin
\[
\|\Delta_E^{1/2}L_p\Delta_E^{1/2}\|
\le(1+\zeta)(1-x^2)
\le(1+\zeta)(1-\gamma^2)<1 .
\]
The extended terminal density is therefore strongly log-concave in the
whitened increment coordinates. In the true phase the extension is used only
to estimate the transition; the output is the true child. If smoothing is
needed, mollify the extension and pass to the limit at fixed \(n\).
Inside the buffered agreement region the mollifications converge to the
original smooth potential with its derivatives.
We next record precisely how canonical estimates pass to a terminal
transition. Let \(V_p\) be the potential actually convolved, either
\(\bar\phi_p\) in an auxiliary step or the true \(\phi_p\), and set
\[
F(y)=V_p(y)-q_{p,\alpha}^{\rm fr}(y).
\]
For fixed parent data and child field, the exact likelihood ratio is
\begin{equation}\label{cap:bridge-density-ratio}
\frac{d\mathsf T}{d\mathsf G}(y)
=\frac{e^{F(y)}}{\E_{\mathsf G}e^F},
\qquad \mathsf G=\mathsf G_{c,p}^{h_c}.
\end{equation}
Let \(\widehat\alpha=\alpha_p(y)\) and take a shell
\(\mathcal S_\tau=\{|\widehat\alpha-\alpha|\le\tau p^2\}\) lying inside
the buffered agreement region. Under \(\mathsf G\), its complement has
probability at most \(e^{-a_\tau k_p}\) for each fixed \(\tau>0\), by
\eqref{cap:canonical-bridge} and the saddle derivative estimate. On
\(\mathcal S_\tau\), the static comparison or its calibrated auxiliary
version, together with \eqref{cap:off-saddle-density}, gives, for a separately
chosen normalization shell \(\mathcal S_{\tau_0}\) with \(0<\tau_0<\tau\),
\begin{equation}\label{cap:shell-likelihood}
\sup_{\mathcal S_\tau}\frac{d\mathsf T}{d\mathsf G}
\le \mathsf G(\mathcal S_{\tau_0})^{-1}
\exp\{[2\varepsilon_v+C(\tau^2+\tau_0^2)]k_p+O_x(1)\}.
\end{equation}
The denominator is bounded below by integrating \(e^F\) on the narrower
shell. Choosing \(\tau\), then \(\tau_0\), and then the comparison errors makes the exponent
at most \(\eta k_p\) for any prescribed fixed \(\eta>0\).
The shell complement requires its own estimate. Write
\(d(y)=|\|y\|-\ell_p(\alpha)|/(p^2M_p)\). The canonical bridge has
\[
\E_{\mathsf G}\|Y\|^2=\ell_p(\alpha)^2,
\qquad \|\Cov_{\mathsf G}(Y)\|\le C\gamma^{-2}p^2.
\]
The first identity follows from
\(\E\|R_pY\|^2=\|a_c\|^2+\Tr(R_c-R_p)=n-\Tr R_p\),
since \(R_p\) is scalar on \(H_p\). Gaussian concentration for the norm,
together with
\(0\le\sqrt{\E\|Y\|^2}-\E\|Y\|\le
\sqrt{\|\Cov(Y)\|}\), therefore gives
\begin{equation}\label{cap:bridge-distance-tail}
\Pr_{\mathsf G}\{d(Y)>v\}\le 2e^{-c\gamma^2k_pv^2},
\qquad v\ge \frac{C}{\gamma\sqrt{k_p}}.
\end{equation}
In particular this estimate applies at every fixed shell threshold once
\(n\) is sufficiently large. For the extended potential, a support from
the frozen typical sphere, with the calibrated value there,
bounds \(F(y)\) above by a fixed reference level plus
\[
\eta_{\rm err}k_p+\eta_{\rm small}k_p d(y)^2.
\]
Leaving \(\mathcal S_\tau\) requires
\(d(Y)\ge c_0\tau\) on the comparable-gap range. Choose
\(\eta_{\rm small}<c\gamma^2/2\), then choose \(\tau_0\) and the comparison
errors so that the normalization loss
\(2\varepsilon_v+C\tau_0^2\), together with \(\eta_{\rm err}\), is less than
\((c\gamma^2-\eta_{\rm small})c_0^2\tau^2/2\).
The narrower shell has canonical probability tending to one for every fixed
\(\tau_0>0\). Integrating the tail against the upper support with this
normalization budget
gives, for every fixed integer \(j\),
\[
\E_{\mathsf T}\bigl[(1+d(Y))^j\mathbf1_{\{Y\notin\mathcal S_\tau\}}\bigr]
\le C_{j,\tau}e^{-a'_\tau k_p}.
\]
For a true transition, on each bounded field range the static value
comparison and the frozen numerator norm-density upper bound control
\(F\) by its calibrated level plus \(\varepsilon_vk_p+C\). The same
normalization budget and \eqref{cap:bridge-distance-tail} then control the
bounded part of the shell complement. Beyond that range, the projected \(v^6\) tail at
parent zero field is multiplied by at most \(e^{Ck_p(1+v^2)}\) from the
added quadratic, so \(e^{-ck_pv^6+Ck_p(1+v^2)}\) bounds the large-length
contribution and all fixed polynomial moments. Gaussian observation noise
has the corresponding quadratic tail. These estimates make the error from
replacing a true transition by its extension exponentially small in scaled
covariance units. Indeed the two unnormalized terminal densities agree on
\(S_1\). If each normalized law assigns at most \(\varepsilon\) to its
complement, their normalization ratio lies between \(1-\varepsilon\) and
\((1-\varepsilon)^{-1}\); the complementary moment bounds then control the
difference of their means and covariances.
The same statements hold for \(x\in[\gamma,1)\) whenever the frozen
parameter lies inside parent agreement with fixed slack; one use is
\(|\alpha|\le2\sigma(\gamma p)^2\). As \(x\uparrow1\), the canonical
increment covariance shrinks proportionally to \(1-x\), so the probability
of leaving agreement is at most \(C e^{-a k_p/(1-x)}\). The support and
projection estimates give the same bound for the relevant true-transition
errors. This exponential factor pays every fixed power of \(1/(1-x)\).
For derivative integrals on a true transition we use the projected trace
bound
\[
\Tr_{H_p}K_p(y)
\le \E_{\mu_{p,y}}\|P_pX\|^2.
\]
The projected moment tails therefore leave only polynomial factors in
\(k_p\), rather than an ambient rank factor.
\subsection{Upper and lower covariance transport}
We use the Brascamp--Lieb variance inequality~\cite{BrascampLieb1976};
its relation to log-concavity and logarithmic Sobolev inequalities is
discussed in~\cite{BobkovLedoux2000}. The paired upper and lower covariance
bounds also have the form used in~\cite[Lemma~7 and Appendix~A]{ChewiPooladian2023Entropic}.
The output Hessian is obtained from the covariance of the terminal field.
We bound this covariance from both sides. Brascamp--Lieb keeps the inverse
curvature correction and will preserve the useful negative radial term.
The Fisher test is linear in the parent correction and supplies the lower
bound needed for transverse and mixed blocks. We perform the lower test with
cutoffs inside agreement, so uncontrolled curvature of the extension is not
integrated.
Differentiating the logarithmic convolution in a direction of \(U\) gives
the exact output Hessian for the potential \(V_p\) and its corresponding
transition \(\mathsf T\):
\begin{equation}\label{cap:child-hessian}
K_c=\Delta_U^{-1}
[\operatorname{Cov}_{\mathsf T}(Y)]_{UU}
\Delta_U^{-1}-\Delta_U^{-1}.
\end{equation}
The covariance is first computed on \(E\). For Brascamp--Lieb, temporarily
take a smooth mollification \(V_p^{\rm ext}\) of the extension and denote
its transition by \(\mathsf T^{\rm ext}\). Put
\[
K_p^{\rm ext}(Y)=\nabla^2 V_p^{\rm ext}(Y),\qquad
M_{\rm ext}(Y)=\Delta_E^{-1}-[K_p^{\rm ext}(Y)]_{EE},\qquad
C_0=(\Delta_E^{-1}-R_p|_E)^{-1}
=\Delta+\Delta R_c\Delta .
\]
The extension margin makes \(M_{\rm ext}\) positive. Brascamp--Lieb on \(E\) gives
\(\operatorname{Cov}_{\mathsf T^{\rm ext}}(Y)
\preceq\E_{\mathsf T^{\rm ext}}M_{\rm ext}(Y)^{-1}\).
Combining this with \eqref{cap:child-hessian} preserves the inverse before
compression to \(U\).
For the lower estimate under this smooth extension law, let
\(\rho_{\rm ext}\) be its density and
\(\mathfrak s=-\nabla_E\log\rho_{\rm ext}\) be the
score. Integration by parts gives
\(\E[(Y-\E Y)\mathfrak s^\top]=I_E\) and
\(\E[\mathfrak s\mathfrak s^\top]=\E M_{\rm ext}\). The nonnegativity of the covariance
of \(Y-T\mathfrak s\) therefore gives, for every constant matrix \(T\) on \(E\),
\begin{equation}\label{cap:fisher-variational}
\operatorname{Cov}_{\mathsf T^{\rm ext}}(Y)
\succeq T+T^\top-T\,\E_{\mathsf T^{\rm ext}}[M_{\rm ext}(Y)]\,T^\top.
\end{equation}
With \(T=C_0\), the right-hand side is
\(C_0+C_0\E_{\mathsf T^{\rm ext}}[K_p^{\rm ext}-R_p]_{EE}C_0\), which is linear in the
parent correction.
We use a localized version when only the agreement region has controlled
classical curvature. This identity applies separately to either smooth
law with density \(\rho\), using its own score \(\mathfrak s\), precision
\(M=-\nabla_E^2\log\rho\), and expectations. Let \(\chi\) be a smooth cutoff supported there and,
for constant \(t,w\in E\), put
\(b_\chi=\chi\,t^\top\mathfrak s-t^\top\nabla\chi\).
Integration by parts yields
\[
\E[(w^\top(Y-\E Y))b_\chi]=\E\chi\,w^\top t,\qquad
\E b_\chi^2=\E\chi^2 t^\top M t+\E(t^\top\nabla\chi)^2.
\]
The inequality \(\E(w^\top(Y-\E Y)-b_\chi)^2\ge0\) becomes
\begin{equation}\label{cap:localized-fisher}
\operatorname{Var}(w^\top Y)
\ge2\E\chi\,w^\top t-\E\chi^2t^\top Mt
-\E(t^\top\nabla\chi)^2 .
\end{equation}
Choose \(\chi=1\) on a narrower shell and supported inside agreement.
For an output test \(v\in U\), take \(w=\Delta_U^{-1}v\) and
\(t=C_0w=Nv\), where
\(N=(p^2+\alpha)/(c^2+\alpha)\). Only the scalar inverse \(\Delta_U^{-1}\)
appears in this test. The shell estimates above, including their polynomial
moment bounds and the improved \(e^{-a k_p/(1-x)}\) rate, make the cutoff
errors exponentially small after any fixed powers of \(k_p\) and
\((1-x)^{-1}\). Thus \eqref{cap:localized-fisher} applies to the true
transition as well as a smoothed extension without integrating uncontrolled
curvature outside agreement.
On the shell subtract the frozen cap resolvent, writing
\[
D^\circ(y)=p^2\bigl(K_p(y)-(p^2+\alpha)^{-1}I_{H_p}\bigr).
\]
If \(\widehat\alpha=\alpha_p(y)\), then exactly
\[
D^\circ(y)-D(y)
=\frac{p^2(\alpha-\widehat\alpha)}
{(p^2+\widehat\alpha)(p^2+\alpha)}I_{H_p}.
\]
This is \(O(\tau)\) on \(\mathcal S_\tau\). In the following resolvents
\(D^\circ\) is restricted to \(E\); the restriction is suppressed in the
notation. Put
\[
S=p^{-2}C_0=p^{-2}\Delta(I-R_p\Delta)^{-1}.
\]
On the agreement shell, where the limiting extension Hessian is the parent
Hessian, put \(M=C_0^{-1}-p^{-2}D^\circ\). The exact identities there are
\[
M^{-1}=C_0+p^{-2}C_0D^\circ(I-SD^\circ)^{-1}C_0,\qquad
\Delta_U^{-1}(C_0)_{UU}\Delta_U^{-1}-\Delta_U^{-1}=R_c|_U.
\]
Together with \(C_0\Delta_U^{-1}=N I_U\), these identities give the
extension bound. Pass to the mollification limit and, for a true output,
use the preceding transition comparison. The cutoff and change-of-law
errors are included in \(\eta_{\rm err}\). The resulting target bound is
\begin{equation}\label{cap:bl-recursion}
c^2(K_c-R_c|_U)
\preceq x^2N^2\,\E_{\mathsf T}
[D^\circ(I-SD^\circ)^{-1}]_{UU}
+\eta_{\rm err}I_U .
\end{equation}
The constant Fisher test gives the
corresponding lower bound \(x^2N^2\E_{\mathsf T}[D^\circ]_{UU}\), up to the
same type of errors. Testing only \(v_\perp\), now transverse to the child
field, gives that transverse lower block with zero mixed block and radial
block \(-c^2(\Delta_U^{-1}+R_c|_U)\). The global upper curvature controls
the rare-event remainder in \eqref{cap:bl-recursion}; all expansions of
\(D^\circ\) use the good shell.
The matrix map in the upper bound preserves Loewner order on its positive
domain. On \(E\),
\begin{equation}\label{cap:resolvent-order}
D(I-SD)^{-1}
=S^{-1/2}\bigl[(I-S^{1/2}DS^{1/2})^{-1}-I\bigr]S^{-1/2}.
\end{equation}
It is increasing whenever \(I-S^{1/2}DS^{1/2}\succ0\). The extension
margin supplies this condition for the actual correction. It will also hold
for the upper comparison below: its positive part is \(O(\delta)+
\eta_{\rm err}\), whereas \(\|S\|\) is bounded by a fixed power of
\(\gamma^{-1}\). The stated tolerance order leaves a strict margin.
\subsection{Regeneration of the radial deficit}
Set \(A=D_{\perp\perp}\), embedded with zero radial block, and
\(z=D_{\perp e}\). Completing the square in the mixed block, using a small
fixed part of the radial deficit, gives
\begin{equation}
D^\circ\preceq A+Czz^\top-d_*ee^\top+\eta_{\rm err}I,
\end{equation}
where \(d_*<d_0\) can be chosen as close to \(d_0\) as desired and
\(C\) depends on \(d_0-d_*\). The order identity
\eqref{cap:resolvent-order} permits this substitution in
\eqref{cap:bl-recursion}. To see the effect of the negative rank, write
\(\mathcal F_S(D)=D(I-SD)^{-1}\) and
\(B=A+Czz^\top+\eta_{\rm err}I\). Sherman--Morrison gives the exact formula
\[
\mathcal F_S(B-d_*ee^\top)
=\mathcal F_S(B)-
\frac{v_Bv_B^\top}
{d_*^{-1}+e^\top S(I-BS)^{-1}e},
\qquad v_B=(I-BS)^{-1}e.
\]
The matrices lie in the positive domain specified above.
Since \(\|B\|=O(\delta)+\eta_{\rm err}\) and \(\|S\|=O_\gamma(1)\),
\(\mathcal F_S(B)=A+Czz^\top+O_\gamma(\delta^2)+\eta_{\rm err}I\),
\(v_B=e+O_\gamma(\delta)+\eta_{\rm err}\), and the denominator differs
from \(d_*^{-1}+e^\top Se\) by \(O_\gamma(\delta)+\eta_{\rm err}\).
After choosing the tolerances, the resulting upper bound retains the
subtraction
\begin{equation}
-(1-\eta_{\rm small})\,\frac{ee^\top}{1/d_*+e^\top S e}
\end{equation}
up to \(O_\gamma(\delta^2)+\eta_{\rm err}\) in operator norm.
Put \(m=R_pY\) and \(T=R_c-R_p\). On \(H_p\), \(R_p\) is scalar, so
\(e=m/\|m\|\). Under the canonical bridge, \(m\) has mean \(a_c\) and
covariance \(T\). The shell likelihood and its complementary moment bounds
transfer the corresponding linear and quadratic tail tests to
\(\mathsf T\), with arbitrarily small errors in scaled units. In particular
on the shell \(\|m\|^2=(b+o(1)+O(\sigma)+O(\tau))np\),
\(\E_{\mathsf T}m\) is arbitrarily
close to \(a_c\), and
\(\E_{\mathsf T}m^\top Tm\) is at most
\(a_c^\top Ta_c+\Tr T^2\) plus an arbitrarily small scaled error.
The linear second-moment bounds are used separately for each tested
direction in \(U\), with numerical constants uniform in that direction.
At \(\alpha=0\), \(S=p^2T\). The denominator of the rank term after replacing
\(e\) by \(m/\|m\|\) is
\(\|m\|^2/d_*+p^2m^\top Tm\). The following calculation bounds its
expectation and hence the radial subtraction.
Since $R_p=p^{-2}I$ on the support of $T$,
\begin{align}
\Tr T&=(b+o(1))n(p-c),\nonumber\\
\Tr T^2&=(2b+o(1))n(1/c-1/p)
-2p^{-2}(b+o(1))n(p-c).\label{cap:trace-difference-square}
\end{align}
The rank denominator before the factor $x^{-2}$ is
$\|m\|^2/d_*+p^2m^\top Tm$. Its expectation is bounded above by
\begin{equation}
\frac{bnp}{d_*}
+p^2\bigl(a_c^\top T a_c+\Tr T^2\bigr)+o(n p^2/c).
\label{cap:radial-denominator}
\end{equation}
For each test vector $v$, Cauchy--Schwarz in the form $\E[(v^\top
m)^2/Q]\ge(v^\top\E m)^2/\E Q$ turns this into a rank-one matrix bound.
Dividing the denominator by $bn p^2/c=x^{-2}\|a_c\|^2(1+o(1))$ gives
successively $x/d_*$, $1-x^2$, and $2(1-x)^2$. This proves
that the radial deficit in output units is at least
\begin{equation}\label{cap:radial-regeneration}
\frac{1-\eta_{\rm small}}
{x/d_*+(1-x^2)+2(1-x)^2+\eta_{\rm small}}.
\end{equation}
In the stated interior-parameter regimes, \(x\ge\gamma^2\) and
\(|\alpha|\le\sigma p^2\), so
\(|\alpha|/c^2\le\sigma\gamma^{-4}\). The high-power choice of
\(\sigma\) therefore includes the resulting perturbation in
\(\eta_{\rm small}\).
Cross-direction losses in making this a purely radial deficit can be
arbitrarily small by the transverse mean and second-moment bounds above.
Formula \eqref{cap:radial-regeneration} exceeds 0.26 throughout, and exceeds
0.31 at \(x=\gamma\). The small constant losses can be chosen to keep these
strict inequalities. Remaining terms in the radial block are negligible for
this estimate.
\subsection{Compression under the fresh refinement}\label{cap:compression-proof}
For the inductive transverse block at \(x=\gamma\), first fix the complete
orientation \(O\) and a child field. Expectations in this paragraph are under
its target transition \(\mathsf T_O\). Fix a smooth function
\(0\le\chi_0\le1\), equal to one on \([-1/2,1/2]\) and zero outside
\((-1,1)\), and use
\begin{equation}\label{cap:field-cutoff}
\chi_h(y)=\chi_0\!\left(
\frac{\alpha_p(y)-\alpha_c(h)}{\tau p^2}\right).
\end{equation}
Here \(\tau\) is a sufficiently small fixed power of \(\gamma\), and
the support lies inside agreement with fixed slack. At the current child
field write \(\chi=\chi_h\). Write
\(A_{\rm par},z_{\rm par}\) for the parent quantities denoted by \(A,z\)
above, and set \(A=\chi A_{\rm par}\), \(z=\chi z_{\rm par}\) in the
compression estimates. The shell moment bounds make this truncation
arbitrarily small. Put
\[
g(y)=\nabla_{H_p}f_p(y)-(p^2+\alpha)^{-1}y,\qquad
Q_s=\mathbf1_{\{\Delta_E\ge sp^2\}} .
\]
On the support of \(\chi\),
\(\nabla_{H_p}g=D^\circ/p^2\) and
\(\|g\|\le\varepsilon_{\rm mac}M_p\), where the frozen-saddle error is
included in \(\varepsilon_{\rm mac}\). With the score
\(\mathfrak s=-\nabla_E\log\mathsf T_O\), integration by parts gives the
matrix identity
\begin{equation}\label{cap:nuclear-ibp}
\E_{\mathsf T_O}[\chi D^\circ Q_s]
=p^2\E_{\mathsf T_O}
[g(\chi Q_s\mathfrak s-Q_s\nabla\chi)^\top].
\end{equation}
The nuclear norm of this expectation is bounded by
\[
p^2\bigl(\E_{\mathsf T_O}\mathbf1_{\operatorname{supp}\chi}\|g\|^2\bigr)^{1/2}
\bigl(\E_{\mathsf T_O}
\|\chi Q_s\mathfrak s-Q_s\nabla\chi\|^2\bigr)^{1/2}.
\]
Summing the localized score identity used in
\eqref{cap:localized-fisher} over an orthonormal basis of \(Q_sE\) gives
\[
\E\|\chi Q_s\mathfrak s-Q_s\nabla\chi\|^2
=\E\chi^2\Tr(Q_sMQ_s)+\E\|Q_s\nabla\chi\|^2
\le C_s d_p/p^2+C_s/(p^4M_p^2)
\le C_sM_p^2.
\]
Here \(\Delta_E^{-1}\le(sp^2)^{-1}\) on \(Q_sE\), the invariant bounds
\(-K_p\) by \(C/p^2\) on the cutoff support, and
\(\|\nabla\chi\|\le C/(p^2M_p)\) with fixed buffer constants.
Thus \eqref{cap:nuclear-ibp} has nuclear norm at most
\(C_s\varepsilon_{\rm mac}p^2M_p^2
=C_s\varepsilon_{\rm mac}k_p\).
The columns omitted by \(Q_s\) have rank \(O(s)k_p+o(k_p)\), by the edge
count in the strip \(0\le\Delta<sp^2\), including the zero-increment
boundary coordinates; their operator norm is bounded.
Changing \(D^\circ\) to \(D\) costs \(C\tau d_p\) in nuclear norm on the
shell. Finally, \(D-A_{\rm par}\) has rank at most two for each field and bounded
nuclear norm. Its radial direction may change with the field, but
\[
\|\E_{\mathsf T_O}[\chi(D-A_{\rm par})]\|_*
\le\E_{\mathsf T_O}\|\chi(D-A_{\rm par})\|_*\le C.
\]
This uses the nuclear triangle inequality, without assigning a fixed rank
to the average. Choosing \(s\) first, then the macroscopic and shell errors,
and finally \(k_p\) large proves
\begin{equation}\label{cap:nuclear-average}
\frac{\|\bar A_O\|_*}{k_p}\le\eta_{\rm err},
\qquad \bar A_O=\E_{\mathsf T_O}A .
\end{equation}
The norm is outside the expectation. In particular the signed normalized
trace is small.
The radial correction has the required sign, but a direct operator-norm
estimate on the transverse block would be enlarged by the change of scale. The
small signed average just proved allows a better estimate. We sample finitely
many independent terminal fields, show that their average has small Frobenius
norm, and compress it using the remaining Haar rotation. The sampled field
directions are exceptional because each matrix depends on its own field; we
estimate their span separately.
We need
\begin{equation}
\|[\E_{\mathsf T_O}A]_{U_\perp U_\perp}\|
\le o_{\gamma\to0}(1)\,\gamma^2\delta,\qquad
\E_{\mathsf T_O}\|P_U z\|^2\le C K_0^2\delta\, H\gamma^3 ,
\label{cap:compression}
\end{equation}
apart from tiny terms, where \(H=C_1\log(1/\gamma)\) with a sufficiently large
fixed coefficient. These assertions hold simultaneously for child fields with
high probability over the refinement. For the moment, keep \(O\) and the child
field fixed and sample \(Y_1,\ldots,Y_l\) independently from \(\mathsf T_O\).
The deterministic function of these fields is
\(\mathcal A(Y_1,\ldots,Y_l)=l^{-1}\sum_jA(Y_j)\). Independence gives exactly
\begin{align}\label{cap:sample-frobenius}
\E_{\mathsf T_O^l}\|\mathcal A\|_F^2
&=l^{-1}\E_{\mathsf T_O}\|A\|_F^2
+\frac{l-1}{l}\|\bar A_O\|_F^2\nonumber\\
&\le \delta^2d_p/l+\delta\|\bar A_O\|_* .
\end{align}
The last inequality uses \(\|A\|\le\delta\) and
\(\|\bar A_O\|_F^2\le\|\bar A_O\|\|\bar A_O\|_*\).
By \eqref{cap:nuclear-average}, the quotient of
\eqref{cap:sample-frobenius} by \(k_p\) is
\(O(\delta^2/l)+\eta_{\rm err}\). We will combine this fixed-\(O\) target-law
estimate with a Haar event first proved under canonical sampling.
\paragraph{Conditional Haar laws before tilting.}
We spell out the conditional laws because the actual transition is not itself
Haar after its terminal fields have been sampled. Fix $\mathscr F_p$ and a
child field in spectral coordinates, denoted $u\in U_0$. Its image is
$h_c=Ou$, and $U=OU_0$. Sample $l$ independent canonical terminal fields
$\xi_1,\ldots,\xi_l$ in spectral coordinates according to
\eqref{cap:canonical-bridge}, independently of $O$. Their means are parallel
to \(u\). Denote this joint law by
\[
\mathbb P_{\rm can}(dO,d\xi_1\cdots d\xi_l)
=\operatorname{Haar}(dO)\prod_{j=1}^l
\mathsf G_{u,\rm diag}(d\xi_j).
\]
Its physical fields are \(Y_j=O\xi_j\), whose conditional law for fixed
\(O\) is \(\mathsf G_O^l\). Conditional on these spectral fields, and on the images under \(O\)
of $u$ and any selected collection of the $\xi_j$, the remaining map on the
orthogonal complements is Haar. Conditional on the image of $\xi_j$, the
matrix $A_j=A(O\xi_j)$ is fixed: the parent potential, its radial splitting,
and its derivatives are all measurable with respect to $\mathscr F_p$ and
that image. Conditional on two images, $A_iA_j$ is fixed. The same statement
applies to $z_j=z(O\xi_j)$, with $z_j\perp O\xi_j$.
Set $e_j=O\xi_j/\|\xi_j\|$, and let $e_0=h_c/\|h_c\|$. The child annulus is
separated from zero field, so $e_0$ is defined. All dimensions below are
comparable to $k_p$ or $k_c$ by the spectral count. The elementary Haar
estimate we use is
\begin{equation}\label{cap:haar-compression}
\|P_V O^\top T O P_V-\tfrac{\Tr T}{d}P_V\|
\le C\left(\frac{\|T\|_F}{d}\sqrt{r+a}
+\frac{\|T\|}{d}(r+a)\right),
\end{equation}
with failure probability at most $2e^{-a}$, when $O$ is Haar in dimension $d$,
$V$ is a fixed $r$-dimensional subspace, and $T$ is symmetric and fixed.
Replacing $r+a$ by a fixed larger multiple absorbs the size of a
fixed-resolution unit net in $V$. To prove the estimate for a fixed unit
vector, write its Haar image as $g/\|g\|$, diagonalize $T$, and use
\begin{equation}
\log\E e^{t\sum_i\lambda_i(g_i^2-1)}
\le C t^2\sum_i\lambda_i^2,
\qquad |t|\max_i|\lambda_i|\le\tfrac14.
\label{cap:gaussian-quadratic-tail}
\end{equation}
The same bound for $\|g\|^2$ handles its normalization. Polarization gives
the bilinear and nonsymmetric versions used for products.
On the complement of the span of $e_0,e_1,\ldots,e_l$, apply
\eqref{cap:haar-compression} with $T=\mathcal A$ and $a$ a sufficiently large
multiple of $Hk_c$. The deviation from its scalar trace is at most
\begin{equation}\label{cap:ordinary-compression}
C\left(\frac{\|\mathcal A\|_F}{\sqrt{k_p}}
\sqrt{H\gamma^3}
+\delta H\gamma^3\right).
\end{equation}
Conditioning only on the image of $\xi_j$, the vector $z_j$ is fixed and
perpendicular to $e_j$. The residual projection of $U$ has rank at most
$\dim U$, so the same Haar estimate gives
$\|P_Uz_j\|^2\le C K_0^2\delta H\gamma^3$.
Its failure rate can be $2Hk_c$ after increasing absolute constants.
\paragraph{The span of sampled directions.}
Write $W$ for the span of the $e_j$ after projection onto $e_0^\perp$.
Under canonical sampling, the centered Gaussian vectors before direction
normalization have squared norm comparable to one in field units, limiting
orthogonal Gram matrix, $U_0$-projection squared norm of order $\gamma$,
and covariance operator norm at most $C/(\gamma^2k_p)$. Gaussian quadratic
tails and nets in the fixed $l$-dimensional coefficient space therefore give
\begin{equation}\label{cap:exceptional-span-geometry}
\|P_{U\cap e_0^\perp}P_W\|\le C\sqrt{H\gamma},\qquad
\sum_{j=1}^l |\langle e_j,w\rangle|^2\le C\|w\|^2
\quad(w\in W).
\end{equation}
The projected generating vectors have a synthesis map and an inverse on its
range bounded by absolute constants. Any pair of the original directions has
bounded Gram conditioning. These conclusions hold at the same $2Hk_c$ rate.
Although $l$ will be a large power of $\gamma^{-1}$, it is fixed before
$n\to\infty$, so the fixed-dimensional nets cost no additional multiple of
$k_p$ in the limit.
Let \(\bar a=\Tr\mathcal A/d_p\). Then
\begin{align}
\|P_W(\mathcal A-\bar a I)P_W\|
&\le C\delta\bigl(\sqrt{H\gamma^3}+l^{-1/2}\bigr),
\label{cap:exceptional-first-moment}\\
\|P_W\mathcal A^2P_W\|
&\le C\|\mathcal A\|_F^2/k_p
+C\delta^2\bigl(\sqrt{H\gamma^3}+l^{-1/2}\bigr).
\label{cap:exceptional-second-moment}
\end{align}
To account for the dependence of each matrix on its own sampled direction,
expand a test vector in the bounded generating frame of \(W\). For
the summand $A_j$, condition on $e_0,e_j$ and apply
\eqref{cap:haar-compression} to the other generating directions in their
orthogonal complement. The scalar term is $\Tr A_j/d_p$ up to a
fixed-dimensional trace correction, which is $o(1)$. Terms with a row or
column in $\operatorname{span}(e_0,e_j)$ have average bounded by
\[
\frac{C\delta}{l}\sum_j\|P_{\operatorname{span}(e_0,e_j)}w\|
\le C\delta l^{-1/2}\|w\|,
\qquad w\in W,
\]
using $w\perp e_0$ and \eqref{cap:exceptional-span-geometry}. This proves
\eqref{cap:exceptional-first-moment} by polarization and a fixed-dimensional
net. For the second estimate expand $\mathcal A^2=l^{-2}\sum_{i,j}A_iA_j$.
Condition on $e_0,e_i,e_j$. The same Haar test gives scalar term
$\Tr(A_iA_j)/d_p$; the sum of these terms is $\Tr\mathcal A^2/d_p$. The
exceptional-row contribution is bounded by $C\delta^2l^{-1/2}$, since
\[
l^{-2}\sum_{i,j}
\|P_{\operatorname{span}(e_0,e_i,e_j)}w\|^2
\le C l^{-1}\|w\|^2.
\]
The same bound includes the diagonal pairs. Summing the scalar terms and the
exceptional-row errors proves \eqref{cap:exceptional-second-moment}. Before
applying the quadratic Haar test to a product, symmetrize it or use
polarization. The argument therefore does not require an individual $A_iA_j$
to be positive.
Conditional on all the sampled images, the residual rotation gives a factor
$C\sqrt{H\gamma^3}$ on the nonexceptional side of the block from $W$ to its
complement. Together with \eqref{cap:exceptional-span-geometry}, this is the
product $C\sqrt{H\gamma}\sqrt{H\gamma^3}$, not merely the first factor. There
is no child radial contribution, since the target tests lie in $U\cap
e_0^\perp$. Combining the three blocks gives, in addition to the scalar trace,
\begin{align}
C\biggl[&
\frac{\|\mathcal A\|_F}{\sqrt{k_p}}\sqrt{H\gamma^3}
+\delta H\gamma^3
+\delta H\gamma\bigl(\sqrt{H\gamma^3}+l^{-1/2}\bigr)\nonumber\\
&+H\gamma^2
\left(\frac{\|\mathcal A\|_F}{\sqrt{k_p}}
+\delta(H\gamma^3)^{1/4}+\delta l^{-1/4}\right)
\biggr].\label{cap:compression-remainder}
\end{align}
Choose \(l=\lceil\gamma^{-16}\rceil\). Up to this point,
\eqref{cap:compression-remainder} is a deterministic bound on the function
\(\mathcal A(Y_1,\ldots,Y_l)\) outside a bad event under the joint canonical
law \(\mathbb P_{\rm can}\). The fixed-\(O\) estimates
\eqref{cap:nuclear-average} and \eqref{cap:sample-frobenius} concern
\(\mathsf T_O^l\). We transfer the geometric event to that law before using
those expectations.
\paragraph{Transfer to actual transitions and a field net.}
So far the Haar estimates concern canonical Gaussian samples. We next
convert them into estimates for the actual transition while keeping track
of the loss in exponential rate. On a narrow parent shell the density ratio
costs an arbitrarily small fixed multiple of the parent dimension for each
sampled field. We choose that loss only after fixing the sample count, so
the remaining rate pays for the child field net. Shell complements already
have uniform conditional bounds and are added separately.
For fixed \(O\), let \(F_O\) denote \(V_p-q_{p,\alpha}^{\rm fr}\) in
\eqref{cap:bridge-density-ratio}. Its normalizing denominator may depend on
the remaining orientation even after the terminal images have been fixed.
Thus the residual Haar statement above is used only under
\(\mathbb P_{\rm can}\). On the shell,
\eqref{cap:shell-likelihood} bounds each likelihood ratio by \(e^{\eta k_p}\).
For the canonical bad event \(\mathcal E\), the joint change of law gives
\[
\E_O\mathsf T_O^l(\mathcal E\cap\mathcal S_\tau^l)
\le e^{l\eta k_p}\Pr_{\rm can}(\mathcal E).
\]
Choose \(\eta\ll H\gamma^3/l\). A canonical bound \(e^{-2Hk_c}\)
then makes the right-hand side at most \(e^{-3Hk_c/2}\).
Markov's inequality in \(O\) gives the explicit conditional form
\[
\Pr_O\left\{
\mathsf T_O^l(\mathcal E\cap\mathcal S_\tau^l)>e^{-Hk_c/2}
\right\}\le e^{-Hk_c}.
\]
The complement of the narrow shell is handled separately. Its conditional
probability and all bounded polynomial moment errors are exponentially small,
uniformly at every child field, by the deterministic upper support bound and
Gaussian distance tails already proved. This uniform estimate is not obtained
by a union bound over orientations or field points, and its exponent need not
be \(2Hk_c\). The scalar signed-trace and sample independence calculations
are also applied anew conditionally at each field.
For an orientation outside the displayed exceptional set, we may now take
the \(\mathsf T_O^l\)-expectation of \eqref{cap:compression-remainder}.
Equation~\eqref{cap:sample-frobenius} bounds the expected Frobenius term by
\(O(\delta/\sqrt l)+\eta_{\rm err}\), and
\eqref{cap:nuclear-average} bounds the expected scalar trace.
The bad-event and shell-complement contributions are small because the
matrices have bounded operator norm there. After division by
\(\gamma^2\delta\), the displayed remainder terms are bounded by constant
multiples of \(H\gamma\), \(H^{3/2}\gamma^{1/2}\),
\(H^{5/4}\gamma^{3/4}\), and powers of \(\gamma\) from
\(l^{-1/4}\) and \(l^{-1/2}\). They tend to zero as \(\gamma\to0\).
This proves the first estimate in \eqref{cap:compression}, with an
arbitrarily small additive error. Applying the same change of law to the
one-vector event gives the stated bound for
\(\E_{\mathsf T_O}\|P_Uz\|^2\).
We extend this estimate to all fields by comparing geometric bad events,
before taking their signed matrix expectations. Take a net of resolution
\(\rho\) in child field units, with \(\rho\) a sufficiently high fixed
power of \(\gamma\). Its size is at most
\(\exp\{C k_c\log(1/\gamma)\}\). Fix an orientation and neighboring
fields with \(\|h-h'\|\le\rho c^2M_c\), and keep the physical terminal
samples fixed. Take \(\rho\) small enough that their joining segment
still has its frozen parameter inside parent agreement. The untruncated
parent matrices are then unchanged. The
scaled saddle Lipschitz bound gives
\begin{equation}\label{cap:cutoff-continuity}
\sup_y|\chi_h(y)-\chi_{h'}(y)|
\le C\gamma^2\rho/\tau.
\end{equation}
Thus the cutoff-weighted \(A\) changes by at most
\(C\delta\gamma^2\rho/\tau\) in operator norm, and its Frobenius norm
divided by \(\sqrt{k_p}\) changes by the same order. The corresponding
bound for \(z\) has \(K_0\sqrt\delta\) in place of \(\delta\).
The child radial projection changes by \(C\rho\), since the annulus is
separated from zero. On the retained Gram event the generating frames and
their inverses have bounded norms. Consequently the geometric inequalities
leading to \eqref{cap:compression-remainder}, with fixed slack in their
constants, persist under these changes with an additional
\(C_\gamma\rho\delta\) error. The same holds for the one-vector test.
The constant \(C_\gamma\) is at most a fixed power of \(\gamma^{-1}\):
the sample count, cutoff width, and frame bounds have already been fixed.
The transition densities require an exponential comparison. For a fixed
orientation, on \(\|Y\|\le C p^2M_p\),
\begin{equation}\label{cap:field-likelihood-comparison}
\left|\log\frac{d\mathsf T_h}{d\mathsf T_{h'}}(Y)\right|
\le C_\gamma\rho k_c.
\end{equation}
Indeed the potential \(V_p(Y)\) cancels in the ratio, and the Gaussian
log ratio uses only \(\Delta_U^{-1}\). The normalizing factor obeys the
same bound: differentiating the convolution gives
\(\nabla_U\phi_c=\Delta_U^{-1}(\E_{\mathsf T_h}Y_U-h)\),
whose norm is \(O_\gamma(M_p)\) by the parent shell and complementary
moment bounds. Integrating this derivative along the short field segment
proves the normalization assertion. This argument uses only the conditioned
parent hypotheses, not a separate good event at the child.
Choose \(\rho\) so that the perturbation of the matrix tests is much
smaller than \(\gamma^2\delta\), and
\(lC_\gamma\rho<H/8\). Prove the net estimates with the stated strict
slack. Outside their orientation exceptions, the conditional probability of
a relaxed geometric failure at a neighboring field, with all samples in a
narrower typical shell, is at most
\(e^{lC_\gamma\rho k_c}e^{-Hk_c/2}\le e^{-3Hk_c/8}\).
The cutoff and radial continuity above justify this event inclusion; the
narrower shell stays in the shell used at the net field. The shell
complements and large lengths have the separate uniform conditional moment
bounds already proved. At the neighboring field apply
\eqref{cap:nuclear-average} and \eqref{cap:sample-frobenius} anew, and then
take the expectation of the relaxed geometric inequality. This retains
their signed cancellation and proves \eqref{cap:compression} everywhere
on the annulus.
Choose the fixed exponents in \(\tau,\rho\) and the sample count first,
then \(C_1\) in \(H=C_1\log(1/\gamma)\) to pay for the net, and then
\(\gamma\) small. The separate uniform shell bounds are added without a
field union bound. The resulting failure probabilities have the form
\(e^{-a k_p}+e^{-a'k_c}\) with fixed positive constants and are summable
on the geometric list down to \(k_p\asymp\sqrt{\log n}\).
\subsection{Completion of one inductive step}
We can now assemble the matrix estimates. Compression makes the
transverse contributions in both transport bounds smaller than the
allowed transverse error. Radial regeneration supplies strict slack in the
upper radial block, and the Fisher test supplies a bounded lower radial
block. Positive-matrix Cauchy--Schwarz between these two bounds then controls
the actual mixed block. This closes the four parts of the invariant at the
child scale. The intermediate-scale estimates use the same deterministic
transport calculations; they do not require another random compression.
By \eqref{cap:compression}, \eqref{cap:bl-recursion} has transverse surplus
\(o(1)\delta\), since the mixed-square term gains \(\gamma^3 H\). For the
Fisher lower bound, let \(e_0\) be the child radial direction. The canonical
mean \(a_c\) is parallel to \(e_0\), and
\(v^\top(R_c-R_p)v\le C_\gamma p^{-2}\) for each unit
\(v\in U\cap e_0^\perp\). The shell norm and transferred second moments
from the radial calculation therefore give
\[
\sup_{\substack{v\in U\cap e_0^\perp\\\|v\|=1}}
\E_{\mathsf T}(v^\top e)^2
\le \eta_{\rm err}+C_\gamma/k_p.
\]
The bounded radial entry of \(D\) is negligible on these tests. Its mixed
terms are negligible by Cauchy--Schwarz and \(\|z\|\le K_0\sqrt\delta\),
after choosing the comparison error and then \(k_p\). The transverse
compression thus gives a Fisher deficit \(o(1)\delta\) as well, including
the fixed factor \(x^2N^2\). The transverse-only Fisher test gives a block lower matrix
with bounded radial deficit, and the Brascamp--Lieb upper matrix has cross
error \(O_\gamma(\delta)+\eta_{\rm err}\) after taking a purely radial gain as
above. The two Loewner bounds therefore force an actual cross block of size
\(O(\sqrt{o(1)\delta})+O_\gamma(\delta)\), by positive-matrix Cauchy--Schwarz.
This proves \eqref{cap:invariant} inductively, with slack. At intermediate
scales in the stated interior-parameter regime the same calculations without
\eqref{cap:compression} give transverse error and actual mixed block bounded
by a quantity tending to zero (at fixed \(\gamma\)), bounded total Hessian,
and radial deficit at least 0.26. For the lower mixed estimate near \(x=1\)
use the full Fisher test; its predicted cross block is small by the same
bridge projections. These intermediate bounds are deterministic and uniform
under the parent hypotheses.
\subsection{Deterministic evaluation at other cap ratios}
Once a parent satisfies the invariant, evaluation at another scale uses
its deterministic support and covariance bounds. No transverse compression
is needed if the target is an upper curvature estimate with prescribed
fixed error, rather than a renewed invariant with the same error.
\begin{lemma}[Evaluation with the induction extension]
\label{cap:deterministic-evaluation}
Under the parent hypotheses of Proposition~\ref{cap:curvature-induction},
the deterministic intermediate-scale conclusions hold also for
\(x=c/p\in[\gamma^2,\gamma]\), whenever the frozen parameter is inside
parent agreement with fixed slack. At fixed \(\gamma\), the transverse
and mixed errors tend to zero with the parent errors, the scaled Hessian
is bounded, and the radial deficit is at least \(0.26\).
\end{lemma}
\begin{proof}
The same extension has a positive convolution margin because
\[
\|\Delta^{1/2}L_p\Delta^{1/2}\|
\le(1+\gamma^4/100)(1-\gamma^4)<1.
\]
All inverse bounds in the shell estimates and the upper and lower
covariance transports remain bounded by fixed powers of
\(\gamma^{-1}\). Thus those deterministic estimates apply with enlarged
constants. Formula~\eqref{cap:radial-regeneration} still gives the stated
radial deficit, while the other blocks tend to zero with the input errors.
The proof uses no additional orientation event and does not extend the
probabilistic induction statement beyond ratio \(\gamma\).
\end{proof}
For a bounded child field ball, the saddle parameter can occupy a wider
range in child units. We choose a coarser parent and center a new support
annulus at that frozen parameter. This is a separate evaluation of the
true potential; its extension is not propagated as the next inductive
potential.
\begin{lemma}[Supports centered at a frozen parameter]
\label{cap:frozen-evaluation-support}
Fix a child field radius \(R\) before choosing the curvature and
comparison tolerances. Suppose a true parent at scale \(p\) satisfies
the invariant and the static comparisons. There is a fixed \(b_1>0\),
depending on \(R,\sigma\), such that the following holds for every
\(x\in[\gamma b_1,b_1]\), \(c=xp\), and
\(h\in H_c\) with \(\|h\|\le R c^2M_c\). Put
\(\alpha=\alpha_c(h)\). With sufficiently small fixed tolerances, there
is an upper-semiconcave extension agreeing with the true parent on a
neighborhood of the frozen typical length sphere. Its convolution has a
strict margin, and its terminal mean and covariance differ from those of
the true transition by exponentially small errors in the corresponding
scaled units. Consequently,
\begin{equation}\label{cap:ball-evaluation-bound}
K_c(h)\preceq C(R)c^{-2}I_{H_c},
\end{equation}
where \(C(R)\) is independent of the chosen ratios and sharp curvature
tolerances. The construction and estimates are uniform on the indicated
fixed ratio interval, with \(n\) taken large after those choices.
\end{lemma}
\begin{proof}
The static saddle bounds give \(\kappa_R,C_R>0\), depending only on
the field radius, such that
\[
-1+\kappa_R\le\alpha/c^2\le C_R.
\]
Choose \(b_1\) so that
\(\max(1,C_R)b_1^2<\sigma/2\). Then \(|\alpha|<\sigma p^2/2\)
throughout the ratio interval. Choose fixed numbers \(e,w>0\), also
satisfying the tail and scalar-margin bounds below, with
\[
C(w+\delta)\ll e\ll\kappa_R(\gamma b_1)^2,
\qquad w<\sigma/4,
\]
and define nested shells
\[
S_j^\alpha=\{y\in H_p:|\alpha_p(y)-\alpha|\le t_jwp^2\},
\qquad (t_0,t_1,t_2)=(1/2,3/4,1).
\]
They lie inside the parent invariant annulus. The radius derivative
computed above gives buffers comparable to \(wp^2M_p\) between them.
Set
\[
L_\alpha=\bigl((p^2+\alpha)^{-1}+ep^{-2}\bigr)I_{H_p}.
\]
On \(S_2^\alpha\), the invariant's positive-part bound gives
\[
K_p(y)\preceq
\bigl((p^2+\alpha)^{-1}+C(w+\delta)p^{-2}\bigr)I_{H_p}
\prec L_\alpha.
\]
This proves the upper-support inequality for pairs in \(S_1^\alpha\)
whose joining segment remains in \(S_2^\alpha\).
For longer pairs, freeze the own saddle at the support point and use
\eqref{cap:frozen-mgf}, the value comparison, and the gradient comparison
as in \eqref{cap:parent-upper-support}. If the buffer divided by
\(p^2M_p\) is \(\beta\asymp w\), the excess over the frozen
\(\alpha\) quadratic is at most
\[
\frac{\|y-z\|^2}{2p^2}
\left[Cw+
2\left(\frac{2\varepsilon_v+C_{\rm dens}/k_p}{\beta^2}
+\frac{\varepsilon_g}{\beta}\right)\right].
\]
Take \(\varepsilon_v+C_{\rm dens}/k_p\ll ew^2\) and
\(\varepsilon_g\ll ew\). The bracket is smaller than \(e\), so the
upper-support inequality holds for every pair in \(S_1^\alpha\).
Define
\[
\bar\phi_{p,\alpha}(y)=\inf_{z\in S_1^\alpha}
\left\{\phi_p(z)+\nabla_{H_p}\phi_p(z)^\top(y-z)
+\tfrac12(y-z)^\top L_\alpha(y-z)\right\}.
\]
The argument of \eqref{cap:parent-extension} shows that this function
agrees with \(\phi_p\) on \(S_1^\alpha\), has at most quadratic
growth, and satisfies \(\nabla^2\bar\phi_{p,\alpha}\preceq L_\alpha\)
distributionally. Its dependence on the child field is through the scalar
\(\alpha\), determined by the eigenvalues and \(\|h\|\). No child
angular refinement enters its construction. For this evaluation hold the
selected extension fixed when differentiating its convolution at \(h\).
Different fields may select different extensions; no derivative of this
selection, or of its parameter \(\alpha\), enters the Hessian formula.
We verify the complementary moments before comparing covariances. Let
\(B_\alpha\) denote the logarithm of the frozen norm density at a field
on the sphere \(\|y\|=\ell_p(\alpha)\). It is constant on that sphere.
For \(d(y)=|\|y\|-\ell_p(\alpha)|/(p^2M_p)\), the nearest support
point on the sphere gives
\[
\bar\phi_{p,\alpha}(y)-q_{p,\alpha}^{\rm fr}(y)
\le B_\alpha+k_p\bigl(\varepsilon_v+
\varepsilon_gd(y)+\tfrac12ed(y)^2\bigr)
\le B_\alpha+k_p\bigl(\varepsilon_v+
\varepsilon_g^2/e+ed(y)^2\bigr).
\]
The canonical calculation leading to \eqref{cap:bridge-distance-tail}
now gives, with constants depending only on the bounded child saddle range,
\[
\Pr_{\mathsf G}\{d(Y)>v\}\le2e^{-c_Rx^2k_pv^2},
\qquad v\ge C_R/(x\sqrt{k_p}).
\]
Require also \(e<c_R(\gamma b_1)^2/4\). Normalize the
extended transition on a shell
\(|\alpha_p(Y)-\alpha|\le\tau_0p^2\) with
\(\tau_0\ll\gamma b_1w\). Its canonical probability tends to one.
The off-saddle identity \eqref{cap:off-saddle-density} and centered density
bounds give a logarithmic normalization loss at most
\(\varepsilon_vk_p+C\tau_0^2k_p+O(1)\) relative to \(B_\alpha\).
Choose the comparison errors and then \(n\) so that
\[
2\varepsilon_v+\varepsilon_g^2/e+C\tau_0^2+O(1)/k_p
\ll (\gamma b_1)^2w^2.
\]
Leaving \(S_1^\alpha\) requires \(d(Y)\ge c w\). Integrating the
Gaussian distance tail against the upper support therefore bounds every
fixed polynomial moment of \(1+d(Y)\) on this complement by
\(C_j e^{-a k_p}\), for a fixed \(a>0\), uniformly in \(x,h\).
The true transition needs its own complementary estimate. On each fixed
bounded parent field range, static comparison and the frozen numerator
density upper bound give
\[
\phi_p(y)-q_{p,\alpha}^{\rm fr}(y)
\le B_\alpha+\varepsilon_vk_p+C_{\rm dens}.
\]
The preceding normalization shell and Gaussian distance tail control the
bounded part of the complement. For the remaining tail, sample the child
spin by tilting the parent zero-field spin law with
\(\exp\{h^\top X+X^\top\Delta X/2\}\), then set
\(Y=h+\Delta X+N(0,\Delta)\). The tilt normalizer is at least one by
zero-field symmetry and \(\Delta\succeq0\). With
\(v=\|P_pX\|/M_p\), its numerator is at most
\(e^{C_R k_p(1+v^2)}\). The parent projection tail
\(e^{-ck_pv^6}\) survives this factor, and the observation noise has
a Gaussian tail in parent field units. Increasing the fixed outer radius
therefore makes the outer contribution and all its fixed polynomial
moments exponentially small. This argument uses the true parent projection
tails, not the extension's Hessian bound.
The two unnormalized transition laws agree on \(S_1^\alpha\). Their
normalization ratio and their complementary first and second moments thus
show that their means and covariances differ exponentially in parent scaled
units, as in the earlier transition comparison. Converting a covariance
error in units \((p^2M_p)^2\) to child Hessian units multiplies it by
\[
c^2\|\Delta_U^{-1}\|^2(p^2M_p)^2
=\frac{x^2k_p}{(1-x^2)^2}.
\]
The exact formula \eqref{cap:child-hessian} therefore leaves an error in
\(c^2K_c\) tending to zero uniformly on the fixed ratio interval.
Finally put \(a=\alpha/c^2\) and
\(q=(1+x^2a)^{-1}+e\). On \(U\), the increment is
\((1-x^2)p^2I\), while the global support curvature is \(q/p^2\).
Impose also
\(e\le\kappa_R(\gamma b_1)^2/[4(1+\sigma/2)]\).
Then the convolution has a strict margin. Brascamp--Lieb, first for smooth
approximations and then by passage to the limit, and
\eqref{cap:child-hessian} give
\[
c^2K_c^{\rm ext}
\preceq
\frac{1+(1+x^2a)e}
{(1+a)-(1-x^2)(1+x^2a)e/x^2}\,I_U
\preceq \frac{8}{3\kappa_R}I_U.
\]
The denominator is at least \(3\kappa_R/4\), and the numerator is at
most two. The exponentially small replacement error is absorbed by
enlarging this constant once. This proves \eqref{cap:ball-evaluation-bound}
with no dependence of its coarse constant on \(x\) or the sharp
tolerances. All shell widths, comparison errors, and the ratio interval are
fixed before \(n\to\infty\); no estimate here requires taking
\(x\to0\) while keeping these choices fixed.
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-multiscale.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-static.tex ====
\section{Spectral caps and comparison with spherical spins}\label{cap:static}
The cap route first compares partition functions and projected means with a
Gaussian measure conditioned on a sphere. The comparison is uniform over
macroscopic balls of edge fields. Its role is to control the values and slopes
needed when differentiating a Gaussian observation convolution.
\subsection{Observation and scale conventions}
For a symmetric matrix $A$ and field $h$, write
\begin{equation}\label{cap:partition-definition}
Z_A(h)=2^{-n}\sum_{x\in\{-1,1\}^n}
\exp\{\tfrac12 x^\top A x+h^\top x\},\qquad
\phi_A(h)=\log Z_A(h).
\end{equation}
We write $Z_p=Z_{J_p}$ once $J_p$ has been defined, and use $Z_A^{\rm sp}$ for
the analogous integral against uniform probability on the sphere of radius
$\sqrt n$. Gradients and Hessians of $\phi_A$ are the Gibbs mean and
covariance.
Here are the conventions for all cap sections. Every tolerance described as
arbitrarily small is a fixed positive number chosen before $n\to\infty$. We
write $\eta_{\rm err}$ for such a tolerance, which can be decreased at later
finite stages of the parameter selection; $\eta_{\rm small}$ has the same
meaning when a separate structural slack is needed. Neither symbol denotes a
rate in $n$. Once the scale ratios, field ranges, and error tolerances have
been fixed, the upper scale is chosen small enough, and only then is $n$ taken
large. Uniform $o(1)$ terms refer to this order of limits. High probability,
unless otherwise specified, only means probability tending to one. Variables
called fields are in energy units. A Gaussian observation with precision \(B\)
is the field increment \(Bx+N(0,B)\). Its effect on posterior log weights is
to add the field and subtract \(B\) from the interaction matrix. One can use
increasing precisions and independent noise increments (including Brownian
observations). The logarithms of partition functions before an increment are
obtained from those after it by log heat convolution. Additive constants in
potentials don't affect gradients, heat-tilted transition probabilities or
Hessians. We use Euclidean matrix and vector norms; empirical averages have
brackets, and sometimes we use \(|v|_n=|v|/\sqrt n\).
The edge coordinates are eigenvectors with gap \(d_i=2-\lambda_i\), in order
of increasing gap. We condition in spectral-coordinate arguments on the
eigenvalues, leaving the frame Haar. The observation scheme is given by the
spectral functions
\begin{equation}
\label{cap:edge-definitions}\quad P_p=1_{\{d \le p^2\}},\qquad k_p=np^3,\qquad M_p=\sqrt{np},
\qquad B_p=(p^2-d)_+,\qquad J_p=J-B_p .
\end{equation}
Write \(H_p=\operatorname{ran}P_p\) and \(d_p=\dim H_p=\operatorname{Tr}P_p\);
the spectral estimates below give \(d_p\asymp k_p\). We use fields in
\(H_p\), and occasionally write \(P_p\) for this space. Field length
units there are \(p^2 M_p\); free-energy and projected-spin length units are
\(k_p,M_p\). The parameter \(p\) is small. Conditionally on \(J_p\) and the
eigenvalues, refinements determining smaller caps use a fresh Haar rotation
within \(P_p\).
\subsection{Spectral estimates at the bottom scale}
The resolvent event of Lemma~\ref{prof:lem:spectrum} covers the entire
cap range. Indeed, if $np_{\min}^3\asymp\sqrt{\log n}$, then
\[
\frac{p_{\min}}{d_0}\asymp
\frac{(\log n)^{1/6}}{\log\log(n+1000)}\longrightarrow\infty.
\]
Thus the lemma applies at $d=\sqrt u$ for every $u\ge c p^2$, where
$p\ge p_{\min}$ and $c>0$ is fixed. In particular, on one ordinary
disorder event of probability tending to one,
\[
d_1\ge-o(p^2),\qquad \|J\|\le3,
\qquad \#\{i:d_i\le u\}\le Cnu^{3/2}\quad(p^2\le u\le4).
\]
The count follows by applying the lemma on a dyadic grid in $\sqrt u$
and using monotonicity between grid points. The ordinary empirical
spectral measure also converges to the semicircle law.
Write $m_J(u)=n^{-1}\operatorname{Tr}(d+u)^{-1}$. The same lemma and
$b_{\rm sc}(2+u)=1-\sqrt u+O(u)$ give, uniformly for small
$u\ge cp^2$,
\[
m_J(u)=b_{\rm sc}(2+u)
+o\!\left(\frac{\sqrt u}{\sqrt{nu^{3/2}}}\right)
=1-\bigl(1+o(1)+O(\sqrt u)\bigr)\sqrt u.
\]
Here $nu^{3/2}\ge c^{3/2}np_{\min}^3\to\infty$.
We will also use the precise edge counting measure. Put
\[
\nu_p=\frac1{k_p}\sum_i\delta_{d_i/p^2}.
\]
For fixed $v,w>0$, its transform difference is exactly
\[
\int\left(\frac1{x+v}-\frac1{x+w}\right)d\nu_p(x)
=\frac{m_J(vp^2)-m_J(wp^2)}p
=\sqrt w-\sqrt v+o(1)+O(p).
\]
The limiting expression equals the same integral against
$\pi^{-1}\sqrt x\,\mathbf1_{\{x>0\}}dx$. The count bound above controls
the tail of the integrand, which is $O(x^{-2})$, by $O(R^{-1/2})$ beyond
$R$. Since the negative support lies in $[-o(1),0]$, the measures
$(1+x)^{-2}\nu_p$ are tight. Uniqueness of their Stieltjes transforms
therefore yields
\[
\nu_p\ \longrightarrow\ \pi^{-1}\sqrt x\,
\mathbf1_{\{x>0\}}dx
\quad\hbox{on bounded intervals}.
\]
All these statements are uniform in the cap range in the convention
that $n\to\infty$ after the fixed upper cutoff on $p$ is chosen, and
then that cutoff may be decreased. For example, a failure of uniformity
would give a sequence of cap scales on which the uniform resolvent
estimate and the same tightness argument give the displayed limit,
a contradiction.
The spectral event is imposed once for all scales. Later failures of
size $e^{-a k_p}$ on a geometric cap list are also summable:
\[
\sum_{j\ge0}e^{-a\gamma^{-3j}k_{\min}}=o(1),
\qquad k_{\min}\asymp\sqrt{\log n}.
\]
\subsection{Spherical saddles, norm densities, and tails}
We compare several times with spins uniform on the Euclidean sphere of radius
\(\sqrt n\), with superscript \({\rm sp}\) for this change. Write
\(\mu_{p,h}\) and \(\mu_{p,h}^{\rm sp}\) for the cube and spherical Gibbs
probabilities normalized by \(Z_p(h)\) and \(Z_p^{\rm sp}(h)\).
At the cap \(p\) with field \(h\in H_p\) in any fixed bounded range of field
units, write
\begin{equation}
\label{cap:saddle}\quad R_{p,\alpha}=(\max(d,p^2)+\alpha)^{-1},\qquad
a_{p,\alpha}=R_{p,\alpha}h,\qquad
n={\rm Tr}\,R_{p,\alpha}+\|a_{p,\alpha}\|^2 .
\end{equation}
This fixes the spectral parameter \(\alpha=\alpha_p(h)\). At this solution
we abbreviate \(R_p=R_{p,\alpha}\) and \(a_p=a_{p,\alpha}\). For any admissible
\(\alpha\), let \(f_{p,\alpha,h}^{\rm norm}\) denote the density of
\(\|N(R_{p,\alpha}h,R_{p,\alpha})\|^2\). The spherical posterior is
\(N(a_p,R_p)\) conditioned on squared norm \(n\). One has
\(-1+c\le\alpha/p^2\le C\) by the spectral facts. Conversely lengths for
\(\alpha=0\) satisfy, with \(b=4/(3\pi)\),
\begin{equation}
\label{cap:trace-constants}\quad n-{\rm Tr}\,R_p=(b+o(1))np,\qquad {\rm Tr}\,R_p^2=(2b+o(1))n/p .
\end{equation}
Changing \(\alpha\) by a small fraction of \(p^2\) perturbs these by relative
small amounts. For bounded arguments in scaled units the saddle and its
inverse (squared field length as function of parameter, where nonnegative) are
uniformly Lipschitz in scaled variables.
We record density-estimate details used below. Changing quadratic and linear
tilts that retain a comparable spectral band, at fixed \(\alpha\), uses
ordinary Gaussian mgfs times the
ratio of squared-norm densities at \(n\), by polar coordinates. At the cap
saddle the density at \(n\) is comparable to \(\sqrt{p/n}\). Indeed the
norm-square variance is of order \(n/p\); in standard deviation units the
characteristic function converges locally to \(e^{-t^2/2}\) after centering
(Taylor expansion, max Gaussian coordinate variance divided by norm-square
standard deviation \(O(1/\sqrt{k_p})\)). Absolute values are bounded by \((1+c
t^2/k_p)^{-c k_p}\) from top components alone, also in the noncentral case.
This proves the assertion by Fourier inversion. The sphere restriction is
imposed by conditioning a Gaussian squared norm. To transfer an
exponential-moment estimate through this conditioning, we need a lower density
at the original saddle and an upper density after the additional tilt, both on
the same scale. The characteristic function below supplies these two bounds.
Its central limit expansion gives the density near the mean, while an entire
spectral band supplies an integrable bound away from the origin.
In diagonal coordinates, for $Y_i=a_i+\sqrt{r_i}g_i$ the formula used is
\begin{equation}\label{cap:norm-characteristic}
\left|\E e^{it\sum_iY_i^2}\right|
=\prod_i(1+4t^2r_i^2)^{-1/4}
\exp\left\{-\sum_i\frac{2t^2r_i a_i^2}{1+4t^2r_i^2}\right\}.
\end{equation}
A band of order $k_p$ coordinates with $r_i\asymp p^{-2}$ supplies the
integrable majorant, and the remaining factors have absolute value at most
one. The variance formula $\operatorname{Var}(\sum_iY_i^2)=2\Tr
R_p^2+4a_p^\top R_pa_p$ sets the scale $n/p$. This proves the centered lower
density bound. It also gives the noncentral upper bound for a linear tilt,
and for a quadratic tilt whenever the new covariance retains a band with the
same size and comparable eigenvalues.
The same argument applies if an extra scalar multiple of \(p^2 P_p\) is added
to the interaction keeping comparable gaps at the saddle. Upper bounds suffice
without the exact trace constraint or centering.
We will use the corresponding bound for a fixed number \(m\) of replicas.
Let independent spectral rows \(Y_i\in\mathbb R^m\) have laws
\(N(a_i,C_i)\) with \(C_i\succ0\), and put
\(\mathcal Q_j=\sum_iY_{ij}^2\). For \(T_t=\operatorname{diag}(t_1,\ldots,t_m)\),
\(H_i=C_i^{1/2}T_tC_i^{1/2}\), and \(\beta_i=C_i^{-1/2}a_i\), the Gaussian
quadratic formula gives
\[
\left|\E e^{iY_i^\top T_tY_i}\right|
=\det(I+4H_i^2)^{-1/4}
e^{-2\beta_i^\top H_i^2(I+4H_i^2)^{-1}\beta_i}
\le \det(I+4H_i^2)^{-1/4}.
\]
Suppose \(L>2\) rows satisfy \(c s^{-1}I_m\preceq C_i\preceq C s^{-1}I_m\).
The singular values of \(H_i\) are at least \(c s^{-1}\) times those of
\(T_t\), even when \(T_t\) has both signs. Independence therefore yields
\[
\left|\E e^{it\cdot\mathcal Q}\right|
\le\prod_{j=1}^m(1+c t_j^2/s^2)^{-L/4}.
\]
Each one-dimensional integral is at most \(Cs/\sqrt L\). Fourier inversion
then proves
\begin{equation}\label{cap:joint-norm-density}
\sup_q f_{\mathcal Q}(q)\le C_m(s/\sqrt L)^m
\le C_m(s^{1/4}/\sqrt n)^m
\quad\text{if }L\asymp ns^{3/2}.
\end{equation}
Independent replicas at their original cap saddles have joint density at
\(n\mathbf1\) at least \(c_m(p/n)^{m/2}\). Thus an additional Gaussian
tilt that preserves the displayed band costs at most
\(C_m(\sqrt s/p)^{m/2}\) in the norm-density ratio.
The density in this last comparison must be the density under the
normalized tilted law. Indeed, for \(0\le F\le1\) with \(\E F>0\),
\[
\E[F\mid\mathcal Q=n\mathbf1]
=\E F\,\frac{f_{\mathcal Q}^{F}(n\mathbf1)}
{f_{\mathcal Q}(n\mathbf1)},
\qquad d\mathbb P_F=\frac{F\,d\mathbb P}{\E F},
\]
where \(f_{\mathcal Q}^{F}\) is the norm density under \(\mathbb P_F\);
at \(n\mathbf1\) we use the representatives given by polar coarea.
For a tail \(F=\mathbf1_{\{\mathcal T\ge b\}}\), exponential Markov gives
the useful polar-conditioning form
\begin{equation}\label{cap:conditioned-tail}
\Pr(\mathcal T\ge b\mid\mathcal Q=n\mathbf1)
\le e^{-\lambda b}\E e^{\lambda\mathcal T}
\frac{f_{\mathcal Q}^{(\lambda)}(n\mathbf1)}
{f_{\mathcal Q}(n\mathbf1)},\qquad
d\mathbb P^{(\lambda)}
=\frac{e^{\lambda\mathcal T}d\mathbb P}{\E e^{\lambda\mathcal T}} .
\end{equation}
Here the conditional quantities are defined by polar disintegration.
For a rowwise linear or quadratic \(\mathcal T\), the tilted law remains
Gaussian: a factor \(e^{Y_i^\top A_iY_i/2+v_i^\top Y_i}\) changes the
covariance to \((C_i^{-1}-A_i)^{-1}\). A strict positive precision margin
that keeps a full band comparable to \(s^{-1}I_m\) permits
\eqref{cap:joint-norm-density} for \(f_{\mathcal Q}^{(\lambda)}\).
For a general projected test, its spherical failure probability will instead
be an explicit hypothesis in the test-transfer argument below.
We also record the exact linear moment bound at a cap saddle. Polar
conditioning with the parameter frozen at \(\alpha=\alpha_p(h)\) gives
\begin{equation}\label{cap:frozen-mgf}
\log\E_{\mu_{p,h}^{\rm sp}}e^{v^\top X}
=a_p^\top v+\tfrac12v^\top R_pv+
\log\frac{f_{p,\alpha,h+v}^{\rm norm}(n)}
{f_{p,\alpha,h}^{\rm norm}(n)}
\le a_p^\top v+\tfrac12v^\top R_pv+C_{\rm dens}.
\end{equation}
The denominator uses the centered lower bound and the numerator the
noncentral upper bound with the same covariance. Thus \(C_{\rm dens}\)
is independent of \(n,p\) on the fixed bounded ranges used here.
Here are some tail estimates in this convention. For a fixed finite number of
spherical replicas at a bounded cap tilt as above (also allowing the
just-mentioned scalar modification), scaled projection norms \(\|P_p x\|/M_p\)
above large \(v\) cost \(\exp(-c k_p v^6)\), up to a decrease in \(c\); pair
overlaps above \(r\ge C p\), \(r\) small, cost \(\exp(-c n r^3)\). The same
norm tail applies for the completely uncapped zero-field model. For detail,
change the Gaussian parameter upward by \(s=\epsilon r^2\) for overlaps (take
\(r=v^2p\le1\) for projection norms), \(\epsilon\) small. The cost in log
normalizer for this is \(O(n s^{3/2})\) for \(s\gg p^2\), by integrating the
trace deficit; bounded cap fields add at most \(O(k_p)\). At uncapped zero
field one can start at parameter \(\rho p^2\), fixed small \(\rho>0\): the
cost in the lower partition bound via its norm density is at most \(C k_p\) in
addition to the log of the density scale \(\sqrt{p/n}\) (fill the \(O(np)\)
deficit with one top Gaussian square, the other squares concentrating by
Chebyshev).
In the changed Gaussian the mean contributes negligibly and the desired tails
cost \(c n r^3\epsilon\), by exponential Markov with parameter \(c's\) in the
quadratic form (trace-square \(O(n/\sqrt s)\)).
For this exponential tilt the replica precision on a band of order
\(ns^{3/2}\) rows remains comparable to \(sI\), if the Markov parameter
\(c's\) is a sufficiently small multiple of \(s\). Equations
\eqref{cap:conditioned-tail} and \eqref{cap:joint-norm-density} therefore
bound the joint norm-density ratio for \(m\) replicas by
\(C_{m,\epsilon}(r/p)^{m/2}\); for a pair it is \(C_\epsilon r/p\).
This introduces no separate factor in \(n\) when \(r\asymp p\).
For \(r\ge p\) its logarithm is absorbed by the tail exponent because
\(nr^3=k_p(r/p)^3\) and \(k_p\to\infty\). More precisely, for fixed \(a>0\),
\[
C_{m,\epsilon}(r/p)^{m/2}e^{-a nr^3}
\le C_{m,\epsilon,a}e^{-a nr^3/2}
\]
uniformly once \(k_p\) is large. Taking \(\epsilon\) small and then \(C\)
large gives the assertions. For overlaps at cap saddles with
zero field one also has below order \(p\) cost \(\exp(O(1)-c n p r^2)\) by the
same density/mgf argument at the saddle. For angular-moment comparisons at
overlaps bounded away from zero we give details separately below.
\subsection{Uniform comparison of values and means}
Comparing the Ising and spherical models by rotation has precedents in
Comets~\cite{Comets1996Spherical}; critical free-energy and overlap
comparisons also appear in Du--Huang~\cite{DuHuang2026CriticalFluctuations,
DuHuang2026CriticalOverlap}. The statement below includes the field and
posterior-test uniformity required by the covariance argument.
\begin{proposition}[Static cap comparison]\label{cap:static-comparison}
Fix a field radius $R<\infty$, an error tolerance $\eta>0$, and a geometric
ratio in $(0,1)$. There is a sufficiently small upper scale $p_0>0$ such that,
with probability tending to one, simultaneously over a geometric list of caps
$p\le p_0$ with $np^3\ge c\sqrt{\log n}$ and all
$h\in P_p\mathbb R^n$ satisfying $\|h\|\le R p^2 M_p$, one has
\begin{equation}\label{cap:static-values}
|\log Z_p(h)-\log Z_p^{\rm sp}(h)|\le \eta k_p,\qquad
\|P_p m_p(h)-a_p(h)\|\le \eta M_p,
\qquad m_p(h)=\nabla\log Z_p(h).
\end{equation}
The projected norm tails proved above hold, with a decreased positive
exponential constant, under these cube posteriors and under the uncapped
zero-field cube posterior. The statement remains valid for any fixed finite collection of auxiliary
geometric cap lists. The zero-field concentration and density estimates
used in the proof also allow bounded scalar modifications of the cap
interaction whose spherical saddle gaps remain comparable to $p^2$.
\end{proposition}
\begin{proof}
It suffices to prove the value comparison on a slightly larger prescribed
ball, with an error \(\varepsilon_v k_p\) chosen below the requested value
error. To see the mean implication, let \(u\in H_p\) be a unit vector and
choose \(t>0\) so that \(h\pm tu\) stay in the larger ball. The two value
bounds and \eqref{cap:frozen-mgf} give
\[
\log\E_{\mu_{p,h}}e^{\pm t u^\top(X-a_p)}
\le 2\varepsilon_v k_p+\tfrac12t^2u^\top R_pu+C_{\rm dens}.
\]
Jensen's inequality in the two signs yields
\[
|\langle u,P_pm_p(h)-a_p\rangle|
\le \tfrac t2u^\top R_pu+
\frac{2\varepsilon_v k_p+C_{\rm dens}}t.
\]
Set \(t=\theta p^2M_p\), and take \(u\) along the projected discrepancy.
The saddle separation gives \(p^2\|R_p|_{H_p}\|\le C\), while
\(p^2M_p^2=k_p\). Hence
\[
\frac{\|P_pm_p(h)-a_p\|}{M_p}
\le C\theta+\frac{2\varepsilon_v+C_{\rm dens}/k_p}{\theta}.
\]
Choose \(\theta\) from the desired mean tolerance, then
\(\varepsilon_v\ll\theta^2\), and finally \(k_p\) large. The larger ball
provides the same fixed margin at boundary fields. We now prove the value
comparison and the stated test transfers.
\paragraph{Replica Gram comparison.}
Haar first moments agree exactly for cube and sphere. For \(l\)-th moments, \(l\) fixed, the angular average once the replica Gram matrix is given is again the same. At off-diagonal normalized overlaps bounded by small \(\rho\) the cube Gram law is bounded by the spherical law, with factor
\begin{equation}
\exp(C_l+C_l n\rho^4),
\label{cap:gram-comparison}
\end{equation}
using comparison cells of radius \(O_l(1/n)\). Indeed the off-diagonal
products of the signs in a row form a bounded lattice vector in fixed
dimension \(l(l-1)/2\), with identity covariance. Exponential tilting gives
near the origin a local upper probability bound \(C_l n^{-l(l-1)/4}\) times
the Cramér exponential. The prefactor follows by Fourier inversion (off
neighborhoods of the finite dual-lattice points in the torus one has strict
decay, uniformly in small tilts). Its entropy rate agrees with
\(-\frac12\log\det Q\), \(Q\) the Gram matrix with diagonal one, up to and
including degree three: the only nonzero third moments are triangles, giving
the cubic term minus the sum of triangle monomials. The spherical density
proportional to \((\det Q)^{(n-l-1)/2}\), by successive projection of unit
vectors, has the reciprocal-width prefactor. Cells of sidelength comparable to
\(1/n\) can be taken disjoint. Cholesky coordinates on the angular integral
have bounded derivatives near the identity so change of the exponential
weights on the cells costs \(O_l(1)\) in logarithm. Inequalities on projected
samples used as restrictions there can correspondingly be thickened by
\(O_l(1/n)\) in unit-sphere norm.
For two replicas and \(\rho\le n^{-0.27}\), the binomial masses equal
spherical masses on overlap cells of width \(2/n\) to relative \(o(1)\). At
zero field the angular integrand is even increasing in absolute overlap (the
sum/difference orthogonal coordinates have weights with coefficients
\(1+r,1-r\)); thus these cell comparisons can then use monotonicity instead of
a constant-factor loss.
\paragraph{Zero field and overlaps bounded away from zero.}
Fix the eigenvalues and let \(\E_O\) denote Haar averaging of the eigenframe.
At uncapped zero field, and also for a small zero-field cap with the bounded
scalar modifications above and comparable saddle separation, we have the
following formulas. Here \(A\) denotes the uncapped interaction \(J\) or
the indicated capped interaction.
\[
\E_O Z_A(0)=Z_A^{\rm sp}(0),\qquad
\frac{\E_O Z_A(0)^2}{(Z_A^{\rm sp}(0))^2}=1+o(1).
\]
For the second assertion, outside \(n^{-0.27}\) up to a small constant,
\eqref{cap:gram-comparison} costs less than the spherical tail gain on dyadic
bands. For the uncapped sphere use the preceding overlap argument with
lower-normalizer scale \(p\) taken minimal. At overlaps bounded away from
zero the net exponential is strictly negative even with binomial weights.
Here is one verification. In sum and difference orthogonal coordinates,
upper exponents are bounded by the spherical free energies at \(1+|r|\) and
\(1-|r|\); restriction to the orthogonal hyperplane does not change the
bound by interlacing. In the limit at zero cutoff the spherical upper bound
\(F(t)\), exact at \(1\), has \(F'(t)=t/2\) for \(0<t\le1\) and
\(F'(t)=1-1/(2t)\) above \(1\). This follows by Gaussian change of density
with precision \(t(w-J)\), giving per-site bound
\((tw-1-\log t-n^{-1}\log\det(w-J))/2\), optimized at \(w=t+1/t\) below
one and at \(w=2+0^+\) above. The per-site density correction for the upper
bound at strictly positive separation is \(o(1)\), also in the hyperplane;
equality at \(1\) follows by the deficit-filling lower bound. Hence
\(F(1+r)+F(1-r)-2F(1)<r^2/2\), whereas the binomial cost is at least
\(r^2/2\). Continuity handles endpoints, and a small operator-norm change
of the matrices preserves these strict bounds.
All errors giving convergence of the moment ratio are summable over our
geometric list (Stirling errors negative powers of \(n\); tails for \(|r|\ge
n^{-0.27}\) exponentially small in a power of \(n\)). This proves zero-field
concentration and transfers projection tails, by the first moment and Markov.
\paragraph{Positive moments and uniform upper bounds.}
The zero-field comparison supplies a denominator and removes pairs with
macroscopic overlap. At a nonzero field, we first keep the replicas in the
small-overlap region and compare their joint Gram law. The resulting
high-moment estimates have two uses: they control the upper value of the
partition function, and they transfer a posterior test whose spherical
failure has a positive exponential rate. The latter use requires an
additive estimate on discarded replicas after the denominator has been
bounded below.
For bounded field lengths, high positive moments \emph{restricted to small
overlaps among all replicas} have normalized exponents as small as desired in
units \(k_p\), by \eqref{cap:gram-comparison} and the spherical tail at
overlaps above \(C p\); the relative error \(O_l(np^4)\) here suffices. The
restriction discards a negligible relative contribution with high probability
in the quenched model, uniformly: pairs at overlap above fixed small cutoff
have exponentially small probability at cap zero field by the preceding
argument, and the field costs only \(O(n p^{5/2})\). One may take the upper
scale small after fixing \(l\).
Here is the normalized form of the test transfer used later. Fix the
eigenvalues and a field \(u\) in the spectral copy of \(H_p\); in physical
coordinates \(J_p(O)=O J_p^{\rm diag}O^\top\) and \(h_O=Ou\).
Let \(E_u\) be one of the projected-configuration events described by the
Gram-cell comparison, with fixed geometric slack. Write \(E_u^+\) for its
specified thickening: whenever a configuration satisfies \(E_u\), all
configurations in the same Gram cell satisfy \(E_u^+\). Define the tested,
unnormalized sum
\[
Z_{p,E,O}(u)=2^{-n}\sum_x
e^{x^\top J_p(O)x/2+h_O^\top x}\,
\mathbf1_{E_u}(O^\top x).
\]
Its ratio to \(Z_{p,O}(u)=Z_p(h_O)\) is the posterior tested probability.
Assume that the spherical probability of \(E_u^+\) is at most \(e^{-c k_p}\).
For \(l\) independent copies, let
\(\mathcal Z_{E,l}^{\rm small}(O,u)\) be the sum defining
\(Z_{p,E,O}(u)^l\), restricted to pair overlaps below the fixed small
cutoff. The Gram comparison and the spherical tail give
\begin{equation}\label{cap:tested-moment}
\frac{\E_O\mathcal Z_{E,l}^{\rm small}(O,u)}
{(Z_p^{\rm sp}(h_O))^l}
\le \exp\{(-lc/2+\eta_l)k_p+o_n(k_p)\}.
\end{equation}
For fixed \(l\), the exponent loss \(\eta_l>0\) can be made as small as
prescribed by decreasing the upper scale; this includes the
\(O_l(np^4)\) Gram error. The remaining \(o_n(k_p)\) is taken afterward.
The spherical denominator depends only on the spectral field \(u\).
On overlap bands larger than \(Cp\), Cauchy--Schwarz divides the spherical
test cost by two and the remaining spherical overlap cost pays
\eqref{cap:gram-comparison}. This is the source of the factor \(1/2\).
Let \(b_l(O,u)\) be the posterior \(l\)-replica probability that some pair
exceeds the fixed overlap cutoff. Directly from the decomposition of the
replica sum,
\[
\left(\frac{Z_{p,E,O}(u)}{Z_{p,O}(u)}\right)^l
\le \frac{\mathcal Z_{E,l}^{\rm small}(O,u)}{Z_{p,O}(u)^l}
+b_l(O,u).
\]
On the quenched high-probability zero-field event, the pair estimate above,
followed by the bounded-field change of density, makes \(b_l\) exponentially
small in \(n\), uniformly after the upper scale is chosen small for fixed
\(l\). Once the value lower bound
proved below gives \(Z_{p,O}(u)\ge e^{-\varepsilon_v k_p}Z_p^{\rm sp}(h_O)\),
Markov's inequality applied to \eqref{cap:tested-moment} shows that
\begin{align}\label{cap:tested-transfer}
&\Pr_O\left\{
\frac{Z_{p,E,O}(u)}{Z_{p,O}(u)}>2e^{-c'k_p},\
Z_{p,O}(u)\ge e^{-\varepsilon_v k_p}Z_p^{\rm sp}(h_O),\
b_l(O,u)\le e^{-2lc'k_p}\right\}\nonumber\\
&\hspace{25mm}\le
\exp\{[-l(c/2-c'-\varepsilon_v)+\eta_l]k_p+o_n(k_p)\}.
\end{align}
Choose \(c'<c/2\) and \(\varepsilon_v<c/2-c'\). Increasing the fixed
replica number \(l\), and only then decreasing the upper scale, gives any
prescribed fixed Haar failure rate. The separately bounded event for \(b_l\)
is added after normalization, as the displayed decomposition requires.
The untested positive moments give the upper value bounds.
The reusable test statement takes the spherical rate as a hypothesis.
Linear and quadratic projection tests, including bounded spectral
transformations, supply that rate through \eqref{cap:conditioned-tail}
when their exponential tilt retains the required precision margin.
When a later concrete test also uses Gaussian observation noise, each
replica receives an independent noise mark. Its application must check the
same cellwise inclusion with those marks fixed; a small spherical probability
alone does not extend the statement to an arbitrary bounded test.
Upper value bounds extend uniformly by convexity from a fine constant mesh on
a larger ball. The zero-field bound (and evenness) then gives a Lipschitz
constant in scaled units on inner balls. Posterior tests with slack and
positive-rate tails can likewise use nets, by restricting the change of
density to bounded projection lengths first (the removed large-length tails
follow from the zero-field version and absorption). We only use nets of size
\(\exp(O(k_p))\).
\paragraph{Angular lower bounds and gradients.}
Upper moments alone do not provide a lower value at every field direction.
We first obtain a set of directions on each length sphere where the value
is large and the projected gradient is close to its prediction. The scalar
modification of the cap turns a zero-field partition comparison into an
average over such a sphere. A second-moment estimate then controls how
small this set of directions can be.
First, on each fixed length sphere in field space, up to arbitrarily small log
losses in scaled units its angular partition average has the predicted lower
bound with high probability. To see this at nonzero length add a scalar
interaction on \(P_p\) to \(J_p\) so that, at the parameter \(\alpha\)
predicted for the given length, the eigenvalue of \(R_p\) on \(P_p\) increases
by \(\|a_p\|^2/{\rm Tr}P_p\). This restores the trace to \(n\), with
comparable separation. Decouple the addition by a centered Gaussian. Under the
integrated spherical weight the field length thus generated concentrates
exponentially at the specified length (the unconditioned generated field using
the modified saddle is Gaussian, with mean zero and expected squared length
precisely \(\|h\|^2\); the norm constraint has at most polynomial cost in
\(k_p\), by density bounds, or by a small quadratic tilt with the bounds
above). By first moment and the zero-field two-moment argument at the modified
cap the same shell has nearly all the integrated cube weight. Narrowing the
shell to any fixed tolerance and using the scaled Lipschitz bounds gives the
claim.
Angular second moments of \(Z/Z^{\rm sp}\) are at most \(e^{\eta_{\rm
err}\,k_p}\) with high probability by the small-overlap comparison. Moreover
the two-replica numerator cost is \(e^{-c_\tau k_p}\) for
\begin{equation}
|(P_p x-a_p)^t(P_p x'-a_p)|>\tau M_p^2
\end{equation}
as follows. In the two independent canonical Gaussians, the exponential
factor \(e^{\pm\theta p^2(P_pX-a_p)^\top(P_pX'-a_p)}\) changes each
two-replica precision by an off-diagonal block of size \(\theta p^2\) on
\(H_p\). For sufficiently small \(\theta\) the cap band remains comparable
to \(p^2I_2\), and the Gaussian log moment is at most \(C\theta^2 k_p\).
Equation~\eqref{cap:conditioned-tail} then gives exponent
\(-\theta\tau k_p+C\theta^2k_p+O(1)\); choosing \(\theta\) as a small
multiple of \(\tau\) gives the stated spherical rate. The Gram comparison
transfers it to the angular numerator. Polynomial weights in the scaled
projections are handled by the projected norm tails.
Thus there is a set of angular measure at least \(e^{-\eta k_p}\), for
arbitrarily small prescribed \(\eta\), where the lower value holds up to error
\(\eta k_p\) and gradients differ from \(a_p\) in projection by any prescribed
small fraction of \(M_p\). Indeed discard the points with low value and those
with large normalized mean discrepancy in the angular first moment (use the
two-replica bound and Cauchy-Schwarz), then use the angular second moment.
Losses in these arguments can be chosen in order after the tolerances. Length
grids suffice. At these directions, the value and gradient estimates define
supporting planes. Evaluating those planes at another field gives a lower
bound there. The Gaussian moment-generating-function bound and the bounded
saddle range show that the extra scaled loss is a small multiple of the
distance plus a constant multiple of its square.
\paragraph{Refinement from a coarser cap.}
The remaining task is to reach every child field from the large-value sets
just constructed. Freeze the child saddle parameter and use its canonical
Gaussian observation as a reference bridge to a coarser cap. Under this
reference law the parent length is predictable. Fresh Haar refinement makes
the reference field close to the useful parent directions, and their
supporting planes convert that distance estimate into a lower bound for
the child value. The gain must pay for a net at the child scale; this is why
the additional scale ratio is chosen small.
For this comparison it is useful to retain the exact normalization of the
frozen Gaussian quadratic. For any admissible \(\alpha\), define
\begin{equation}\label{cap:frozen-potential}
\begin{split}
C_{n,\alpha}
&=\log\!\left(2^{n/2}\Gamma(n/2)n^{1-n/2}\right)
+\tfrac n2(2+\alpha),\\
q_{u,\alpha}^{\rm fr}(h)
&=C_{n,\alpha}+\tfrac12\log\det R_{u,\alpha}
+\tfrac12h^\top R_{u,\alpha}h .
\end{split}
\end{equation}
The determinant is on the ambient space. Coarea in squared radius gives,
for every such \(\alpha\),
\begin{equation}\label{cap:sphere-frozen-density}
\phi_u^{\rm sp}(h)
=q_{u,\alpha}^{\rm fr}(h)+\log f_{u,\alpha,h}^{\rm norm}(n).
\end{equation}
This identity also follows by writing the Gaussian density on
\(\|x\|^2=n\): its factor independent of \(x\) is
\((2\pi)^{-n/2}(\det R_{u,\alpha})^{-1/2}
e^{-n(2+\alpha)/2-h^\top R_{u,\alpha}h/2}\), and the squared-radius
coarea factor is \(\pi^{n/2}n^{n/2-1}/\Gamma(n/2)\).
To obtain a uniform lower bound at \(c\), write \(c=rp\) for a very small
fixed ratio \(r\), using an additional coarser cap list. Put
\(\Delta=B_p-B_c\), freeze \(\alpha=\alpha_c(h_c)\), and abbreviate
\(R_u=R_{u,\alpha}\), \(a_c=R_ch_c\). Since
\(R_c^{-1}=R_p^{-1}-\Delta\), Gaussian integration gives
\[
q_{c,\alpha}^{\rm fr}(h_c)
=\log\int e^{q_{p,\alpha}^{\rm fr}(h_c+z)}N(0,\Delta)(dz).
\]
The corresponding probability law for the parent field is
\begin{equation}\label{cap:static-canonical-bridge}
\begin{split}
\mathsf G_{c,p}^{h_c}(dy)
&=e^{q_{p,\alpha}^{\rm fr}(y)-q_{c,\alpha}^{\rm fr}(h_c)}
N(h_c,\Delta)(dy)\\
&=N(h_c+\Delta a_c,\ \Delta+\Delta R_c\Delta)(dy),\qquad
R_pY\sim N(a_c,R_c-R_p).
\end{split}
\end{equation}
Equivalently, sample \(X\sim N(a_c,R_c)\) without norm conditioning and set
\(Y=h_c+\Delta X+N(0,\Delta)\). Then
\(X\mid Y=y\sim N(R_py,R_p)\). Conditioning this joint law on
\(\|X\|^2=n\) gives the spherical transition and the exact density ratio
\begin{equation}\label{cap:spherical-bridge-ratio}
\frac{d\mathsf T_{c,p}^{\rm sp}}{d\mathsf G_{c,p}^{h_c}}(y)
=\frac{f_{p,\alpha,y}^{\rm norm}(n)}
{f_{c,\alpha,h_c}^{\rm norm}(n)} .
\end{equation}
The numerator here is generally evaluated away from the parent's own
saddle. Let \(\widehat\alpha=\alpha_p(y)\). From
\eqref{cap:sphere-frozen-density},
\[
\log\frac{f_{p,\alpha,y}^{\rm norm}(n)}
{f_{p,\widehat\alpha,y}^{\rm norm}(n)}
=q_{p,\widehat\alpha}^{\rm fr}(y)-q_{p,\alpha}^{\rm fr}(y).
\]
The first \(\alpha\)-derivative of \(q_{p,\alpha}^{\rm fr}(y)\) is
\(\tfrac12(n-\Tr R_{p,\alpha}-\|R_{p,\alpha}y\|^2)\), which vanishes at
\(\widehat\alpha\). Its second derivative is
\(\tfrac14\operatorname{Var}_{p,\alpha,y}(\|X\|^2)\). Thus, with
\(d_\alpha=\alpha-\widehat\alpha\),
\begin{equation}\label{cap:off-saddle-density}
\log\frac{f_{p,\alpha,y}^{\rm norm}(n)}
{f_{p,\widehat\alpha,y}^{\rm norm}(n)}
=-\frac{d_\alpha^2}{4}\int_0^1(1-t)
\operatorname{Var}_{p,\widehat\alpha+t d_\alpha,y}(\|X\|^2)\,dt .
\end{equation}
On bounded ranges with comparable gaps this variance is \(O(n/p)\).
Consequently a shell \(|\widehat\alpha-\alpha|\le\tau p^2\) costs at most
\(C\tau^2k_p\) in the logarithm. The centered parent and child densities
have ratio of order \(\sqrt{p/c}\), so their logarithmic cost is \(O_r(1)\).
This is a statement about the ratio: each individual density contains the
factor \(n^{-1/2}\).
The true heat convolution, rewritten using \(\mathsf G=\mathsf G_{c,p}^{h_c}\),
is
\[
\phi_c(h_c)-q_{c,\alpha}^{\rm fr}(h_c)
=\log\E_{\mathsf G}e^{\phi_p(Y)-q_{p,\alpha}^{\rm fr}(Y)} .
\]
Jensen and \eqref{cap:sphere-frozen-density} therefore give
\begin{equation}\label{cap:frozen-jensen}
\phi_c(h_c)-\phi_c^{\rm sp}(h_c)
\ge \E_{\mathsf G}\bigl[\phi_p(Y)-\phi_p^{\rm sp}(Y)\bigr]
+\E_{\mathsf G}\log
\frac{f_{p,\alpha,Y}^{\rm norm}(n)}
{f_{c,\alpha,h_c}^{\rm norm}(n)} .
\end{equation}
We must bound both expectations from below, including the reference fields
outside any fixed shell. We first prove that the normalization term satisfies
\begin{equation}\label{cap:averaged-frozen-density}
-C_{R,r}\le
\E_{\mathsf G}\log
\frac{f_{p,\alpha,Y}^{\rm norm}(n)}
{f_{c,\alpha,h_c}^{\rm norm}(n)}
\le0 .
\end{equation}
Here $R$ is the prescribed child field radius. Since $r$ is fixed and
$k_c=r^3k_p\to\infty$, this loss is $o(k_c)$ even at the bottom scale.
\paragraph{Averaged normalization loss.}
The child saddle bounds give $-1+\kappa_R\le\alpha/c^2\le C_R$,
with constants depending on $R$ before $r$ is chosen. Taking $r$ small
makes $|\alpha|/p^2$ small. Put
\[
v=(p^2+\alpha)^{-1},\qquad D=n-\Tr R_p.
\]
Then $d_p\asymp k_p$, $v\asymp p^{-2}$, $D\asymp np$, and
$\Tr R_p^2\asymp n/p$. For the deficit, changing the parameter from
zero to $\alpha$ changes the trace by at most
$C|\alpha|n/p=O_R(r^2np)$, while \eqref{cap:trace-constants} gives
the positive deficit $(b+o(1))np$ at zero. The same estimates hold in
a sufficiently small fixed $p^2$-neighborhood of $\alpha$.
We need a density lower bound valid for every $y\in H_p$. Split the
squared norm of $N(R_py,R_p)$ as $S_a+S_b$, where the independent
active and bulk parts have laws
\[
S_a=\|N(vy,vI_{H_p})\|^2,\qquad
S_b=\|N(0,R_p|_{H_p^\perp})\|^2.
\]
Write $D_b=n-\E S_b=D+d_pv\asymp np$. Since
$\Var S_b\le Cn/p$, Chebyshev gives
\[
\Pr\{|S_b-\E S_b|\le D_b/2\}\ge1-C/k_p\ge\tfrac12.
\]
On this event $t=n-S_b\in[D_b/2,3D_b/2]$. Spherical integration
in the active space, followed by Jensen with a uniform unit vector
$\omega\in H_p$, gives
\[
f_{S_a}(t)
=e^{-v\|y\|^2/2}f_{v\chi_{d_p}^2}(t)
\E_\omega e^{\sqrt t\,y^\top\omega}
\ge e^{-v\|y\|^2/2}f_{v\chi_{d_p}^2}(t).
\]
Here $t/(d_pv)$ lies in a fixed compact subset of $(0,\infty)$.
Stirling's bound for the chi-square density therefore gives
$f_{v\chi_{d_p}^2}(t)\ge C^{-1}(v\sqrt{d_p})^{-1}e^{-Ck_p}$.
Convolving over the bulk event proves
\begin{equation}\label{cap:global-frozen-density}
f_{p,\alpha,y}^{\rm norm}(n)
\ge C^{-1}\sqrt{p/n}\,
\exp\left\{-Ck_p\left(1+
\frac{\|y\|^2}{p^4M_p^2}\right)\right\}
\qquad(y\in H_p).
\end{equation}
The child density is comparable to $\sqrt{c/n}$, and the noncentral
upper bound from \eqref{cap:norm-characteristic} bounds the parent
density by $C\sqrt{p/n}$ for every $y$. Their common $n^{-1/2}$ factor
thus cancels before taking logarithms. In particular the logarithmic
ratio in \eqref{cap:averaged-frozen-density} is integrable under the
Gaussian bridge.
For its sharper average bound, set $T=R_c-R_p$ and
$W=\|R_pY\|^2$. The bridge and the child saddle equation give
\[
\E_{\mathsf G}W=\|a_c\|^2+\Tr T=D,\qquad
\Var_{\mathsf G}W=2\Tr T^2+4a_c^\top Ta_c\le C_Rn/c.
\]
Indeed $\Tr R_c^2\le C_Rn/c$, $\|R_c\|\le C_Rc^{-2}$, and
$\|a_c\|^2\le C_Rnc$. Fix a sufficiently small $\delta>0$ and put
$\mathcal C=\{|W-D|\le\delta np\}$. The Gaussian quadratic moment
formula bounds $\log\E e^{t(W-D)}$ by $C_Rt^2n/c$ for
$|t|\le c_Rc^2$. Exponential Markov in both signs consequently gives
$\mathsf G(\mathcal C^c)\le C_{R,r,\delta}e^{-a_{R,r,\delta}k_p}$.
On $\mathcal C$, monotonicity of the saddle equation places
$\widehat\alpha=\alpha_p(Y)$ in the comparable-gap neighborhood above:
the derivative of $\Tr R_{p,s}+\|R_{p,s}Y\|^2-n$ has negative
magnitude at least $c_2n/p$ there. Integrating that derivative to each
endpoint first locates the root, and then gives
\[
|\widehat\alpha-\alpha|\le C(p/n)|W-D|.
\]
Along the intervening parameter segment, the norm-square variance is
at most $Cn/p$. The centered density bounds and
\eqref{cap:off-saddle-density} imply on $\mathcal C$ that
\begin{align*}
\log\frac{f_{p,\alpha,Y}^{\rm norm}(n)}
{f_{c,\alpha,h_c}^{\rm norm}(n)}
&\ge-C_{R,r}-C(n/p)|\widehat\alpha-\alpha|^2,\\
\E_{\mathsf G}\!\left[
\mathbf1_{\mathcal C}(n/p)|\widehat\alpha-\alpha|^2\right]
&\le C(p/n)\Var W\le C_Rp/c.
\end{align*}
For the complement, $\|Y\|^2/(p^4M_p^2)\asymp W/(np)$ has bounded
second moment. Cauchy--Schwarz and the preceding exponential tail show
that the negative contribution allowed by
\eqref{cap:global-frozen-density} is at most
$C_{R,r,\delta}k_pe^{-a_{R,r,\delta}k_p/2}=o(1)$.
This proves the lower bound in \eqref{cap:averaged-frozen-density}.
Integrability and \eqref{cap:spherical-bridge-ratio} identify its
expectation with $-D_{\rm KL}(\mathsf G\Vert\mathsf T_{c,p}^{\rm sp})$,
which proves the upper bound.
\paragraph{Averaging the parent supporting planes.}
On a finite length grid near the typical parent length, let $\mathcal U$
be the closed set of fields $z$ satisfying
\[
\phi_p(z)-\phi_p^{\rm sp}(z)\ge-\varepsilon_ak_p,
\qquad \|P_pm_p(z)-a_p(z)\|\le\varepsilon_gM_p.
\]
Choose $\varepsilon_a,\varepsilon_g$ using the preceding angular
argument, so that these sets have angular measure at least
$e^{-\eta k_p}$ on each grid sphere, with $\eta$ as small as required.
The set $\mathcal U$ uses only parent data and the prescribed radii,
so it is fixed under the fresh refinement. These anchors lie in a fixed
bounded parent field range. Convexity gives
a lower supporting plane for $\phi_p$ at $z$. On the spherical side,
\eqref{cap:frozen-mgf}, frozen at the own saddle of $z$, gives
\[
\phi_p^{\rm sp}(y)\le\phi_p^{\rm sp}(z)
+a_p(z)^\top(y-z)
+\tfrac12(y-z)^\top R_{p,\alpha_p(z)}(y-z)+C_{\rm dens}.
\]
This holds for every $y\in H_p$: the denominator density is centered at
$z$, and the numerator has the same covariance, so its noncentral upper
bound is independent of $y$. Subtracting the two inequalities yields the
global estimate
\begin{equation}\label{cap:global-anchor-support}
\phi_p(y)-\phi_p^{\rm sp}(y)
\ge-\varepsilon_ak_p-\varepsilon_gM_p\|y-z\|
-Cp^{-2}\|y-z\|^2-C_{\rm dens}
\qquad(y\in H_p,\ z\in\mathcal U).
\end{equation}
Put $d(y)=\operatorname{dist}(y,\mathcal U)/(p^2M_p)$ and
$D_2=\E_{\mathsf G}d(Y)^2$. Choosing a nearest anchor and averaging
\eqref{cap:global-anchor-support} gives
\begin{equation}\label{cap:averaged-anchor-support}
\E_{\mathsf G}[\phi_p(Y)-\phi_p^{\rm sp}(Y)]
\ge-k_p\{\varepsilon_a+\varepsilon_g\sqrt{D_2}+CD_2\}
-C_{\rm dens}.
\end{equation}
In particular this average includes all Gaussian length tails.
To bound this distance, put $\ell_*=(p^2+\alpha)\sqrt D$.
The child saddle $\alpha$ depends only on the eigenvalues and
$\|h_c\|$, since $h_c\in H_c$; thus $\ell_*$ is independent of the
fresh orientation. The preceding variance estimate gives
\[
\E_{\mathsf G}\left[
\frac{(\|Y\|-\ell_*)^2}{p^4M_p^2}\right]
\le\frac{C\Var W}{Dnp}=O_{R,r}(k_p^{-1}).
\]
Let the grid mesh be at most $\xi p^2M_p$. Under Haar averaging the
direction of $Y$ is uniform conditional on its length. Choose a grid
radius within $\xi p^2M_p$ of $\ell_*$ and apply spherical
concentration to its anchor set of angular measure at least
$e^{-\eta k_p}$. The radial bound and the triangle inequality give
\[
\E_O D_2\le C_R(\eta+\xi^2)+O_{R,r}(k_p^{-1}),
\]
where $O$ denotes the fresh refinement. Hence the Haar average of
$\sqrt{D_2}$ is as small as prescribed. Its Frobenius Lipschitz constant is
at most $C\sqrt r+O_r(k_p^{-1/2})$. Indeed the reference field in parent
units has mean norm $O(\sqrt r)$ and covariance norm $O_r(1/k_p)$;
the Gaussian $L^2$ norm of its change between two rotations bounds the
change of its $L^2$ distance to the fixed set $\mathcal U$.
Haar Lipschitz concentration therefore gives
$D_2\le\epsilon r^3+o(1)$ except with probability
$\exp(-c_3\epsilon r^2k_p)$ (proper rotations suffice).
The constants before choosing $r$ depend only on the prescribed child
field range. First choose $\epsilon$ from the requested child value
error $\varepsilon_v$, then choose $r$ sufficiently small that the
failure rate pays for the fixed fine child net, whose size is
$\exp(O(k_c))=\exp(O(r^3k_p))$. Next choose
$\varepsilon_a,\varepsilon_g,\eta,\xi$ so that the Haar mean of
$\sqrt{D_2}$ is smaller than $\tfrac12\sqrt{\epsilon r^3}$ and the
coefficient $\varepsilon_a+\varepsilon_g\sqrt{D_2}+CD_2$ is a
sufficiently small multiple of $\varepsilon_vr^3$ on the retained event.
Then choose the upper cap scale sufficiently small for the preceding
angular estimates, and finally take $n$ large with all parameters fixed.
Equations
\eqref{cap:frozen-jensen}, \eqref{cap:averaged-frozen-density}, and
\eqref{cap:averaged-anchor-support} give the child lower bound throughout
the net. Extend by the scaled Lipschitz bound.
All finitely many grids and extra lists, with positive-rate failures per
$k_p$ or the summable zero-field errors above, retain high probability.
This proves \eqref{cap:static-values}.
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//cap-static.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//dynamics.tex ====
\section{From the functional inequalities to mixing}\label{sec:dynamics}
We now turn the two functional endpoints into worst-start mixing
bounds. Both deductions use the unscaled heat-bath form and keep the
factor $n$ between the two clocks explicit. The support-profile proof
is the finite-state truncation mechanism of
\cite[Theorem~1.1 and Lemma~2.1]{GMT2006}. The classical logarithmic
Sobolev proof uses the hypercontractive smoothing of
\cite[Theorems~3.5 and~3.7]{DSC1996}, followed by spectral-gap decay.
\subsection{The support-profile route}
\label{prof:mixing-transfer}
\begin{lemma}\label{lem:profile-to-mixing}
Let $\mu$ be a full-support law on the cube, with heat-bath form
$\D$, generator $\mathcal L$, and one-attempt kernel $P$. Suppose
that for some $\alpha>0$ and every nonnegative $f$ with support mass
$0<p=\mu(f>0)\le1/2$,
\begin{equation}
\D(f)\ge \alpha\,\mu(f^2)\log(1/p).
\label{eq:common-support-profile}
\end{equation}
Write $t_{\mathrm{mix}}^{\mathrm{ct}}(\varepsilon)$ and
$t_{\mathrm{mix}}^{\mathrm{dt}}(\varepsilon)$ for the worst-start
total-variation mixing times of $e^{t\mathcal L}$ and $P^k$, respectively.
Put
\[
\mu_{\min}=\min_x\mu(x),\qquad
M=\max\{8,\mu_{\min}^{-1}\},\qquad
k_0=\left\lceil\frac{2n}\alpha
\log\frac{\log(M/4)}{\log2}\right\rceil.
\]
Then, for every $0<\varepsilon<1/2$,
\begin{align}
t_{\mathrm{mix}}^{\mathrm{ct}}(\varepsilon)
&\le\frac1\alpha\log\frac{\log(M/4)}{\log2}
+\frac1{2\alpha\log2}\log\frac7{4\varepsilon^2},
\label{prof:continuous-mixing}\\
t_{\mathrm{mix}}^{\mathrm{dt}}(\varepsilon)
&\le k_0+
\left\lceil\frac n{\alpha\log2}
\log\frac7{4\varepsilon^2}\right\rceil.
\label{prof:discrete-mixing}
\end{align}
In particular, if $\log(1/\mu_{\min})\le Cn$, then for every fixed
$\varepsilon>0$ the worst-start mixing time is
$O_{C,\varepsilon}(\alpha^{-1}\log(n+1))$ in per-site time and
$O_{C,\varepsilon}(n\alpha^{-1}\log(n+1))$ attempts.
\end{lemma}
\begin{proof}
First obtain a Poincare inequality, which will handle the final part
of mixing. Subtract a median from a real function and take the positive
and negative parts $f_+,f_-$. Both supports have mass at most $1/2$.
For each edge of the cube, splitting into these two parts can only
decrease the sum of squared increments. The two-point representation
of each conditional variance therefore gives
$\D(f_+)+\D(f_-)\le\D(f)$. Applying
\eqref{eq:common-support-profile} to the two parts yields
\[
\Var_\mu(f)\le\mu[(f-\operatorname{med}_\mu f)^2]
\le\frac{\D(f)}{\alpha\log2}.
\]
Thus the gap is at least $\alpha\log2$.
If $g\ge0$ is a probability density relative to $\mu$ and $V=\mu(g^2)$,
let $u=(g-V/4)_+$. Markov's inequality and the elementary inequality
$(x-a)_+^2\ge x^2-2ax$ for $x,a\ge0$ give
\[
\mu(u\ne0)\le4/V,\qquad \mu(u^2)\ge V/2.
\]
The truncation is a contraction, so $\D(g)\ge\D(u)$.
For $V\ge8$, \eqref{eq:common-support-profile} consequently implies
\begin{equation}
\D(g)\ge\frac\alpha2 V\log(V/4)
\ge\frac{\alpha V}{6}\log V.
\label{prof:density-energy}
\end{equation}
For the continuous-time semigroup with generator
$\mathcal L=\sum_{i=1}^n(Q_i-I)$, where $Q_i$ is conditional expectation given
$X_{-i}$, the density from a point evolves by $g_t=e^{t\mathcal L}g_0$.
Reversibility gives $V'_t=-2\D(g_t)$. While $V_t\ge8$, setting
$z_t=\log(V_t/4)$ in \eqref{prof:density-energy} yields
$z'_t\le-\alpha z_t$. Put
\[
t_0=\alpha^{-1}\log\frac{\log(M/4)}{\log2}.
\]
Since $V_0=1/\mu(x)\le M$, by time $t_0$ we have $V_t\le8$ for every
starting point.
Afterwards the gap bound gives
$V_{t_0+t}-1\le7e^{-2\alpha(\log2)t}$.
The inequality $\norm{g\mu-\mu}_{\TV}\le\tfrac12\sqrt{V-1}$
therefore proves \eqref{prof:continuous-mixing}.
For attempted updates we obtain the squared-norm decrease directly
from positivity of the average of conditional-expectation projections.
The transition operator is $P=n^{-1}\sum_iQ_i=I+\mathcal L/n$.
Each $Q_i$ is an orthogonal projection in $L^2(\mu)$, hence
$0\preceq P\preceq I$ and $P^2\preceq P$.
If $g_{k+1}=Pg_k$, then
\[
V_k-V_{k+1}
=\langle g_k,(I-P^2)g_k\rangle_\mu
\ge\langle g_k,(I-P)g_k\rangle_\mu
=\frac1n\D(g_k).
\]
While $V_k\ge8$, this and \eqref{prof:density-energy} give
$V_{k+1}\le V_k[1-\alpha\log(V_k/4)/(2n)]$.
If the next value is still at least $8$, taking logarithms gives
\[
z_{k+1}\le(1-\alpha/(2n))z_k
\le e^{-\alpha/(2n)}z_k,
\qquad z_k=\log(V_k/4).
\]
The factor is positive: the gap bound and $-\mathcal L\preceq nI$ imply
$\alpha\log2\le n$. In addition, whenever this recurrence is used its
bracket must be positive since it bounds the positive value $V_{k+1}$.
Thus after at most $k_0$ steps the squared density norm is at most $8$.
On centered functions the same positive-operator inequality and the gap
imply
$V_{k+1}-1\le(1-\alpha\log2/n)(V_k-1)$.
The total-variation inequality above now gives
\eqref{prof:discrete-mixing}. This proves the transfer for deterministic
numbers of attempts.
If $\log(1/\mu_{\min})\le Cn$, then
$\log(\log(M/4)/\log2)=O_C(\log(n+1))$.
The two explicit bounds give the stated big-$O$ consequences for fixed
$0<\varepsilon<1/2$. For $\varepsilon\ge1/2$, the bound at accuracy $1/4$
suffices by monotonicity of mixing time in the accuracy threshold.
\end{proof}
The support bound in \Cref{prof:thm-profile} holds simultaneously
for all tests on its good-disorder event, so it applies throughout the
density evolution in the lemma. Fixed constants and logarithmic losses
are absorbed by choosing a slightly smaller exponent tolerance first.
For the big-$O$ consequence used here, the remaining hypothesis is a lower
bound on point masses. On
$\norm J\le3$, the Hamiltonian oscillation is at most $3n$, and
\begin{equation}\label{eq:minimum-mass}
\mu_{J,\min}\ge\exp[-(3+\log2)n].
\end{equation}
This proves the asserted worst-start upper exponents through the
support-profile route.
\subsection{The logarithmic Sobolev route}
The classical inequality gives a second, independent smoothing
argument. With our convention
$\Ent_\mu(f^2)\le C_{\mathrm{LS}}\D(f)$,
finite-chain hypercontractivity takes the form
\begin{equation}\label{eq:hypercontractivity}
\norm{e^{t\mathcal L}f}_{q(t)}\le\norm f_p,
\qquad q(t)-1=(p-1)e^{4t/C_{\mathrm{LS}}},\quad p>1.
\end{equation}
Here the factor four is the reversible-chain normalization in
\cite[Theorem~3.5(ii)]{DSC1996}: its energy-to-entropy constant is
$1/C_{\mathrm{LS}}$ for our unscaled form.
For the density of a point mass, choose
$p=1+1/\max(1,\log(1/\mu_{\min}))$. Its $L^p$ norm is at most $e$.
The time at which $q(t)=2$ is
$O(C_{\mathrm{LS}}\log(2+\log(1/\mu_{\min})))$.
Expansion of the logarithmic Sobolev inequality at $f=1+u g$, with
$\mu g=0$, gives
\begin{equation}\label{eq:lsi-poincare}
2\Var_\mu(g)\le C_{\mathrm{LS}}\D(g).
\end{equation}
The remaining $L^2$ decay to any fixed TV accuracy therefore costs
$O_\varepsilon(C_{\mathrm{LS}})$.
To transfer this bound to attempted updates, compare the two
semigroups in $L^2$. If $\theta\in[0,1]$ is an eigenvalue of $P$, then
$\theta^{2k}\le e^{-2k(1-\theta)}$. Expanding a centered initial
density in an orthonormal eigenbasis gives
\begin{equation}\label{eq:discrete-l2-comparison}
\norm{P^k(g-1)}_2
\le\norm{e^{(k/n)\mathcal L}(g-1)}_2.
\end{equation}
Thus the same argument, with a factor $n$ in time, proves the discrete
upper bound. Using \Cref{cap:thm-lsi} and \eqref{eq:minimum-mass}
establishes both upper exponents by the second route as well.
\subsection{Sharp exponents and preservation of the lower quantifiers}
The linear tests already give lower bounds for the full functional
constants: by definition $R_J\ge R_{\mathrm{lin},J}$.
Applying \Cref{thm:accepted-critical} and \eqref{eq:lsi-poincare} gives
\[
C_{\mathrm{LS},J}\ge2R_J\ge2R_{\mathrm{lin},J}
=n^{2/3+o(1)}
\]
as a lower-exponent statement in disorder probability. The median
consequence of \Cref{prof:support-profile} gives the upper bound for
$R_J$, while \Cref{cap:thm-lsi} gives the upper bound for
$C_{\mathrm{LS},J}$ (and, by \eqref{eq:lsi-poincare}, another upper bound
for $R_J$). Intersecting these finitely many fixed-tolerance disorder
events gives the functional assertions of \Cref{thm:main}.
For a lower mixing bound at every fixed threshold below $1/2$,
choose a nonconstant eigenfunction $g$ of $-\mathcal L_J$ with
eigenvalue $1/R_J$. Center it and normalize $\norm g_\infty=1$.
At a point where $|g|=1$, its expectation decays exactly as
$e^{-t/R_J}$, whereas its stationary expectation is zero. Testing TV
against this bounded function gives
\[
\max_x\norm{e^{t\mathcal L_J}(x,\cdot)-\mu_J}_{\TV}
\ge\tfrac12 e^{-t/R_J}.
\]
The corresponding eigenvalue of $P_J$ is $1-1/(nR_J)$, so the discrete
lower bound is $\tfrac12(1-1/(nR_J))^k$.
For every fixed $\varepsilon<1/2$, these inequalities give mixing
lower bounds of order $R_J$ and $nR_J$, respectively. Together with
the upper bounds, they prove \Cref{thm:main}.
The realized-start assertions in \Cref{thm:accepted-critical} apply to
every deterministic little-oh time sequence. Its covariance and
linear-Rayleigh assertions apply to every diverging deterministic
sequence $M_n$. These quantifiers are retained from the companion;
no stronger upper quantifier or cutoff statement is inferred from
the new exponent estimates.
\subsection{Uniform relative-entropy decay}
The spectral-gap and classical logarithmic Sobolev bounds together
determine the critical time scale for uniform relative-entropy decay.
We state the normalization
explicitly, since constants called modified logarithmic Sobolev constants
use several conventions. For the per-site generator, define
\[
\mathcal E_J(u,v):=-\mu_J[u\mathcal L_Jv]
=\sum_i\mu_J\!\left[\Cov_{\mu_J}(u,v\mid X_{-i})\right],
\qquad \mathcal E_J(f,f)=\D_J(f),
\]
and let
\[
C_{\mathrm{EP},J}:=
\sup_{\substack{g>0,\ \mu_Jg=1\\ \Ent_{\mu_J}(g)>0}}
\frac{\Ent_{\mu_J}(g)}{\mathcal E_J(g,\log g)}.
\]
This is the inverse constant for the logarithmic entropy-production
inequality. The standard comparison and semigroup interpretation appear
in \cite[Propositions~3.5--3.6 and Theorem~2.4]{BobkovTetali2006Modified};
we give the short derivation in the present normalization.
\begin{corollary}[Entropy-production exponent]\label{cor:entropy-production}
For every realization of the disorder,
\[
\frac{R_J}{2}\le C_{\mathrm{EP},J}\le\frac{C_{\mathrm{LS},J}}4.
\]
Consequently, for the critical zero-field Gaussian model,
$C_{\mathrm{EP},J}=n^{2/3+o(1)}$ in disorder probability, with the fixed
exponent-slack meaning of \Cref{thm:main}. For every initial probability
law $\nu$ on the cube and every $t\ge0$,
\[
\operatorname{KL}(\nu e^{t\mathcal L_J}\Vert\mu_J)
\le e^{-t/C_{\mathrm{EP},J}}\operatorname{KL}(\nu\Vert\mu_J)
\le e^{-4t/C_{\mathrm{LS},J}}\operatorname{KL}(\nu\Vert\mu_J).
\]
\end{corollary}
\begin{proof}
For positive $a,b$,
$(a-b)(\log a-\log b)\ge4(\sqrt a-\sqrt b)^2$.
Apply this inequality to the two-point conditional representation of
each term of $\mathcal E_J$. The classical inequality then gives
\[
\mathcal E_J(g,\log g)\ge4\D_J(\sqrt g)
\ge\frac4{C_{\mathrm{LS},J}}\Ent_{\mu_J}(g),
\]
which proves the upper comparison. For a centered real function $f$,
$g=1+uf$ is a positive density for sufficiently small $|u|$. On the
finite cube,
\[
\Ent_{\mu_J}(1+uf)=\tfrac12u^2\Var_{\mu_J}(f)+O(u^3),\qquad
\mathcal E_J(1+uf,\log(1+uf))=u^2\D_J(f)+O(u^3).
\]
Letting $u\to0$ and then taking the supremum over $f$ proves
$C_{\mathrm{EP},J}\ge R_J/2$. For a requested exponent slack $\delta$,
apply \Cref{thm:main} with slack $\delta/2$ and then absorb the fixed
factors for sufficiently large $n$.
If $g_t$ is the density of $\nu e^{t\mathcal L_J}$ relative to $\mu_J$,
reversibility gives $g_t=e^{t\mathcal L_J}g_0$ and
\[
\frac{d}{dt}\Ent_{\mu_J}(g_t)
=-\mathcal E_J(g_t,\log g_t).
\]
The chain is irreducible, so $g_t>0$ for $t>0$ even when $g_0$ vanishes
at some states. The defining inequality and Gronwall give the displayed
decay; continuity includes $t=0$. Conversely, differentiating any
uniform exponential entropy bound at zero for positive initial
densities recovers its entropy-production inequality. Thus
$1/C_{\mathrm{EP},J}$ is the optimal rate $c$ in bounds of the displayed
form $\operatorname{KL}(\nu e^{t\mathcal L_J}\Vert\mu_J)
\le e^{-ct}\operatorname{KL}(\nu\Vert\mu_J)$ that hold for every initial
law and every time.
\end{proof}
If $\operatorname{KL}(\nu\Vert\mu_J)\le H$ for a fixed $H$, Pinsker's
inequality therefore gives total-variation accuracy $\varepsilon>0$ in
$O_{H,\varepsilon}(C_{\mathrm{LS},J})$ per-site time. A point mass has initial
relative entropy $\log(1/\mu_J(x))=O(n)$ under
\eqref{eq:minimum-mass}, recovering the logarithmic factor in the
worst-start upper bound. The exponent for $C_{\mathrm{EP},J}$ concerns
this uniform rate; it does not assert that every initial law relaxes at
the slowest rate.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//dynamics.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//introduction.tex ====
\section{Introduction}\label{sec:introduction}
We determine the mixing exponent of single-site heat-bath dynamics
at the critical temperature of the zero-field Gaussian
Sherrington--Kirkpatrick model. Starting from the worst configuration,
the dynamics mixes in time $n^{2/3+o(1)}$ when each site has rate one,
or after $n^{5/3+o(1)}$ attempted uniform-site updates. The same
per-site exponent governs the relaxation time and the inverse classical
logarithmic Sobolev constant.
At criticality, equilibrium fluctuations are much larger than for
independent spins. An upper mixing bound must hold from every initial
configuration, while the functional bounds must hold for every test
function. We obtain these uniform controls in two ways. The first relates
a function's heat-bath energy to the mass of its support. The second proves
a classical logarithmic Sobolev inequality. Each argument has its own
initialization and entropy estimates.
Let $J^\circ$ be a symmetric matrix with zero diagonal and independent
$J^\circ_{ij}\sim N(0,1/n)$ for $i<j$. On $\Omega_n=\{-1,1\}^n$, put
\begin{equation}\label{eq:gibbs-law}
\mu_J(x)=Z_J^{-1}\exp\!\left(\frac12x^{\mathsf T}J^\circ x\right).
\end{equation}
This is Gaussian SK at inverse temperature one and zero external field.
Write $Q_i$ for conditional expectation under $\mu_J$ given $x_{-i}$,
and set
\begin{equation}\label{eq:clock-conventions}
\mathcal L_J=\sum_{i=1}^n(Q_i-I),\qquad
P_J=\frac1n\sum_{i=1}^nQ_i,\qquad
\D_J(f)=\sum_{i=1}^n\mu_J\!\left[\Var_{\mu_J}(f\mid x_{-i})\right].
\end{equation}
Thus $\mathcal L_J=n(P_J-I)$, and a discrete attempt may leave the spin
unchanged. Let $t_{\mathrm{mix},J}(\varepsilon)$ and
$k_{\mathrm{mix},J}(\varepsilon)$ denote worst-start total-variation
mixing times for $e^{t\mathcal L_J}$ and $P_J^k$, respectively.
The default threshold is $1/4$.
Define the relaxation time and the classical logarithmic Sobolev
inverse constant by
\begin{align}
R_J&=\sup_{\Var_{\mu_J}(f)>0}\frac{\Var_{\mu_J}(f)}{\D_J(f)},
\label{eq:relaxation-definition}\\
C_{\mathrm{LS},J}&=\sup_{\Ent_{\mu_J}(f^2)>0}
\frac{\Ent_{\mu_J}(f^2)}{\D_J(f)},
\label{eq:lsi-definition}
\end{align}
where $\Ent_\mu(g)=\mu(g\log g)-\mu(g)\log\mu(g)$ for $g\ge0$.
Both constants use the unscaled sum of heat-bath conditional variances.
Thus $R_J$ is the relaxation time for the rate-one-per-site generator;
for one attempted update the relaxation time is $nR_J$. The conditional
variance weights in $\D_J$ are retained in every comparison below.
\Needspace{11\baselineskip}
\begin{theorem}[Critical mixing and functional inequalities]
\label{thm:main}
For every fixed $\varepsilon\in(0,1/2)$, in probability over the Gaussian
disorder,
\begin{align*}
t_{\mathrm{mix},J}(\varepsilon)&=n^{2/3+o(1)},&
k_{\mathrm{mix},J}(\varepsilon)&=n^{5/3+o(1)},\\
R_J&=n^{2/3+o(1)},& C_{\mathrm{LS},J}&=n^{2/3+o(1)}.
\end{align*}
In each expression, $X_n=n^{a+o(1)}$ means that for every fixed
$\delta>0$, $\Pprob\{n^{a-\delta}\le X_n\le n^{a+\delta}\}\to1$.
\end{theorem}
The two functional constants describe complementary forms of relaxation.
For every centered observable $f$, the spectral theorem gives
$\Var_{\mu_J}(e^{t\mathcal L_J}f)\le e^{-2t/R_J}\Var_{\mu_J}(f)$,
with equality for a slowest eigenfunction. The classical logarithmic
Sobolev inequality supplies hypercontractive smoothing for arbitrary
densities and implies the Poincar\'e inequality. \Cref{sec:dynamics}
derives the worst-start bounds from these controls. Let $\operatorname{KL}$
denote relative entropy. \Cref{cor:entropy-production} also gives the
same per-site exponent for the inverse of the best rate $c$ such that
$\operatorname{KL}(\nu e^{t\mathcal L_J}\Vert\mu_J)
\le e^{-ct}\operatorname{KL}(\nu\Vert\mu_J)$
for every initial probability law $\nu$ on the cube and every $t\ge0$.
The contribution of this paper is the upper bounds. To state precisely
how they combine with the existing lower bounds, we record the critical
companion's realized-start and linear-observable theorems
\cite[Theorems~1.2 and~1.3]{AcceptedCritical}. Their quantifiers are
stronger than fixed exponent bounds and will be retained throughout.
\begin{theorem}[Realized equilibrium starts and linear observables]
\label{thm:accepted-critical}
For every deterministic sequence $t_n\ge0$ with $t_n=o(n^{2/3})$,
\[
\mu_J\!\left\{x:
\norm{e^{t_n\mathcal L_J}(x,\cdot)-\mu_J}_{\TV}>\frac14\right\}
\longrightarrow1
\]
in probability over disorder. For every deterministic nonnegative integer
sequence $k_n=o(n^{5/3})$, the same assertion holds with $P_J^{k_n}$.
Let $\Sigma_J=\Cov_{\mu_J}(X)$ and
\[
R_{\mathrm{lin},J}=\sup_{a\in\R^n\setminus\{0\}}
\frac{\Var_{\mu_J}(a^{\mathsf T}X)}{\D_J(a^{\mathsf T}X)}.
\]
For every positive deterministic $M_n\to\infty$, with probability
tending to one both $\norm{\Sigma_J}$ and $R_{\mathrm{lin},J}$ belong to
$[n^{2/3}/M_n,M_n n^{2/3}]$.
\end{theorem}
In the realized-start assertion, one samples an initial configuration
from $\mu_J$ and holds it fixed before taking the transition law and its
TV distance. Averaging that transition law over the initial configuration
would instead give $\mu_J$ at every time. The last assertion concerns
covariance and linear Rayleigh quotients. Its arbitrary-diverging-factor
upper bound does not extend here to the full relaxation or logarithmic
Sobolev constant: the upper estimates in \Cref{thm:main} have every
fixed positive exponent slack.
For other fixed TV thresholds below $1/2$, the lower bound in
\Cref{thm:main} follows by testing the slowest gap eigenfunction, as
shown in \Cref{sec:dynamics}. The realized-start assertion itself
retains its stated threshold $1/4$.
The result concerns exactly the critical, zero-field Gaussian model.
It gives the mixing exponent and does not assert a cutoff, a cutoff
window, or a leading constant. The two upper-bound arguments retain
their separate intermediate conclusions and parameter choices.
\paragraph{The development of the dynamical question.}
Sherrington and Kirkpatrick introduced the infinite-range spin-glass
model in 1975 \cite{SherringtonKirkpatrick1975}. Their later analysis
with Gaussian couplings studied single-spin Glauber relaxation through
a linearized mean-field calculation and Monte Carlo simulations
\cite[Section~VII]{KirkpatrickSherrington1978}. A parallel dynamical
mean-field program described correlation and response functions in
soft-spin Langevin models; the infinite-range specialization was
developed by Sompolinsky and Zippelius
\cite{SompolinskyZippelius1982}. These studies examined relaxation near
the transition through correlation and response. The finite-size mixing
question asks how long the full distribution, started from the most
difficult configuration, takes to approach equilibrium.
Numerical work later examined how the critical time scale grows with the
number of spins. Billoire's study of slow samples used binary couplings
$J_{ij}=\pm n^{-1/2}$ \cite{Billoire2011SlowSamples}. Billoire and
Campbell's subsequent critical study measured the equilibrium
autocorrelation
$q(t)=n^{-1}\sum_i\langle X_i(0)X_i(t)\rangle$ after equilibration.
For heat-bath updates, with time measured in updates per spin, it tested
the finite-size scaling form
$q_c(n,t)\sim n^{-1/3}F(t/n^{2/3})$
\cite[equation~(7)]{BilloireCampbell2011CriticalDynamics}.
That critical preprint does not restate the coupling law. We use its
reported collapse as numerical evidence about equilibrium autocorrelation,
keeping it separate from the Gaussian worst-start question here.
\paragraph{Rigorous high-temperature dynamics.}
For comparison with the critical law \eqref{eq:gibbs-law}, inverse
temperature $\beta$ multiplies its exponent by $\beta$; the resulting
interaction entries have variance $\beta^2/n$. The SK statements below
hold with high probability over disorder. The inverse temperature
$\beta>0$ is fixed inside each stated range, and
mixing bounds use fixed total-variation accuracy unless $\varepsilon$
is displayed.
Spectral criteria gave functional inequalities well inside the
high-temperature phase. Bauerschmidt and Bodineau proved a logarithmic
Sobolev inequality under a spectral-width condition that applies to SK
for fixed $\beta<1/4$ \cite[Theorem~1 and Corollary~2]{BauerschmidtBodineau2019}.
For Ising spins their energy is the unweighted full spin-flip gradient.
The heat-bath form $\D_J$ instead retains conditional variance weights,
so that result is not the heat-bath inequality used in this paper.
Eldan, Koehler, and Zeitouni obtained a heat-bath spectral gap uniform
over external fields under the corresponding shifted spectral condition.
Their field-dependent worst-start bound gives polynomial mixing for
zero-field SK in the range $\beta<1/4$
\cite[Theorems~1 and~11, Section~5]{EldanKoehlerZeitouni2022}.
Anari, Jain, Koehler, Pham, and Vuong then obtained
$O_\beta(n\log n)$ random-site attempts in that range, uniformly over
external fields, through a modified logarithmic Sobolev inequality
\cite[Theorem~12]{AnariJainKoehlerPhamVuong2021}.
Subsequent work increased the temperature range. Anari, Koehler, and
Vuong derived a criterion governed by a
Volterra equation; it gives $O(n\log n)$ attempted updates for SK below
a reported numerical threshold of approximately $\beta=0.295$
\cite[Theorem~3 and Section~1.3]{AnariKoehlerVuong2024}.
Wang proved $O_\beta(n\log(n/\varepsilon))$ attempted updates for every
fixed $\beta<1/2$, with high probability simultaneously over external
fields and $\varepsilon\in(0,1)$
\cite[Theorem~1.1]{Wang2026OptimalMixing}.
At zero field, Boban, Li, and Oveis Gharan obtained a constant per-site
spectral gap and an $O(n^2)$ attempted-update bound for
$\beta<1/2+\varepsilon_0$, with a universal $\varepsilon_0>0$
\cite[Theorem~1.5 and Corollary~1.6]{BobanLiOveisGharan2026}.
The companion spectral-gap theorem covers each fixed zero-field
$0<\beta<1$, with a constant that may depend on $\beta$
\cite[Theorem~1.1]{AcceptedGap}. None of these fixed-subcritical bounds
supplies a bound at $\beta=1$ by taking a limit. At that endpoint the
critical companion identifies the slow linear and realized-start scales
in \Cref{thm:accepted-critical}; the present upper bounds match their
exponents for the full dynamics.
\paragraph{Static comparisons and the two critical mechanisms.}
The equilibrium geometry behind the critical scale is accessible through
comparison with spins on a sphere. Comets conditioned on the eigenvalues
of a GOE completion and averaged its eigenframe, obtaining an exact
conditional first-moment identity between cube and spherical partition
functions \cite[equations~(2.1)--(2.2)]{Comets1996Spherical}.
At criticality, Du and Huang developed sharper comparisons for free
energy and overlap. Their local overlap-cell estimates compare
partition-function moments
\cite[Theorem~1.6 and Section~3.3]{DuHuang2026CriticalFluctuations},
and their mixed cube--sphere tensor calculation cancels leading terms
in a squared norm
\cite[Section~3.2, Lemma~3.5]{DuHuang2026CriticalOverlap}.
These are static comparisons. The upper bounds here need corresponding
control after observations have changed the field and interaction,
together with probability estimates strong enough for arbitrary test
functions.
Gaussian stochastic localization supplies the framework for revealing
those observations while tracking the remaining variance or entropy
\cite{Eldan2013StochasticLocalization,ChenEldan2025Localization}.
The root calculation also develops conditional Kac--Rice methods used
for TAP landscapes by Fan, Mei, and Montanari, and by Celentano, Fan, and
Mei \cite{FanMeiMontanari2021,CelentanoFanMei2023}. Their cited
applications have different positive-overlap or fixed-signal hypotheses;
the shrinking-overlap critical estimates are proved here.
The isotropic route extends the signed spin--Gaussian comparison to
observation-dependent fields, then improves a typical-orientation estimate
along the observation path. The cap route keeps the reduction of covariance
in the field direction while fresh rotations reduce errors in orthogonal
directions inside the active spectral subspace. The entropy arguments
use these estimates only on their stated path or field domains and pay
for their exceptions. \Cref{sec:proof-overview} describes the inputs
and outputs of the two mechanisms in detail.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//introduction.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//model.tex ====
\section{Conventions and Gaussian observations}\label{sec:model}
For a full-support law $\nu$ on $\Omega_n$, we write
\[
\D_\nu(f)=\sum_i\nu\!\left[\Var_\nu(f\mid X_{-i})\right].
\]
Conditional expectation given all spins except one is an orthogonal
projection on $L^2(\nu)$. The average of these projections therefore has
spectrum in $[0,1]$. Its Dirichlet form is $\D_\nu/n$, a fact used later
to pass directly from continuous time to deterministic numbers of
attempted updates.
For $f(x)=a^{\mathsf T}x$,
\begin{equation}\label{eq:linear-energy}
\D_\nu(f)=\sum_i a_i^2\nu\!\left[\Var_\nu(X_i\mid X_{-i})\right]
\le\norm a^2.
\end{equation}
All vector norms are Euclidean and all matrix norms are operator norms
unless another norm is indicated. Empirical averages have a factor $1/n$.
We use the orthogonal invariance of a GOE completion. Adjoin independent
$J_{ii}\sim N(0,2/n)$ to $J^\circ$ and call the completed matrix $J$.
The added energy is $\frac12\sum_iJ_{ii}$ because $x_i^2=1$; it is
constant on the cube. Hence the Gibbs law, transition kernels, and all
functional constants are unchanged. We may condition on the eigenvalues
of $J$ and use its Haar-distributed eigenframe in intermediate estimates.
Once the conclusion is expressed in these intrinsic quantities, it is
a statement about the original off-diagonal disorder.
For matrix $A$ and field $h$, a tilted spin law is proportional to
$\exp(x^{\mathsf T}Ax/2+h^{\mathsf T}x)$. A Gaussian observation of
positive semidefinite precision $B$ has value
\[
Y=BX+B^{1/2}G,
\]
where $G$ is standard Gaussian independent of $X$. Expanding the
Gaussian likelihood shows that the posterior adds $Y$ to the field and
subtracts $B$ from the interaction. Density calculations take place on
the range of $B$; null directions carry no observation. In the isotropic
case $B=sI$, the subtraction is diagonal and the posterior is simply a
field tilt of the original spin law. Spectral caps use the same identity
with a matrix precision.
For increasing deterministic precisions, independent Gaussian increments
give the same experiment. At time $s$ let $\mu_s$ be the posterior law,
$m_s=\mu_s X$ and $\Sigma_s=\Cov_{\mu_s}(X)$. The isotropic experiment
is $Y_s=sX+B_s$, with $B_s$ standard Brownian motion. Its innovation
$I_s:=Y_s-\int_0^s m_u\,\dd u$ is Brownian in the observation filtration.
The finite-prior filtering identity is
\begin{equation}\label{eq:common-filtering}
\dd(\mu_s f)=\Cov_{\mu_s}(f,X)^{\mathsf T}\dd I_s.
\end{equation}
Conditioning also gives
\begin{equation}\label{eq:common-energy-contraction}
\E\D_{\mu_s}(f)\le\D_{\mu_0}(f).
\end{equation}
This is the finite-prior form of Gaussian stochastic localization
\cite{Eldan2013StochasticLocalization}. The posterior formula is given
in \cite[Section~2.4.2, Fact~14 and Equation~(6), arXiv version~2]{ChenEldan2025Localization};
the entropy drift and associated stability criteria appear in its
Equation~(27), Proposition~39 and Lemma~40. The contraction
\eqref{eq:common-energy-contraction} is the heat-bath Dirichlet-form
supermartingale of
\cite[Lemma~9, arXiv version~2]{EldanKoehlerZeitouni2022}.
The gap companion records these temperature-free identities with the
same normalization \cite[Section~2]{AcceptedGap}.
The critical task is to bound the posterior quantities in these
identities on the observation paths actually used. Each route proves
those estimates, including the cost of exceptional observations. In
particular, a covariance estimate on one random path is not an
all-tilts stability theorem, and the classical logarithmic Sobolev
inequality will require its own entropy and layer argument.
For a nonnegative density $g$ with $\mu g=1$, $\KL(g\mu\Vert\mu)
=\Ent_\mu(g)$. The classical inequality studied here is
$\Ent_\mu(f^2)\le C_{\mathrm{LS}}\D_\mu(f)$ for all real $f$.
Here the energy acts on $f$, while the entropy acts on $f^2$.
An estimate using the logarithmic entropy-production form of $g$ would
be a different inequality and does not replace this one.
For a fixed law on the finite cube every real function is admissible.
Unless a varying parameter is explicitly specified, all constants and
tolerances are chosen before $n$ tends to infinity.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//model.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Critical mixing in the Sherrington--Kirkpatrick model}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
At the critical inverse temperature $\beta=1$, we determine the worst-start
total-variation mixing exponent for single-site heat-bath dynamics in the
zero-field Gaussian Sherrington--Kirkpatrick model. The mixing time is
$n^{2/3+o(1)}$ when every site has rate one, or $n^{5/3+o(1)}$ attempted
uniform-site updates, in probability over the disorder.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/model}
\input{sections/accepted-inputs}
\input{sections/proof-overview}
\input{sections/profile-spectral}
\input{sections/profile-roots}
\input{sections/profile-matrices}
\input{sections/profile-orbits}
\input{sections/profile-local}
\input{sections/profile-amplification}
\input{sections/profile-positive}
\input{sections/profile-entropy}
\input{sections/cap-static}
\input{sections/cap-multiscale}
\input{sections/cap-classical}
\input{sections/cap-initialization}
\input{sections/cap-entropy}
\input{sections/dynamics}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//paper.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//preamble.tex ====
\usepackage[T1]{fontenc}
\usepackage{lmodern,microtype}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,bm,mathrsfs}
\usepackage{graphicx,booktabs,array,enumitem,needspace}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,calc,fit}
\usepackage[numbers,sort&compress]{natbib}
\usepackage{xurl}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\usepackage[nameinlink,noabbrev]{cleveref}
\hypersetup{pdftitle={Critical mixing in the Sherrington--Kirkpatrick model},pdfauthor={OpenAI}}
\pdfgentounicode=1
\pdfglyphtounicode{parenleftbig}{0028}
\pdfglyphtounicode{parenleftBig}{0028}
\pdfglyphtounicode{parenleftbigg}{0028}
\pdfglyphtounicode{parenleftBigg}{0028}
\pdfglyphtounicode{parenrightbig}{0029}
\pdfglyphtounicode{parenrightBig}{0029}
\pdfglyphtounicode{parenrightbigg}{0029}
\pdfglyphtounicode{parenrightBigg}{0029}
\pdfglyphtounicode{bracketleftbig}{005B}
\pdfglyphtounicode{bracketleftBig}{005B}
\pdfglyphtounicode{bracketleftbigg}{005B}
\pdfglyphtounicode{bracketleftBigg}{005B}
\pdfglyphtounicode{bracketrightbig}{005D}
\pdfglyphtounicode{bracketrightBig}{005D}
\pdfglyphtounicode{bracketrightbigg}{005D}
\pdfglyphtounicode{bracketrightBigg}{005D}
\pdfglyphtounicode{floorleftbig}{230A}
\pdfglyphtounicode{floorleftBig}{230A}
\pdfglyphtounicode{floorleftbigg}{230A}
\pdfglyphtounicode{floorleftBigg}{230A}
\pdfglyphtounicode{floorrightbig}{230B}
\pdfglyphtounicode{floorrightBig}{230B}
\pdfglyphtounicode{floorrightbigg}{230B}
\pdfglyphtounicode{floorrightBigg}{230B}
\pdfglyphtounicode{ceilingleftbig}{2308}
\pdfglyphtounicode{ceilingleftBig}{2308}
\pdfglyphtounicode{ceilingleftbigg}{2308}
\pdfglyphtounicode{ceilingleftBigg}{2308}
\pdfglyphtounicode{ceilingrightbig}{2309}
\pdfglyphtounicode{ceilingrightBig}{2309}
\pdfglyphtounicode{ceilingrightbigg}{2309}
\pdfglyphtounicode{ceilingrightBigg}{2309}
\pdfglyphtounicode{braceleftbig}{007B}
\pdfglyphtounicode{braceleftBig}{007B}
\pdfglyphtounicode{braceleftbigg}{007B}
\pdfglyphtounicode{braceleftBigg}{007B}
\pdfglyphtounicode{bracerightbig}{007D}
\pdfglyphtounicode{bracerightBig}{007D}
\pdfglyphtounicode{bracerightbigg}{007D}
\pdfglyphtounicode{bracerightBigg}{007D}
\pdfglyphtounicode{angbracketleftbig}{27E8}
\pdfglyphtounicode{angbracketleftBig}{27E8}
\pdfglyphtounicode{angbracketleftbigg}{27E8}
\pdfglyphtounicode{angbracketleftBigg}{27E8}
\pdfglyphtounicode{angbracketrightbig}{27E9}
\pdfglyphtounicode{angbracketrightBig}{27E9}
\pdfglyphtounicode{angbracketrightbigg}{27E9}
\pdfglyphtounicode{angbracketrightBigg}{27E9}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbb R}
\newcommand{\E}{\mathbb E}
\newcommand{\Pprob}{\mathbb P}
\newcommand{\D}{\mathcal D}
\newcommand{\TV}{\mathrm{TV}}
\newcommand{\dd}{\mathrm d}
\newcommand{\1}{\mathbf 1}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\abs}[1]{\left\lvert#1\right\rvert}
\newcommand{\av}[1]{\left\langle#1\right\rangle}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\Ent}{Ent}
\DeclareMathOperator{\KL}{KL}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\osc}{osc}
\numberwithin{equation}{section}
\setlength{\emergencystretch}{2em}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//preamble.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-amplification.tex ====
\section{Amplification along observations}
\label{prof:amplification}
\subsection{A compact-interval improvement of overlap tails}
\label{prof:sec-bridge}
We next improve the probability of the static covariance bounds. The first
ingredient concerns overlaps about the exact field root on a compact interval
of small positive observation times. Throughout this section,
\(\mathcal G\) denotes the spectral restriction already imposed,
\(r=\sqrt s\), and \(k=nr^3\). Probabilities with the indicator of
\(\mathcal G\) are integrated probabilities under the planted law; no claim
about the planted probability of \(\mathcal G\) is needed. We write
\(\Sigma_s=\operatorname{Cov}_{\mu_{h_s}}X\) and
\(m_s^{\mathrm{post}}=\mu_{h_s}(X)\). Thus the subscripted vector
\(m_s^{\mathrm{post}}\) is distinct from the posterior probability measure
\(\mu_{h_s}\), and from the exact root \(m\).
\begin{lemma}[Overlap improvement on compact intervals]
\label{prof:lem-compact-bridge}
There are \(r_0>0\) and \(a_*>0\), with \(a_*\) independent of a lower
endpoint \(r_1\), such that the following holds. For every fixed
\(0<r_1\le r_0\) and \(\chi>0\), there is \(c=c(r_1,r_0,\chi)>0\)
for which, at each \(r\in[r_1,r_0]\),
\[
\mu_{h_s}^{\otimes2}
\bigl( |(X-m)^T(X'-m)|>\chi nr\bigr)\le e^{-cn}
\]
outside an integrated planted exceptional set of mass at most
\(e^{-a_*k}\), with \(\mathcal G\) imposed. The root is the unique
regular root on the retained event.
\end{lemma}
\begin{proof}
Use the root orbits, their three projection columns \(P\), and the reference
partition integral \(Z_*\) from the local comparison. We write
\(\vartheta\) for the normalized projection coordinate, so that
\(\delta=\operatorname{diag}(\sqrt q,q,q^{3/2})\vartheta\), and put
\(z=X-m\), \(N=nq^3\). For different replicas let
\[
t_{ij}=n^{-1}(z_i-P\delta_i)^T(z_j-P\delta_j).
\]
On this compact interval all polynomial bounds may depend on \(r_1\).
First, outside Haar mass \(e^{-Bk}\), with any prescribed fixed \(B\),
the full tilted spin integral is at least \(Z_*e^{-o(n)}\).
Indeed \eqref{prof:mass-anchor} supplies a fixed positive fraction of
orientations with full mass at least \(Z_*/2\). The simultaneous
covariance and mean bounds in \eqref{prof:eq13} give a bounded raw
operator second moment of \(z\) on this compact interval.
Jensen's inequality, evaluated at such an anchor, bounds the
loss under a rotation at fixed Frobenius distance by
\(C\sqrt n+O(1)\): the linear energy increment pairs with a raw
second-moment matrix of Hilbert--Schmidt norm \(O(\sqrt n)\), and the
quadratic increment costs \(O(1)\).
Concentration of distance to the anchor
set excludes distances greater than a sufficiently large fixed constant
with probability \(e^{-Bk}\). The determinant weight changes this estimate
only by the already established arbitrarily small loss in its rate.
Fix a small absolute \(D_*>0\). For any fixed replica number \(l\),
consider the unnormalized tilted spin integral on
\begin{equation}
D_*\le \sum_{i=1}^l|\vartheta_i|^2
+\sum_{i<j}|t_{ij}/q|^2\le4D_*.
\label{prof:eq-bridge-annulus}
\end{equation}
Its Haar mean, divided by \(Z_*^l\), is at most
\(e^{-cD_*N}\), where \(c>0\) is independent of \(l\), although the
threshold in \(n\) may depend on \(l\). We verify this uniformity because
the replica number will depend on \(\chi\).
Under the tilted reference, the projection coordinates
\(\sqrt N\,\vartheta_i\) have bounded Gaussian covariance before length
conditioning. Every linear combination with unit Euclidean coefficients
of the off-diagonal coordinates \(nt_{ij}/\sqrt{n/q}\) is a centered
Gaussian quadratic form whose matrix has bounded Hilbert--Schmidt norm and
operator norm at most \(C/\sqrt N\). These constants are independent of
\(l\). Gaussian exponential Markov bounds, followed by a fixed-mesh net,
therefore give cost at least \(c_0D_*N\) for
\eqref{prof:eq-bridge-annulus}. The net and conditioning on the finitely
many lengths cost only polynomial factors for fixed \(l\); the latter
follows either from the length-density bounds or from buffered shells.
The annulus penalty just obtained is under the \emph{physically
tilted} reference shell, with covariance \(C_o\) before length
conditioning. The change of prior uses a different density:
the \emph{unweighted} \(o\) prior has uniform complement directions
after its projections and lengths are fixed. The common angular energy
integral is multiplied by the spin Gram density divided by this reference
Gram density. We estimate this quotient using polynomial boxes and
Chernoff upper bounds.
The single-projection density ratios in~\eqref{prof:eq14} cost at most
\(CN D_*^{3/2}+o(n)\). At fixed projections the unweighted reference conditional
Gram density has exponent
\[
-\frac n2\sum_{i<j}\frac{t_{ij}^2}{B_iB_j}
+O(ND_*^{3/2}),
\qquad
B_i=b-2\sqrt q\,(\delta_i)_1-|\delta_i|^2,
\]
up to polynomial factors. At the independent spin projection saddles
write the site deviations as \(\xi_i+d_i^0\), with independent centered
bounded \(\xi_i\).
Set \(R_{ij}=t_{ij}/(B_iB_j)\) and \(R_{ii}=0\).
The quadratic log moment satisfies
\[
\log\mathbb E\exp(\xi^TR\xi/2)
=\tfrac12\operatorname{Var}(\xi^TR\xi/2)
+O(\|R\|_{\mathrm{HS}}^3),
\]
with an absolute constant when \(\|R\|_{\mathrm{HS}}\) is small.
For completeness, decouple the off-diagonal quadratic form using a random
bipartition. Jensen's inequality bounds its exponential moment by that
of four times the corresponding directed cross sum. Integrating one
side by the bounded-variable Gaussian moment bound produces a squared
linear-image penalty; introducing a white Gaussian and integrating the
other side gives a uniform exponential moment for a small multiple of
\(|\xi^TR\xi|/\|R\|_{\mathrm{HS}}\). Taylor's formula then gives the
displayed third-order remainder.
The linear terms involving \(d_i^0\) are treated by H\"older's inequality
with relative inflation \(O(q\sqrt{D_*})\) in the centered quadratic
moment. The remaining averaged log cost is at most
\[
C\bigl(1+(q\sqrt{D_*})^{-1}\|R\|\bigr)
\|R\|q^3D_*.
\]
Here we used the response bounds~\eqref{prof:eq17} for \(d_i^0\) and linear sub-Gaussianity.
The site-averaged quadratic coefficient for each pair differs from
\(B_iB_j\) by \(O(q^2)\). To see that the constant is independent
of \(l\), note on the annulus that
\[
\|R\|_{\rm HS}\le Cq\sqrt{D_*},\qquad
\frac1n\sum_{\text{sites}}\sum_{i=1}^l|d_i^0|^2
\le Cq^3D_*.
\]
The latter inequality is \eqref{prof:eq17} summed over replicas.
The displayed log-moment bound is first applied at each site and
then averaged. Its cubic remainder costs
\(n\|R\|_{\rm HS}^3\le CN D_*^{3/2}\), the variance-profile
discrepancy costs
\(nq^2\|R\|_{\rm HS}^2\le CNqD_*\), and the displayed
H\"older cost, after multiplication by \(n\), is at most
\(CNqD_*^{3/2}\). These estimates use only sums of squares and
the Hilbert--Schmidt norm. Their constants therefore do not grow
with the fixed replica number. After the Chernoff subtraction, the
total relative cost is bounded by
\(CN D_*(q+\sqrt{D_*})\), besides polynomial factors. Choosing first
\(D_*\), then the upper bound on \(q\), absorbs this cost into
\(c_0D_*N\). Angular integrals on these boxes have controlled variation;
their constants may depend on \(l,r_1\).
The annulus rate is independent of the fixed replica count. This lets
us turn a requested overlap accuracy into a sufficiently large, but
still fixed, number of replicas. A first-crossing selection will then
find an annulus subset without allowing a single large statistic to
account for the whole crossing.
We need an exponential posterior bound for each individually large
statistic, not just the polynomial cutoff loss used in the local
comparison. Let \(\mathcal R_s\) be the domain where all root/scalar
diagnostics retained for the local comparison hold. On this domain the
root is unique and regular, \(q>0\), and the three axes are nondegenerate.
Its excluded complement, including nonunique-root fields, has separate charge
\[
\Pr_{\rm pl}(\mathcal G\cap\mathcal R_s^c)\le n^Ce^{-a_0k},
\]
after taking the fixed \(a_0>0\) below the preceding exclusion rates.
Only on \(\mathcal R_s\), define
\(E_1=\{|\vartheta_1|^2>D_*/4\}\) for one fresh posterior replica and
\(E_2=\{|t_{12}/q|^2>D_*/4\}\) for two. The plant and retained
root/scalar data remain fixed inside \(\mu_{h_s}^{\otimes j}\).
The scalar-root estimates after equivariant plant replacement, and
the plant--replica and replica--replica overlap estimates, give
\[
\int_{\mathcal G\cap\mathcal R_s}
\mu_{h_s}^{\otimes j}(E_j)\,d\Pr_{\rm pl}
\le n^Ce^{-a_0k},\qquad j=1,2,
\]
after decreasing \(a_0\) if necessary. Conditional Markov on this
retained domain gives
\[
\Pr_{\rm pl}\!\left\{\omega\in\mathcal G\cap\mathcal R_s:
\mu_{h_s}^{\otimes j}(E_j)>e^{-a_0k/2}\right\}
\le n^Ce^{-a_0k/2}.
\]
Relabeling \(E_1,E_2\) and taking a finite union covers every
individual statistic in any fixed replica family. The complement of
\(\mathcal R_s\) remains the separate charge displayed above.
Suppose the conclusion of the lemma fails. Draw a sufficiently large
fixed number \(l_{\rm pair}\) of disjoint violating pairs, depending
on \(\chi\). It uses \(2l_{\rm pair}\) replicas; the preceding estimates
apply to every subset with \(l\) equal to that subset's replica count. Since
\[
n^{-1}z_i^Tz_j=t_{ij}+\delta_i\cdot\delta_j,
\]
their statistics force the total in \eqref{prof:eq-bridge-annulus} to cross
\(D_*\). Requiring all \(l_{\rm pair}\) pairs costs at most
\(l_{\rm pair}c n\) in log likelihood. Choose \(c\) so that
\(l_{\rm pair}c n\le a_0k/4\), possible uniformly on the compact
interval because \(k\ge nr_1^3\).
The preceding exponential conditional bounds then make all
individually large statistics, including those across pairs,
negligible even after this conditioning. This choice explains the
dependence of \(c\) on \(r_1,\chi\), while the field-rate constant
remains fixed.
A subset of the selected replicas then satisfies
\eqref{prof:eq-bridge-annulus}. Indeed take the first prefix crossing
\(D_*\). If its new row contributes too much, retain enough earlier
indices to make that row contribution comparable to \(D_*\); each of its
terms is at most \(D_*/4\), while the earlier internal sum is below
\(D_*\). A finite union over subsets, the denominator lower bound, and
the preceding Haar estimate exclude the violating orientations at a rate
bounded below independently of the number of replicas. Returning to the
counting weight costs \(o(k)\) on this compact interval. The initially
excluded orbits had a fixed small rate as well. This proves the lemma.
\end{proof}
\subsection{A conditional weak Poincar\'e inequality}
\label{prof:sec-conditional-variance}
The next lemma transports variance one short step backwards in the
observation filtration. This Gaussian posterior interpolation belongs
to the stochastic-localization framework of
\citet{Eldan2013StochasticLocalization,ChenEldan2025Localization};
the exceptional-set variance estimate needed here is proved below.
Given \((J,h_s)\), let \(u=(1+d)s\),
\(\tau=u-s\), with \(d>0\) fixed and small, and write
\[
g=\frac{h_u-h_s-\tau m_{\mathrm{sh}}}{\sqrt\tau}.
\]
The shift \(m_{\mathrm{sh}}\) is measurable at time \(s\) and has
polynomially bounded norm. Let \(P\) be the conditional law of \(g\),
\(G\) standard Gaussian measure, and \(L_P=\log(dP/dG)\).
The Gaussian-mixture formula gives convexity of \(L_P\) and
\[
\nabla^2\log\frac{dP}{dg}=-I+\tau\Sigma_u,
\]
where \(dP/dg\) is the Lebesgue density.
\begin{lemma}[Conditional variance estimate]
\label{prof:lem-conditional-variance}
Fix an observation bound \(T<\infty\) and positive constants
\(\alpha,\beta\). Suppose \(0<s<u=(1+d)s\le T\),
\(k=ns^{3/2}\), and
\begin{equation}
P\{\tau\|\Sigma_u\|>1/4\}\le e^{-\beta k},
\qquad
\log\int e^{2L_P}\,dG\le\alpha k,
\label{prof:eq20}
\end{equation}
where \(\alpha/\beta\) is less than a sufficiently small absolute
constant. For every fixed \(D>0\) and smooth test \(\phi\) whose values
and required derivatives have fixed polynomial bounds,
\begin{equation}
\operatorname{Var}_P\phi
\le3\mathbb E_P\|\nabla_g\phi\|^2+n^{-D},
\label{prof:eq21}
\end{equation}
provided \(\beta k/\log n\) is sufficiently large in terms of those
bounds and \(D\). Bounded tests with polynomial cutoffs are included.
\end{lemma}
\begin{proof}
We first transfer small exceptional sets from \(P\) to \(G\). Since
\(\int e^{L_P}dG=1\), the second moment in \eqref{prof:eq20} and
Paley--Zygmund give
\[
G\{L_P\ge-1\}\ge(1-e^{-1})^2e^{-\alpha k}
\ge e^{-2\alpha k}
\]
for sufficiently large \(\alpha k\); enlarging harmless constants covers
the other case. Also
\(G\{L_P>\beta k/8\}\le e^{-\beta k/8}\).
Convexity implies that
\begin{equation}
G\{L_P<-\beta k/4\}\le e^{-c\beta k}
\label{prof:eq-likelihood-lower-tail}
\end{equation}
for an absolute \(c>0\).
Here is a proof that does not presume regular
boundaries of the two level sets. Use Gaussian endpoint conditioning to
drive two Brownian motions with the same noise into
\(\{L_P\ge-1\}\) and a proposed low set of mass at least
\(e^{-c\beta k}\), with joint success probability greater than \(0.8\).
Positive smooth approximations to the conditioning densities and energy
stopping, in the entropy--drift representation of
\citet[Proposition~1 and Section~2.5]{Lehec2013Entropy}, give total
expected squared drift energy at most
\(C(\alpha+c\beta)k\).
Reflect the second control about the first.
The third endpoint is the reflected point, so convexity forces its
likelihood above \(\beta k/8\) on the joint hit, for large \(k\).
Its relative entropy with respect to \(G\) is at most
\(C'(\alpha+c\beta)k\), by Girsanov and data processing. Binary entropy
applied to \(\{L_P>\beta k/8\}\) contradicts this bound if \(c\) and
\(\alpha/\beta\) are small enough. This proves
\eqref{prof:eq-likelihood-lower-tail}. Consequently, if
\(P(A)\le e^{-\beta k/2}\), then
\[
G(A)\le e^{-\beta k/4}+e^{-c\beta k}.
\]
The likelihood comparison has transferred rare starting sets to the
Gaussian reference. We now obtain contraction along typical Langevin
paths and telescope short-time variances to get the gradient term.
A final comparison through the Gaussian reference will remove the
remaining terminal variance.
Run stationary Langevin diffusion for \(P\), with generator
\(\Delta+\nabla\log(dP/dg)\cdot\nabla\), for
\(T'=C_D\log n\). Apart from probability
\(\operatorname{poly}(n)e^{-\beta k}+e^{-n^2}\), its log-density Hessian
is at most \(-2I/3\) all along the path. Here are the buffers needed
for this path statement. Let \(M_n\) be a polynomial bound for the
Lipschitz constant of the Hessian, supplied by the bounded-spin
formulas for posterior derivatives, and take
\(\rho=(100M_n)^{-1}\). At every point of a polynomial time grid,
stationarity and \eqref{prof:eq20} give the upper bound \(-3I/4\)
outside total probability \(\operatorname{poly}(n)e^{-\beta k}\).
The drift has linear growth with a polynomial additive bound.
Choose the grid mesh polynomially small enough that, on a
polynomial ball, its drift and Brownian increments are less than
\(\rho/4\) between successive grid points. Gaussian increment
bounds and the tails of the Gaussian mixture \(P\) make the
remaining probability at most \(e^{-n^2}\). The Hessian Lipschitz
bound then preserves a strict upper bound below \(-2I/3\)
throughout the tube of radius \(\rho\) about the path.
The fixed observation bound and fixed \(\beta\) give
\(\beta k=O(n)\). Thus these Gaussian-tail errors remain negligible
after the exponential reweightings below. The lemma is only asserted
on this compact observation domain.
This tube also controls neighboring flows for that same future
noise. Start a synchronous flow within \(\rho/4\) of the path's
initial point and stop at the first time its distance from the
path reaches \(\rho/2\). Before that time, the mean Hessian along
the line segment between the flows is at most \(-2I/3\), so their
distance contracts. It cannot reach the stopping boundary.
Differentiating this stopped comparison and keeping a little slack
gives Jacobian norm at most \(e^{-v/2}\) for every \(v\le T'\).
This is a neighborhood of a good random path, not a claim about
every initial point.
Let \(K_v\) be the Langevin semigroup. Choose a polynomially small
\(\Delta t\) and an integer \(N_t\) with \(T'=N_t\Delta t\),
adjusting \(T'\) negligibly. Stationarity gives the exact telescope
\[
\operatorname{Var}_P\phi-\operatorname{Var}_P K_{T'}\phi
=\sum_{j=0}^{N_t-1}\E_{x\sim P}
\operatorname{Var}_{K_{\Delta t}(x,\cdot)}
(K_{j\Delta t}\phi).
\]
Every short transition \(K_{\Delta t}(x,\cdot)\) has Poincar\'e constant
at most \((2+o(1))\Delta t\), uniformly in \(x\). Indeed the global
upper Hessian bound is polynomial, hence the flow Jacobian is
\(1+o(1)\) up to this time; the conditional heat variance identity
integrated over the short interval gives the assertion.
For a variance term involving \(K_v\phi\), Jensen in the common future
noise bounds
\(\operatorname{Var}_{K_{\Delta t}(x,\cdot)}K_v\phi\) by the average
variance of \(y\mapsto\phi(Z_v(y))\), where \(Z_v\) is that future
Langevin flow. On a good path from \(x\), cut this test off on the
inverse-polynomial ball just described. Its gradient there is bounded by
\(e^{-v/2}\|\nabla\phi(Z_v(y))\|\). Crossing and cutoff-annulus errors
are negligible by short-time Gaussian tails after making \(\Delta t\)
small enough.
The good-path event is determined from the future path started at
\(x\), and the cutoff test has polynomial bounds on this ball.
Bad paths, averaged over stationary \(x\), cost only a polynomial
times their probability. On good paths, drop the event indicator
from the resulting nonnegative gradient integral. Then
\(x\sim P\), \(y\sim K_{\Delta t}(x,\cdot)\), and independent
future noise make \(Z_v(y)\) stationary. Each variance term is
therefore at most
\[
(2+o(1))\Delta t\,e^{-v}
\int\|\nabla\phi\|^2dP
\]
plus an arbitrarily small inverse-polynomial error. Summing over the polynomial number of time steps preserves the
negligible error. The gradient contributions sum to at most
\((2+o(1))\int\|\nabla\phi\|^2dP\).
It remains to bound \(\operatorname{Var}_P K_{T'}\phi\).
Let \(b(x)\) be the conditional probability that the future path
from \(x\) fails the preceding tube test. Its \(P\) mean is at
most \(\operatorname{poly}(n)e^{-\beta k}+e^{-n^2}\). Markov at
\(e^{-\beta k/3}\), using \(\beta k\gg\log n\), leaves a set of
\(P\)-mass at most \(e^{-\beta k/2}\). Off that set, averaging the
common-noise coupling gives local differences bounded by
\(e^{-T'/2}\operatorname{Lip}(\phi)\) times distance, plus an
arbitrarily small inverse-polynomial error. Transfer the exceptional
starting set to \(G\) using \eqref{prof:eq-likelihood-lower-tail}.
For independent \(g_0,g_1\sim G\), use a polynomial mesh on
\(\cos\theta\,g_0+\sin\theta\,g_1\), \(0\le\theta\le\pi/2\).
Every vertex has law \(G\), and Gaussian norm tails put adjacent
vertices within the local ball. A union bound over the vertices
leaves Gaussian exceptional probability
\(\operatorname{poly}(n)e^{-c\beta k}\).
The local estimates telescope along this polygon. To reweight its
endpoints to two independent \(P\)-points, Cauchy--Schwarz and
\eqref{prof:eq20} bound the exceptional probability by
\[
e^{\alpha k}
\bigl(\operatorname{poly}(n)e^{-c\beta k}\bigr)^{1/2}.
\]
It is smaller than any required inverse polynomial when
\(\alpha/\beta\) is sufficiently small and \(\beta k/\log n\)
is sufficiently large. Taking \(C_D\) large gives
the desired residual variance bound. Absorb the remaining errors in
the slack between \(2+o(1)\) and \(3\).
\end{proof}
\subsection{Backward amplification to logarithmic observation scale}
\label{prof:sec-amplification}
We use two positive-time inputs: for every compact
\([s_1,T]\subset(0,\infty)\), posterior covariance is bounded by a
constant except at integrated planted probability \(e^{-c(s_1,T)n}\);
and at a sufficiently large fixed \(T\), the posterior unscaled gap is
bounded below with the same type of exception. The critical endpoint
verification of these inputs is given in the positive-time argument.
Their use here is only with the rotation-invariant restriction
\(\mathcal G\), and only on fixed positive intervals.
\begin{proposition}[Amplified covariance estimate]
\label{prof:prop-amplified-covariance}
For every sufficiently small fixed \(\epsilon>0\), there are
\(s_0,a>0\) and \(K_{\log}<\infty\) such that, at every deterministic
time in the indicated range,
\begin{equation}
\|\Sigma_s\|\le\frac{1+\epsilon}{s},
\qquad 0<s\le s_0,\qquad ns^{3/2}\ge K_{\log}\log n,
\label{prof:eq22}
\end{equation}
outside an integrated planted exceptional set, intersected with
\(\mathcal G\), of mass at most \(e^{-ans^{3/2}}\).
For fixed \(1/2<\gamma<\gamma'<1\), the same exceptional set may be
used for
\begin{equation}
\begin{split}
\|A_0\Sigma_sA_0\|&\le r^{-\gamma},\qquad A_0=2I-J,\\
v_s&:=\sup_{\substack{H^T=-H\\ \|H\|_{\mathrm{HS}}\le1}}
\operatorname{Var}_{\mu_{h_s}}
\bigl((X-m_s^{\mathrm{post}})^T[H,J](X-m_s^{\mathrm{post}})\bigr)
\le C_v s^{-1-\gamma'/2}.
\end{split}
\label{prof:eq23}
\end{equation}
\end{proposition}
\begin{proof}
Each backward step first uses later observations to obtain coarse
covariance and moment bounds. These control derivatives along rotation
orbits, where the static majority estimates reset covariance to its sharp
threshold. The initial positive-time band uses the same reset; its coarse
bounds come from the separate fixed-positive-time and terminal inputs.
We carry all three estimates together. First fix
\(0<\iota<\gamma'/2\), with the target covariance slack
\(\epsilon\) small enough for the variance calculation below.
That calculation gives a constant \(C_\iota'\) independent of
\(L_0\) and of the relative step size. Choose \(L_0\) large enough
that \(C_\iota'L_0^{\iota-\gamma'/2}<1/4\), and then choose
\(s_0\) small enough for the static medians and
Lemma~\ref{prof:lem-compact-bridge}.
The quadratic-variance constant
\(C_v\) and the auxiliary fourth-moment constants below may depend on
these choices, but will not depend on the eventual relative step size.
The initial band is \([s_0/L_0,s_0]\). Let \(F_b^+\) denote the
positive-time covariance failure, including the terminal gap failure
when \(b=T\), and define on this band
\[
Q_s^+=\mathbf1_{F_s^+}+\Pr_s(F_T^+)
+\int_s^T(1+b^{-C})\Pr_s(F_b^+)\,db.
\]
Here and below \(\Pr_s\) conditions on \((J,h_s)\), and the fixed
\(C\) dominates the weights in the moment transfers. Tonelli and
Markov on the fixed interval \([s_0/L_0,T]\) give a fixed
\(c_{\rm init}>0\) such that \(Q_s^+\le e^{-c_{\rm init}n}\)
outside integrated mass \(e^{-c_{\rm init}n}\). On this one event,
the bounded-covariance martingale transfer to \(T\) and its gap
bound give all the initial quadratic and fourth-moment bounds proved
below. No conditional variance lemma is needed for this initial
transfer. Orientation concentration will reset the covariance
thresholds on the same band.
Let \(c_+>0\) be a positive-time failure rate on \([s_0,T]\),
and let \(a_{\rm stat}>0\) be smaller than the fixed root,
orbit, diagnostic, and spherical-width exclusion rates at the
accuracies used below. The compact bridge has its endpoint-independent
rate \(a_*\). Choose \(a>0\) once, with
\[
8a<\min\{a_{\rm stat},a_*,
c_{\rm init}/s_0^{3/2},c_+/s_0^{3/2},1\},
\]
and small enough for the final orbit-gradient condition whenever
its concentration exponent is at most \(2a\). That condition uses
only the moment constants and the already fixed \(s_0,L_0\).
\paragraph{A conditional event with room in its probability exponent.}
For \(s<s_0/L_0\), put \(u=(1+d)s\) and \(t=L_0s\), and suppose the
induction estimates hold at all required times at least \(u\).
The active \(d\) will be one of two fixed values specified below;
the rate \(a\) will not be decreased after those choices.
Let \(F_b\) be the union of the three inductive failures at
\(b\le s_0\), and the positive-time covariance failure at
\(s_0<b\le T\). Include the terminal gap failure in \(F_T\).
Define
\[
Q_s=\Pr_s(F_u)+\Pr_s(F_t)+\Pr_s(F_T)
+\int_u^T(1+b^{-C})\Pr_s(F_b)\,db.
\]
Choose the fixed exponent \(C\) large enough to dominate the
polynomial weights in the variance and fourth-moment estimates
below. All these weights are polynomial in \(n\) on the stated
time range. Tonelli's theorem and the deterministic-time induction
give
\[
\E_{\rm pl}[\mathbf1_{\mathcal G}Q_s]
\le \operatorname{poly}(n)e^{-ak(u)}
+\operatorname{poly}(n)e^{-c_+n}.
\]
Here \(\mathcal G\) is measurable at time \(s\); the equality
behind Tonelli therefore retains its indicator. Above \(s_0\) we
used the positive-time inputs on the fixed interval \([s_0,T]\).
Set \(\beta=c'da\), with a sufficiently small absolute \(c'>0\).
Markov at \(Q_s=e^{-\beta k}\) gives, before polynomial costs, the
exponent
\[
\bigl(a(1+d)^{3/2}-\beta\bigr)k.
\]
For small fixed \(d\) this strictly exceeds \(ak\).
The other Markov term is
\(\operatorname{poly}(n)e^{-c_+n+\beta k}\); the prior choice of
\(a\), and \(k\le ns_0^{3/2}\), also place it strictly above
the required rate. We next intersect this test with the likelihood
and diagnostic events needed for the derivative bounds.
For sufficiently small \(d\), the covariance hypothesis in
\eqref{prof:eq20} follows from \(Q_s\le e^{-\beta k}\).
The likelihood hypothesis
follows from the exact Gaussian identity
\[
\int e^{2L_P}dG
=\mathbb E_s\exp\{\tau(X^1-m_{\mathrm{sh}})^T
(X^2-m_{\mathrm{sh}})\}.
\]
The initial band is already fixed, so the back-step must be chosen
without shrinking that band again. Two likelihood estimates provide
this separation of choices: the spherical log-moment estimate near zero,
and the compact overlap bridge on the remaining interval.
There are two parameter ranges. At sufficiently small \(r\), use the
spherical shift \(m_{\mathrm{sh}}=m_o\). Choose
\(d_{\rm near}>0\) so that \(Cd_{\rm near}^2\) is a
sufficiently small multiple of \(d_{\rm near}a\). Apply the
off-diagonal replica log-moment bound~\eqref{prof:eq5} to this
fixed two-replica test, with requested failure rate \(4a\).
Given \(\varepsilon_{\rm like}>0\), it gives
\[
\log\int e^{2L_P}dG
\le Cd_{\rm near}^2k+\varepsilon_{\rm like}k+C_2\log n
\]
on a sufficiently small upper range. Choose
\(\varepsilon_{\rm like}\ll d_{\rm near}a\), then its upper
endpoint \(r_c>0\), and finally take \(K_{\log}\) large enough
that \(C_2\log n\ll d_{\rm near}ak\). Shrinking \(r_c\)
alone does not absorb this logarithmic term.
Only this selected test is needed, not simultaneous control for
all prospective step sizes. The width exclusion has the larger
rate fixed above. On the remaining compact
interval use \(m_{\mathrm{sh}}=m\) and
Lemma~\ref{prof:lem-compact-bridge}, with \(\chi\ll a\).
Choose a second step \(d_{\rm comp}>0\) on this compact interval.
The good-pair contribution to the second moment is at most
\(e^{d_{\rm comp}\chi k}\), while the exceptional-pair
contribution is at most
\[
e^{-cn+4d_{\rm comp}s n},
\]
using \(\|X-m\|\le2\sqrt n\). Choose \(d_{\rm comp}\) after
the compact-tail constant \(c\) so that this term is negligible.
Then \(\alpha\le 2d_{\rm comp}\chi\) for large \(n\), while
\(\beta=c'd_{\rm comp}a\). In either range,
\(\alpha/\beta\) in \eqref{prof:eq20} is as small as required.
Thus no choice of \(s_0\) depends on \(a\) or on these small step sizes.
Choose \(K_{\log}\) last to absorb the polynomial costs of
Lemma~\ref{prof:lem-conditional-variance}.
For each active step let \(\mathcal L_s\) be its likelihood event
just constructed, and let \(\mathcal D_s\) retain the diagnostics
and root or spherical-width events used in its orbit comparison.
The full event used from now on is
\[
\mathcal C_s=\{Q_s\le e^{-\beta k}\}
\cap\mathcal L_s\cap\mathcal D_s.
\]
The first two factors are tests of \((J,h_s)\), and are the only
ones used to apply the conditional variance lemma under \(\Pr_s\).
The diagnostic factor may also use the plant and root-orbit data;
it restricts the subsequent orbit integration without further
conditioning the posterior spin law. Put
\[
\Delta=\min_{d\in\{d_{\rm near},d_{\rm comp}\}}
\{a((1+d)^{3/2}-1)-c'da\}>0.
\]
The preceding estimates give
\[
\Pr_{\rm pl}(\mathcal G\cap\mathcal C_s^c)
\le\operatorname{poly}(n)e^{-(a+\Delta)k}
+\operatorname{poly}(n)e^{-4ak}.
\]
All assertions remain integrated with \(\mathcal G\) imposed;
none is a bound conditional on every spectrum. On the initial band
define \(\mathcal C_s\) instead as
\(\{Q_s^+\le e^{-c_{\rm init}n}\}\cap\mathcal D_s\),
whose excluded mass has a fixed rate larger than \(4a\).
\paragraph{Coarse covariance and a longer variance transfer.}
On \(\mathcal C_s\), total covariance gives
\[
\Sigma_s=\mathbb E_s\Sigma_u+
\operatorname{Cov}_s(m_u^{\mathrm{post}}).
\]
Apply \eqref{prof:eq21} to every linear projection of
\(m_u^{\mathrm{post}}\). Its derivative in \(g\) is
\(\sqrt\tau\,\Sigma_u\), so the second term is at most
\(3\tau\mathbb E_s\Sigma_u^2\), up to an arbitrarily small
inverse-polynomial operator error. Apply the same argument after
multiplication by \(A_0\). This gives both covariance bounds with
fixed coarse factors.
For any smooth polynomially bounded test of the later field,
\begin{equation}
\operatorname{Var}_s\bigl(\phi(h_t)\bigr)
\le C_\iota tL_0^{1+\iota}
\mathbb E_s\|\nabla\phi(h_t)\|^2+n^{-D}.
\label{prof:eq24}
\end{equation}
Here \(\iota>0\) is arbitrarily small if \(\epsilon\) is chosen
sufficiently small, and \(C_\iota\) is independent of \(L_0,d\).
To prove this, the posterior-expectation martingale on \([u,t]\) has
diffusion coefficient
\[
\nabla\mathbb E_b\phi(h_t)
=\mathbb E_b\nabla\phi(h_t)
+\operatorname{Cov}_b(X,\phi(h_t)).
\]
This follows directly from the Gaussian likelihood of the future
observation increment given \(X\).
The observations have unit
diffusion and posterior-mean drift in their own filtration. If
\(R_b=\mathbb E_s\operatorname{Var}_b(\phi(h_t))\), their martingale
variance identity and Cauchy--Schwarz yield
\[
-R_b'\le \frac{1+\iota}{b}R_b
+C_\iota\mathbb E_s\|\nabla\phi(h_t)\|^2
+\operatorname{poly}(n)\Pr_s(\text{failure at }b).
\]
In the covariance term use
\(\|\operatorname{Cov}_b(X,Y)\|^2\le
\|\Sigma_b\|\operatorname{Var}_bY\), and absorb the mixed term
by the slack in \(\iota\).
Integrating backwards from \(R_t=0\) gives explicitly
\[
R_u\le C_\iota t(t/u)^{1+\iota}
\E_s\|\nabla\phi(h_t)\|^2
+\int_u^t (b/u)^{1+\iota}
\operatorname{poly}(n)\Pr_s(F_b)\,db.
\]
The last term is negligible on the conditional event. For the first interval
\([s,u]\), apply \eqref{prof:eq21} to \(\mathbb E_u\phi(h_t)\)
and use the same gradient estimate. It adds at most a fixed multiple
of \(R_u+\tau\E_s\|\nabla\phi(h_t)\|^2\), since \(\tau/u\le d\).
As \(t/u\le L_0\), this proves \eqref{prof:eq24} with a constant
independent of \(L_0,d\).
When covariance is bounded on a fixed positive band beginning at \(s\),
this same martingale argument works directly from \(s\) up to \(T\),
with fixed constants in place of the displayed powers.
\paragraph{Quadratic and fourth moments.}
Let \(M=[H,J]=-[H,A_0]\), with \(\|H\|_{\mathrm{HS}}\le1\), and
put \(D_t=m_t^{\mathrm{post}}-m_s^{\mathrm{post}}\). Total covariance
gives \(\mathbb E_sD_tD_t^T\preceq\Sigma_s\). At time \(t\),
the centered quadratic from time \(s\) splits into a centered quadratic
there and the linear term
\(2D_t^TM(X-m_t^{\mathrm{post}})\), plus its conditional mean.
The averaged variance of this linear term is
\(O_{L_0}(s^{-1-\gamma/2})\). Indeed expand
\(M=-HA_0+A_0H\). On successful descendants the two terms obey
\[
\begin{split}
\E_s[\mathbf1_{\rm good}D_t^THA_0\Sigma_tA_0H^TD_t]
&\le Ct^{-\gamma/2}\operatorname{Tr}(H^T\Sigma_sH)
\le Ct^{-\gamma/2}s^{-1},\\
\E_s[\mathbf1_{\rm good}D_t^TA_0H\Sigma_tH^TA_0D_t]
&\le Ct^{-1}\operatorname{Tr}(H^TA_0\Sigma_sA_0H)
\le Ct^{-1}s^{-\gamma/2}.
\end{split}
\]
The norm bound at \(t\) is applied before averaging, and
\(\E_sD_tD_t^T\preceq\Sigma_s\) is then used in the trace.
No independence between \(D_t\) and \(\Sigma_t\) is needed.
Polynomially bounded errors on failed descendants are negligible.
The conditional mean of the whole quadratic is
\(\operatorname{Tr}(M\Sigma_t)+D_t^TMD_t\). As a function of the
field at \(t\), its gradient is its conditional covariance, as a spin
test, with \(X\). On successful descendants its squared norm is thus
at most \((1+\epsilon)/t\) times its conditional variance. Elsewhere
use the polynomial deterministic bound. Total variance,
\eqref{prof:eq24}, and splitting sums with a factor two give
\[
v_s\le C_\iota' L_0^{1+\iota}C_vt^{-1-\gamma'/2}
+C_{L_0}s^{-1-\gamma/2}+o(1).
\]
The coefficient of the first term relative to
\(C_vs^{-1-\gamma'/2}\) is
\(C_\iota'L_0^{\iota-\gamma'/2}<1/4\) by our earlier choice.
Choose \(C_v\) large. The strict
inequality \(\gamma<\gamma'\) absorbs the second term. This closes
the quadratic-variance induction.
On the initial band, apply the bounded-covariance version of the transfer
to \(T\). At \(T\), the gap input bounds the centered quadratic
variance because its single-site heat-bath energy is at most a constant
times
\(\operatorname{Tr}(M\Sigma_TM)+\|M\|_{\mathrm{HS}}^2\),
as follows by expanding a single spin flip. Coarse covariance on the
initial band comes directly from the positive-time input.
We also obtain uniformly in unit \(e\)
\[
\mathbb E_s|e^T(X-m_s^{\mathrm{post}})|^4\le Cr^{-4},\qquad
\mathbb E_s|e^TA_0(X-m_s^{\mathrm{post}})|^4\le Cr^{-2\gamma}.
\]
Integrate the posterior-mean martingale from \(u\) to \(T\), whose
diffusion coefficient is \(\Sigma_b\). The respective directional
quadratic variations have second moments bounded by the squares of
the integrals of \(Cb^{-2}\) and \(Cb^{-1-\gamma/2}\) on small
scales; use bounded covariance later. These estimates follow from
\(\|\Sigma_b\|^2\) and
\(\|\Sigma_bA_0\|^2\le
\|\Sigma_b\|\|A_0\Sigma_bA_0\|\), with deterministic polynomial
bounds on exceptions.
Martingale fourth-moment inequalities apply.
Terminal fourth moments follow by applying the gap first to a linear
test and then to its centered square. Finally apply
\eqref{prof:eq21} to a centered linear projection of
\(m_u^{\mathrm{post}}\) and to its square; their gradients involve
\(\Sigma_u\). This treats
\(m_u^{\mathrm{post}}-m_s^{\mathrm{post}}\) without constants
depending on \(d\). On the initial band start the mean martingale
directly at \(s\).
\paragraph{Restoring the sharp covariance thresholds.}
On each root orbit, monitor the two covariance norms and the two mean
deviations in the static comparison, in their own normalized units.
Use tolerance \(\epsilon/2\) for the sharp covariance median and
\(\epsilon\) for its failure threshold, with fixed slack also for
the other quantities. In particular the mean thresholds are
\(r^{-1-\eta}\) and \(r^{-\gamma/2}\), with an arbitrarily small
\(\eta>0\). Let a Lipschitz ramp be zero below the median thresholds
and one when any failure threshold is reached.
On the conditional
events already constructed, its gradient in the transition region is
bounded by
\[
D_{\mathrm{grad}}=Cr^{-1-\gamma''/2},\qquad
\gamma'<\gamma''<1.
\]
To check this, a root-preserving rotation has score \(z^TMz/2\), with
\(m\) fixed. Its centered quadratic variance is bounded by
\eqref{prof:eq23}. The additional linear term about
\(m_s^{\mathrm{post}}\) has variance at most
\(Cr^{-2-\gamma-2\eta}\), by the coarse covariance bounds and the
transition thresholds.
Choose \(\eta\) sufficiently small.
Cauchy--Schwarz with the fourth and second moments just proved then
controls derivatives of directional centered squares and means.
The derivative of \(A_0\) itself costs at most
\(Cr^{-1-\eta}\) in these units. Outside these conditional events
the ramp has a polynomial global gradient bound. This calculation
does not differentiate the root.
When \(K_{\log}\log n\le k<\log^{10}n\), use instead unconstrained
orbits that co-rotate the field and matrix, centered at \(m_o\).
The polylogarithmic edge comparison and the spherical shell estimate
give the required medians, after excluding the small-rate event on
which the spherical width squared differs from \(s\) by more than a
fixed relative tolerance \(\epsilon'\ll\epsilon\). Here is the
conversion from raw moments. In this band
\(r=n^{-1/3}\log^{O(1)}n\), so the unweighted error
\(n^{1/2+o(1)}\) in \Cref{prof:prop:polylog} is \(o(s^{-1})\).
On the width event, \eqref{prof:eq2} bounds the spherical raw
moment about \(m_o\) by
\((1+O(\epsilon')+o(1))s^{-1}I\). The cube covariance is bounded
by its raw moment about \(m_o\), giving the sharp covariance
median. The weighted raw moment is at most \(n^{1/6+o(1)}\),
which is \(o(r^{-\gamma})\) because \(\gamma/3>1/6\).
Jensen bounds the squares of the ordinary and weighted mean
deviations by their respective raw moment operators. These give
the two looser mean medians \(r^{-1-\eta}\) and
\(r^{-\gamma/2}\). All four statements concern a Haar majority
on the orbit that co-rotates this field.
The extra linear
score coefficient is
\[
Mm_o+Hh=(\lambda-J)Hm_o.
\]
Since \(\|m_o\|=O(\sqrt k/r)\), its standard deviation has an
additional multiplier at most \(C\sqrt k\) in the preceding bounds.
Differentiating the center costs the same allowance. Thus
\(D_{\mathrm{grad}}\) may be enlarged by a fixed power of \(\log n\)
in this range. The unconstrained partition-ratio comparison~\eqref{prof:eq4} transfers
the orbit weights.
The conditional derivative bound has exceptions whose masses depend
on the orbit. We need a concentration estimate that is linear in those
masses, so that it can be averaged over the orbit parameters. A Haar ramp zero on a
fixed majority, with gradient at most \(D_{\mathrm{grad}}\) off a set
of mass \(p\), and polynomially bounded gradient everywhere, has
plateau mass at most
\[
e^{-bk}+\operatorname{poly}(n)p,
\]
provided \(D_{\mathrm{grad}}\sqrt{bk/n}\) is sufficiently small.
Indeed split the upper ramp levels at tail probabilities successively
halving from \(1/2\) to \(e^{-bk}\).
If the plateau exceeds both
this last probability and a large polynomial times \(p\), truncate
and rescale the ramp across one such quantile interval of width \(w\)
and probability level \(p'\). The entropy of its square is at least
\(cp'\log(1/p')\), whereas its gradient-square integral is at most
\[
w^{-2}\bigl(2D_{\mathrm{grad}}^2p'
+\operatorname{poly}(n)p\bigr).
\]
The rotation-group heat log-Sobolev inequality gives
\(w\le CD_{\mathrm{grad}}/\sqrt{n\log(1/p')}\).
Strict supports and inclusive plateaux handle atoms or zero-width
intervals. Summing these widths bounds the full rise by
\(CD_{\mathrm{grad}}\sqrt{bk/n}\), a contradiction.
In particular the orbit-dependent exceptional masses enter linearly
and can be averaged without taking a fractional power.
Take \(b=a+\Delta/2<2a\). The gradient condition uses only
\(L_0,s_0,C_v\), the moment constants and \(\gamma''<1\), and
holds by the earlier choice of \(a\). It also
holds throughout the polylogarithmic band despite the extra logarithms.
On regular root anchors, the determinant-weight comparison at the
concentration radius costs
\[
O\bigl(\sqrt k+k/(nq^2)\bigr)=o(k),
\]
so it does not require shrinking \(r\) after the rate has been chosen.
We display the two measure changes. Let \(p(\omega)\) be the
orbit mass of \(\mathcal C_s^c\), including its likelihood and
diagnostic exclusions, and let \(h(\omega)\) be
its ramp plateau mass. The quantile estimate and
\(\int w_0\,d\pi\le n^C\) give
\[
\int w_0h\,d\pi
\le e^{-bk}\int w_0\,d\pi
+\operatorname{poly}(n)\int w_0p\,d\pi.
\]
The first comparison in \Cref{prof:orbits} bounds
\(\int w_0p\,d\pi\) by the counting derivative-failure mass
times \(e^{\xi k}\), plus larger-rate exclusions. Returning the
last display to the counting law by the reverse comparison costs
a second \(e^{\xi k}\). Thus the derivative term retains exponent
at least \(a+\Delta-2\xi\), and the direct plateau term retains
exponent at least \(b-\xi\), before polynomial factors. Choose
\(0<\xi<\Delta/8\). The excluded
diagnostic and base-orbit terms have already been assigned larger
rates. Finally the degree identity charges excess root multiplicity
to the excluded negative-Jacobian roots when returning from counts
to planted fields.
For unconstrained orbits choose partition-ratio accuracies relative to
the small-range step size, then enlarge \(K_{\log}\). Every reset
cost and discarded event therefore fits within \(e^{-ak}\).
The same reset works on the initial band with its fixed constants
and the larger initial-event rate already reserved; \(a\) is unchanged.
For clarity, the order of choices is: the target accuracies,
\(\iota<\gamma'/2\), and then large \(L_0\); small \(s_0\) for the static medians, the fixed outer cuts,
and the compact bridge; then \(C_v\) and all moment constants,
including those on the initial band; then \(a\); then the two
positive relative step sizes \(d_{\rm near},d_{\rm comp}\); and
finally \(K_{\log}\). Put
\(d_{\min}=\min(d_{\rm near},d_{\rm comp})>0\).
On successive backward bands with ratio \(1+d_{\min}\), every
active \(u=(1+d(s))s\) lies in an earlier band, as does \(t\).
Induction on these bands proves \eqref{prof:eq22}--\eqref{prof:eq23}
for each deterministic \(s\). Tonelli supplies the time integrals
used above from these pointwise statements; no union over
uncountably many observation times is taken. No positive-time input
is required uniformly down to an endpoint depending on \(a\).
\end{proof}
\subsection{Covariance below the logarithmic threshold}
\label{prof:sec-bottom-scale}
The backward induction stops at logarithmic \(k\). Below that threshold
we prove a separate, coarser covariance bound with an arbitrarily large
polynomial probability exponent. The required probability cannot be
obtained by merely repeating the second-moment comparison. The Haar Gram comparison and the mixed
cube--sphere tensor cancellation are closely related to
\citet[Section~3.3]{DuHuang2026CriticalFluctuations} and
\citet[Lemma~3.5]{DuHuang2026CriticalOverlap}. Here the full
finite-replica expansion keeps track of every normalization factor to
obtain the required probability exponent along the observation fields.
\begin{proposition}[Bottom-scale covariance]
\label{prof:prop-bottom-covariance}
For every fixed \(B>0\) and \(K_{\log}<\infty\), at each deterministic
time satisfying \(ns^{3/2}\le K_{\log}\log n\),
\begin{equation}
\|\Sigma_s\|\le Cn^{2/3}\ell^C,
\qquad \ell=\log\log(n+1000),
\label{prof:eq25}
\end{equation}
except on an integrated planted event, intersected with
\(\mathcal G\), of mass at most \(n^{-B}\).
\end{proposition}
\begin{proof}
Set \(L=n^{2/3}\). The spherical raw second moment about \(m_o\)
already satisfies the bound in \eqref{prof:eq25}. The hypotheses of the
polylogarithmic edge comparison, namely
\[
\lambda-\lambda^0\le n^{-2/3}\log^C n,
\qquad \|h\|\le\sqrt n\,r\log n,
\]
may be imposed with arbitrarily small inverse-polynomial failure by
testing the spherical observation at a larger polylogarithmic edge
width. We prove a higher-moment version of that comparison. Only
unweighted raw second moments are needed here.
On an unconstrained Haar orbit divide every integral by \(Z_o(h)\).
Let \(\Delta\) be either the signed cube-minus-sphere mass, or the signed
matrix integral of \((x-m_o)(x-m_o)^T/L\), with this common denominator.
For a fixed even integer \(p\), expand \(\|\Delta\|_{\mathrm{HS}}^p\)
(the mass case means \(|\Delta|^p\)). A subset \(B'\subseteq[p]\)
specifies which of the \(p\) draws have sign priors; the other draws have
spherical priors. The common kernel is respectively \(1\), or
\[
\prod_{a=1}^{p/2}
\left(\frac{(x_{2a-1}-m_o)^T(x_{2a}-m_o)}L\right)^2.
\]
The sign of that term is \((-1)^{p-|B'|}\).
\paragraph{Truncating the moment before annealing.}
In each term first restrict the internal sign overlaps
\(Q_{ij}=x_i^Tx_j/n\), \(i,j\in B'\), to
\(|Q_{ij}|\le\eta_p\), where \(\eta_p>0\) is fixed and small.
This avoids assuming an unrestricted high moment of the partition
function. Outside exponentially exceptional orientations the truncated
alternating sum differs negligibly from the desired power: the
unnormalized large-overlap sign-pair integral is exponentially small by
the global pair bound, while the total normalized cube mass is at most
\(e^{c_{\mathrm{tiny}}n}\) by its first moment. Choose
\(c_{\mathrm{tiny}}\) sufficiently small after \(p,\eta_p\).
The polynomial kernel costs can also be absorbed.
To discard the exceptional orientations inside an alternating sum,
we need a bound on each term separately. Let \(T_{B'}(O)\ge0\)
be the normalized, individually truncated term at orientation \(O\),
including its mass or product-of-squares kernel. We claim
\[
\E_{\rm Haar}T_{B'}^2\le e^{o(n)}.
\]
To verify this, bound the polynomial insertions crudely
and integrate out the spherical priors. Two groups of sign vectors
remain, with nearly identity internal Grams. If \(R\) is their matrix
of cross overlaps, the empirical-site-type entropy cost is at least
\[
\frac{\|R\|_{\mathrm{HS}}^2}{2(1+C_p\eta_p)}
+c_p\|R\|_{\mathrm{HS}}^4.
\]
Indeed let \(E\) be the first-group sign vector in the empirical
site-type law, and let \(b_j(E)\) be the conditional bias of the
\(j\)th sign in the second group. Then
\[
R_{ij}=\E[E_i b_j(E)],\qquad
\sum_j\E b_j(E)^2
\ge \frac{\|R\|_{\rm HS}^2}{1+C_p\eta_p},
\]
by projection onto the first-group coordinates and their Gram bound.
Conditional relative entropy dominates the sum of marginal
conditional entropies. The binary entropy inequality
\(I(b)\ge b^2/2+c b^4\), followed by Jensen and the fixed number
of coordinates, gives the displayed quadratic and quartic costs.
On the other hand, diagonalizing the
full Gram and applying the semicircle angular bound successively shows
that the angular energy gain above the independent baseline is at most
\(n/4\) times the squared Hilbert--Schmidt norm of the full off-diagonal
Gram, plus \(o(n)\). The cross block \(R\) occurs twice in that
norm, so the angular gain is bounded above by
\[
\frac n2\|R\|_{\rm HS}^2+O_p(n\eta_p^2)+o(n).
\]
The leading quadratic halves cancel against the entropy cost.
The remaining positive terms are
\(C_p n\eta_p\|R\|_{\rm HS}^2+O_p(n\eta_p^2)\), whereas the
quartic term is \(-c_p n\|R\|_{\rm HS}^4\). Thus cross overlaps outside any sufficiently small
fixed neighborhood of zero have a strict negative rate after choosing
\(\eta_p\) sufficiently small. Within that neighborhood the small-Gram
comparison and the spherical length-deficit tails give \(e^{o(n)}\).
This proves the second-moment bound. Cauchy--Schwarz applied to
each \(T_{B'}\) removes an exponentially exceptional orientation
set at cost \(e^{o(n)}e^{-cn/2}\). It may therefore be removed
before the signed sum, even though that sum need not be nonnegative
on the exceptional set.
The same estimates restrict each term's Haar integral, to arbitrarily
high inverse-polynomial accuracy, to
\begin{equation}
|Q_{ij}|\le (L/n)\log^{C_p}n\quad(i\ne j).
\label{prof:eq-bottom-window}
\end{equation}
For internal sign overlaps, use the small-Gram comparison until this
window and choose its logarithmic power large enough that the
length-deficit decay absorbs all losses. Once those overlaps are in the
window, all overlaps involving spherical priors have the same tails:
conditional on the internal sign Gram, the rotated prior is exactly the
spherical prior conditioned on that Gram. Polynomially fine boxes have
controlled angular variation on these nonsingular internal Grams.
The central cutoffs may be smooth. The large-overlap truncation has
therefore disappeared before the alternating cancellation below.
\paragraph{The finite-label expansion.}
The overlap window is now common to all terms. The only remaining
prior ratio is the internal sign Gram mass, in lattice density units,
divided by the spherical internal Gram density on the label set \(B'\).
We expand this ratio with every normalization factor included. Each
coefficient must have an explicit finite set of replica labels: the
alternating sum cancels exactly those terms that omit a label.
For \(B'\subseteq[p]\), put \(v=|B'|\), \(d_v=\binom v2\), and let
\(Q_{B'}\) be the Gram with diagonal one. Zero-dimensional densities
and empty products are one. Let \(\rho_{B'}\) be the preceding density
ratio. To every fixed expansion order it has the form
\begin{equation}
\rho_{B'}(Q,n)=
\exp\left\{n\sum_{b\ge4}F_{B',b}(Q)\right\}
\sum_{b,j\ge0}n^{-j/2}A_{B',b,j}(Q),
\qquad A_{B',0,0}=1.
\label{prof:eq-label-expansion}
\end{equation}
Here the displayed coefficients are homogeneous of degree \(b\) in the
off-diagonal overlaps.
Every degree-\(b\) rate coefficient is a sum of
terms involving at most \(6b\) replica labels; every amplitude coefficient
of degree \(b\) and order \(n^{-j/2}\) is such a sum involving at most
\(6(b+j)\) labels. A term involving a label set \(S\) means a universal
summand indexed by labels in \(S\), including all internally summed
indices, whose value is independent of whether further labels belong to
\(B'\). This qualification includes the normalized central density
prefactors, rather than merely the visible overlap monomial. On
\eqref{prof:eq-bottom-window}, the expansion and its exponential can be
truncated with uniform relative error \(O(n^{-A})\), for any prescribed
fixed \(A\).
We prove these assertions, starting with the normalizations. For one
sign site let
\(Y=(\varepsilon_a\varepsilon_b)_{a<b,\ a,b\in B'}\).
If \(\delta(S)\) denotes the cut vector of a subset of labels, then
\(Y=\mathbf1-2\delta(S)\). Its difference lattice is
\(\Lambda_v=\operatorname{span}_{\mathbb Z}\{Y-\mathbf1\}\).
Distinguish one label, denoted \(i_\bullet\) in this lattice calculation.
A basis consists of
\[
\{2\delta(\{i\}):i\ne i_\bullet\}
\ \cup\
\{4e_{ij}:i<j,\ i,j\ne i_\bullet\}.
\]
Indeed
\[
\delta(S)=\sum_{i\in S}\delta(\{i\})
-2\sum_{\{i,j\}\subseteq S}e_{ij},\qquad
\delta(\{i\})+\delta(\{j\})-\delta(\{i,j\})=2e_{ij}.
\]
Ordering the edges incident to \(i_\bullet\) first makes the basis block
triangular, so its covolume is
\[
c_v=2^{v-1}4^{\binom{v-1}2}=2^{(v-1)^2}\quad(v\ge1),
\qquad c_0=1.
\]
The sample overlap belongs to \(\mathbf1+n^{-1}\Lambda_v\), whose
cell volume is \(c_vn^{-d_v}\).
Thus the sign density in lattice units is
\[
f^S_{n,B'}(Q)=\frac{n^{d_v}}{c_v}
\Pr\left\{n^{-1}\sum_{i=1}^nY_i=(Q_{ab})_{a<b}\right\}.
\]
The reciprocal lattice, using the \(2\pi\) convention, is
\[
\Lambda_v^\vee=\frac\pi2
\left\{z\in\mathbb Z^{d_v}:
\sum_{b\ne a}z_{ab}\equiv0\pmod2
\text{ for every }a\right\}.
\]
Equality in the triangle inequality shows that the characteristic
function of \(Y\) has modulus one precisely on this reciprocal lattice.
The same is true under every sufficiently small real exponential tilt,
whose support is unchanged. There is consequently a single central
saddle on the reciprocal torus, and uniform contraction away from it.
Boundary representatives of that one saddle are identified on the torus.
For the spherical density, use independent Gaussian coordinates and
the one-site statistics
\[
\left((g_ag_b)_{a<b},\ ((g_a^2-1)/\sqrt2)_{a\in B'}\right).
\]
Their covariance is the identity in dimension \(d_v+v\); the sign
statistics also have identity covariance in dimension \(d_v\).
Denote the two log transforms by \(\Psi_G,\Psi_S\), and their Legendre
transforms by \(I_G,I_S\). At target \((Q_{ab},0)\),
\(I_G=-\tfrac12\log\det Q_{B'}\). If \(f^G\) is the full Gaussian
statistic density, conditioning on the zero normalized-length targets
gives exactly
\[
f^O_{n,B'}(Q)=f^G_{n,B'}(Q,0)/f^{\mathrm{len}}_{n,B'}(0).
\]
There is no extra Jacobian because both densities use the same normalized
length coordinates.
With \(z_n=n/2\), their exact length-only factor is
\[
f^{\mathrm{len}}_{n,B'}(0)
=\left(\frac n{2\pi}\right)^{v/2}b_n^v,
\qquad
b_n=\frac{\sqrt{2\pi}\,z_n^{z_n-1/2}e^{-z_n}}{\Gamma(z_n)},
\qquad \log b_n=-\frac1{6n}+O(n^{-3}).
\]
The saddle densities have forms
\[
\begin{split}
f^S_{n,B'}(Q)&=\left(\frac n{2\pi}\right)^{d_v/2}
e^{-nI_S(Q)}\mathcal A^S_{B'}(Q,n),\\
f^G_{n,B'}(Q,0)&=\left(\frac n{2\pi}\right)^{(d_v+v)/2}
e^{-nI_G(Q,0)}\mathcal A^G_{B'}(Q,n).
\end{split}
\]
Their amplitudes at zero target and leading order equal one.
Hence
\[
\rho_{B'}=
e^{n(I_G-I_S)}\,
\mathcal A^S_{B'}b_n^v/\mathcal A^G_{B'}.
\]
Every power of \(n/(2\pi)\), and the entire lattice covolume, have
cancelled before expanding. As a check, the exact spherical Gram density is
\[
\frac{\Gamma(n/2)^v}
{\pi^{d_v/2}\prod_{j=0}^{v-1}\Gamma((n-j)/2)}
(\det Q_{B'})^{(n-v-1)/2},
\]
whose central factor has the same leading value
\((n/(2\pi))^{d_v/2}\).
The sign transform through degree three is
\[
\Psi_S(t)=\frac12\sum_{a<b}t_{ab}^2
+\sum_{a<b<c}t_{ab}t_{ac}t_{bc}+O(|t|^4).
\]
Indeed the only nonzero third products are triangles, each with value
one. Its Legendre transform therefore has quadratic term
\(\tfrac12\sum Q_{ab}^2\) and cubic term
\(-\sum Q_{ab}Q_{ac}Q_{bc}\). These agree with the expansion of
\(-\tfrac12\log\det Q_{B'}\). Thus the rate difference starts in
degree four, as in \eqref{prof:eq-label-expansion}.
Here is the label count for all subsequent terms. Each statistic
coordinate carries at most two labels, including the single-label
Gaussian length coordinates. A zero-tilt cumulant tensor with \(M\)
coordinate slots therefore involves at most \(2M\) labels, and its
value is unaffected by additional replicas. Formal inversion of
\(\nabla\Psi(t)=Q\) produces rooted trees with nonlinear vertex valences
\(m\ge3\). A degree-\(b\) Legendre coefficient satisfies
\(\sum(m-2)=b-2\). Since \(m\le3(m-2)\), it uses at most
\(3(b-2)\) cumulant slots, hence at most \(6b\) labels. Setting the
Gaussian diagonal targets to zero preserves this bound.
For the central amplitude, perform saddle Fourier inversion on the
\(n^{-1/2}\) frequency scale against an identity-covariance Gaussian.
This expands the determinant instead of leaving an implicit
factor depending on dimension. A higher Fourier vertex of degree
\(m\ge3\) carries weight \(n^{-(m-2)/2}\); at order
\(n^{-j/2}\), their total Fourier-slot count is at most \(3j\).
A vertex from the perturbed Hessian has two Fourier slots and at least
one tilt slot. If the initial Taylor expansion has \(T\) tilt slots,
all its slots therefore number at most \(3j+3T\).
Substitute the
rooted-tree expansion for the saddle, with a total of \(b\) target
leaves. This adds at most \(3(b-T)\) slots. There are thus at most
\(3(j+b)\) slots, or \(6(j+b)\) labels. Gaussian pairings identify
slots and introduce no further labels; unused coordinates integrate
to one. This counts the determinant, inverse-Hessian contractions,
and every higher correction. Odd \(j\) vanish by Fourier parity,
although allowing them in the bookkeeping is harmless.
The factor \(b_n^v\) obeys the same rule by multiplying its
single-label expansions. Products and reciprocals of series with
leading constant one preserve the bound, because the weights \(b+j\)
add. This proves the asserted support rule even for coefficients
that depend on \(|B'|\) after their label sums are evaluated.
For fixed \(p\), the transforms are analytic near zero with uniformly
positive Hessian there. The saddles on
\eqref{prof:eq-bottom-window} lie in this neighborhood. Taylor
expansion under Gaussian damping on logarithmic frequency balls gives
any required inverse-polynomial relative error. The sign Fourier
tails are negligible by the reciprocal-lattice contraction just
proved; the Gaussian quadratic determinant formula controls its
remaining Fourier tails. This proves the uniform saddle remainders.
Moreover \(nQ^4=O(n^{-1/3}\log^{C_p}n)=o(1)\) on the window, so
the exponential in \eqref{prof:eq-label-expansion} can also be
expanded to arbitrary inverse-polynomial precision.
\paragraph{A common integration measure and high-order quadrature.}
Fix the orbit and let \(\mu_o\) be its normalized spherical Gibbs
law, equivalently the Gaussian law below conditioned on length.
Let \(K(x_1,\ldots,x_p)\) be either \(1\) or the product of
squares displayed above, and write
\(\mathcal U(x)=(x_a^Tx_b/L)_{a<b}\). Define the nonnegative
common kernel density \(G_p\) by
\[
\int_A G_p(U)\,dU
=\E_{\mu_o^{\otimes p}}
[K(x_1,\ldots,x_p)\mathbf1_{\{\mathcal U(x)\in A\}}]
\]
for Borel \(A\) in the interior Gram region. This definition
includes the common normalization \(Z_o(h)^{-p}\). We now bound
the density and its derivatives.
Use \(p\) independent Gaussian representatives
\(x_a=m_o+D^{1/2}\eta_a\), with \(D=(\lambda-J)^{-1}\), and impose
all their lengths exactly. Fourier-invert the length statistics
\((\|x_a\|^2-n)/L\) and overlaps \(U_{ab}=x_a^Tx_b/L\).
For a symmetric Fourier matrix \(T\), the tilted white covariance is
\((I-2iT\otimes D/L)^{-1}\), up to fixed conventions on off-diagonal
coefficients, and has norm at most one. The tilted mean is bounded
by \(C_p\|T\|\|D^{1/2}m_o\|/L\).
The polylogarithmic edge inputs give
\[
\|D/L\|+\operatorname{Tr}((D/L)^2)
+\|D^{1/2}m_o\|/L\le\log^C n,
\]
with logarithmically bounded reciprocals of the exact length densities
in these units. The spectral spread supplies arbitrarily many
covariance eigenvalues of inverse-polylogarithmic size. As a
nonzero \(T\) has an eigenvalue at least \(\|T\|/\sqrt p\), the
quadratic determinant is integrable against every fixed frequency
power with only a logarithmic loss; the noncentral characteristic
factor has modulus at most one.
The matrix-moment kernel is a product of squares of
\(\eta_a^T(D/L)\eta_b\) with \(a\ne b\). Its Wick expansion under
this complex tilt also costs only powers of \(\log n\) and \(T\).
A contraction loop of length at least two is bounded by powers of
\(\|D/L\|\) times \(\operatorname{Tr}((D/L)^2)\). A loop of length
one uses an off-diagonal replica block of the tilted covariance,
whose identity term vanishes; the resolvent identity supplies an
additional \(D/L\), giving the same bound with a frequency factor.
Open contractions end in the already bounded means. Differentiation
of the resulting density adds only frequency powers. Consequently
\(G_p(U)\) has all
fixed derivatives bounded by powers of \(\log n\); on the common
smooth window their \(L^1\) norms have the same bounds. This holds
for both the mass and matrix kernels.
The density and all required derivatives are now controlled. This is
what allows high-order quadrature on each subset's actual measure.
For the subset \(B'\), only its internal coordinates initially lie on
a lattice. In \(U\) units that lattice has covolume
\(c_v/L^{d_v}\) and a fixed-shape basis of size \(O_p(L^{-1})\).
Its mixed lattice--continuous integral is exactly the quadrature of
\(\rho_{B'}((L/n)U,n)G_p(U)\), with that cell-volume factor.
For every finite smooth expansion term, repeated quadrature after
the fixed basis change, or Poisson summation, gives error
\(O_{p,A}(L^{-A}\log^C n)\), uniformly over affine lattice shifts.
The relative expansion remainder also has a uniform absolute bound
on the common window. Indeed the rate transforms agree through
degree three, and their analytic amplitudes are one at zero, so
for fixed \(p\)
\[
\rho_{B'}(Q,n)
=\exp\{O_p(n\max_{a<b}|Q_{ab}|^4)\}
[1+O_p(\max_{a<b}|Q_{ab}|+n^{-1})]=1+o(1).
\]
Here \(\max|Q_{ab}|=O(n^{-1/3}\log^{C_p}n)\), so the
exponent tends to zero uniformly over all subsets \(B'\).
Thus a relative remainder \(O_{p,A}(n^{-A})\) is also an
absolute remainder of that order. It requires no derivatives, but
is still evaluated on its own mixed lattice--continuous measure.
Apply the same high-order quadrature to the smooth windowed
\(G_p\) itself. Its derivative bounds show that its mixed sum is
at most \(\log^C n+O_{p,A}(L^{-A}\log^C n)\), uniformly in the
affine shift, because its continuous mass has that bound.
Multiplying this mixed-sum bound by the uniform absolute remainder
controls the remainder on its actual measure. We may therefore convert all subset terms to one continuous
integral, with arbitrarily high inverse-polynomial error and with
all covolumes included. The different subset lattices need not have
matching supports before conversion.
\paragraph{Cancellation and the probability bound.}
Quadrature has put every subset term on one continuous integration
measure. We can now take the alternating sum without identifying the
original lattice supports. A surviving term must use every replica
label, and the coefficient-count rule turns that requirement into a
power of the dimension that grows with the moment order.
A term in the fully expanded expression has rate factors of degrees
\(b_1,\ldots,b_t\ge4\), and an amplitude of degree \(b\) and order
\(n^{-j/2}\). Put \(W=b_1+\cdots+b_t+b+j\).
Its coefficient involves at most \(6W\) labels. On the window,
\[
nQ^{b_i}=O(n^{-b_i/12}\log^{C_p}n),\qquad
n^{-j/2}Q^b=O(n^{-b/3-j/2}\log^{C_p}n),
\]
where the first estimate uses \(b_i/3-1\ge b_i/12\) for
\(b_i\ge4\). The whole term thus costs at least
\(n^{-W/12}\), up to logarithms. A diagram whose label set
\(S\) is a proper subset of \([p]\) cancels exactly, since
\[
\sum_{B':\,S\subseteq B'\subseteq[p]}(-1)^{p-|B'|}=0.
\]
Every surviving term has \(6W\ge p\), and is bounded by
\(n^{-p/72}\) before logarithmic losses.
Choose the expansion
order sufficiently large in terms of \(p\) and the desired error
power. The logarithmic kernel bounds and integration, followed by
an enlargement of the threshold in \(n\), give the safe absolute
bound \(n^{-cp}\), for example with \(c=1/200\).
We now return from the expansion to probability. The calculation bounds
the Haar expectation of the truncated alternating power.
The separate second-moment bounds remove the exponentially exceptional
orientations even though the truncated sum may be signed there. On their complement it agrees with the nonnegative
quantity \(\|\Delta\|_{\mathrm{HS}}^p\) up to negligible error.
For the mass choose threshold \(n^{-c/2}\); Markov then gives Haar
failure at most \(n^{-cp/2}\). For the matrix choose threshold one
in Hilbert--Schmidt norm; its Haar failure is at most \(n^{-cp}\).
The planted likelihood \(R=Z(h)/Z_o(h)\) has bounded second
moment by the two-replica comparison and pair tails.
Cauchy--Schwarz therefore leaves planted failure exponents at
least \(cp/4\) and \(cp/2\), respectively. A sufficiently large
fixed even \(p\) gives any requested \(B\), with room for the
earlier width exclusions.
Let \(M_{\rm cube}\) and \(M_{\rm sphere}\) be the normalized raw
second moments about \(m_o\). On the retained event,
\[
R=1+O(n^{-c/2}),\qquad
R M_{\rm cube}-M_{\rm sphere}=L\Delta_{\rm matrix},
\qquad \|\Delta_{\rm matrix}\|\le
\|\Delta_{\rm matrix}\|_{\rm HS}\le1.
\]
The spherical bound is
\(\|M_{\rm sphere}\|\le CL\ell^C\). Thus
\(\|M_{\rm cube}\|\le CL\ell^C\), and
\(\Sigma_s\preceq M_{\rm cube}\). This proves \eqref{prof:eq25}.
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-amplification.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-entropy.tex ====
\section{Entropy localization and the support profile}
\label{prof:entropy}
We now convert the posterior estimates into an inequality for every test
function of small support. All final typical-environment assertions concern
the original, unplanted disorder. The passage from the planted estimates to
these statements will be made before choosing a test function; this order is
essential for the uniformity of the resulting profile.
Write $\mu=\mu_J$ for the critical zero-field Gibbs measure. For a probability
measure $\pi$ on the cube, use the unscaled heat-bath form
\[
\mathcal D_\pi(f)
=\sum_{i=1}^n\pi\!\left[\operatorname{Var}_\pi
(f\mid X_{-i})\right].
\]
Thus $\mathcal D_J$ is the form at $\mu_J$, and $\mathcal D_b$ will denote the
same form for the posterior law at observation parameter $b$.
When used for heat-bath dynamics, this unscaled form corresponds
to rate one at each site. The observation parameter \(b\) instead
measures Gaussian information with the precision rate defined below.
We use the following static conclusions established above, with exactly their
stated probability qualifications: the sharp small-time covariance estimate
\eqref{prof:eq22}, the estimate below the logarithmic threshold
\eqref{prof:eq25}, and the positive-time assertions in
Section~\ref{prof:positive-inputs}. The latter include bounded covariance on
each compact positive-time interval, a terminal observation gap, the
stable-field implication \eqref{prof:eq26}, and the square-reweighting
estimates \eqref{prof:eq27} and \eqref{prof:eq27-coarse}.
In particular, the small-positive-time failure
rate for stability can be any prescribed fixed multiple of $n b^{3/2}$, and
the probability of an approximate zero with $q>u$ is at most
$\exp(-cnu^3)$ for \(C\sqrt b\le u\le u_*\), with a fixed
small \(u_*>0\). Constants and residual tolerances may
depend on a fixed compact positive-time interval. The proofs in
Section~\ref{prof:positive-inputs} supply these statements at the critical
parameter itself; no subcritical spectral-gap theorem is being applied at
its endpoint.
\subsection{The observation identities}
The use of Gaussian observations here is a stochastic localization argument
in the sense of Eldan~\cite{Eldan2013StochasticLocalization}.
The variance and entropy transfers are related to the localization methods
of Eldan, Koehler, and Zeitouni~\cite{EldanKoehlerZeitouni2022} and the
framework of Chen and Eldan~\cite{ChenEldan2025Localization}.
We give the finite-state identities and the estimates needed here explicitly.
Choose the sufficiently large fixed terminal time $T$ supplied by the static
inputs. On the spectral event $\|J\|\le3$, fix $c_+\ge3$ and put
$R=J+c_+I\succeq0$. Continue the scalar observations after $T$ for one unit
of time with precision $R$. Set $E=T+1$ and
\[
\Lambda_b=I\quad(0\le b\le T),\qquad
\Lambda_b=R\quad(T<b\le E).
\]
More explicitly, conditional on $X$, the field has independent increments
with drift $\Lambda_bX\,db$ and covariance $\Lambda_b\,db$. The accumulated
quadratic term in its likelihood removes the interaction by time $E$:
at $b=T+\lambda$, the off-diagonal interaction is $(1-\lambda)J^\circ$, and
$\mu_E$ is a product law.
For a real function $f$ with $\mu(f^2)=1$, let $\nu=f^2\mu$. Use $\mu_b$
and $\nu_b$ for the conditional spin laws given the observation history,
under the respective initial laws. For a fixed $J$, write
$\mathbb P_J^\pi$ for the observation-history law with initial spin law
$\pi$, and $\mathbb E^\pi$ for expectation under this conditional law.
Define
\[
\begin{gathered}
w_b=\mu_b(f^2),\qquad m_b^{\mathrm{post}}=\mu_b(X),\qquad
\Sigma_b=\operatorname{Cov}_{\mu_b}(X),\\
a_b=\nu_b(X)-\mu_b(X),\qquad
H_b=\mathbb E^\nu\operatorname{KL}(\nu_b\Vert\mu_b).
\end{gathered}
\]
Histories with $w_b=0$ have zero probability under $\nu$ and can be omitted.
The likelihood ratio of the two field-history laws is $w_b$, and
$\nu_b=f^2\mu_b/w_b$. Consequently
\[
H_b=\mathbb E^\mu\operatorname{Ent}_{\mu_b}(f^2),\qquad
\operatorname{KL}(\text{field law under }\nu
\Vert\text{field law under }\mu)=H_0-H_b.
\]
The Gaussian filtering identities give, at every time away from the
precision change and in integral form across it,
\begin{equation}
\begin{split}
H'_b&=-\frac12\mathbb E^\nu
\|\Lambda_b^{1/2}a_b\|^2,\\
a_b&=\mathbb E^\nu\left[
a_E+\int_b^E\Sigma_v\Lambda_v a_v\,dv
\,\middle|\,\mathcal F_b\right].
\end{split}
\label{prof:eq28}
\end{equation}
Here $\mathcal F_b$ is the observation history, including $J$. To verify
the first identity, the field drifts under the two laws differ by
$\Lambda_ba_b$; the expected logarithm of their likelihood ratio therefore
has derivative $\mathbb E^\nu\|\Lambda_b^{1/2}a_b\|^2/2$.
Subtract this from the preceding entropy decomposition. For the second
identity, the $\nu$-posterior mean is a martingale under $\nu$, whereas the
$\mu$-posterior mean has drift $\Sigma_b\Lambda_ba_b$ under that law.
Integrating their difference to $E$ proves the assertion. These arguments
also follow directly by It\^o's formula for the finite posterior sums.
Two further identities will be used repeatedly. Since $\mu_E$ is product,
binary Pinsker on each coordinate and entropy tensorization from below give
\begin{equation}
\mathbb E^\nu\|a_E\|^2\le2H_E.
\label{prof:product-mean}
\end{equation}
Also, conditioning additionally on the observations can only decrease an
averaged conditional variance. Applied separately with $X_{-i}$ fixed, this
gives
\begin{equation}
\mathbb E^\mu\mathcal D_b(f)\le\mathcal D_J(f),\qquad
\mathbb E^\nu\frac{\mathcal D_b(f)}{w_b}
=\mathbb E^\mu\mathcal D_b(f)\le\mathcal D_J(f).
\label{prof:energy-under-observation}
\end{equation}
\subsection{A profile with a bounded test likelihood}
\begin{proposition}[Restricted support profile]
\label{prof:restricted-profile}
For every fixed $\omega>0$ there are $c_\omega>0$ and disorder events
$\mathcal G_n(\omega)$ with probability tending to one such that, on
$\mathcal G_n(\omega)$, simultaneously for every real $f$ satisfying
\[
\mu(f^2)=1,\qquad
0<p:=\mu(f\ne0)\le\tfrac12,\qquad
\|f\|_\infty^2\le100p^{-2},
\]
one has
\begin{equation}
\mathcal D_J(f)\ge c_\omega n^{-2/3-\omega}\log(1/p).
\label{prof:eq29}
\end{equation}
\end{proposition}
\begin{proof}
Choose the covariance accuracy \(\epsilon>0\) sufficiently small
in terms of the requested \(\omega\), and obtain the corresponding
\(a,s_0,K_{\log}\) from \Cref{prof:prop-amplified-covariance}.
This same accuracy is used in all three entropy ranges below.
Put $h=\log(1/p)$. Entropy relative to the support and the upper bound on
$f^2$ imply
\begin{equation}
h\le H_0=\mu(f^2\log f^2)\le2h+\log100.
\label{prof:initial-entropy}
\end{equation}
On $\|J\|\le3$, every atom of $\mu$ is at least
$2^{-n}e^{-3n}$, so $h\le(3+\log2)n$. We treat three ranges of $h$.
\paragraph{Moderate entropy.}
For constants \(K'\) and \(c'\) fixed in the order below, first suppose
\[
K'\log n\le h\le c'n,
\qquad r_h=(h/n)^{1/3},\qquad t_h=r_h^2.
\]
The environment restrictions below are imposed on dyadic bins for $h$;
using an endpoint of a bin in place of $h$ only changes fixed constants.
Fix the low-likelihood exponent \(C_2>0\) and a desired negligible
error order \(n^{-M}\). Choose \(C_{\rm rate}\) with fixed slack
beyond the likelihood exponent \(4+C_2\) and the subsequent
Markov losses. The remaining constants will be chosen after the
following static consequences are stated.
For every required fixed rate $C_{\mathrm{rate}}$, the preceding static
estimates have the following consequences. All integrated probabilities
in this list use planted disorder and reference observations with
the spectral restriction imposed; the first also draws two
conditionally independent replicas from \(\mu_b\).
\begin{enumerate}
\item For $0\le b\le c_1t_h$, where $c_1\le1$, a reference posterior pair
has $|\langle X^1,X^2\rangle|\le C_1nr_h$, except with integrated
probability at most $e^{-C_{\mathrm{rate}}h}$.
Here $C_1$ need not increase
as $c_1$ decreases.
\item On each fixed-ratio window $[c_1t_h,L_1t_h]$, with
$0<c_1<L_1<\infty$, the replica log-moment estimate
\eqref{prof:eq5} holds with time $b$ and scale $nb^{3/2}$, with the required
small off-diagonal tilts and any prescribed fixed relative error, outside
integrated probability $e^{-C_{\mathrm{rate}}h}$.
\item By choosing $L_1$ large enough, the sharp bound
$\|\Sigma_b\|\le(1+\epsilon)/b$ on $[L_1t_h,s_0]$ has the same failure
rate. Beyond $s_0$, use the bounded positive-time covariance, including the
terminal extension. At $E$, the product field satisfies
$\max_i|(h_E)_i|\le C'\sqrt h$ with that rate, also when the exceptional
probability is weighted by $1+\max_i|(h_E)_i|$.
\end{enumerate}
For completeness, the first assertion does not need a lower spherical
width bound at very early times. Impose the upper width bound at scale
$t_h$ with an arbitrarily large fixed rate. The spherical length-deficit
estimate then controls the pair integral relative to $Z_o(h_b)^2$, using
\eqref{prof:eq6}, the field-norm bound, and the large-overlap comparison.
Exact shell conditioning costs only a polynomial by the spectral and
spherical estimates; translations improve the required small-ball bound.
The trace loss is estimated at shifts above the upper width bound.
Orientations with $Z(h_b)/Z_o(h_b)<e^{-C_{\mathrm{rate}}h}$ have at most
that planted mass conditional on the eigen-coordinates, so division by the
partition function is absorbed by an increase of the rate budget. The
second and third assertions follow from the spherical width tails,
\eqref{prof:eq5}, \eqref{prof:eq22}, and the positive-time inputs. For the
last field maximum, Gaussian tails and a union bound apply; under planting,
the $RX$ term has Gaussian site marginals with bounded translates. Gaussian
moment bounds give the assertion with the extra maximum-field factor.
Complete the fixed parameter choices before constructing the disorder
event. Obtain \(C_1\) at the chosen rate and then take \(c_1\) small enough
for \eqref{prof:early-entropy}; \(C_1\) does not increase under this choice.
Next enlarge \(L_1\). At the start of its sharp window the covariance
exponent is
\[
an(L_1t_h)^{3/2}=aL_1^{3/2}h,
\]
so \(L_1\) can pay the required rate. Then choose the fixed replica number
and its relative accuracy for the intermediate window. Finally decrease
\(c'\): a positive-time exception \(e^{-cn}\) has exponent at least
\((c/c')h\), and this choice also keeps \(L_1t_h\le s_0\) and all scales
in their allowed ranges. Increase \(K'\) to absorb polynomials and the
logarithmic threshold. We may require all discarded errors below to be
$O(n^{-M})$ for the fixed sufficiently large $M$ chosen above.
With these choices fixed, choose the disorder event on which the support
profile will hold. Entropy levels have been placed in dyadic bins; in each
bin integrate the conditional exceptions over its three time windows.
This construction uses the bin and fixed tolerances, before any test
function is chosen. For a field failure \(B_b\), write
\(q_J(b)=\mathbb P_J^\mu(B_b)\) for its reference-field probability
conditional on \(J\). For a pair failure, \(q_J(b)\) denotes the
conditional field average of its \(\mu_b^{\otimes2}\) probability.
The ordinary high-probability
lower bound $Z(0)\ge n^{-C}e^{n/4}$, together with
$\mathbb EZ(0)=e^{n/4}$, transfers planted exceptions on the spectral event
to ordinary exceptions at polynomial cost. Indeed, with
\(L(J)=Z(0)/\mathbb EZ(0)\),
\[
\E_{\rm ord}\!\left[\mathbf1_{\{\mathcal G,\,L\ge n^{-C}\}}
q_J(b)\right]
\le n^C\E_{\rm pl}[\mathbf1_{\mathcal G}q_J(b)].
\]
Integrate over each specified time window, apply Markov with unused rate
slack, and take a union over the $O(\log n)$ bins.
This produces one typical disorder event, independent of
$f$, on which all required conditional time-integral exceptions hold.
There is enough unused rate to absorb the later likelihood factors.
The chosen $K'$ makes their products with any specified polynomial in $n$
and $e^{C_0h}$ negligible, for each of the finitely many constants $C_0$
used below. Endpoint tests are treated in the same way.
On the early window, the pair bound implies
\begin{equation}
\mathbb E^\nu\|a_b\|^2\le Cnr_h
+\text{an error negligible in time integral}.
\label{prof:early-drift}
\end{equation}
Indeed the square of a posterior mean is bounded by the posterior
expectation of the absolute pair inner product. To apply this also to
$\nu_b^{\otimes2}$, discard $w_b<e^{-C_2h}$: its probability under $\nu$
is at most $e^{-C_2h}$, and its squared-drift contribution is at
most \(4ne^{-C_2h}\). On the complement, the density of the
\(\nu\)-field and two \(\nu_b\) replicas relative to the
reference field and two \(\mu_b\) replicas is exactly
\[
w_b\,\frac{f^2(X^1)}{w_b}\frac{f^2(X^2)}{w_b}
=\frac{f^2(X^1)f^2(X^2)}{w_b}
\le10^4e^{(4+C_2)h}.
\]
The ordinary posterior mean evaluated under the
$\nu$-field law is controlled in the same way. Taking $c_1>0$ sufficiently
small, \eqref{prof:eq28}, \eqref{prof:initial-entropy}, and
$nr_ht_h=h$ now give
\begin{equation}
H_{c_1t_h}\ge h/2.
\label{prof:early-entropy}
\end{equation}
On the intermediate window, use $l$ replicas in \eqref{prof:eq5} with
off-diagonal matrix entries $\delta/l$, where $\delta>0$ is sufficiently
small and independent of $l$. The entropy variational inequality applied
to $\nu_b^{\otimes l}$ gives a multiple of
$b\|\nu_b(X)-m_o\|^2$ bounded by a constant times
$\operatorname{KL}(\nu_b\Vert\mu_b)$ and the log-moment cost divided by
$\delta(l-1)$. Indeed the expectation of the tilt is exactly
$b\delta(l-1)\|\nu_b(X)-m_o\|^2/2$, whereas the relative entropy is
$l\operatorname{KL}(\nu_b\Vert\mu_b)$.
The squared Hilbert--Schmidt norm
of the tilt matrix is bounded by $\delta^2$, uniformly in $l$. Apply the
same inequality to ordinary replicas, whose entropy cost is zero, and use
$\|a_b\|^2\le2\|\nu_b(X)-m_o\|^2+2\|\mu_b(X)-m_o\|^2$.
After averaging over the $\nu$-field law, this yields
\begin{equation}
b\mathbb E^\nu\|a_b\|^2
\le C_\delta H_b+\zeta nb^{3/2}
+\text{negligible integrated error}.
\label{prof:intermediate-drift}
\end{equation}
Here $\zeta>0$ can be as small as desired by first taking a sufficiently
large fixed $l$ and then the relative log-moment error sufficiently small.
All exceptional field likelihoods are paid for by the same sup-norm bound.
Gronwall on this fixed-ratio window, followed by a sufficiently small
choice of $\zeta$, gives
\begin{equation}
H_{L_1t_h}\ge c h.
\label{prof:middle-entropy}
\end{equation}
The positive constant may depend on the fixed ratio $L_1/c_1$, which causes
no loss of a power of $n$.
For later times let $c_v$ be the good covariance bound:
$c_v=(1+\epsilon)/v$ up to $s_0$, and a fixed constant afterwards. Set
$M_v=\mathbb E^\nu\|\Lambda_v^{1/2}a_v\|^2$. Conditional Jensen in
\eqref{prof:eq28}, then Cauchy--Schwarz including its endpoint term, gives
\begin{equation}
\begin{split}
(\mathbb E^\nu\|a_b\|^2)^{1/2}
&\le \sqrt{2H_E}
+\int_b^E c_v\|\Lambda_v\|^{1/2}\sqrt{M_v}\,dv+n^{-M}\\
&\le \sqrt{2H_b}
\left(1+\int_b^E c_v^2\|\Lambda_v\|\,dv\right)^{1/2}
+n^{-M}.
\end{split}
\label{prof:backward-entropy}
\end{equation}
We used $2H_E+\int_b^E M_v\,dv=2H_b$.
The deterministic bounds
$\|\Sigma_v\|\le n$, $\|a_v\|\le2\sqrt n$ control the discarded
integrals at polynomial cost. Squaring introduces only another negligible
error, since all remaining factors are polynomially bounded on this range.
As $\Lambda_b=I$ below $s_0$, the coefficient in the resulting differential
inequality for $H_b$ is at most
\[
\frac{(1+\epsilon)^2}{b}+C\qquad(L_1t_h\le b\le s_0),
\]
and is bounded afterwards. Gronwall and \eqref{prof:middle-entropy} imply,
with $\kappa=(1+\epsilon)^2$,
\begin{equation}
H_E\ge c t_h^\kappa h-n^{-M+O(1)}.
\label{prof:entropy-at-product}
\end{equation}
For a Bernoulli law with smaller atom $z$, the elementary two-point
inequality is
\[
\operatorname{Ent}(u^2)
\le C\log(1/z)\operatorname{Var}(u).
\]
One may first replace $u$ by $|u|$. For comparable nonnegative values use
$\operatorname{Ent}(u^2)\le\operatorname{Var}(u^2)/\mathbb E u^2$;
if the values are not comparable, bound the entropy by the larger value
squared times binary entropy and compare the difference of the values.
For a spin of field $v$, $\log(1/z)\le C(1+|v|)$. Tensorizing at the
product endpoint and using \eqref{prof:energy-under-observation} therefore
gives
\begin{equation}
H_E\le C\mathbb E^\mu\left[
(1+\max_i|(h_E)_i|)\mathcal D_E(f)\right]
\le C'\sqrt h\,\mathcal D_J(f)+n^{-M}.
\label{prof:product-entropy}
\end{equation}
To justify the correlated exceptional term, put
\(B_E=\{\max_i|(h_E)_i|>C'\sqrt h\}\). Pointwise,
\(\mathcal D_E(f)\le n\mu_E(f^2)=nw_E\le n\|f\|_\infty^2\).
Hence its contribution is at most
\begin{equation}\label{prof:product-exception}
n\|f\|_\infty^2\,
\E^\mu\!\left[(1+\max_i|(h_E)_i|)\mathbf1_{B_E}\right].
\end{equation}
The weighted endpoint exception and the restricted supremum bound
make this negligible. On \(B_E^c\), pull out \(C'\sqrt h\) and
use \eqref{prof:energy-under-observation}.
Combining \eqref{prof:entropy-at-product} and
\eqref{prof:product-entropy} yields
\[
\mathcal D_J(f)\ge c n^{-2\kappa/3}h^{1/2+2\kappa/3}.
\]
At $\kappa=1$ the right side is $c n^{-2/3}h^{7/6}$.
The initial choice of \(\epsilon\) in terms of \(\omega\) proves
\eqref{prof:eq29} in this range, since $h\ge1$ and the extra exponent of
$n$ can be less than $\omega$.
\paragraph{Small entropy.}
In this range the test likelihood is only polynomially large, so the
bottom-scale estimate can absorb its exceptional contribution. We use
variance rather than entropy; the accumulated covariance bound has the
leading exponent needed for the support profile.
For $h<K'\log n$, use
$V_b=\mathbb E^\mu\operatorname{Var}_{\mu_b}(f)$ up to $T$.
Cauchy--Schwarz on the support gives $V_0\ge1-p\ge1/2$.
Filtering and the covariance bound imply
\[
-V'_b=\mathbb E^\mu
\|\operatorname{Cov}_{\mu_b}(f,X)\|^2
\le c_bV_b+\text{negligible integrated error}.
\]
Here the exceptional errors can be made arbitrarily small polynomially:
the test bound is polynomial in $n$, \eqref{prof:eq25} has any required
inverse-polynomial failure rate, and \eqref{prof:eq22} is used only above
a sufficiently large logarithmic threshold.
Specifically take
$b_*=(K''\log n/n)^{2/3}$, with $K''$ enlarged as necessary. Below $b_*$
use $c_b\le Cn^{2/3}(\log\log(n+1000))^C$ from
\eqref{prof:eq25}; its integral is $o(\log n)$. Between $b_*$ and $s_0$
use $(1+\epsilon)/b$, and then the bounded positive-time input. Thus
\[
\int_0^T c_b\,db
\le\tfrac23(1+\epsilon)\log n+o(\log n).
\]
The same transfer and time-integrated Markov argument supplies one typical
environment event for these assertions. Gronwall gives
$V_T\ge n^{-2/3-O(\epsilon)-o(1)}$. The terminal gap and
\eqref{prof:energy-under-observation} give
$V_T\le C\mathcal D_J(f)+n^{-M}$. The initial choice of $\epsilon$ leaves
enough power slack to absorb $h\le K'\log n$ and obtain
\eqref{prof:eq29}.
\paragraph{Entropy of order $n$.}
It remains to treat $h>c'n$. The maximum test likelihood no longer
controls the fixed positive-time exceptions. Instead, we charge those
exceptions to the relative entropy spent in changing the observation law.
The resulting differential inequality forces that spent entropy to
vanish at leading order. We show that, for
every fixed $C_*>0$, with probability tending to one no test function in
this range can have $\mathcal D_J(f)\le C_*n^{1/3}$.
Taking $C_*=1$ implies the desired estimate because
$h\le(3+\log2)n$.
The active class throughout this paragraph is still
\[
\mu(f^2)=1,\qquad p=\mu(f\ne0)\le\tfrac12,\qquad
\|f\|_\infty^2\le100p^{-2},\qquad h=\log(1/p)>c'n.
\]
We choose the diagnostics before selecting a violating function.
Take a countable list of compact positive-time bands, integer rate
multipliers \(A_*\), the fixed dyadic outer-threshold grids from
\Cref{prof:positive-inputs}, and residual and mean tolerances
tending to zero on each band. For each fixed finite collection,
the integrated planted bounds, the size-bias lower bound, and
Markov with unused rate slack give an ordinary disorder event of
probability tending to one. Intersect it with the finite ordinary word
diagnostics, the original spectral restriction used to define \(R\), and
the ordinary disorder event in \Cref{prof:terminal-extension}. This
intersection still has ordinary probability tending to one, and all its
restrictions are independent of \(f\). On this event the corresponding
conditional reference-field bounds hold outside time sets whose
Lebesgue measure tends to zero.
If the desired conclusion failed with probability bounded away
from zero, diagonal selection would therefore give sizes tending
to infinity, environments satisfying successively more of these
finite collections, and a violating restricted function on each.
All diagnostics in this selection precede that function. Work
along such a sequence and define
\[
e_n(b)=\frac{H_0-H_b}{n},\qquad 0\le b\le T.
\]
These functions vanish at zero, are nondecreasing, and are $2$-Lipschitz by
\eqref{prof:eq28} and $\|a_b\|\le2\sqrt n$. Passing to a subsequence,
assume $e_n\to e$ uniformly. We prove $e=0$.
For the fixed environment \(J\) and \(b\in[0,T]\), let
\(B\in\mathcal F_b\) be an event of the observation history through \(b\)
with \(\mathbb P_J^\mu(B)\le e^{-nw}\), where \(w>0\). The relative
entropy of the restrictions of the two field laws to \(\mathcal F_b\),
conditional on this same \(J\), followed by binary relative entropy, gives
\begin{equation}
\mathbb P_J^\nu(B)
\le\frac{e_n(b)+n^{-1}\log2}{w}.
\label{prof:binary-transfer}
\end{equation}
Indeed this restricted field relative entropy is $ne_n(b)$, and its projection to the
indicator of $B$ is at least $\mathbb P_J^\nu(B)nw-\log2$.
At small fixed positive times apply \eqref{prof:eq26} with any fixed large
rate multiplier $A_*$. Its bad event has $w=A_*b^{3/2}$ in
\eqref{prof:binary-transfer}, after using rate slack in the environment
transfer. On its complement, \eqref{prof:eq26} implies the
whole-region stability at \(A=V_y\) required by
\eqref{prof:eq27}; that mean estimate uses no overlap lower bound.
It makes
$\mathbb E^\nu[\|a_b\|^2/n]$ arbitrarily small as $n\to\infty$ and
then its tolerance tends to zero. To see the error scale explicitly, the
additive energy error in that estimate has expectation at most
\[
C_\zeta n^{-1/2}(1+\mathcal D_J(f))=O_{\zeta,C_*}(n^{-1/6})
\]
by \eqref{prof:energy-under-observation}. Since
$\|a_b\|/\sqrt n\le2$, this error is charged linearly when estimating
the squared drift.
On the bad event use the approximate-zero estimate
\eqref{prof:eq27-coarse}.
Write $Q_b$ for the supremum of $q(y)$ over the sufficiently-small-residual
points occurring there. It yields, in squared units, an upper bound
$CQ_b$ plus the same vanishing error. For
\(C\sqrt b\le u\le u_*\), the outer estimate supplies
\(\mathbb P_J^\mu(Q_b>u)\le e^{-cnu^3}\).
On a compact positive-time band, and with finitely many thresholds, one
may use a single residual tolerance smaller than all those provided:
an approximate zero at this tolerance remains one at each larger tolerance.
Put \(R_b=C'A_*^{1/3}\sqrt b\), taking \(C'\) large enough
for the outer range and the initial interval short enough that
\(R_b\le u_*\). If \(B_b\) is the stability failure, then
\(0\le Q_b\le1\) gives
\[
\begin{split}
\E^\nu[Q_b\mathbf1_{B_b}]
\le{}&R_b\mathbb P_J^\nu(B_b)
+\sum_{\substack{j\ge0\\2^jR_b<u_*}}
2^{j+1}R_b\,\mathbb P_J^\nu(Q_b>2^jR_b)\\
&+\mathbb P_J^\nu(Q_b>u_*).
\end{split}
\]
On a fixed compact band the moving thresholds are bracketed by
the finite fixed dyadic grid described in
\Cref{prof:positive-inputs}, losing at most a fixed factor in
their cubic rates. Binary transfer makes the first term at most
\(CR_be(b)/(A_*b^{3/2})\) in the limit, while the sum is at most
\[
Ce(b)\sum_{j\ge0}
\frac{2^jR_b}{(2^jR_b)^3}
\le Ce(b)/R_b^2.
\]
The final saturated tail uses the rate at \(u_*\) and contributes
at most \(Ce(b)/u_*^3\le C_{u_*}e(b)/R_b^2\).
Thus no outer rate beyond the supported small range is used.
Consequently, in integral form on every compact band in the initial
interval,
\begin{equation}
e'(b)\le C A_*^{-2/3}\frac{e(b)}{b}.
\label{prof:macroscopic-drift}
\end{equation}
The constant \(C\) is independent of \(A_*\). To see this,
first fix \(A_*\), then take \(n\to\infty\), and then send the
mean tolerance to zero. The stability margins and the refined
mean constants, which may depend on \(A_*\), disappear in this
order. The remaining coefficient comes from the absolute leading
constant \(2\) in \eqref{prof:eq27-coarse}, the absolute outer-tail
constants, and the fixed \(u_*\).
Choose $A_*$ so large that
$\theta=CA_*^{-2/3}<1$. For $0<a<b$ in this interval, Gronwall and
$e(a)\le2a$ imply
$e(b)\le2b^\theta a^{1-\theta}$; letting $a\downarrow0$ proves
$e(b)=0$. On the remaining positive times, the fixed-rate version of
\eqref{prof:eq26}, the good-field estimate \eqref{prof:eq27}, and
\eqref{prof:binary-transfer} give $e'\le Ce$ on each compact band.
Therefore $e=0$ through $T$.
The selection at the start of this paragraph licenses these
inequalities in integral form on every selected compact band.
The omitted time sets have vanishing measure, and the uniform
Lipschitz bound on \(e_n\) makes their contribution vanish.
Thus the limiting conclusion \(e=0\) applies to the selected
sequence of violating functions.
As $e(T)=0$, \eqref{prof:initial-entropy} gives $H_T\ge c'n-o(n)$.
During the terminal extension the bounded covariance input holds with
exponentially small exceptions even for an arbitrary input spin law.
It therefore applies directly to the $\nu$-field law. The proof of
\eqref{prof:backward-entropy} on this fixed interval gives
$H_E\ge cH_T-o(n)\ge c''n$. At the product endpoint it is enough to use
$\max_i|(h_E)_i|\le C_B\sqrt n$, with exception at most $e^{-Bn}$ for any
fixed $B$. This follows from $\|R\|=O(1)$, $\|X\|=\sqrt n$, and Gaussian
tails.
Choose \(B\) to pay the restricted bound \(f^2\le100e^{2h}\).
The deterministic estimate in \eqref{prof:product-exception},
now with threshold \(C_B\sqrt n\), and the corresponding weighted
Gaussian tail control the correlated exceptional energy. The same product entropy estimate now
gives
\[
H_E\le C\sqrt n\,\mathcal D_J(f)+o(1)
\le CC_*n^{5/6}+o(1)=o(n),
\]
a contradiction. This completes all three ranges and the proposition.
\end{proof}
\Needspace{12\baselineskip}
\subsection{Removing the bound on the test function}
\begin{theorem}[Uniform support profile]
\label{prof:support-profile}
\label{prof:thm-profile}
For every fixed $\omega>0$ there are $c_\omega>0$ and disorder events of
probability tending to one on which, simultaneously for every nonzero real
function $f$ with $p=\mu_J(f\ne0)\le1/2$,
\begin{equation}
\mathcal D_J(f)
\ge c_\omega n^{-2/3-\omega}\log(1/p)\,\mu_J(f^2).
\label{prof:full-profile}
\end{equation}
In particular the spectral gap for rate-one-per-site heat-bath dynamics is
at least $c_\omega(\log2)n^{-2/3-\omega}$ on the same event.
\end{theorem}
\begin{proof}
The restricted profile holds for every admissible test on one disorder
event. We may therefore apply it to all dyadic pieces of an arbitrary
function on that same event. Pieces too small to meet the likelihood
bound will lose only a fixed fraction of the squared norm, and their
combined heat-bath energy is bounded by the original energy.
It suffices to consider $f\ge0$: replacing $f$ by $|f|$ preserves its
support and squared norm and cannot increase any conditional-variance
form. For every integer $j$ put
\[
g_j=\min\{(f-2^j)_+,2^j\},\qquad
A_j=\mu(g_j^2),\qquad p_j=\mu(f>2^j).
\]
For a scalar $u\ge0$ the same truncations satisfy
\begin{equation}
\frac{u^2}{4}\le
\sum_{j\in\mathbb Z}\min\{(u-2^j)_+,2^j\}^2
\le u^2.
\label{prof:dyadic-norms}
\end{equation}
For $2^m\le u\le2^{m+1}$ the sum is
$4^m/3+(u-2^m)^2$, which verifies both bounds; $u=0$ is immediate.
Thus $S:=\sum_jA_j$ is finite and lies between $\mu(f^2)/4$ and
$\mu(f^2)$.
The truncation increments also obey, for $u\ge v\ge0$,
\[
d_j:=\min\{(u-2^j)_+,2^j\}
-\min\{(v-2^j)_+,2^j\}\ge0,
\qquad \sum_jd_j=u-v.
\]
The sum identity follows by integrating the indicators of the disjoint
intervals $(2^j,2^{j+1})$. Hence $\sum_jd_j^2\le(u-v)^2$.
Using the representation of conditional variance by two independent
conditional samples gives
\begin{equation}
\sum_j\mathcal D_J(g_j)\le\mathcal D_J(f).
\label{prof:dyadic-energies}
\end{equation}
All sums are justified by nonnegativity and monotone convergence.
Discard indices with $A_j<4^jp_j^2/100$, as well as $p_j=0$.
On $\{f>2^j\}$ the preceding piece $g_{j-1}$ is saturated, so
$A_{j-1}\ge4^{j-1}p_j$. Since $p_j\le p\le1/2$, the discarded pieces
satisfy
\[
\sum_{j\,\mathrm{discarded}}A_j
\le\sum_j\frac{4^jp_j^2}{100}
\le\frac1{50}\sum_jA_{j-1}=\frac S{50}.
\]
Every remaining normalized piece $g_j/\sqrt{A_j}$ has squared supremum at
most $4^j/A_j\le100/p_j^2$. Proposition~\ref{prof:restricted-profile},
on its single event valid for all test functions, therefore gives
\[
\begin{split}
\mathcal D_J(f)
&\ge c_\omega n^{-2/3-\omega}
\sum_{j\,\mathrm{remaining}}A_j\log(1/p_j)\\
&\ge\frac{49}{200}c_\omega n^{-2/3-\omega}
\log(1/p)\,\mu(f^2).
\end{split}
\]
Renaming the constant proves \eqref{prof:full-profile}.
Finally, for an arbitrary real $u$ choose a $\mu$-median $m$, so that
$u_+=(u-m)_+$ and $u_-=(m-u)_+$ both have support mass at most $1/2$.
For each pair of values the squared increments of these two functions sum
to at most the squared increment of $u$. Thus the support profile gives
\[
\mathcal D_J(u)\ge\mathcal D_J(u_+)+\mathcal D_J(u_-)
\ge c_\omega n^{-2/3-\omega}\log2\,\mu((u-m)^2)
\ge c_\omega n^{-2/3-\omega}\log2\,
\operatorname{Var}_\mu(u).
\]
This is the asserted gap bound.
\end{proof}
The support profile and its Poincar\'e consequence complete the first
functional route. Section~\ref{sec:dynamics} converts this endpoint to
worst-start mixing in both time conventions.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-entropy.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-local.tex ====
\section{Comparison on root-preserving rotation orbits}\label{prof:local}
The next comparison transfers continuous-prior estimates to spins on
most orientations of each retained root orbit. Its essential quantitative
point is the signed cancellation between two spin and two Gaussian
replica integrals. An absolute bound on the matrix insertion would be too
large for the weighted covariance estimate.
The mixed cube--sphere tensor cancellation of
\citet[Lemma~3.5]{DuHuang2026CriticalOverlap} is the close predecessor of
this comparison. We retain the observation-dependent projection constraints
and a weighted signed matrix kernel throughout the calculation below.
\begin{proposition}[Covariance estimates on most root orientations]
\label{prof:median-covariance}
Let $\mathcal S$ be suitable base orbit data retained by the preceding
exclusions in \Cref{prof:orbits}. On $\mathcal S$, use the disintegration
$d\pi\,d\mathrm{Haar}$ before the determinant weight $w$.
Here $d\pi$ includes the scalar root factors and the imposed
rotation-invariant spectral restriction, $\mathrm{Haar}$ is normalized
measure on rotations fixing $\mathcal P$, and $w_0$ is the median
reference weight defined there.
Fix $1/2<\gamma<1$ and positive accuracies $\epsilon,\eta$, together
with the requested implied small constants below. Choose a sufficiently
small fixed rate $a_{\mathrm{orb}}>0$, then the upper bound on $r$
sufficiently small. At each scale $k=nr^3\ge\log^{10}n$, for sufficiently
large $n$, one may discard a further set $\mathcal E\subseteq\mathcal S$
satisfying
\[
\int_{\mathcal E}w_0\,d\pi\le Cn^C e^{-a_{\mathrm{orb}}k}.
\]
On each remaining orbit, a set of Haar orientations of mass $>3/4$
satisfies simultaneously
\begin{equation}\label{prof:eq13}
\begin{split}
\|\Sigma_s\|&\le (1+\epsilon)/s,\qquad
\|A_0 \Sigma_s A_0\|\ll r^{-\gamma},\\
\|m_s^{\mathrm{post}}-m\|&\ll r^{-1-\eta},\qquad
\|A_0(m_s^{\mathrm{post}}-m)\|\ll r^{-\gamma/2}.
\end{split}
\end{equation}
Here $m_s^{\mathrm{post}}$ and $\Sigma_s$ are the posterior mean and
covariance. Every implied small constant can be prescribed in advance by
shrinking the upper bound on $r$ and then taking $n$ sufficiently large.
The previously excluded unsuitable data retain their preceding
integrated bounds; $w_0$ is used only on $\mathcal S$. No bound uniform
conditional on each spectrum is asserted.
\end{proposition}
\begin{proof}
\par\smallskip\noindent\textit{Projection cuts and the three priors.}\enspace The plant is gauged to $\mathbf1$ as above. Let $\mathsf P$ be the
$n\times3$ matrix whose columns are the three orthogonalized profiles spanning
$\mathcal P$, normalized by $n^{-1}\mathsf P^T\mathsf P=I_3$.
Its first column is $m/\sqrt q$, and its second is the normalized vector
$\mathbf1-(M/q)m$; its third is the normalized residual of $y$ after
projection onto those first two columns. We write $\mathsf P_j$ for column
$j$, and empirical contractions of matrices always include the factor $n^{-1}$.
Put \(z=X-m,\ N=nq^3,\ L=\sqrt{n/q}\), and use the statistics
\[
a=\sqrt n\,\delta,\quad \delta=\langle \mathsf P^T z\rangle=Q \vartheta,\quad Q=\operatorname{diag}(\sqrt q,q,q^{3/2}),\quad W=\sqrt N\, \vartheta.
\]
Here \(\vartheta\) denotes the scaled projection coordinate. For a pair write \(t=\langle(z-\mathsf P\delta)(z'-\mathsf P\delta')\rangle,\ U=n t/L\), so \(t=qU/\sqrt N\). The three-column notation always uses empirical contractions over sites. The exact length constraint is \(\langle z^2\rangle=b-2\sqrt q\,\delta_1\). We tilt by the common energy \(z^T(J-b)z/2\); with \(p\) the product of independent spins of means \(m\) (translated by \(-m\)), this gives precisely the posterior by the root equation.
We cut \(p\) smoothly to \(|\vartheta|\le\epsilon_0\) (one below
\(\epsilon_0/2\)), \(\epsilon_0>0\) an arbitrarily small fixed constant.
Except at a small rate in the true counting law with good spectrum, the
cut omits negligible normalized posterior mass (\(n^{-D}\), \(D\) any
fixed constant). Indeed a posterior spin can replace the plant in tests
with the equivariant axes \(m,y-lm\) by uniqueness and
\eqref{prof:eq9}, so
\(|\langle m z\rangle|\ll q,\ |\langle(y-lm)z\rangle|\ll q^3\).
The overlaps of plant and replica concentrate about \(q\) to any fixed
tolerance times \(q\) by \eqref{prof:eq1}, the spherical observation
estimates, \eqref{prof:eq4}–\eqref{prof:eq6} and \(q/r\simeq1\).
Thus \(|\langle z\rangle|\) (overlap with the gauged plant minus \(M\))
is bounded by arbitrarily small fixed times \(q\), and the subtraction
\((M/q)\langle mz\rangle\) in the \(e\) axis is likewise small.
Orthogonalizing the cubic axis uses \(|\langle y-lm\rangle|=O(q^3)\)
and residual norm asymptotic, in empirical units up to constant factors,
to \(q^{3/2}\), by the clipped Gaussian moment diagnostics and the
tail-square bound. Here multi-root fields do not affect counting claims:
the count of positive-sign regular roots is at most one plus the number
of negative-sign roots by degree, whose expectation on good spectrum is
exponentially small. Null singular fields may be ignored. The orbit
weight comparisons thus allow us to discard further orbits at a small
integrated rate measured by \(w_0\,d\pi\), so that the cut loss is
negligible on most Haar orientations.
Use two continuous auxiliary priors, not necessarily of mass one.
We fix their length-density units here. Put
\[
E=\mathsf P/\sqrt n,\qquad
\mathscr L(z)=\|z\|^2+2m^Tz,\qquad
S_{\mathsf P}=E^TVE,\qquad \kappa_2=\langle v^2\rangle.
\]
For a Gaussian prior, the notation
\(\delta_{\rm D}(\mathscr L(z)-nb)\,dz\) means the coarea density
at the raw squared length \(nb\). Since this shell is
\(\|z+m\|^2=n\), it is surface measure divided by \(2\sqrt n\).
In particular, if \(T=(\mathscr L-nb)/L\), then
\(\delta_{\rm D}(\mathscr L-nb)=L^{-1}\delta_{\rm D}(T)\).
The spin prior already lies on this shell.
With \(\varphi_C\) the density of \(N(0,C)\), define the uncut
pre-energy measures
\[
\begin{split}
g_0(dz)&=\mathfrak c_{g_0}\mathfrak r_{g_0}(a)
\varphi_V(z)\delta_{\rm D}(\mathscr L(z)-nb)\,dz,\\
o(dz)&=\mathfrak c_o\mathfrak r_o(a)
\varphi_{bI}(z)\delta_{\rm D}(\mathscr L(z)-nb)\,dz,
\end{split}
\]
where
\[
\mathfrak r_{g_0}(a)=e^{qa_1^2/\kappa_2},\qquad
\mathfrak r_o(a)=
e^{qa_1^2/b^2-\frac12a^T(S_{\mathsf P}^{-1}-b^{-1}I)a}.
\]
The constants \(\mathfrak c_{g_0},\mathfrak c_o\) are fixed by
the zero-projection saddle normalization below. Multiplication by
the common energy \(e^{z^T(J-bI)z/2}\) changes the underlying
Gaussian precisions to \(K\) and \(K_o\), respectively.
For \(Z\sim N(0,C_o)\), \(C_o=K_o^{-1}\), put
\(T_o(Z)=(\mathscr L(Z)-nb)/L\). The reference mass used
throughout is projection-uncut but still shell-constrained:
\[
Z_*:=\int e^{z^T(J-bI)z/2}\,o(dz)
=\mathfrak c_o\det(bK_o)^{-1/2}\frac{f_{T_o}(0)}{L}.
\]
Thus its normalized physical law is exactly the density
disintegration of \(N(0,C_o)\) conditional on \(T_o=0\).
Both factors in the last formula are invariant under rotations
fixing \(\mathcal P\), so \(Z_*\) is an orbit invariant.
We cut \(g_0\) by the same projection cutoff as \(p\), and multiply
it there by a projection-only factor to form \(g\). All cuts below
are separable across replicas.
The three priors serve different parts of the comparison. The reference
prior has an orbit-invariant tilted mass; the variable-variance Gaussian
has the precision needed for covariance control. The projection
multiplier aligns its single-replica marginal with the spin marginal.
What remains is a pair-density calculation in which the signed sum
cancels common errors and errors additive in the replica types.
\par\smallskip\noindent\textit{Single-prior projection saddles.}\enspace
We begin by defining the single-prior saddle functions. Write
\(p_\ell=\mathsf P_{\ell\bullet}^T\in\mathbb R^3\) for the
profile row at site \(\ell\). The independent site laws are
\[
\begin{split}
\gamma_{p,\ell}&=\frac{1+m_\ell}{2}\delta_{1-m_\ell}
+\frac{1-m_\ell}{2}\delta_{-1-m_\ell},\\
\gamma_{g_0,\ell}&=N(0,v_\ell),\qquad
\gamma_{o,\ell}=N(0,b).
\end{split}
\]
Their constraint vectors and per-site targets are
\[
\begin{array}{c|c|c|c}
i&D_{i,\ell}(z)&c_i(\delta)&d_i=\dim D_{i,\ell}\\ \hline
p&p_\ell z&\delta&3\\
g_0,o&(p_\ell z,z^2)&(\delta,b-2\sqrt q\,\delta_1)&4.
\end{array}
\]
For a spin, \(z_\ell^2+2m_\ell z_\ell=v_\ell\) identically,
so its shell constraint is automatic; adding a square coordinate
would make the constraint covariance singular. For Gaussians the
square coordinate and its target are equivalent to the translated
length coordinate by a determinant-one triangular change.
Set
\[
\Lambda_i(\theta)=\frac1n\sum_{\ell=1}^n
\log\int e^{\theta^TD_{i,\ell}(z)}\,\gamma_{i,\ell}(dz).
\]
On the local interior branch let
\(\nabla\Lambda_i(\theta_i)=c_i(\delta)\) and
\(\mathsf H_i=\nabla^2\Lambda_i(\theta_i)\).
The branch and its positive Hessian are established on the small
projection cut below. With \(\mathfrak c_p=\mathfrak r_p=1\),
define
\[
f_i^a(a)=\mathfrak c_i\mathfrak r_i(a)\,
n^{3/2}(2\pi n)^{-d_i/2}(\det\mathsf H_i)^{-1/2}
e^{n[\Lambda_i(\theta_i)-\theta_i^Tc_i(\delta)]}.
\]
This is a smooth saddle function per ordinary \(d^3a\), not an
exact density for an atomic spin projection. The factor \(n^{3/2}\)
converts the three raw projection sums \(\sqrt n\,a\) to \(a\);
the Gaussian raw length-density coordinate keeps its original
units. Primitive lattice covolumes are excluded here and will
belong to the mixed counting measures.
At zero projection the constraint Hessians are
\(S_{\mathsf P}\), \(\operatorname{diag}(S_{\mathsf P},2\kappa_2)\),
and \(\operatorname{diag}(bI_3,2b^2)\). Hence the choices
\[
\mathfrak c_{g_0}=\sqrt{4\pi n\kappa_2},\qquad
\mathfrak c_o=\sqrt{4\pi n b^5/\det S_{\mathsf P}}
\]
make all three saddle functions agree at zero. These are saddle
normalizations, not divisions by the exact shell masses.
The additional multiplier defining \(g\) on its cut is
\(f_p^a/f_{g_0}^a\).
These constant normalizations cost only polynomial for the unconditioned product Gaussian density integrals. On the cut we have
\begin{equation}\label{prof:eq14}
\log(f_i^a/f_o^a)=O(|\vartheta|+N |\vartheta|^3),
\end{equation}
also with the corresponding Taylor bounds after up to two scaled derivatives (bounded derivatives for the prefactor log, and \(O(N |\vartheta|^{3-j})\) for the rate-part log).
We include details of the expansions and bounds; all constants may use the scalar diagnostics above. In particular \(S_{\mathsf P}=I-O(q)\); \(Q^{-1}S_{\mathsf P} Q\) and its inverse are bounded. At typical sites, meaning the clipped region of those diagnostics with cutoff exponent sufficiently tiny, the profiles \(\mathsf P\) have bounded empirical absolute moments through any prescribed fixed degree, and \(y\) such moments at scale \(\sqrt q\). Outside that region \(\langle y^2{\bf1}_{\rm tail}\rangle=O(q^4)\); the scaled profiles \(\mathsf P Q\) are bounded there by \(C |y|\).
These bounds follow directly from \eqref{prof:eq9}, \(l=1+O(q)\), and the residual cubic normalization just noted. Centered moments involving \(\mathsf P\) and exponentially damped site fluctuations of order through fixed \(j\ge2\) thus have tail bounds for their profile factors up to \(C_j(1+q^{4-3j/2})\).
For \(p\) solve the linear-projection saddle by tilting by \(\mathsf P\theta\), \(\theta=O_Q(|\vartheta|)\), where \(O_Q\) denotes a bound in \(Q^{-1}\) units. Indeed on that parameter ball the response beyond linear \(v\mathsf P\theta\) has empirical \(L^2\) size \(O(q^{3/2}|Q^{-1}\theta|^2)\). Its scaled derivatives of order \(j\) through fixed orders have the bound \(C_j q^{3/2}|Q^{-1}\theta|^{\max(2-j,0)}\). This follows by the tanh derivatives (second bounded by a constant times the absolute small argument on typical sites, using the moment bounds), and on the tail the derivatives are exponentially damped since the new field differs from \(y\) by at most a small fraction of \(|y|\).
Thus scaled smooth inversion applies at uniform small radii. Evaluating the Legendre exponent at \(\theta=S_{\mathsf P}^{-1}\delta\) instead of the minimizer costs \(O(q^3 |\vartheta|^4)\) per site, by the ordinary Hessian bounds. The density exponent per site, apart from the Gaussian projection term \(-\delta^T S_{\mathsf P}^{-1}\delta/2\), is
\begin{equation}\label{prof:eq15}
-\sqrt q\,\delta_1^3-\delta_1^4/4+O(q^3|\vartheta|^3).
\end{equation}
Indeed the third log moment term is \(-\langle mv (\mathsf P\theta)^3\rangle/3\), with \(111\) profile average \(\langle mv \mathsf P_1^3\rangle=3\sqrt q+O(q^{3/2})\), \(112\) average \(O(q)\) by \eqref{prof:eq9}.
The fourth term is \(-\langle v(1-3m^2)(\mathsf P\theta)^4\rangle/12\), with \(1111\) average \(3+O(q)\), \(1112\) average \(O(\sqrt q)\). The fifth coefficient has a factor \(O(m)\), and the sixth remainder is bounded with tail damping. All other scaled monomials thus cost \(O(q^3)\) by typical moments and the tail bounds. In the two leading powers \(\theta_1\) can be replaced by \(\delta_1\). The estimates retain their Taylor derivative orders: the nonlinear Legendre correction does also by scaled smooth inversion and the stated derivative bounds (the unscaled shift in the minimizing coefficient has norm \(O(q^{3/2}|\vartheta|^2)\) with the corresponding derivatives).
For \(g_0\), first optimize the projection coefficients in the
formal saddle and then optimize the length coefficient. The
Gaussian projection exponent is exact. At this canonical projection
saddle, the length target differs from the mean by
\(-2\sqrt q\,\delta_1-\langle(v\mathsf P S_{\mathsf P}^{-1}\delta)^2\rangle\)
per site. This is a saddle mean, rather than the trace of an
exactly projection-conditioned finite-dimensional Gaussian.
The length curvature in the partial log moment is
\(2\langle v^2\rangle+O(q|\vartheta|^2)\). Thus the additional
exponent is the negative square of that deficit divided by
\(4\langle v^2\rangle\), to \(O(q^3|\vartheta|^3)\); its leading
quadratic cancels the explicit rank factor.
Since \(\langle(v\mathsf P S_{\mathsf P}^{-1}\delta)^2\rangle=|\delta|^2+O(q^3|\vartheta|^2)\), this gives exactly \eqref{prof:eq15}. These expansions use only the analytic bounded-variance Gaussian length tilt, adjusting the projection tilt by the corresponding three-dimensional linear solve. Length coefficients are \(O(q |\vartheta|)\), and scaled Taylor derivative estimates are unchanged. The calculation for \(o\) replaces the variance profile by constant \(b\), giving the same terms. The Gaussian central prefactors for all three priors have uniformly bounded log derivatives in \(\vartheta\) by the nonsingular saddle Hessians and the profile bounds. This proves \eqref{prof:eq14}.
\par\smallskip\noindent\textit{The signed pair comparison.}\enspace
We next form the pair saddle functions on
\(|\vartheta|,|\vartheta'|\le\epsilon_0\) and
\(|t|/q\le\epsilon_1\), where \(\epsilon_1\) is fixed small
and \(\epsilon_0\) may be much smaller. For types
\(i,j\in\{p,g_0,o\}\), put
\[
\begin{split}
D_\ell^{ij}(z,z')&=(D_{i,\ell}(z),D_{j,\ell}(z')),\\
c^{ij}&=(c_i(\delta),c_j(\delta')),\\
h_\ell(z,z')&=(z-p_\ell^T\delta)(z'-p_\ell^T\delta'),
\end{split}
\]
so \(n^{-1}\sum_\ell h_\ell=t\) at the projection constraints.
Define
\[
\Lambda_{ij}(\theta,\beta)=\frac1n\sum_\ell
\log\iint e^{\theta^TD_\ell^{ij}+\beta h_\ell}
\,\gamma_{i,\ell}(dz)\gamma_{j,\ell}(dz').
\]
Keep the targets and the \(\delta\)'s inside \(h_\ell\) fixed
when differentiating. On the local interior branch let
\(\nabla_\theta\Lambda_{ij}(\theta_\beta,\beta)=c^{ij}\), and put
\[
F_{ij}(\beta)=
\Lambda_{ij}(\theta_\beta,\beta)-\theta_\beta^Tc^{ij}
-\bigl[\Lambda_{ij}(\theta_0,0)-\theta_0^Tc^{ij}\bigr].
\]
Thus \(F_{ij}(0)=0\); it is the partial infimum over the
single-constraint coefficients, minus its value at zero cross
tilt. It is a formal optimized log moment, not the conditional
log moment of an unsmoothed atomic event.
At the optimizing law write
\[
D_2=\langle\operatorname{Cov}(D_\ell^{ij},D_\ell^{ij})\rangle,
\quad v_\beta=\langle\operatorname{Cov}(D_\ell^{ij},h_\ell)\rangle,
\quad w_\beta=\langle\operatorname{Var}(h_\ell)\rangle.
\]
These are averages of sitewise covariances. Implicit differentiation
and the envelope identity give
\[
\theta_\beta'=-D_2^{-1}v_\beta,\qquad
F_{ij}'(\beta)=\langle\E_\beta h_\ell\rangle,\qquad
F_{ij}''(\beta)=w_\beta-v_\beta^TD_2^{-1}v_\beta.
\]
In particular the last expression is the variance after regressing
the cross statistic on the single constraints.
Let \(\beta_{ij}(t)\) solve \(F_{ij}'(\beta_{ij})=t\), with
\(t=LU/n\). The formal joint function in ordinary
\(d^3a\,d^3a'\,dU\), retaining every raw Gaussian length density, is
\[
\begin{split}
\widehat f_{ij}^{\,a,a',U}
={}&\mathfrak c_i\mathfrak c_j
\mathfrak r_i(a)\mathfrak r_j(a')\,
n^3L(2\pi n)^{-(d_i+d_j+1)/2}\\
&\times[\det D_2^{ij}(\beta_{ij})F_{ij}''(\beta_{ij})]^{-1/2}
e^{n[\Lambda_{ij}(\theta_{\beta_{ij}},\beta_{ij})
-\theta_{\beta_{ij}}^Tc^{ij}-\beta_{ij}t]}.
\end{split}
\]
The factor \(n^3L\) converts the six raw projection sums and the
raw cross sum \(nt=LU\). Dividing this formula by its two single
saddle functions, and converting \(U\) back to \(t\), gives
\[
\widehat h_{ij}^{\,t}(t\mid a,a')
=\sqrt{\frac n{2\pi}}
\left[\frac{\det D_2^{ij}(0)}
{\det D_2^{ij}(\beta_{ij})F_{ij}''(\beta_{ij})}\right]^{1/2}
e^{n[F_{ij}(\beta_{ij})-\beta_{ij}t]}.
\]
The determinant quotient uses each pair's own constraint dimension:
\(6,7,8\) for \(pp\), mixed pairs, and two Gaussian replicas.
The Gaussian shell units and all explicit rank factors cancel in
this quotient. Define the dimensionless formal ratio
\[
\mathcal R_{ij}=
\widehat h_{ij}^{\,t}/\widehat h_{oo}^{\,t}.
\]
It is equally the ratio in \(U\) units, since both densities then
acquire the factor \(L/n\). The full formal pair ratio is
\[
\frac{\widehat f_{ij}^{\,a,a',U}}
{\widehat f_{oo}^{\,a,a',U}}
=\frac{f_i^a(a)f_j^a(a')}{f_o^a(a)f_o^a(a')}\mathcal R_{ij}.
\]
This is an algebraic quotient of saddle formulas. The local-limit
argument below relates it to the actual mixed statistic measures.
After the multiplier defining \(g\), both effective single factors
are \(f_p^a\); the projection cut does not change the conditional
calculation.
The signed comparison and its verification below use the four ordered
pairs \(i,j\in\{p,g_0\}\). With signs \(s_p=1,s_{g_0}=-1\) and
\(E_*=1+|W|+|W'|+|U|\), the full signed pair-density comparison needed is
\begin{equation}\label{prof:eq16}
\frac{f_p^a(a)f_p^a(a')}{f_o^a(a)f_o^a(a')}
\sum_{i,j\in\{p,g_0\}}s_i s_j\,\mathcal R_{ij}
=
\frac{q}{\sqrt N}\,e^{c_2 q U^2}(c_1 U+c_3' U^3)
+O\!\left(\frac qN E_*^C e^{\zeta E_*^2}\right).
\end{equation}
Here cuts can multiply both sides; the coefficients are bounded and independent of the statistics. The constant \(\zeta\) can be made as small as needed by reducing the projection cut and the upper \(r\). Central-prefactor relative errors \(O(q/N)\) are allowed.
Here is a verification. Abbreviate \(F_{ij}\) by \(F\) for the
pair under discussion. At \(\beta=0\) the canonical laws are
independent single-replica laws. Write their site means and
variances as \(\lambda_i,\sigma_i\), where \(\sigma_i\) denotes
variance rather than its square root, and put
\(d_i^0=\lambda_i-\mathsf P\delta\), with the primed target for
the second replica.
Uniformly with scaled \(\vartheta\) derivatives,
\begin{equation}\label{prof:eq17}
\begin{split}
\|d_i^0-(v\mathsf P S_{\mathsf P}^{-1}\delta-\mathsf P\delta)\|_{2,\mathrm{emp}}
&\le C q^{3/2}|\vartheta|^2,\qquad
\|d_i^0\|_{2,\mathrm{emp}}\le C q^{3/2}|\vartheta|,\\
\|\sigma_i-\langle\sigma_i\rangle\|_{2,\mathrm{emp}}&\le Cq,\qquad
\langle\sigma_i\rangle=B(\vartheta)-\|d_i^0\|_{2,\mathrm{emp}}^2,\quad
B(\vartheta)=b-2\sqrt q\,\delta_1-|\delta|^2 .
\end{split}
\end{equation}
Indeed the common linear response and nonlinear bounds were just obtained (for Gaussian the shifted length coefficient changes the response only nonlinearly).
The variance assertions use tanh or Gaussian differentiation as above; the last identity uses the exact target, automatic also for spins. The first bounds hold in any needed fixed typical-site empirical moment; on the tail \(d_i^0\) and its scaled derivatives are bounded by \(C |y|\).
Consequently
\[
F'(0)=\langle d_i^0 d_j^0\rangle,\qquad
F''(0)=\langle\sigma_i\sigma_j\rangle+O(q^3(|\vartheta|+|\vartheta'|)^2).
\]
In the conditional curvature the centered-fluctuation product is orthogonal to the constraints, and the two linear remainder terms are regressed on them, proving the second formula, also to differentiated orders. For the size bounds with scaled projection derivatives used here, \(\sigma_i=v\) at zero, and \(\sigma_i\) and its scaled derivatives after empirical mean subtraction have \(L^2\) size \(O(q)\): the logistic first derivative of variance brings a factor of the magnetization, and higher Taylor terms in the site shift or the Gaussian length coefficient already pay order \(q\) (use the same parameter estimates as for the single saddles).
On the tail all changes of the site fields and their scaled derivatives cost \(O(1+|y|)\) besides bounded quadratic coefficients, against fluctuation moments damped exponentially. Constraint covariance matrices and their needed scaled derivatives at the independent saddles are bounded, with eigenvalues bounded below using just bounded-profile sites. Regression coefficients for the two linear remainders and their derivatives therefore use ordinary unscaled constraint units (profiles \(\mathsf P\), squares for Gaussian), with size \(O(q^{3/2})\) by \eqref{prof:eq17}.
Thus \(F'(0)\) to quadratic order, of size \(O(q^3(|\vartheta|+|\vartheta'|)^2)\), is common to all four pairs. The curvature discrepancy from \(B(\vartheta)B(\vartheta')\) and its derivatives are \(O(q^2)\); it is common at zero projections, and to linear order varies by an \emph{additive} sum of single-type terms. The third derivative is \(O(q)\) at \(\beta=0\), with \(O(q)\) projection derivatives. Indeed it is the third cumulant averaged over sites of the regression-adjusted cross observable.
At zero cross tilt the product of centered site fluctuations contributes the average product of individual third moments, \(O(q)\) with derivatives. The linear corrections have coefficients \(d_i^0,d_j^0\), and regression coefficients on site constraints \(D=(\mathsf P z,\mathsf P z',\text{applicable squares})\) of size \(O(q^{3/2}(|\vartheta|+|\vartheta'|))\). Their cumulant contributions and derivatives satisfy the stated bound by typical moments in \eqref{prof:eq17}; on tails they use at most exponentially damped polynomials in \(|y|+1\), of tail average \(O(q^3)\).
For clarity, fourth cross derivatives stay bounded up to \(|\beta|\le C\epsilon_1 q\). The constraint-coefficient shifts have first derivatives of norm \(O(q^{3/2}(|\vartheta|+|\vartheta'|)+|\beta|)\), second derivatives bounded. Indeed bootstrap their first derivatives at \(O(q)\), so shifts are \(O(q^2)\), giving only small parameter changes even in \(Q^{-1}\) units. The constraint Hessian \(D_2\) (averaged covariance of \(D\)) stays well-conditioned. Centered site fluctuations of \(z,z'\) remain exponentially damped in \(|y|\) on tails, with bounded means: for a coupled Gaussian integrate the small quadratic tilt exactly, and for spins the conditional odds or Gaussian-integrated odds still have fields \(y+O(1)+o(1)|y|\).
Thus the regression-adjusted observable \(H=(z-\mathsf P\delta)(z'-\mathsf P\delta')+\theta'_\beta D\), with \(\theta_\beta\) the optimizing single coefficients, has bounded empirical centered fourth moment, using the \(O(q)\) coefficient times the worst \(\mathsf P_3\) profile. Implicit differentiation then bounds \(\theta''_\beta\) by \(C\|\langle\kappa(D,H,H)\rangle\|\le C'\), closing the bootstrap (or use continuation at a larger threshold and small \(\epsilon_1\)); \(\kappa\) denotes joint cumulant.
The fourth derivative of \(F\) uses only the fourth cumulant of \(H\) and the constraint regression of its square. Similarly \(\partial_\beta D_2\) is bounded (using at most two constraint factors and one \(H\)), and \(\partial_\beta^2 D_2=O(1+q^{-1/2})\): with two \(H\)'s the worst four-profile tail costs \(Cq^2 q^{4-6}\), while the \(\theta_\beta''\) term costs a third constraint moment. At zero cross tilt \(\partial_\beta D_2\) varies with bounded scaled projection derivatives by the same estimates. Hessian lower bounds throughout use just typical sites; in particular \(F''\) stays bounded below. All these statements apply to mixed constraint dimensions without dummy square variables for spins.
The pair conditional exponent is therefore
\[
n\left[-\frac{(t-F'(0))^2}{2F''(0)}
+\frac{F'''(0)(t-F'(0))^3}{6F''(0)^3}+O((t-F'(0))^4)\right].
\]
The preceding calculations give the cross mean, curvature, and third
cumulant at the independent saddle, with their projection derivatives.
We now convert them into a conditional density. Both the prefactor and
the Legendre exponent must be retained, because each contributes to the
odd term in the signed comparison.
For the unweighted reference, the exact conditional law uses two
independent uniform directions in dimension \(n-3\). Writing
\(B=B(\vartheta)\), \(B'=B(\vartheta')\), its density is
\[
h_{oo}^{t,\mathrm{ex}}(t\mid a,a')
=\frac{c_{n-3}}{\sqrt{BB'}}
\left(1-\frac{t^2}{BB'}\right)^{(n-6)/2},
\qquad
c_d=\frac{\Gamma(d/2)}{\sqrt\pi\,\Gamma((d-1)/2)}
\]
on \(|t|<\sqrt{BB'}\). The formal \(\widehat h_{oo}^{\,t}\)
has the same power and replaces \(c_{n-3}\) by
\(\sqrt{n/(2\pi)}\); their ratio is \(1+O(1/n)\), a permitted
relative error. The reference Legendre rate exponent is
\(\frac n2\log(1-t^2/(BB'))=-nt^2/(2BB')+O(nt^4)\).
The remaining factor \((1-t^2/(BB'))^{-3}\) belongs to the
reference saddle prefactor. The pair conditional prefactor is
\((F''(\beta)\det D_2(\beta)/\det D_2(0))^{-1/2}\) at the dual
\(\beta\), while the reference prefactor is
\((BB')^{-1/2}\) up to \(O(t^2+1/n)\) relative error.
The curvature part at \(\beta=0\) thus has a common constant,
additive linear log terms of size
\(O(q^2(|\vartheta|+|\vartheta'|))\), and error
\(O(q^2(|\vartheta|+|\vartheta'|)^2)\). The log change at
\(\beta\) is a projection-independent linear term in \(t\),
up to \(O(qE_*^C/N)\), by the preceding derivative bounds and
dual inversion.
We organize the exponent difference by its dependence on the pair
types. Common terms and terms additive in one type cancel after the
signed sum. The remaining leading terms are odd in the normalized
overlap; quadratic products of the smaller terms fit the remainder.
In the exponent difference, the \(t^2\) terms give a common \(c_2 q U^2\) plus additive single-type terms \(O(q U^2 (|W|+|W'|)/\sqrt N)\) and error \(O(q E_*^C/N)\). The term linear in \(F'(0)\) is common up to that error, of size \(O(q E_*^3/\sqrt N)\); its square contribution to the rate is negligible. The cubic term at leading order is odd, of order \(qU^3/\sqrt N\), with coefficient at zero projections. All fourth-order remainders cost \(Cq E_*^C/N\). Indeed \(nt^2=q^{-1}U^2,\ nt^3=U^3/\sqrt N,\ nt^4=q U^4/N\).
In \(F''(0)-B B'\) the terms through projection degree one are just from the empirical covariance of the two site-variance profiles (both \(v\) at zero). Hence the inverse-curvature discrepancy has a common constant and single-replica-summed linear terms, with remainder \(O(q^2 E_*^2/N)\). In \(ntF'(0)\), the response product has a common bilinear part, leaving error \(O(q E_*^C/N)\) after multiplication. The log-prefactor shift uses \(\beta=(t-F'(0))/F''(0)+O((t-F'(0))^2)\); even its \(O((1+q^{-1/2})\beta^2)\) Taylor remainder fits.
Products of two of the small terms cost \(O(q E_*^C/N)\). Exponentiating (with the small Gaussian-growth envelope stated, using small cutoffs), the common and additive terms cancel from the signed sum, leaving just the claimed odd terms. Finally \eqref{prof:eq14} allows the outside projection factor to be replaced by one on those odd terms to the stated remainder. We check the remainder sizes term by term, including at the ends of
the cut window. Writing \(\mathcal W=|W|+|W'|\), the linear-mean term \(ntF'(0)/F''(0)\) has common leading polynomial of size \(C q |U|\mathcal W^2/\sqrt N\) and remainder \(C q |U|\mathcal W^3/N\), while \(nF'(0)^2\le C q^3\mathcal W^4/N\).
The inverse curvature difference from \(1/(B(\vartheta)B(\vartheta'))\) has quadratic-and-higher Taylor error \(C q^2\mathcal W^2/N\). The cubic coefficient discrepancy from its value at zero projections is \(Cq\mathcal W/\sqrt N\), and replacing \(t-F'(0)\) by \(t\) inside that cube also fits the stated error. Hence besides the common \(c_2 q U^2\) and projection-independent log-prefactor constant (common because both variances are \(v\) at zero), log contributions retained are of size \(C q E_*^3/\sqrt N\), common, single-type summed, or odd as indicated.
In exponentiating, both the leading polynomials in the log and the full log deviations (using \(n|t-F'(0)|^4\) for the Legendre remainder) have absolute upper bounds by constants plus \(Cq E_*^2\) on these cuts, before any outside projection log which adds \(O(|\vartheta|+|\vartheta'|+(|\vartheta|+|\vartheta'|)E_*^2)\). This yields the stated Gaussian-growth envelope even at window ends, and the common and single-type-summed terms indeed drop out after multiplying by the common projection weights and summing both signs.
The zero-projection coefficients and the exact common/additive
cancellation are also written explicitly in
Lemma~\ref{prof:surviving-coefficients}. This proves the smooth expansion
\eqref{prof:eq16}. Individually the full ratios have at most the same Gaussian-growth envelope, and differ from one on bounded normalized statistics by \(O(q)+o(1)\).
\par\smallskip\noindent\textit{Density inversion, tails, and the common angular kernel.}\enspace
The four pair laws have different statistic measures. We keep their
lattice coordinates and translated lengths exact, and smooth only the
other coordinates. Each remainder is estimated on its own measure.
The smooth leading terms are transferred to common volume before we
perform the signed cancellation.
\begin{itemize}
\item On the central region just used, one can evaluate orbit integrals as lattice sums and smooth densities with the saddle formulas, each to relative remainder \(O(q/N)\). Use lattice variables \(\sum x\) for each sign replica and \(\sum xx'\) for a pair of signs, on the translated lattice generated by differences of sign patterns. One can use \(a_1,a_3\) as the remaining continuous-approximation projection coordinates for sign replicas. Smooth all nonlattice variables at inverse arbitrarily high polynomial scale with compactly supported smooth kernels, \emph{except} exact Gaussian length constraints \(\sum(z^2+2mz)=nb\).
Changing an angular test by this smoothing costs an arbitrarily small inverse polynomial absolute error times the corresponding positive partition integrals (on slightly enlarged regions). Indeed projections and residual Gram factor parametrize the complement rotations, with Gram nonsingular here, so energies, pre-factors (in log), smooth cutoffs and the bounded-degree polynomial tests have polynomial variation. Only the Gaussian length itself needs remain imposed on the original site integral; inverse site variances need not be bounded.
Projection multipliers also have polynomial log derivatives. When lattice variables are present keep them exact in this procedure. For instance for two signs one may hold \((\sum x,\sum x',\sum xx')\) and smooth the two nonconstant projection coordinates for each sign; \(a_2,a'_2,U\) are recovered using those constant-coordinate sums, the other \(a\)'s and known site data. These are nonsingular affine or triangular coordinate changes (equivalently use the three raw statistics in site units together with the nonconstant profiles).
For a mixed pair the raw lattice variable is just \(\sum x\) for the sign, and one also smooths the nonlattice overlap. Deterministic statistic Jacobians are understood in saddle densities and in converting lattice mass units. The tilt optimization using projected deviations rather than raw cross products is an affine change at the fixed projection constraints, with the same partial Legendre rate.
\end{itemize}
For the density estimation, tilt to the canonical product saddle at the evaluation point; variation of tilting factors across the mollifier costs negligible relative error. Fourier inversion there gives the Gaussian prefactor, including lattice covolume, to relative \(O((1+q^{-2})/n)\). Indeed covariance in site-sum units (normalized by \(\sqrt n\)) is bounded positive definite. We can use the well-conditioned equivalent raw coordinates (including \(zz'\) or \(xx'\) rather than the projection-subtracted statistic).
Maximum centered site variance in any such direction is \(O(q^{-3})\), average third absolute and fourth moments \(O(1+q^{-1/2})\), \(O(1+q^{-2})\), by the bounds above. In the central Fourier expansion up to small times \(\sqrt N\) in normalized frequency units, each site factor stays close to one. The cubic log term integrates to zero at first order by oddness, and the fourth remainder and squared cubic term, with Gaussian damping, give the indicated error. One can evaluate at the exact mean here even with smoothing (or keep its transform, which changes central frequencies negligibly).
For more detail about Fourier tails, bounded typical sites damp through small constant frequencies in unnormalized (single-site) units by nonsingular covariance on that subset. Up to arbitrarily high polynomial frequencies thereafter there is exponential-in-size damping off the central region of the reciprocal-lattice fundamental cell. For a sign pair, phase comparisons on the four patterns test the two nonconstant profile combinations plus constants by single-spin flips (with both signs for the other spin).
Thus if nonconstant coefficients for either replica are not tiny, the nonlattice diagnostics give damping; with tiny coefficients on the bounded typical sites, only the reciprocal lattice of the constant three-pattern statistics can escape the test. More explicitly these statistics are the coefficients of \(x,x',xx'\), and equality of the four phases characterizes exactly that dual lattice. Typical-site pattern probabilities are bounded below. The single sign test works identically. For Gaussian variables on typical sites condition on any site signs first.
Quadratic Gaussian phases separated from zero give contraction irrespective of the linear part. With tiny quadratic coefficients, Gaussian linear phases separated from zero on a fixed fraction also give contraction; for a mixed sign pair one tests both sign choices to detect the cross coefficient. The Gaussian profile columns are well-conditioned on the bounded typical sites. Their conditional real means are bounded there. Thus failure of contraction from these tests requires all Gaussian-related coefficients tiny, when on this subset they only perturb the sign phase test by a small amount.
This argument can take successively small fixed cutoffs, or pass to sequences where contraction vanishes (quadratic phases must tend to zero, then Gaussian linears for each sign choice, then the empirical sign-phase dispersions). Beyond polynomial frequencies the mollifiers suffice; in unsmoothed length directions the Gaussian quadratic factors on typical sites themselves give a bound \((1+c|\omega_{\rm lengths}|^2)^{-c'n}\), also in the joint inversion. Here length direction means the single-site quadratic coefficients; the linear parts do not affect this bound.
Thus one is approximating ordinary joint constraint densities in the Gaussian length dimensions, and possibly smoothed/lattice joint masses elsewhere. Factoring a result into single prefactors and a conditional saddle expression \eqref{prof:eq16} just uses determinant Schur factorization and rate subtraction at the level of these formulas; no division by tiny unsmoothed atomic projection probabilities is required. These estimates justify inversions in the different constraint dimensions and also the denominator calculations for conditional densities.
\begin{itemize}
\item Overlap tails outside \(|t|\le\epsilon_1 q\), with the projection cuts, are negligible (\(e^{-cN}\) times polynomial) in each pair orbit integral relative to \(Z_*^2\), even for polynomial tests. First take \(\epsilon_1 q<|t|<c_*\) with \(c_*\) small fixed. Conditional density ratios can here be estimated as upper bounds on polynomially small boxes. At independent single canonical laws use exponential Markov on \((z-\mathsf P\delta)(z'-\mathsf P\delta')\) up to tilts \(C c_*\), without having to keep its constraint regressions optimized.
Its mean is \(O(q^3)\); unconditional curvature differs from \(B B'\) by \(O(q^2)\) by \eqref{prof:eq17}; third cumulant is \(O(q)\), fourth derivatives bounded. Indeed single saddle parameters remain fixed in this tilt and the damping on tails is preserved (no large new profile coefficients, \(|\mathsf P\delta|\le C\epsilon_0 |y|\) there, and site constants do not enter higher cumulants). Thus relative to the reference conditional overlap density the log cost, besides polynomial losses, is at most
\end{itemize}
\[
C n(q^3|t|+q^2t^2+q|t|^3+t^4).
\]
Imposing exact Gaussian length densities costs at most polynomial under these tilts by bounded typical-site Gaussian smoothing as in the inversion bound.
Projection marginals relative to reference cost at most \(\exp(C\epsilon_0^3 N)\) besides constants by \eqref{prof:eq14}, including our modification. Boxes can straddle lattice coordinates without needing any matching lower bound there for the numerator. In the physically tilted reference \(C_o\), the length deficit of one of \((z\pm z')/\sqrt2\) is at least \(n|t|/2\) from trace, on taking \(\epsilon_0\ll\epsilon_1\). This costs \(\exp(-c n |t|^3)\) by negative square tilt of strength small-times \(t^2\) (and conditioning at polynomial cost).
Indeed the covariance loses normalized trace at most \(C\sqrt{\xi}+o(q)\) at added precision \(\xi\ge0\) small by \eqref{prof:eq1} and finite-rank comparison; lengths have central denominators and upper density bounds polynomial under the tests by Gaussian Fourier damping. Equivalently exact shells here can use buffered radial conditioning on the spheres translated by \(m\).
The absorption is uniform down to the moving endpoint
\(|t|=\epsilon_1q\), since
\[
C\left(\frac{q^3}{t^2}+\frac{q^2}{|t|}+q+|t|\right)
\le C\left(\frac q{\epsilon_1^2}+\frac q{\epsilon_1}+q+c_*\right)
\]
is the scalar cost divided by \(n|t|^3\), while the projection cost
has relative size at most \(C(\epsilon_0/\epsilon_1)^3\).
Hold \(\epsilon_1\) fixed, choose \(c_*\) and then the upper bound
on \(q\) small, and finally choose \(\epsilon_0/\epsilon_1\) small.
The reference penalty then leaves a fixed multiple of \(n|t|^3\),
which is at least a fixed multiple of \(N\). Angular comparisons
over these boxes have polynomially controlled variation.
For larger \(|t|\), use directly the angular semicircle pair upper bound from \eqref{prof:eq6} in the complement of \(\mathcal P\), whose Gram eigenvalues here are \(1\pm t+o_{\rm small}(1)\), and ignore projected components at \(o_{\rm small}(n)\) cost. The log gain besides independent baselines is at most \(nt^2/2+o_{\rm small}(n)\). Indeed \(\log Z_*/n\ge1/4-1/2-o_{\rm small}(1)\) from its \(C_o\) formula by central length density, the semicircle log determinant upper bound and \eqref{prof:eq10}; finite-rank and prior normalization changes do not increase the log determinant by a nonvanishing order per spin.
The sign-sign prior pays the strict Cramér slack already used, with biased signs and translations changing bounds by \(o_{\rm small}(n)\). For Gaussian pairs, simultaneous length/overlap Chernoff gives at least the rate \(-\frac12\log(1-t^2)\) in the small-\(q\) limit, strictly larger than \(t^2/2\) throughout this range. For two Gaussian types this uses fixed tilts of the Gram matrix, with log moments tending to the iid unit-variance values (\(v\le1\)); for mixed types condition on the sign vector, of bounded entries after translation and near-unit empirical length, and tilt the one Gaussian length and its projection on that vector, with the same limiting rate.
This follows immediately from the normal moment formulas since \(\langle|v-1|\rangle=q\); the mixed vector entries are bounded. Interior finite tilts suffice including by approaching the endpoints, using compactness and only requesting strict slack. Length targets differ from 1 negligibly, and their exact Gaussian densities again cost polynomial under the tilts. Projection/pre-rank factors cost \(o_{\rm small}(n)\). These estimates also handle singular pair Grams directly, by splitting the fixed overlap range into sufficiently small fixed bins. This proves the tail assertion.
\begin{itemize}
\item Here is the accuracy of kernel integration and of converting lattice sums to ordinary integrals. Under the tilted reference, condition two \(C_o\)-normals independently on their length shells, whose density denominators in \(L\) units are bounded above and below as in \eqref{prof:eq2}. Conditional on projections \(a,a'\) before length conditioning, their white coordinates \(\eta_i\) have fixed projections described by standard-normal coordinates \(\alpha,\alpha'\) (exactly whitened \(a,a'\)), and fresh standard orthogonal complements.
In particular \(|W|\le C|\alpha|\), and raw projection covariance eigenvalues are bounded below since \(K_o\) has bounded norm. Lengths and \(U\) are linear-quadratic forms in the remaining normals. The quadratic parts in a Fourier direction of norm one have operator and Hilbert-Schmidt norms at most \(C/\sqrt N,C\), with spread giving determinant decay \((1+c |v_f|^2/N)^{-c'N}\) at frequency \(v_f\). Indeed they use \(C_o/L\) tensored with a two-by-two symmetric matrix (whose norm in such a direction is bounded below), up to the finite-rank compressions/subtractions, and \eqref{prof:eq10} applies.
The length linear part uses \(\|C_o^{1/2}m\|/L\le C\). Consequently central densities with any fixed derivatives in \(a,a',U\), including the projection density, are bounded by
\end{itemize}
\begin{equation}\label{prof:eq18}
C(1+|\alpha|+|\alpha'|+|U|)^C e^{-c(|\alpha|^2+|\alpha'|^2+U^2)}
\big|\det(\partial\alpha/\partial a)\big|^2 .
\end{equation}
To see the \(U\)-tail factor before inversion, tilt in \(U\) by parameter small-times the target \(U\); up to \(\epsilon_1\sqrt N\) the quadratic norm shift is small. Its centered determinant cost is \(\exp(C\,{\rm parameter}^2)\) (off-diagonal tilt), and the projected translations cost at most \(\exp(o_{\rm small}(1)(|\alpha|^2+|\alpha'|^2))\). The tilted spread is unchanged up to constants.
For the weighted matrix test, subtract $C_o$ times the mass before
forming the Hilbert--Schmidt square. The resulting insertion must be
averaged with its sign: its absolute value would lose the Gaussian
cancellation at the scale required here. Its insertion is
\[
H=(\eta_1^TD\eta_2)^2-\eta_1^TD^2\eta_1-
\eta_2^TD^2\eta_2+\operatorname{Tr}D^2,
\qquad D=C_o^{1/2}A_0^2C_o^{1/2},\quad \|D\|\le C.
\]
The next kernel estimate controls this signed average and, after the
projection variables are integrated out, its smaller odd part.
Lemma~\ref{prof:gaussian-signed-kernel}, applied with $C=C_o$,
$A=A_0$ and $E=\mathsf P/\sqrt n$, proves the signed-density bound
\eqref{prof:eq18} times $Cnq$, including its fixed derivatives.
We verify its hypotheses explicitly. The formula for $K_o$,
$S_{\mathsf P}=I-O(q)$, and $\|J\|\le3$ give $\|K_o\|\le C$;
therefore $C_o\succeq C^{-1}I$. The lower precision bound,
trace-square bound, and edge spread follow from \eqref{prof:eq10}.
Its first projection estimate also gives
\[
\|C_o^{1/2}m\|^2=nq\,u^TC_ou\le Cn/q=CL^2.
\]
Thus
\[
\operatorname{Var}(\|z\|^2+2m^Tz)
=2\operatorname{Tr}C_o^2+4m^TC_om\asymp L^2,
\qquad \operatorname{Tr}C_o=nb+o(L).
\]
The lower variance bound uses the edge spread. The density estimate
\eqref{prof:eq2}, with its Fourier upper bound, now puts both shell
density denominators between fixed positive constants in $L$ units.
After integrating out $a,a'$, the odd parts of both the mass and matrix
densities gain $N^{-1/2}$.
The lemma proves this by comparing the
cross-frequency signs before projection conditioning; hence it does
not assume an artificial symmetry of the translated shells.
Projection cuts multiplying the leading odd term can be removed at
exponentially small cost, since \eqref{prof:eq18} gives Gaussian tails
in the whitened projection variables.
Lattice sums after the density approximation use steps bounded by \(C/\sqrt n\) in the raw \(a\)'s and \(C/L\) in full-overlap \(L\) units (fixing the other projection coordinates first, with affine shifts). Their covolumes are exactly the prefactor ones from Fourier inversion. Passing to \(U\) has bounded derivatives on the cut: besides subtracting \(a\cdot a'/L\), converting a full \(x\)-overlap to a \(z\)-overlap subtracts the \(m\)-linears with derivative \(\sqrt{nq}/L=q\). Converting constant-axis sums to \(a_2\)'s uses \(\mathsf P_2\)'s formula with bounded coefficients.
Thus the centered-cell integration error for smooth leading terms is \(O(1/n)\) times their respective polynomial-moment envelope scale. More explicitly use the individual expansions leading to \eqref{prof:eq16} with second derivatives of the explicit constant/common/additive and odd terms, keeping the projection factors via \eqref{prof:eq14}; in raw \(a,a',U\) units these derivatives cost just polynomials times \(e^{\zeta E_*^2}\). Remainders \(O(q E_*^C e^{\zeta E_*^2}/N)\) require no derivatives.
The type of error matters here. Density-ratio errors, including relative
saddle errors, are scalar functions of the statistics and multiply the
\emph{same angular kernel} as the leading terms. We make the exact
denominator in this assertion explicit. Use the chart \(\Phi\),
Jacobian \(J_\Phi\), mixed measures \(dm_i\), and exact unweighted
reference density \(f_{oo}^{\rm unw}\) from
Lemma~\ref{prof:common-angular-kernel}. For the reference single
projection, isotropy gives the exact formula
\[
f_o^{a,\rm ex}(a)=\mathfrak c_o\mathfrak r_o(a)
\varphi_{bI_3}(a)f_{b\chi^2_{n-3}}(\rho(a)^2),
\qquad \rho(a)^2=nb-2\sqrt{nq}\,a_1-|a|^2.
\]
Its ratio to \(f_o^a\) is a dimension-only factor \(1+O(1/n)\)
by Stirling's formula. Together with the exact conditional overlap
density above, this proves, uniformly on the regular window,
\[
f_{oo}^{\rm unw}=\widehat f_{oo}^{\,a,a',U}(1+\varepsilon_{oo}),
\qquad |\varepsilon_{oo}|\le C/n.
\]
No raw shell density, rank factor, or normalizing constant has
been removed from either side.
For the numerator inversion, first smooth the uncut statistic laws
only in their nonlattice coordinates on a slightly enlarged window,
and then apply the smooth projection cuts and the multiplier defining
\(g\). Let \(\widehat f_i^{\rm mod}\), for
\(i=pp,pg,gp,gg\), denote the corresponding formal pair function
with these factors included. Uniform inversion gives at each pair's
own mixed point
\[
f_i^{\rm sm}(w)
=J_\Phi\widehat f_i^{\rm mod}(\Phi(w))(1+\varepsilon_i(w)),
\qquad |\varepsilon_i(w)|\le Cq/N.
\]
The factors are applied after smoothing, so this identity makes no
relative-error assertion at a zero of a convolved cutoff. Commuting
those factors through a mollifier of radius \(\delta\) instead
changes the tested integral by at most \(\delta n^C\) times the
positive mass on the enlarged window. The multiplier's logarithmic
derivatives and the cutoff derivatives have polynomial bounds there. The
positive-mass estimate below and a sufficiently high inverse-polynomial
choice of \(\delta\) make this separate error negligible.
Write \(\widehat R_i=\widehat f_i^{\rm mod}/\widehat f_{oo}\)
in this chart, and let \(L_i\) be the finite smooth individual
expansion used above for quadrature. Thus
\(\widehat R_i=L_i+\widehat e_i\), where
\(|\widehat e_i|\le C(q/N)E_*^C e^{\zeta E_*^2}\).
Consequently the exact ratios used by the angular disintegration are
\[
\begin{split}
r_i(w)&=\frac{f_i^{\rm sm}(w)}
{J_\Phi f_{oo}^{\rm unw}(\Phi(w))}
=L_i(w)+e_i(w),\\
e_i(w)&=\widehat e_i(w)+\widehat R_i(w)
\frac{\varepsilon_i(w)-\varepsilon_{oo}(\Phi(w))}
{1+\varepsilon_{oo}(\Phi(w))},\qquad
|e_i(w)|\le C\frac qN E_*^C e^{\zeta E_*^2}.
\end{split}
\]
Here \(1/n\le q/N\), and the individual formal ratios have
the Gaussian-growth envelope already proved. Only the smooth
\(L_i\) undergo quadrature. Their signed sum has the leading
odd term in \eqref{prof:eq16}, with its stated envelope remainder.
These are absolute remainder bounds on each
own mixed measure, including cutoff boundaries; no regularity of
\(e_i\) is needed. Bounds \eqref{prof:eq18} integrate or sum them
against the absolute value of the signed angular average, not
against the angular average of the absolute insertion.
Thus the envelope in Lemma~\ref{prof:mixed-coordinate-transfer}
is \(\mathscr E=E_*^C e^{\zeta E_*^2}\).
Holding the length constraints exact ensures that at those points both angular averages use the same residual norms and Gram. Thus a scalar saddle error against an angular kernel is integrated using that kernel's reference signed density at the points, not by integrating absolute Wick polynomials. Mollification changes the angular test itself and therefore requires
a separate estimate. For that change we use absolute bounds: comparison of positive angular integrands at neighboring regular Grams, with polynomial derivative costs relative to base exponential weights.
Such costs times an arbitrarily small inverse polynomial are negligible here. Indeed the positive partition integrals with smooth cut to this region are polynomially bounded in reference units already by the scalar upper saddle bounds against the mass version of \eqref{prof:eq18}, summing also on the meshes; neighboring unsmoothed mass needed on a smaller region obeys the same type of bound by the angular energy comparison itself and unit mollifier mass. One can slightly enlarge regular windows in this justification and use smooth boundaries, without moving length or lattice statistics.
Cut boundaries can all use smooth tapers, neglecting the \(t\)-boundary by the preceding tails or \eqref{prof:eq18}. Lemmas~\ref{prof:common-angular-kernel} and
\ref{prof:mixed-coordinate-transfer} below state this disintegration and
quadrature step in a single explicit chart, including its Jacobian and
primitive lattice covolumes. They justify \eqref{prof:eq16} inside both
mass and signed-kernel tests without identifying the distinct atomic
and continuous statistic measures.
\par\smallskip\noindent\textit{Transfer of moments and final parameter choices.}\enspace
We now finish \eqref{prof:eq13}. All four pair laws have been related
to the same angular kernel, with their individual remainders controlled.
We can therefore square differences of the single-replica integrals.
Under Haar averaging, the prior density ratios multiply the common
angular integral at fixed statistics. When the reference prior
density and the two energy factors in that integral are combined
and divided by \(Z_*^2\), they give the tilted reference shell
kernel. This is the mixed-measure identity made explicit below.
The mass of modified \(g\), divided by \(Z_*\), tends to one in
probability: its first and second moments follow from
\eqref{prof:eq14}, the individual pair comparisons, and the tails.
The Hilbert-Schmidt second moments for the \emph{difference} of the cut \(p\) and \(g\) integrals, still with common denominator, are bounded as follows:
\begin{equation}\label{prof:eq19}
\begin{array}{c|c}
{\rm integrand} & \mathbb E_{\rm Haar}\|\text{difference}\|_{\rm HS}^2\\ \hline
1 & O(q/N)\\
z & O(q L/\sqrt N)\\
zz^T & O(q L^2/N)\\
A_0(zz^T-C_o)A_0 & O(n q^2/N).
\end{array}
\end{equation}
Each row follows by expressing the square as a pair integral and using
\eqref{prof:eq16}, \eqref{prof:eq18}.
The insertion determines whether
an odd-part gain is needed. For the mean the inner product kernel \(L U+a\cdot a'\) needs no odd gain. For the raw second moment its square differs from \(L^2U^2\) by at most \(L^2 E_*^4 C/\sqrt N\), so the leading odd term is harmless. For the last test use exactly the signed \(H\) bounds. All constants here may use a fixed small upper bound on \(\epsilon_1,\zeta,\epsilon_0\) satisfying the preceding absorptions.
On orientations obeying \eqref{prof:eq12}, the uncut \(g_0\) physical law has centered covariance at most \((1+o(1))K^{-1}\) and raw second moment at most \(C K^{-1}\), exactly by the shell argument of \eqref{prof:eq2}. Here displacement of the standardized length target is bounded rather than vanishing, so use centered conditional Gaussian variance in that argument (joint Gaussian limit with any white projection, variance can only decrease). Projection and length density bounds needed for uniform moments follow after any bounded codimension compression by \eqref{prof:eq10}.
The modification defining \(g\) changes these estimates by
\(o(1)\) in covariance units. Indeed \(W\) has uniform sub-Gaussian
tails, \eqref{prof:eq14} gives a factor tending to one with
arbitrarily small Gaussian growth, and the removed tail begins at
order \(\sqrt N\). Cauchy--Schwarz applies uniformly to each
projection moment test.
We spell out the normalization and centering. Write
\[
I_i(F)=Z_*^{-1}\int F(z)e^{z^T(J-bI)z/2}\,i(dz),
\qquad M_i=I_i(1),\qquad i=p,g,
\]
where both priors include their cuts and \(g\) includes its
projection multiplier. Since \(L/\sqrt N=q^{-2}\), the four
root-mean-square errors in \eqref{prof:eq19} are, in order,
\[
O\bigl(\sqrt{q/N}\bigr),\qquad O(q^{-1/2}),\qquad
O(q^{-3/2}),\qquad O(q^{-1/2}).
\]
Choose the failure probabilities in Markov's inequality in advance,
with their sum and the preceding orientation losses below \(1/4\).
For small \(q\) and large \(N\), the resulting orientations have
\(M_p,M_g\) bounded away from zero and close to one. For any matrix
or vector test \(F\),
\[
\frac{I_p(F)}{M_p}-\frac{I_g(F)}{M_g}
=\frac{I_p(F)-I_g(F)}{M_p}
+\frac{M_g-M_p}{M_p}\frac{I_g(F)}{M_g}.
\]
The normalized modified Gaussian \(g\) has raw second moment
\(O(q^{-2})\), so its mean is \(O(q^{-1})\). For \(F=z\) and
\(F=zz^T\), the last display and
the first three errors show that the normalized mean difference is
\(O(q^{-1/2})+o(q^{-1})\), and the normalized raw second-moment
difference is \(O(q^{-3/2})+o(q^{-2})\). Subtracting the two mean
outer products costs at most \(O(q^{-3/2})+o(q^{-2})\).
Consequently the cut spin covariance is bounded above by
\((1+o(1))K^{-1}+o(q^{-2})I\).
For the weighted estimates choose the loss in
\eqref{prof:eq10} separately, denoting it by \(\eta_{\rm mat}\),
with \(2\eta_{\rm mat}<\gamma\). The normalized modified Gaussian
\(g\) has raw weighted second moment \(O(q^{-2\eta_{\rm mat}})\).
Use the last row of \eqref{prof:eq19} in the normalization identity
with \(F=A_0(zz^T-C_o)A_0\). The subtraction is a constant under
each normalized law, so the difference of raw weighted second
moments is
\[
O(q^{-1/2})+o(q^{-2\eta_{\rm mat}}+1)=o(q^{-\gamma}).
\]
Here \(\gamma>1/2\) is exactly what makes the first error smaller
than the target. The cut spin raw weighted second moment is
therefore \(o(q^{-\gamma})\). It bounds the weighted covariance;
Jensen's inequality bounds the squared weighted mean by the same
quantity. In the unweighted case the raw second moment is
\(O(q^{-2})\), so the mean is \(O(q^{-1})=o(q^{-1-\eta})\) for
the independently prescribed \(\eta>0\).
Finally restore the removed spin mass. If it is at most \(n^{-D}\),
then \(\|X-m\|\le2\sqrt n\) bounds the changes in normalized means
and raw second moments by \(Cn^{1/2-D}\) and \(Cn^{1-D}\),
respectively; the same bounds, up to constants, hold after \(A_0\)
since \(\|A_0\|\le5\). Choose \(D\) large. The sharp unweighted
coefficient follows by choosing the precision, shell, transfer, and
\(q/r\) tolerances so that
\[
(1+o(1))(1-\epsilon_{\rm mat})^{-1}s/q^2\le1+\epsilon.
\]
This proves \eqref{prof:eq13} with all the prescribed small
constants.
The same event also provides the mass anchor needed in the next
section. We may require \(M_p\ge1/2\), and positivity of the omitted
integral gives
\begin{equation}\label{prof:mass-anchor}
\int e^{z^T(J-bI)z/2}\,p_{\rm full}(dz)\ge Z_*/2.
\end{equation}
It holds simultaneously with the norm bounds on the retained
orientation set. On any fixed compact observation-time interval
bounded away from zero, contained in the established small-time regime,
and on which \(r\) is bounded below, the translated posterior raw
second moment \(\E[(X-m)(X-m)^T\mid J,h]\) is bounded by a fixed
constant in operator norm on these orientations, using the covariance
and mean estimates.
All choices in this static assertion begin with coarse scalar
tolerances and matrix budgets giving fixed constants in
\eqref{prof:eq9}--\eqref{prof:eq12}, and with the orientation
failure allocations just described. Choose the overlap cutoff and
then the projection cutoff small, including for the Gaussian
envelopes, and use a sufficiently small scalar and overlap rate to
put \(q/r\) close to one and control the projection-cut loss.
Stricter diagnostics do not require increasing the coarse constants. Typical-site moment and tail cutoff exponents can be chosen for the finite derivative orders used. The upper bound on \(r\) is then sufficiently small in terms of these choices, which have not used dynamical probability/moment inputs.
\end{proof}
\subsection{Exact statistic measures and signed cancellation}
The following statements make the measure-theoretic and Gaussian parts
of the preceding comparison explicit. The smooth saddle expansions
\eqref{prof:eq14}--\eqref{prof:eq17} have already supplied the scalar
coefficients and their errors; the first two lemmas specify exactly how
these scalar quantities act on atomic and continuous priors.
\begin{lemma}[Common angular kernel and mixed statistic measures]\label{prof:common-angular-kernel}
Let $0<q<1$, $b=1-q$, $L=\sqrt{n/q}$, and $\mathsf P^T\mathsf P=nI_3$, $m=\sqrt q\mathsf P_1$, $M=n^{-1}\sum m_i$, and
$c=(1-M^2/q)^{1/2}>0$, with
$\mathsf P_2=c^{-1}(\mathbf1-(M/q)m)$.
For $z=x-m$ on $\|z\|^2+2m^Tz=nb$, set
$a=\mathsf P^Tz/\sqrt n$ and $z_\perp=z-\mathsf P a/\sqrt n$. Define primed variables
likewise, and set $U=z_\perp^Tz'_\perp/L$. Then
\[
\rho(a)^2=nb-2\sqrt{nq}\,a_1-|a|^2,\qquad
\Gamma(a,a',U)=
\begin{pmatrix}\rho(a)^2&LU\\LU&\rho(a')^2\end{pmatrix}.
\]
On $\Gamma\succ0$, the statistic $(a,a',U)$ is a complete
invariant for simultaneous proper rotations of
$\operatorname{ran}(\mathsf P)^\perp$, provided $n-3\ge3$.
Thus every integrable physical pair insertion has one Haar angular
average, independent of the choice of prior. To specify the reference
used in the application, put \(\mathbf s=(a,a',U)\), let
\(T(z,z')=\mathbf s\), and define the exact unweighted density
\[
f_{oo}^{\rm unw}=\frac{d\,T_\#(o\otimes o)}{d^3a\,d^3a'\,dU}
\]
on the positive Gram region. It includes both rank factors, both
raw shell densities, and \(\mathfrak c_o^2\). It is distinct
from the formal product of single saddle functions and a
conditional saddle function.
For a pair insertion \(F\), write
\[
\mathcal A_F(\mathbf s)=\int_{\rm Haar}
e^{\mathcal E(gz)+\mathcal E(gz')}F(gz,gz')\,dg,
\qquad \mathcal E(z)=z^T(J-bI)z/2,
\]
using any representative of \(\mathbf s\). All physical matrices
in \(F\) are fixed together in this vector-rotation representation.
Let \(k_F\) be the signed statistic density under two independent
normalized physical reference shells, with insertion \(F\).
Then
\[
Z_*^{-2}f_{oo}^{\rm unw}(\mathbf s)\mathcal A_F(\mathbf s)
=k_F(\mathbf s)
\]
for almost every regular \(\mathbf s\). This includes the mass
insertion and the centered matrix insertion of
Lemma~\ref{prof:gaussian-signed-kernel}.
There is one smooth chart for the four pair priors. Put
$S=\sum x_i$, $S'=\sum x_i'$, and $R=\sum x_ix_i'$, and use
$w=(a_1,a_3,a'_1,a'_3,S,S',R)$. Its map $\Phi:w\mapsto(a,a',U)$ is
\[
a_2=\frac{S-nM}{\sqrt n c}-\frac{M}{\sqrt q c}a_1,\qquad
a'_2=\frac{S'-nM}{\sqrt n c}-\frac{M}{\sqrt q c}a'_1,
\]
\[
U=\frac{R-nq-\sqrt{nq}(a_1+a'_1)-a\cdot a'}{L},
\qquad |\det D\Phi|=\frac{1}{nc^2L}=:J_\Phi.
\]
For $pp$, the raw triple belongs to the affine lattice
$n(1,1,1)+\Lambda$, where
\[
\Lambda=\operatorname{span}_{\mathbb Z}
\{(2,0,2),(0,2,2),(0,0,4)\},\qquad
\operatorname{covol}(\Lambda)=16.
\]
For $pg$ only $S\in n+2\mathbb Z$ is retained as a lattice
coordinate, and for $gp$ only $S'\in n+2\mathbb Z$ is retained;
these covolumes are $2$. The $gg$ measure has no lattice coordinates.
Let $dm_{pp}$ be $16$ times counting measure in its lattice
coordinates times Lebesgue measure in the remaining coordinates;
define $dm_{pg},dm_{gp}$ with factor $2$, and $dm_{gg}=dw$.
Smooth only the nonlattice coordinates, preserving the exact
translated shell constraints. For cut or multiplied priors apply
their smooth factors after this convolution; the preceding
commutator estimate charges the change from the original prior.
If $f_i^{\rm sm}$ denotes the resulting density with respect to $dm_i$, set
\[
r_i(w)=\frac{f_i^{\rm sm}(w)}{J_\Phi f_{oo}^{\rm unw}(\Phi(w))},\qquad
\widetilde k_F(w)=J_\Phi f_{oo}^{\rm unw}(\Phi(w))
\mathcal A_F(\Phi(w)).
\]
Then the smoothed pair integral is exactly
$I_i^{\rm sm}=\int r_i\widetilde k_F\,dm_i$, and
\[
Z_*^{-2}\widetilde k_F(w)=J_\Phi k_F(\Phi(w)).
\]
For a fixed insertion we abbreviate \(\widetilde k_F\) by
\(\widetilde k\) below.
In particular, a scalar density remainder multiplies the same
signed reference kernel at each actual mixed mesh point.
\end{lemma}
\begin{proof}
Two ordered residual pairs with the same positive Gram matrix
are related by an orthogonal map. If its determinant is negative,
compose with a reflection on the orthogonal complement of the
target pair; this fixes the pair and corrects the determinant.
Haar invariance and Fubini therefore give, for every finite pair
measure $\mu$, atomic or continuous, and every integrable \(B\),
\[
\int_G\int B(gz,gz')\,d\mu\,dg
=\int\mathcal A_B(s)\,d(T_\#\mu)(s),
\]
where \(\mathcal A_B\) is its angular average.
For signed insertions use absolute integrability. Moving the
rotation from the interaction matrix to the pair vectors gives
precisely this identity for orbit averages in the application;
all matrix factors in the insertion move together.
For the reference prior, \(o\) is invariant under the rotations:
its isotropic Gaussian density, translated shell, and projection
rank factor are all invariant. Applying this disintegration to
\(e^{\mathcal E(z)+\mathcal E(z')}F(z,z')\,o(dz)o(dz')\),
and then using
\(Z_*^{-1}e^{\mathcal E}o=\operatorname{Law}(Z\mid T_o=0)\),
proves \(f_{oo}^{\rm unw}\mathcal A_F=Z_*^2k_F\).
For a signed insertion absolute integrability justifies Fubini;
the insertion keeps its sign inside the angular average.
The chart formulas follow by expanding $\sum x_i$ and
$x^Tx'$. Its Jacobian is triangular, with the three new diagonal
factors $1/(\sqrt n c),1/(\sqrt n c),1/L$.
For $pp$, the four sign-pattern counts are
\[
\frac{n+S+S'+R}{4},\quad \frac{n+S-S'-R}{4},\quad
\frac{n-S+S'-R}{4},\quad \frac{n-S-S'+R}{4}.
\]
Their integrality gives the stated lattice and determinant $16$.
The nonnegativity boundaries lie outside the central regular
window. In particular the conditional $R$ step is $4$, so three
independent step-$2$ lattices would give the wrong normalization.
The mixed lattices have only the one sign-sum coordinate.
Finally the asserted integral formula follows by multiplying and
dividing the smoothed scalar density by the positive unweighted reference
density evaluated at the same point. This is an identity in the
chosen mixed measures, not a Radon--Nikodym derivative of an
atomic statistic law with respect to a continuous law.
\end{proof}
\begin{lemma}[Separate remainders and common-volume cancellation]\label{prof:mixed-coordinate-transfer}
Suppose $r_i=L_i+e_i$, where $L_i$ extends smoothly to the common
chart, while $e_i$ need only be defined at its own mixed points.
Let $\mathscr E$ be a nonnegative envelope on the chart.
Assume $|e_i|\le\varepsilon\mathscr E$ and
\[
\int\mathscr E|\widetilde k|\,dm_i\le B,\qquad
\left|\int L_i\widetilde k\,dm_i
-\int L_i\widetilde k\,dw\right|\le\delta_i.
\]
Then, with pair signs $+,-,-,+$,
\[
\left|\sum_i s_iI_i^{\rm sm}
-\int\Big(\sum_i s_iL_i\Big)\widetilde k\,dw\right|
\le4\varepsilon B+\sum_i\delta_i.
\]
\end{lemma}
\begin{proof}
Bound the four errors separately on their own meshes, perform
quadrature on the four smooth terms, and only then add their
ordinary-volume integrals. No matching support or regularity of
the residual functions is needed. The relevant absolute value is
$|\widetilde k|$, the absolute value of the signed angular
average; the insertion is not replaced by its absolute value.
\end{proof}
For clarity, centered-cell quadrature supplies the displayed
$\delta_i$ from two derivatives. If $C_\Lambda$ is a centered
fundamental parallelepiped of a fixed-dimensional lattice, Taylor's
formula gives
\[
\left|\operatorname{covol}(\Lambda)\sum_{\ell}F(\ell)
-\int F\right|
\le\frac12\sum_{\ell}\int_{C_\Lambda}|u|^2
\sup_{0\le t\le1}\|\nabla^2F(\ell+tu)\|\,du.
\]
The linear term integrates to zero. In the application,
$c^2=1-M^2/q=1-O(q)$ is bounded below. Raw lattice steps then
change $a,a'$ by $O(n^{-1/2})$ and $U$ by $O(L^{-1})$ on the
projection cut. Hold the nonlattice coordinates
$(a_1,a_3,a'_1,a'_3)$ fixed when differentiating in the raw
lattice coordinates. The only quadratic term of $\Phi$ in those
directions has
$\partial_S\partial_{S'}U=-1/(nc^2L)$.
Thus a summable polynomial--Gaussian
envelope for two derivatives in $(a,a',U)$ costs $O(1/n)$ times
its envelope scale. The covolumes in this estimate are exactly
those already present in the local Fourier inversion.
Mollification is a different error and is estimated before using
signed scalar remainders. For a probability mollifier of radius
$\delta$ in the nonlattice chart coordinates,
\[
\int\mathcal A\,d(\nu*\kappa_\delta)-\int\mathcal A\,d\nu
=\int\!\int[\mathcal A(w+h)-\mathcal A(w)]
\kappa_\delta(h)\,dh\,d\nu(w).
\]
On the enlarged regular Gram region choose a smooth Cholesky
representative and impose
$|\nabla\log\mathcal A_1|\le n^C$,
$|\nabla\mathcal A_H|\le n^{C'}\mathcal A_1$, where
$\mathcal A_1>0$ is the positive angular energy integral.
These inequalities follow by differentiating the representative:
Gram eigenvalues are of order $n$, all physical parameters and
the polynomial insertion are polynomially bounded, and the
exponential derivative is its value times a polynomial. They
preserve the translated lengths by recomputing $\rho(a)$.
A smoothed positive-mass bound then bounds the unsmoothed mass
within $e^{n^C\delta}$, so the signed smoothing error is at most
$\delta n^{C'}e^{n^C\delta}$ times that positive mass.
Taking an arbitrarily high inverse-polynomial smoothing scale
makes this negligible. Smooth enlarged cuts handle boundaries.
No Gaussian length coordinate and no retained lattice coordinate
is convolved in this argument.
\subsection{The centered Gaussian matrix kernel}
\begin{lemma}[Centered Gaussian kernel and its odd part]\label{prof:gaussian-signed-kernel}
Fix a derivative order $h\ge0$. There is a threshold $N_0=N_0(h)$
for the following statement. Let $0<q\le q_0<1$,
$N=nq^3\ge N_0$, and $L=\sqrt{n/q}$.
Let $C$ be positive definite, and let $E$ have a fixed number $d$ of
orthonormal columns. Put $\Pi=I-EE^T$. Assume
\[
c_0I\preceq C,\quad \|C\|\le C_0q^{-2},\quad
\operatorname{Tr}C\le C_0n,\quad \operatorname{Tr}C^2\le C_0n/q,
\]
and that at least $c_0N$ eigenvalues belong to
$[c_0q^{-2},C_0q^{-2}]$. Suppose $m\in\operatorname{ran}E$ and
$\|C^{1/2}m\|/L\le C_0$. Let $A$ be symmetric, and assume
$D=C^{1/2}A^2C^{1/2}$ has norm at most $C_0$.
For independent standard normal vectors $\eta_1,\eta_2$, set
\[
z_i=C^{1/2}\eta_i,\quad
T_i=(\|z_i\|^2+2m^Tz_i-nb)/L,\quad
a_i=E^Tz_i,\quad U=z_1^T\Pi z_2/L.
\]
Assume the density of each $T_i$ at zero belongs to $[c_0,C_0]$.
Write $k_1(a_1,a_2,U)$ for the density conditional on $T_1=T_2=0$,
and $k_H$ for the signed density with insertion
\[
H=(\eta_1^TD\eta_2)^2-\eta_1^TD^2\eta_1
-\eta_2^TD^2\eta_2+\operatorname{Tr}D^2.
\]
Put $\alpha_i=(E^TCE)^{-1/2}a_i$ and $J_E=\det(E^TCE)^{-1}$.
For every multi-index $\beta$ with $|\beta|\le h$, on $|U|\le\epsilon\sqrt N$,
with sufficiently small fixed $\epsilon$, one has
\[
|\partial^\beta k_1|\le C_\beta P_\beta
e^{-c(|\alpha_1|^2+|\alpha_2|^2+U^2)}J_E,\qquad
|\partial^\beta k_H|\le C_\beta nq P_\beta
e^{-c(|\alpha_1|^2+|\alpha_2|^2+U^2)}J_E,
\]
where $P_\beta$ is a fixed polynomial in
$1+|\alpha_1|+|\alpha_2|+|U|$.
After integrating both projection variables, the resulting densities
$\overline k_1,\overline k_H$ obey
\[
|\overline k_1(U)-\overline k_1(-U)|
\le CN^{-1/2}P(U)e^{-cU^2},\qquad
|\overline k_H(U)-\overline k_H(-U)|
\le CnqN^{-1/2}P(U)e^{-cU^2}.
\]
The same bounds hold for derivatives of order at most $h$.
\end{lemma}
\begin{proof}
Both $B=C/L$ and $B_\Pi=C^{1/2}\Pi C^{1/2}/L$ have operator norm
$O(N^{-1/2})$, Hilbert--Schmidt norm $O(1)$, and nuclear norm
$O(\sqrt{nq})$. Their difference has rank at most $d$.
Conditioning on $a_i$ amounts to writing
\[
\eta_i=S\alpha_i+\xi_i,\qquad
S=C^{1/2}E(E^TCE)^{-1/2},\qquad S^TS=I,
\]
where $\xi_i$ is standard Gaussian on $S^\perp$.
Take Fourier transforms in the two lengths and in $U$, allowing a
real cross tilt $u$ with $|u|\le c\sqrt N$. The precision then has
real part bounded above and below by fixed positive constants.
Let $\Sigma,\mu$ denote the complex Gaussian covariance and mean.
The resolvent identity, the preceding three norm bounds, and the
fixed-rank projection correction give
\[
\|\Sigma-I\|_{\rm HS}\le CP,\qquad
\|\Sigma-I\|_1\le C\sqrt{nq}\,P,\qquad \|\mu\|\le CP,
\]
where $P$ is a polynomial in the frequencies, $|u|$, and
$1+|\alpha_1|+|\alpha_2|$. The length linear coefficient is
$C^{1/2}m/L$; projection conditioning adds coefficients of norm
at most $C(|\alpha_1|+|\alpha_2|)/\sqrt N$.
Write $\Sigma_{11}=I+\Delta_1$, $\Sigma_{22}=I+\Delta_2$,
$R=\Sigma_{12}$, and $s=\mu_1^TD\mu_2$. Wick's formula is exactly
\begin{align*}
\mathbb E_{\Sigma,\mu}H={}&
\operatorname{Tr}(D\Delta_2D\Delta_1)
+\operatorname{Tr}(DR^TDR^T)+(\operatorname{Tr}(DR^T))^2\\
&+2s\operatorname{Tr}(DR^T)+s^2
+\mu_1^TD\Delta_2D\mu_1+\mu_2^TD\Delta_1D\mu_2
+2\mu_1^TDR^TD\mu_2.
\end{align*}
All zero-mean terms except the trace square cost a polynomial by
Hilbert--Schmidt bounds. The exceptional term satisfies
\[
|\operatorname{Tr}(DR^T)|^2\le\|D\|^2\|R\|_1^2\le CnqP.
\]
The mean terms cost $C\sqrt{nq}P$. In particular the terms
$\operatorname{Tr}D^2$ and $\operatorname{Tr}(D^2\Delta_i)$
have canceled identically. This is an estimate for the signed
Gaussian expectation, not for the absolute insertion.
For real Fourier variable $v\in\mathbb R^3$, determinant damping is
\[
|\det(I-2iQ)|^{-1/2}\le C(1+c|v|^2/N)^{-c'N}.
\]
Before fixed-rank changes, $Q$ is the tensor product of $C/L$ with
the symmetric two-by-two frequency matrix. The latter has an
eigenvalue of magnitude at least a constant times $|v|$. The spread
assumption supplies $cN$ singular values of size $c|v|/\sqrt N$;
interlacing loses only a fixed number. A real cross tilt conjugates
the Fourier quadratic matrix by bounded invertible operators, so
the same bound holds with changed constants.
Every fixed frequency
polynomial is integrable against this bound, uniformly for large
$N$. Fourier inversion therefore proves the density bounds and
their fixed derivatives. To obtain the Gaussian tail in $U$, shift
the cross contour by $u=c_1U$. Its centered determinant cost is
$O(u^2)$, since the cross tilt has trace zero and bounded
Hilbert--Schmidt norm. The projected mean cost is at most
$C\epsilon(|\alpha_1|^2+|\alpha_2|^2)$ on the stated window.
Choose $c_1$ and then $\epsilon$ small. The factor $e^{-c_1U^2}$
and the projection Gaussian density absorb these costs. Division
by the bounded shell denominators completes these bounds.
The parity estimate uses the unconditioned projection variables.
Integrate them out before Fourier inversion, so that reversing the
cross frequency can be compared at the level of the Gaussian formula.
Reversing $U$ reverses both the cross Fourier frequency and the
real cross tilt. Conjugation by $\operatorname{diag}(I,-I)$
preserves the determinant and all zero-mean contractions of $H$.
The inverse cross block satisfies the sharper operator estimate
\[
\|\Sigma_{12}\|\le CN^{-1/2}(|v|+|u|),
\]
by the block inverse formula.
The length linear coefficients have
bounded norms times $|v|$. Their Gaussian exponential factors
therefore differ by at most $N^{-1/2}$ times a frequency polynomial.
This estimate keeps the determinant damping: the two exponents
and the segment between them have nonpositive real part, because
the linear coefficients are imaginary and the real precision is
positive. For the mass this proves the extra $N^{-1/2}$.
For $H$, its zero-mean Wick expression costs $Cnq$ times this
factor. The mean terms can be bounded directly by
$C\sqrt{nq}P$, which suffices since
$\sqrt{nq}\le nq/\sqrt N$ for $q\le1$.
Fourier inversion with the same real tilt proves the odd-part
bounds and their derivatives.
\end{proof}
\begin{lemma}[The surviving scalar coefficients]\label{prof:surviving-coefficients}
Put $v_i=1-m_i^2$ and
$\kappa_2=\langle v^2\rangle$, $\kappa_3=4\langle m^2v^2\rangle$.
For $i,j\in\{p,g_0\}$, let $F_{ij}(\beta)$ be the per-site cross
log moment after optimizing all single-replica constraints. At zero projection,
\[
F'_{ij}(0)=0,\quad F''_{ij}(0)=\kappa_2,\quad
F'''_{ij}(0)=\kappa_3\mathbf1_{\{i=j=p\}}.
\]
The common quadratic coefficient relative to the reference is
$c_2=(\kappa_2-b^2)/(2q^2\kappa_2b^2)$. Up to the common prefactor
$b/\sqrt{\kappa_2}$ and an error $O(qE_*^Ce^{\zeta E_*^2}/N)$,
the signed zero-projection sum is
\[
\frac{q}{\sqrt N}e^{c_2qU^2}
\left(-\frac{\kappa_3}{2\kappa_2^2}U+\frac{\kappa_3}{6q\kappa_2^3}U^3\right).
\]
\end{lemma}
\begin{proof}
At zero constraints both canonical site laws have mean zero and
variance $v_i$.
Their third centered moments are $-2m_iv_i$ for
signs and zero for Gaussians. The cross observable is orthogonal
to all single-replica constraints. The first derivative of the
optimizing coefficients thus vanishes, so the first three
optimized cross derivatives are its ordinary first three
cumulants. This proves the displayed identities.
Let $D_2(\beta)$ be the constraint covariance. At zero cross tilt
it is block diagonal. Each diagonal block of $D_2'(0)$ vanishes:
the corresponding third cumulant contains a centered factor from
the other replica. Hence
$\left.\partial_\beta\log\det D_2(\beta)\right|_0=0$.
The conditional saddle prefactor is proportional to
$[F''(\beta)\det D_2(\beta)/\det D_2(0)]^{-1/2}$, whose
logarithmic $t$ derivative is therefore $-F'''(0)/(2\kappa_2^2)$.
Its cubic Legendre term is $nF'''(0)t^3/(6\kappa_2^3)$.
Now use $t=qU/\sqrt N$, $nt^2=U^2/q$, and
$nt^3=U^3/\sqrt N$. Reference finite-codimension corrections
$O(t^2+1/n)$ lie within $O(q/N)$ on bounded normalized statistics.
Away from zero the Taylor estimates following \eqref{prof:eq17} organize the logarithms as
$\log R_{ij}=L_0+C_*+A_i+A'_j+B_{ij}+r_{ij}$.
Here $L_0$ is common constant/quadratic, $C_*$ is common,
$A_i,A'_j$ are additive first projection terms, $B_{ij}$ is the
odd term just computed, and the retained small terms have size
$CqE_*^C/\sqrt N$. The remainder has size
$CqE_*^Ce^{\zeta E_*^2}/N$. With $s_p=1,s_{g_0}=-1$,
$\sum s_is_j=\sum s_is_j(C_*+A_i+A'_j)=0$.
Quadratic products from exponentiation cost $q^2/N\le q/N$.
The common projection multiplier differs from one by
$N^{-1/2}E_*^Ce^{\zeta E_*^2}$, so its effect on the surviving
odd term also fits the stated remainder. The projection estimate \eqref{prof:eq14}, the derivative bounds
following \eqref{prof:eq17}, and the mixed local-limit calculation
therefore give precisely this cancellation for the smooth saddle
ratios. Lemma~\ref{prof:mixed-coordinate-transfer} passes from these
ratios to the actual mixed statistic measures.
\end{proof}
\begin{corollary}[Weighted moment comparison]\label{prof:weighted-kernel-transfer}
Use the priors, cuts, and shell-constrained reference mass $Z_*$
of the preceding mixed comparison, whose physical reference covariance
is $C_o=K_o^{-1}$. Let $A$ be symmetric and assume the hypotheses of
Lemma~\ref{prof:gaussian-signed-kernel} with $C=C_o$.
Suppose the mixed-coordinate disintegration and local saddle
estimates give the preceding expansion, with scalar remainder
$O(qE_*^Ce^{\zeta E_*^2}/N)$ at each actual mixed mesh point.
Suppose centered-cell quadrature for the smooth leading terms
costs $O(1/n)$ times the signed kernel envelope. Then, with the
common normalization by $Z_*$,
\[
\mathbb E_{\rm Haar}\left\|
Z_*^{-1}\int A(zz^T-C_o)A\,d(p-g)_{\rm tilted}
\right\|_{\rm HS}^2\le Cnq^2/N.
\]
\end{corollary}
\begin{proof}
There are three errors to account for: scalar remainders, the surviving
odd term, and quadrature.
The scalar remainders cost $(q/N)(nq)$ using
the signed density bound on each mesh separately. After quadrature, the odd leading
term costs $(q/\sqrt N)(nq/\sqrt N)$. Quadrature itself costs
$(nq)/n=q\le nq^2/N$. Removing smooth projection cuts is
exponentially negligible since $|W|\le C|\alpha|$ and the cuts
start at $|W|$ of order $\sqrt N$. Mollification is separately
controlled by an absolute angular derivative bound and an
arbitrarily high inverse-polynomial smoothing scale. The scalar
saddle remainders never require placing $|H|$ inside the angular
integral.
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-local.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-matrices.tex ====
\section{Conditioned matrix estimates}\label{prof:matrices}
The scalar diagnostics now determine the small eigenvalues of the
conditional matrix. The point is to retain the coefficient $1$ in the
lower edge $q^2$, which later yields the coefficient $1$ in $s^{-1}$.
Put
\[
u=\frac{m}{\sqrt{nq}},\qquad E=u^\perp,\qquad
e=\frac{P_E\mathbf1}{\sqrt n},\qquad
c_3=\frac{P_{\operatorname{span}(m,\mathbf1)^\perp}y}
{\|P_{\operatorname{span}(m,\mathbf1)^\perp}y\|}.
\]
The scalar diagnostics make $c_3$ well defined. The plant component
$e$ is not normalized: $\|e\|^2=1-M^2/q$. Let
$\mathcal P=\operatorname{span}(u,e,c_3)$ and let
$V_{\mathcal P}=(P_{\mathcal P}VP_{\mathcal P})|_{\mathcal P}$.
Operators defined on $\mathcal P$ are extended by zero on its
orthogonal complement. Consider
\[
\begin{split}
K&=V^{-1}+bI-J-2q uu^{\mathsf T}/\langle v^2\rangle,\\
K_o&=(b+b^{-1})I-J+
[V_{\mathcal P}^{-1}-b^{-1}I_{\mathcal P}]
-2q uu^{\mathsf T}/b^2,\\
K_T&=V^{-1}+bI-J-2q uu^{\mathsf T}.
\end{split}
\]
\begin{proposition}[Conditioned precision bounds]\label{prof:conditioned-precision}
Fix a sufficiently small $\eta>0$, an accuracy $\epsilon>0$, and a
budget $B>0$. On retained scalar data with the required accuracies,
the following estimates hold with the integrated exclusion specified
below. For $\mathsf K\in\{K,K_o,K_T\}$,
\begin{equation}\label{prof:eq10}
\mathsf K\succeq(1-\epsilon)q^2I,\qquad
\operatorname{Tr}\mathsf K^{-2}\le Cn/q,\qquad
\|A_0\mathsf K^{-1/2}\|\le Cq^{-\eta},
\quad A_0=2I-J.
\end{equation}
For $K_o$ the last bound holds with $\eta=0$. Each precision has at
least $c_A nq^3$ eigenvalues below $C_Aq^2$, and
$\operatorname{Tr}K_o^{-1}=nb+o(\sqrt{n/q})$. For
$\mathsf K\in\{K,K_o\}$, the three projection bounds are
\[
u^{\mathsf T}\mathsf K^{-1}u\le C_Aq^{-2},\qquad
e^{\mathsf T}\mathsf K^{-1}e\le C_Aq^{-1},\qquad
c_3^{\mathsf T}\mathsf K^{-1}c_3\le C_A.
\]
Let $\mathcal S$ be the retained scalar set, let $\mathcal G_{\rm spec}$
be the original spectral event, and let $\mathcal F_y$ be failure of
any estimate above. In the conditional law $\mathsf Q_y$ of
Lemma~\ref{prof:root-count}, the bound is
\[
\int_{\mathcal S}G(y)\,
\mathsf Q_y(\mathcal G_{\rm spec}\cap\mathcal F_y)\,dy
\le n^C e^{-Bk}.
\]
The same bound with the determinant weight follows after increasing
the preliminary budget. The spectral restriction stays inside the
event being estimated; the argument does not condition on its probability.
\end{proposition}
\begin{proof}
\par\smallskip\noindent\textit{Conditional law and Haar concentration.}\enspace
We use Haar concentration in the Frobenius metric on proper rotations:
an $L$-Lipschitz function on $SO(N)$ has tail
$2e^{-cNz^2/L^2}$ about its mean. See
\cite[Sections~5.2--5.3, Proposition~5.13 and Theorem~5.16]{Meckes2019ClassicalGroups}.
A constant-mesh net of size $e^{O(j)}$ then gives an operator bound on
a $j$-dimensional space, using polarization for bilinear forms.
All such tests below condition first on deterministic eigenvalues,
so they also apply to resolvents defined separately on spectral bins.
In $\mathsf Q_y$, the scalar data and the vectors determined by $y$
are fixed. The regression in Lemma~\ref{prof:root-count} gives
\[
J|_E=W_E+ee^{\mathsf T},\qquad
p:=P_EJu=p_0+\sqrt{s/\nu}\,w,\qquad
p_0=\frac{q}{\nu}\frac{P_E(y-d\mathbf1)}{\sqrt{nq}}
+\frac{M}{\sqrt q}e,
\]
where $W_E$ is a centered GOE block and $w\sim N(0,I_E/n)$ is
independent of it. The mean $p_0$ is the projected regression mean of
$Wm+\sqrt s\,g$, plus the plant column. We may next condition on the
eigenvalues of $W_E$ and retain its Haar eigenframe. On the intersection
with $\mathcal G_{\rm spec}$, interlacing of
$W_E$, $W_E+ee^{\mathsf T}=J|_E$, and $J$ supplies the required counts
and $\lambda_{\max}(W_E)\le\lambda_{\max}(J)$. The concentration
probabilities are estimated before that intersection is imposed.
\par\smallskip\noindent\textit{Diagonal truncation and the compensating moments.}\enspace
Compare with $D=(b+b^{-1})I-J$. In $K$ and $K_T$, replace the
positive diagonal entry $v_i^{-1}-1$ by
$\min\{v_i^{-1}-1,q^{1-\eta}\}$. This gives a lower precision.
Let $B$ be the difference of either truncated precision, or of $K_o$,
from $D$, and set $B_E=P_EB|_E$ and $f=P_EBu$.
The scalar moments give the quantities needed by the Schur complement.
With
$\mu_B=n^{-1}\operatorname{Tr}B_E$ and
$\nu_B=n^{-1}\operatorname{Tr}B_E^2$,
\[
\begin{gathered}
\|B\|=O(q^{1-\eta}),\qquad \mu_B,\nu_B=O(q^2),\qquad
\mu_B-\nu_B=o(q^2),\\
u^{\mathsf T}Bu\ge2q^2+o(q^2),\qquad
\|f\|^2=6q^2+o(q^2),\qquad p_0=f/3+O_A(q^{3/2}),\\
e^{\mathsf T}f=O_A(q^2),\quad \|B_Ee\|=O(q),\quad
e^{\mathsf T}B_Ee=O_A(q^2),\quad e^{\mathsf T}p_0=O_A(q^{3/2}).
\end{gathered}
\]
The error in the vector relation is in Euclidean norm.
For the diagonal precisions, the truncated entry expands as
$m^2+m^4+\cdots$ on $|y|\le q^{1/2-\eta_0}$, with
$0<\eta_0<\eta/2$. The missed bounded powers on the complement are
$O(m^2)$ and have average $O_A(q^4)$ by the tail-square diagnostic.
The bulk mean is therefore $3q^2-q^2=2q^2$ to leading order, equal
to the centered variance $\langle(m^2-q)^2\rangle=2q^2+o(q^2)$.
In the root direction,
\[
\frac{\langle m^4+m^6\rangle}{q}=3q+7q^2+o(q^2),\qquad
\frac{2q}{\langle v^2\rangle}=2q+4q^2+o(q^2).
\]
Subtracting the scalar $b^{-1}-1=q+q^2+O(q^3)$ leaves at least
$2q^2+o(q^2)$. Also
\[
\|P_E\operatorname{diag}(m^2)u\|^2
=\frac{\langle m^6\rangle}{q}
-\left(\frac{\langle m^4\rangle}{q}\right)^2
=6q^2+o(q^2).
\]
This is the leading value of $\|f\|^2$.
The root identity $y=m+m^3/3+O(m^5)$ on the clipped region gives
\[
\frac{P_Ey}{\sqrt{nq}}
=\frac13P_E\operatorname{diag}(m^2)u+O_A(q^{3/2}).
\]
The tail contributes $O_A(q^4/q)$ to its squared error. In the formula
for $p_0$, the extra $e$ coefficient is
$(M-(q/\nu)d)/\sqrt q=O_A(q^{3/2})$, giving $p_0=f/3+O_A(q^{3/2})$.
The bounds involving $e$ follow similarly from
$\langle m^3\rangle=3qM+O_A(q^3)$.
For $K_o$, put
$L_B=(P_{\mathcal P}(\operatorname{diag}(m^2)-qI)P_{\mathcal P})|_{\mathcal P}$.
Its norm is $O_A(q)$: in the cubic direction the numerator with an
extra factor $m^2$ is $O(q^4)$ and the normalization is of order
$q^3$. Since $V_{\mathcal P}=bI_{\mathcal P}-L_B$,
\[
V_{\mathcal P}^{-1}
=b^{-1}I_{\mathcal P}+b^{-2}L_B+b^{-3}L_B^2+O_A(q^3).
\]
Here $(L_B)_{uu}=2q-8q^2+o(q^2)$ and
$(L_B^2)_{uu}=10q^2+o(q^2)$. Projection onto $\mathcal P$ in the
second identity loses only $O_A(q^3)$, by the preceding approximation
of the cubic column. Substitution gives the same root coefficient and
column estimates; the $e$ entries again use the third moment.
Finally, the diagonal restored in $K$ and $K_T$ is positive and is
supported where $v_i^{-1}-1>q^{1-\eta}$. The tail-square bound limits
this support to $O(nq^{3+\eta'})$ sites for some fixed $\eta'>0$.
Thus restoring it is a positive perturbation of rank $o(nq^3)$.
\par\smallskip\noindent\textit{Resolvents away from the edge.}\enspace
Augment by one scalar $\beta$ for the negative spike $-ee^{\mathsf T}$,
so the augmented form contains
$\beta^2-2\beta e^{\mathsf T}x_E$. Put
$\Delta_{\mathrm{sp}}=q^{8/5}$ and define
\[
N=\operatorname{ran}\mathbf1_{\{\,2-W_E\le\Delta_{\mathrm{sp}}\,\}},
\qquad H=E\ominus N.
\]
Then $\dim N=O(n\Delta_{\mathrm{sp}}^{3/2})$. The high-block resolvent
for the unperturbed comparison, also after subtracting
$(1-\epsilon)q^2$, is
\(R=(2+O(q^2)-W_E)^{-1}_H\).
Its traces divided by \(n\) for the first and second powers are \(1+o(1),\,\asymp \Delta_{\mathrm{sp}}^{-1/2}\). On the conditional Haar orientation with the spectra satisfying the interlacings, except at arbitrarily high rate,
\[
\|R^{1/2}B_E R^{1/2}\|=o(1).
\]
Indeed dyadic blocks at depths $t_1,t_2\ge\Delta_{\mathrm{sp}}$ give
centered bilinear deviations at most
$Cq^{1-\eta}\max(t_1,t_2)^{3/4}$ by Haar concentration and nets.
After division by $\sqrt{t_1t_2}$, the dyadic sum is bounded by a
logarithmic factor times
$q^{1-\eta}\Delta_{\mathrm{sp}}^{-1/2}
=q^{1/5-\eta}$. The mean costs
$O(q^2/\Delta_{\mathrm{sp}})=O(q^{2/5})$. Both vanish for small
$\eta$; the sum includes a constant-width bulk bin.
\par\smallskip\noindent\textit{Uniform tests on the remaining coordinates.}\enspace
For remaining variables \(\gamma,x_N,\beta\), normalized by \(\gamma^2+\|x_N\|^2+\beta^2/\sqrt{\Delta_{\mathrm{sp}}}\le1\), write \(x=x_N+R(p\gamma+e\beta)\). The needed simultaneous Haar estimates are
\begin{equation}\label{prof:eq11}
\begin{split}
\|x\|&=O_A(1),\quad x^TB_E x=\mu_B\|x\|^2+o(q^2),\quad
f^T x= (2q^2+o(q^2))\gamma+o(q^2),\\
\|R^{1/2}(B_E x+f\gamma)\|^2&=\nu_B\|x\|^2+6q^2\gamma^2+o(q^2).
\end{split}
\end{equation}
Here little-oh can mean any specified fixed leading error.
For a fixed symmetric weight $T$, normalized Gaussian quadratic tails
and polarization give error
\[
C_B\frac{\sqrt k}{n}\sqrt{\operatorname{Tr}T^2}
+C_B\frac{k}{n}\|T\|
\]
for a Haar quadratic test on fixed orthonormal axes, at failure
$e^{-Bk}$. The same bound holds for an independent Gaussian column
with covariance $I_E/n$. A subspace test instead pays its dimension
through a net. Since $q\asymp_A r$, fixed changes of the constants
allow $nq^3$ in place of $k$.
Fix, before sampling $W_E$ and $w$, a fixed-dimensional space $S$
containing $e,c_3,p_0,f$. On $S$, compression of $R$ differs from its
trace mean by
$O_A(q^{3/2}\Delta_{\mathrm{sp}}^{-1/4}+q^3/\Delta_{\mathrm{sp}})$;
compression of $R^2$ has relatively vanishing error. Test $w$
separately, with zero mixed means, so $S$ stays deterministic. Also
$\|P_N|_S\|=O(\Delta_{\mathrm{sp}}^{3/4})$.
These estimates give $\|x\|=O_A(1)$ and
\[
f^{\mathsf T}Rp
=\frac{\operatorname{Tr}R}{n}f^{\mathsf T}p_0+o(q^2)
=2q^2+o(q^2).
\]
The terms $f^{\mathsf T}x_N$ and $\beta f^{\mathsf T}Re$ are $o(q^2)$
on the normalized variable ball. This proves the $f^{\mathsf T}x$
line of \eqref{prof:eq11}. The same tests give
\[
P_Sx=\frac{\operatorname{Tr}R}{n}\,\beta e+O_A(q).
\]
The fixed-axis estimates have identified the projection of the
optimizing vector. We next expose the low frame and the two resolved
columns. Rotations preserving these data control the remaining
components and hence the final weighted quadratic test.
Now expose $N,Rp,Re$ and put
$Y=S+N+\operatorname{span}(Rp,Re)$. Before this exposure, the part of
$Y$ beyond $S$ is invariant under rotations fixing $S$, hence is a
uniform subspace of $S^\perp$. Its compressions of $B_E^j$, $j=1,2$,
have deviations from their scalar means of size
$O_A(q^{j(1-\eta)}\Delta_{\mathrm{sp}}^{3/4})$; mixed compressions
with $S$ have the same bound and mean zero. Together with the estimates
on $B_Ee$, $e^{\mathsf T}B_Ee$, and $P_Sx$, this gives
\[
x^{\mathsf T}B_Ex=\mu_B\|x\|^2+o(q^2),\qquad
\|B_Ex\|^2=\nu_B\|x\|^2+o(q^2),
\]
as well as
$\|P_SB_Ex\|=o(q)$ and
$\|P_{Y\cap S^\perp}B_Ex\|=O_A(q^2)$.
After the exposure, only rotations fixing $Y$ pointwise remain.
Split $R=R_{\rm near}+R_{\rm far}$ at a small fixed depth $\zeta$.
The conditional mean of the far restriction on $Y^\perp$ is its
trace on that complement divided by its dimension. Since
$\dim Y=O(n\Delta_{\mathrm{sp}}^{3/2})$,
$\|R_{\rm far}\|\le C/\zeta$, and
$n^{-1}\Tr R_{\rm near}=O(\sqrt\zeta)$, this scalar is
\[
1+O(\sqrt\zeta)+O(\Delta_{\mathrm{sp}}^{3/2}/\zeta)+o(1).
\]
Residual Haar concentration and the low-dimensional nets make this
the quadratic coefficient on the needed vectors in $Y^\perp$.
The component of $B_Ex$ in $Y$ contributes $o(q^2)$ to the far
quadratic and mixed terms, by its two projection bounds and the fixed
far norm. The mixed term with $Rf$ has zero leading conditional mean;
$f^{\mathsf T}Rf$ is already covered by the fixed-axis test.
For the near part, $\|R_{\rm near}\|\le C/\Delta_{\mathrm{sp}}$ and
its compression to $S$ is small by the fixed-axis tests. Its action
on the $O_A(q^2)$ component in $Y\cap S^\perp$ is negligible at
scale $q^2$. On the remaining vectors, the conditional square-root
norm is bounded by a constant times
\[
\sqrt{\Tr R_{\rm near}/n}
+\sqrt{\dim Y/n}\,\|R_{\rm near}^{1/2}\|
=O(\zeta^{1/4}+\Delta_{\mathrm{sp}}^{1/4}).
\]
This is the Haar Lipschitz bound plus its net cost. First choose
$\zeta$ small, then the upper $q$ scale. These estimates prove the
last line of \eqref{prof:eq11}.
All these failures were estimated under the Gaussian/Haar conditional
law before imposing $\mathcal G_{\rm spec}$. On its intersection,
interlacing supplies the spectra of $W_E$ used in the trace and count
bounds. Integration in $y$ gives the stated exclusion.
\par\smallskip\noindent\textit{Schur complements, counts, and projections.}\enspace
Substitution of the simultaneous tests into the high-block Schur
complements gives the perturbed-minus-unperturbed difference
\[
\gamma^2 u^TB u+2\gamma f^Tx+x^T B_E x-\|R^{1/2}(B_E x+f\gamma)\|^2.
\]
By \eqref{prof:eq11} and $u^{\mathsf T}Bu\ge2q^2+o(q^2)$, the displayed
difference is bounded below on the normalized variable ball by
\[
(2+4-6)q^2\gamma^2+(\mu_B-\nu_B)\|x\|^2-o(q^2)=-o(q^2).
\]
This is the one-sided root-direction cancellation and the bulk cancellation.
To see the remaining positive margin, let
$x_{\rm phys}=\gamma u+x_E$. The shifted augmented base form is exactly
\[
x_{\rm phys}^{\mathsf T}
\bigl(D-(1-\epsilon)q^2I\bigr)x_{\rm phys}
+(\beta-e^{\mathsf T}x_E)^2.
\]
The original spectral bound makes its first term at least a fixed
multiple of $\epsilon q^2\|x_{\rm phys}\|^2$. At the high optimum,
$x_H=R(p\gamma+e\beta)$, and
\[
\|x_H\|^2\ge c\beta^2\|Re\|^2-C\gamma^2\|Rp\|^2,
\qquad \|Re\|^2\asymp\Delta_{\mathrm{sp}}^{-1/2},\quad
\|Rp\|=o(1).
\]
After absorption, the base complement therefore controls
$c\epsilon q^2(\gamma^2+\|x_N\|^2+
\beta^2/\sqrt{\Delta_{\mathrm{sp}}})$. The $o(q^2)$ difference
preserves this margin and proves the lower precision bound.
The one-sided estimate proves positivity. The trace and shell-density
arguments also need enough eigenvalues near the critical edge. Removing
the fixed-rank directions makes the bulk cancellation two-sided, so the
same calculation can now be repeated at dyadic thresholds.
After removing the fixed-rank $u,e$ interactions, the unaugmented
$N$-Schur complement for a truncated precision differs from the base
one by $o(q^2)$ on both sides, because $\mu_B-\nu_B=o(q^2)$.
At a shifted threshold $a\le\Delta_{\mathrm{sp}}/2$ the high block
remains positive. Schur inertia therefore identifies the number of
full eigenvalues below $a$ with the negative-eigenvalue count of the
low complement, up to the removed fixed ranks. The two-sided comparison
bounds this count by the $W_E$ edge counts at thresholds displaced by
$o(q^2)$.
The same estimates hold on a finite dyadic grid
$Cq^2\le a\le\Delta_{\mathrm{sp}}/2$: the shift is at most half the
high gap, so the high resolvents remain comparable and their normalized
traces change by $O(\sqrt{\Delta_{\mathrm{sp}}})$. This gives the
upper count $Cna^{3/2}$ and at least $cnq^3$ eigenvalues below
$Cq^2$ for a sufficiently large fixed $C$. Above a fixed multiple of
$\Delta_{\mathrm{sp}}$, the weighted dyadic form bound gives the same
upper count. Restoring the positive discarded diagonal preserves upper
counts and removes at most its rank $O(nq^{3+\eta'})=o(nq^3)$ from
the lower spread. Dyadic integration of these counts gives all
trace-square bounds.
For $K_o$, the correction to $D$ has fixed rank. At
$b+b^{-1}$, the semicircle transform is exactly $b$;
\eqref{prof:eq1} and the fixed-rank trace error $O(q^{-2})
=o(\sqrt{n/q})$ give its stated inverse trace. For a truncated
precision, write $A_0$ as that precision minus an
$O(q^{1-\eta})$ term and use the lower bound at $q^2$ to obtain
$\|A_0\mathsf K^{-1/2}\|\le Cq^{-\eta}$. Restoring the positive
diagonal preserves this bound by inverse order. The fixed-rank
structure gives the stated $\eta=0$ bound for $K_o$.
It remains to control variances in the distinguished directions. For
projection bounds at \(e,c_3\) first use \(D\). At \(R_1=(b+b^{-1}-W_E)^{-1}\), deterministic fixed-axis compressions have means \(1+O(q)\) and errors \(O_A(q)\), \(\|R_1 e\|^2\asymp_A 1/q\) (lower via modes at sufficiently enlarged width to get any desired rate). Thus \(1-e^T R_1 e\gtrsim_A q\) by the principal-block top eigenvalue bound at \(2+q^2/2\) and shifting. Also \(e^T R_1 p=O_A(q^{3/2})\), \(c_3^TR_1 p=O_A(q)\). Sherman-Morrison for the spike followed by inclusion of \(u\) (whose inverse variance bound after inclusion is the global operator bound) gives the orders stated.
For a truncated precision or $K_o$, the reduced Schur form controls
the base reduced form up to a factor two. View a projection variance
as the square of the dual norm of its physical source in the augmented
positive form. The high precision stays comparable to $R^{-1}$, and
its minimizing vector changes by
$-(R^{-1}+B|_H)^{-1}P_H(B_Ex+f\gamma)$.
For a source at $e$ or $c_3$, \eqref{prof:eq11} and its bounded
$R^{1/2}$ norm bound the change in testing this optimum by
\[
C_Aq\bigl(|\gamma|+\|x_N\|+
\Delta_{\mathrm{sp}}^{-1/4}|\beta|\bigr).
\]
The reduced energy controls $q^2$ times the square of this
parenthesis, so the additional squared dual norm is $O_A(1)$.
The remaining high square also costs $O_A(1)$. Adding these costs to
the base variances preserves the orders $q^{-1}$ for $e$ and $1$ for
$c_3$; the global lower precision bound supplies the $q^{-2}$ order
for $u$. Inverse order again permits restoration of the diagonal.
\end{proof}
\begin{corollary}[Uniqueness of the root on the retained event]
\label{prof:unique-root}
On the retained field event every root is regular and the root is unique.
The excluded root-count mass, and hence the exceptional field probability
under the spectral restriction, have the chosen exponential rate.
\end{corollary}
\begin{proof}
First remove the null set of critical target fields from
Lemma~\ref{prof:root-count}. For the remaining fields every root is
regular, and the probability of an excluded root is bounded by its
expected count. On the complementary field event,
$V^{1/2}K_TV^{1/2}
=I+V^{1/2}(bI-J-2mm^{\mathsf T}/n)V^{1/2}$ is positive.
The Jacobian $D\Phi_J$ is similar to this matrix and thus has positive
determinant at every root.
Properness and the identity homotopy outside a sufficiently large ball
give degree one. Hence the signed root count is one. With all signs
positive there is exactly one root. More generally, for a regular
target the numbers of positive and negative roots obey
$N_+-N_-=1$, so the excess multiplicity is
$N_++N_--1=2N_-$. This identity also bounds multiplicity errors by
the excluded negative-root count.
On the retained event the unique root is a measurable function of
$(J,h)$; extend it by zero elsewhere when a globally defined profile
is needed. Conditional on $(J,h)$ in the planted experiment, the
planted spin has law $\mu_h$. Thus an equivariant test involving this
profile may replace the plant by a posterior replica, with the
excluded event charged separately. This is a conditional posterior
identity under planted disorder; the later likelihood comparison
performs the transfer to ordinary disorder.
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-matrices.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-orbits.tex ====
\section{Changing orbit weights and locating the trace}\label{prof:orbits}
This section changes from the counting measure to Haar measure on an
orbit that fixes the three distinguished root directions. It then places
the Gaussian trace within one length standard deviation, using three
length densities rather than a direct trace expansion.
\subsection{The determinant weight}
Fix the root $y$ and the conditional matrix data modulo proper rotations
that fix $\mathcal P$ pointwise. For a proper rotation $O$ of
$\mathcal P^\perp$, put $R=I_{\mathcal P}\oplus O$. The orbit action is
\[
J\longmapsto RJR^{\mathsf T},\qquad
h=\Phi_J(y)\longmapsto Rh.
\]
It fixes $y,m,\mathbf1$ and preserves the root equation. The original
spectral restriction is constant on the orbit. Before the determinant
weight, disintegrate the root integral as $d\pi(\omega)\,d\mathrm{Haar}(O)$,
where $\omega$ denotes the base data and $d\pi$ includes the scalar
factors and, when imposed, the invariant spectral restriction.
The conditional matrix law is Haar in $O$ because all its fixed means
and axes lie in $\mathcal P$.
Let $\mathcal D_\omega$ be the set of orientations satisfying all
precision, trace, count, weighted, and projection diagnostics of
Proposition~\ref{prof:conditioned-precision} with the chosen tolerances.
This is stronger than mere Jacobian regularity. Call an orbit suitable
when $\mathrm{Haar}(\mathcal D_\omega^c)\le e^{-B_0k}$ for a large
fixed $B_0$. The integrated matrix exclusion and Markov's inequality
remove unsuitable base data at any desired smaller rate, by starting
with a larger raw rate. Lemma~\ref{prof:determinant-square} and
Cauchy--Schwarz give the same conclusion after determinant weighting.
On $\mathcal D_\omega$, $K_T\succeq cq^2I$. Extend scalar $\log t$
linearly below $cq^2$, and let $F(O)$ be the trace of this extended
function of $K_T$. It equals $\log\det K_T$ on $\mathcal D_\omega$.
Concavity of the extension and the trace-norm bound on the second
rotation derivative of $J$ give the upper Taylor bound
\[
F(O')\le F(O)+\langle\nabla F(O),\dot O\rangle
+Cq^{-2}\operatorname{dist}(O,O')^2
\]
along a minimizing geodesic, with the linear term evaluated on its
initial tangent. On diagnostic orientations,
$\|\nabla F\|\le\|[J,K_T^{-1}]\|_{\rm HS}=o(\sqrt n)$.
For the diagonally truncated precision, the commutator is bounded by
$Cq^{1-\eta}\sqrt{n/q}$. Restoring its positive diagonal changes the
inverse by a matrix of the discarded rank; the weighted inverse and
operator bounds give commutator cost at most
$Cq^{-1-\eta}\sqrt{\operatorname{rank}}$. Since that rank is
$O(nq^3)$ and $\eta$ is small, both costs are $o(\sqrt n)$.
Choose a median $m_F$ of $F$ on $\mathcal D_\omega$. Its high and low
anchor sets each have a fixed positive Haar mass. Distance to either
set has a sub-Gaussian tail at scale $n^{-1/2}$. The upper Taylor bound
from a low anchor bounds $F-m_F$ above; applying it from a diagnostic
orientation to a high anchor bounds $F-m_F$ below. At distance
$C_B\sqrt{k/n}$, the two errors are
\[
O_B\left(\sqrt k+\frac{k}{nq^2}\right)=o(k),
\]
and the complementary Haar mass is $e^{-Bk}$. At distance $O(n^{-1/2})$
the errors are bounded. Integrating the same distance tail shows that
$\E_{\rm Haar}e^{t(F-m_F)}$ is bounded for every fixed $t>0$.
This also controls the weighted contribution of exceptional diagnostic
orientations.
Write $w=\mathcal W(y,J)$ and
\[
w_0=e^{-nb^2/2}\det(V)e^{m_F}.
\]
On $\mathcal D_\omega$, $w=e^{-nb^2/2}\det(V)e^{F}$, so the preceding
bounds compare $w$ with the constant orbit weight $w_0$. A fixed
positive Haar fraction has $w\ge c w_0$; the total root-count bound
therefore gives $\int w_0\,d\pi\le n^C$ on suitable base data.
For every $0\le p(O)\le1$, every small fixed $\xi>0$, and every
needed fixed $B$, the comparison yields
\[
\int w_0\E_{\rm Haar}p\,d\pi
\le e^{\xi k}\int\E_{\rm Haar}(wp)\,d\pi
+e^{-Bk}\int w_0\,d\pi.
\]
Conversely,
\[
\int\E_{\rm Haar}(wp)\,d\pi
\le e^{\xi k}\int w_0\E_{\rm Haar}p\,d\pi
+e^{-Bk}\int w_0\,d\pi+\mathcal E_B,
\]
where $\mathcal E_B\le n^Ce^{-Bk}$ is the globally excluded weighted
mass of non-diagnostic orientations, after the raw matrix rate is
chosen. The comparison is integrated
over $\omega$; it does not require a lower weight on each spectral
fiber.
These inequalities permit further Markov exclusions over base data.
When the input is a field probability with a planted spin, the degree
identity in Corollary~\ref{prof:unique-root} charges excess multiplicity
to twice the negative-root count. Its integrated exclusion rate is
already available. Thus the same comparisons apply to those posterior
tests after this multiplicity error is included.
\subsection{Median trace control}
We need one further estimate on most orientations of each suitable
orbit:
\begin{equation}\label{prof:eq12}
|\operatorname{Tr}K^{-1}-nb|\le C\sqrt{n/q}.
\end{equation}
The precision estimates give the fluctuation scale of Gaussian length,
but not the location of its mean. We compare its density at the target
and one fluctuation on either side. The lemma below shows that these
three densities locate the trace.
Put $L=\sqrt{n/q}$ and $y_w=nb+wL$ for $w=0,\pm1$; here $y_w$ is a
scalar squared length. On $\mathcal P^\perp$, let $\gamma_V$ be the
normalized law $N(0,V)$ conditioned on zero projection onto
$\mathcal P$, and let $\gamma_b=N(0,bI_{\mathcal P^\perp})$. Denote
their squared-length densities by $\ell_V$ and $\ell_b$. Define on
every orientation
\[
A_w(O)=\ell_V(y_w)\,
\E_{\gamma_V}\!\left[
e^{z^{\mathsf T}(J-bI)z/2}\mid\|z\|^2=y_w\right].
\]
This is an unnormalized weighted length density. On a diagnostic
orientation the full weighted integral is finite; writing its mass as
$Z_V^\perp(O)$ gives
\[
A_w(O)=Z_V^\perp(O)f_{\widehat C_O}(y_w),\qquad
\widehat C_O=(K|_{\mathcal P^\perp})^{-1},
\]
where $f_C$ denotes the squared-length density under $N(0,C)$.
The fixed-length definition of $A_w$ remains meaningful even on the
other orientations.
For the scalar prior, the normalized weighted covariance is
\[
\widehat C_o=\bigl(((b+b^{-1})I-J)|_{\mathcal P^\perp}\bigr)^{-1}
=(K_o|_{\mathcal P^\perp})^{-1}.
\]
Let $Z_o^\perp$ be its unnormalized weighted mass and
$f_o=f_{\widehat C_o}$. Both are invariant under the remaining
rotation. Haar averaging at each fixed length gives the exact identity
\[
\overline A_w:=\E_O A_w(O)
=Z_o^\perp f_o(y_w)\frac{\ell_V(y_w)}{\ell_b(y_w)}.
\]
The ratios $\overline A_{\pm1}/\overline A_0$ are bounded above and
below. For $f_o$, this follows from
$\operatorname{Tr}\widehat C_o=nb+o(L)$, its trace-square scale $L^2$,
and the Fourier local limit supplied by the spread. For the prior
ratio, both traces are $nb+O(1)$, the top eigenvalues are bounded, and
the trace-square discrepancy is $O(nq^2+1)$. The scalar length-tilt
expansion therefore gives
\[
\log\frac{\ell_V(y_w)/\ell_b(y_w)}
{\ell_V(y_0)/\ell_b(y_0)}
=O\!\left(\frac{L^2q^2}{n}+\frac{L^3}{n^2}+o(1)\right)=O(1).
\]
The normalizer $Z_o^\perp$ is the same at all three lengths.
We next show $\E_O(A_w/\overline A_w)^2\le1+o(1)$. Let
$\rho_V(T\mid y_w,y_w)$ and $\rho_b(T\mid y_w,y_w)$ be the overlap
densities of two independent draws from the corresponding unweighted
priors, conditioned on those lengths. Let $\mu_{o,w}$ be the scalar
prior at length $y_w$, weighted by the common angular energy and
normalized. Haar averaging of a pair at fixed overlap gives
\[
\E_O\left(\frac{A_w}{\overline A_w}\right)^2
=\E_{\mu_{o,w}^{\otimes2}}
\frac{\rho_V(T\mid y_w,y_w)}{\rho_b(T\mid y_w,y_w)},
\qquad T=z_1^{\mathsf T}z_2.
\]
Thus the positive multiplier is a ratio of \emph{prior} densities,
while the expectation and its tails use the \emph{energy-tilted}
reference law.
To estimate the multiplier, tilt each prior length to $y_w$; its
scalar tilt is $O(L/n)$. At zero cross tilt the overlap is symmetric,
and its covariance with either length is zero. Its conditional
curvature is therefore $\operatorname{Tr}\widetilde V^2$, where
$\widetilde V$ is the length-tilted covariance. The two tilted traces
are the same target, and their trace squares differ relatively by
$O(q^2+1/n)$: the variable prior differs from scalar covariance by
normalized Hilbert--Schmidt size $O(q+1/\sqrt n)$, and the scalar tilt
changes eigenvalues by a uniformly Lipschitz function. Analytic
expansion of the three quadratic log moments, with their nondegenerate
Hessian, and Fourier inversion of the central prefactors give, for
$|T|/n$ small,
\[
\log\frac{\rho_V(T\mid y_w,y_w)}{\rho_b(T\mid y_w,y_w)}
\le o(1)+C(q^2+1/n)\frac{T^2}{n}+C\frac{T^4}{n^3}.
\]
Odd cross derivatives vanish by symmetry; the prefactor error is
$O(q+|T|/n)$ in the same small-scale convention. Conditioning the
initial projections at zero changes only a fixed rank.
Here is the tail comparison that makes this local bound integrable.
Put $U=T/L$ and $N=nq^3$. The positive quadratic and quartic errors
become
\[
C\left((q+(nq)^{-1})U^2+\frac{U^4}{nq^2}\right).
\]
Under $\mu_{o,w}^{\otimes2}$, quadratic exponentiation and the two
length-density bounds give a Gaussian tail in $U$ up to
$|U|\le c\sqrt N$. There the quartic error is at most $C c^2qU^2$,
so the Gaussian tail dominates after the upper scale is decreased;
for fixed $U$ the multiplier tends to one. For
$a=|T|/n$ from order $q$ to a small fixed cutoff, the reference
length-deficit estimate of Section~\ref{prof:subsec:replica-integrals}
gives $e^{-cna^3}$. Dividing the positive exponent by $na^3$ leaves
$O(q^2/a+a+1/(na))$, which is small in this range after the cutoff
and upper scale are chosen. These two ranges give uniform integrability
through small overlaps.
For a fixed larger overlap, simultaneous quadratic Chernoff bounds for
the unweighted priors, conditional on both lengths, give at least the
spherical fixed-overlap rate $-\tfrac12\log(1-a^2)$ up to an $o(1)$
loss per coordinate as the upper scale decreases. This rate is strictly
larger than $a^2/2$ away from zero. Near the parallel endpoints, fixed
tilts chosen before $n$ supply the finite strict rate gap needed below.
Exact length densities under these tilts cost at most a polynomial by
Fourier inversion, using damping from typical sites or a bounded-eigenvalue
block; the preceding $O(L)$ prior length deviations cost subexponentially.
The angular pair upper bound of
Section~\ref{prof:subsec:replica-integrals} costs only $a^2/2$ after
normalization: the reference angular integral per draw is at least
$\exp\{n(1/4-o(1))\}$. To verify that lower bound, use the Gaussian
formula on $\mathcal P^\perp$ at precision
$((b+b^{-1})I-J)|_{\mathcal P^\perp}$. Its centered length density
costs only a polynomial, its normalized log determinant is at most
$1/2+o(1)$ by the semicircle bound and fixed-rank interlacing, and
$y_w/n=1+o(1)$. The strict rate gap controls the remaining overlaps.
Thus the displayed second moment is $1+o(1)$.
The first and second moments imply that all three ratios
$A_w/\overline A_w$ are close to one outside as small a fixed Haar
mass as desired. On diagnostic orientations the common factor
$Z_V^\perp(O)$ cancels, so
$f_{\widehat C_O}(y_{\pm1})/f_{\widehat C_O}(y_0)$ is bounded above.
The next lemma now applies. Its operator and trace-square hypotheses
follow from Proposition~\ref{prof:conditioned-precision} after removing
three dimensions. Since $v_i\ge1/2$ outside at most $2nq$ sites, the
coordinate subspace supported on $\{i:v_i\ge1/2\}$ has intersection
with $\mathcal P^\perp$ of dimension at least $n-2nq-3$. The compressed
precision has bounded norm on this intersection, so the min--max principle
gives a fixed positive fraction of covariance eigenvalues bounded below.
Finally, fixed-rank interlacing gives
$|\operatorname{Tr}K^{-1}-\operatorname{Tr}\widehat C_O|
\le Cq^{-2}=o(L)$. The lemma proves \eqref{prof:eq12}.
\begin{lemma}[Three length densities locate a Gaussian trace]
\label{prof:trace-from-three-densities}
Fix positive constants $c_0,C_0,M$. Let $0<q\le q_0$ and
$nq^3\ge\log^{10}n$, and put $L=\sqrt{n/q}$. Let $C$ be a positive
definite matrix of dimension $n-O(1)$ such that
\[
\|C\|\le C_0q^{-2},\qquad
c_0L^2\le\operatorname{Tr}C^2\le C_0L^2.
\]
Assume also that at least $c_0n$ eigenvalues of $C$ exceed $c_0$.
Let $f$ be the density of $\|C^{1/2}g\|^2$, with $g$ standard Gaussian,
and let $x_0\in[c_0n,C_0n]$. If
\[
f(x_0+L)\le M f(x_0),\qquad
f(x_0-L)\le M f(x_0),
\]
then $|\operatorname{Tr}C-x_0|\le C_1L$, where $C_1$ depends only on
the fixed constants. The assertion holds for all sufficiently large $n$.
\end{lemma}
\begin{proof}
Write $m(\lambda)=\operatorname{Tr}(C^{-1}+\lambda I)^{-1}$ on
$\lambda>-\|C\|^{-1}$ and $C_\lambda=(C^{-1}+\lambda I)^{-1}$.
The function $m$ decreases continuously from infinity to zero, and
\[
m'(\lambda)=-\operatorname{Tr}C_\lambda^2.
\]
If $|m(0)-x_0|\le L$, there is nothing to prove. Otherwise set
$\varepsilon=\operatorname{sign}(m(0)-x_0)$ and
$x_1=x_0+\varepsilon L$. There is a unique $\lambda$ with
$m(\lambda)=x_1$, and $\varepsilon\lambda>0$.
Exponential tilting of the squared length gives its density $f_\lambda$
under covariance $C_\lambda$ and the identity
\[
\frac{f(x_1)}{f(x_0)}
=e^{|\lambda|L/2}\frac{f_\lambda(x_1)}{f_\lambda(x_0)}.
\]
This identity treats the two signs with exactly the same inequality.
First we bound the tilt without assuming a local central limit theorem
near the final pole. This coarse bound will place the covariance in a
region comparable to its original value; sharper density estimates can
then improve the logarithmic tilt bound to a constant. Since
$\operatorname{Tr}C_\lambda=x_1\asymp n$, its standard deviation is
at most $\sqrt2x_1\le Cn$. The mean-density bound of
Lemma~\ref{prof:quadratic-mean-density} gives
$f_\lambda(x_1)\ge c/n$.
If $\lambda>0$, the inequality $m(\lambda)\le n/\lambda$ implies
$\lambda\le C$; if $\lambda<0$, every eigenvalue increases.
In both cases at least $c n$ eigenvalues of $C_\lambda$ are bounded
below by a fixed positive constant. The quadratic Gaussian product
formula therefore gives
\[
|\mathbb E e^{it\|C_\lambda^{1/2}g\|^2}|
\le (1+c t^2)^{-cn},
\qquad \sup_x f_\lambda(x)\le Cn^{-1/2}.
\]
Consequently
$M\ge c n^{-1/2}e^{|\lambda|L/2}$, so
$|\lambda|L\le C\log n$.
This is the promised control near a pole: no local central limit theorem
is being assumed there, and the density loss is only polynomial.
Choose $a>0$ so small that $aC_0<1/2$. Because
$q^2L=\sqrt{nq^3}\ge\log^5 n$, the preceding bound forces
$|\lambda|<a q^2$. Thus $C_\theta$ is comparable to $C$ for every
$\theta$ between zero and $\lambda$, and
$\operatorname{Tr}C_\theta^2\asymp L^2$.
We can now use the spectral spread at the tilted covariance. For these
tilts, the largest eigenvalue divided by the square root of
the trace square is $O((nq^3)^{-1/2})=o(1)$.
To check the Fourier bound explicitly, if $a_i$ are the eigenvalues of
$C_\lambda$ and $\sigma^2=2\sum_i a_i^2$, concavity of
$x\mapsto\log(1+ux)$ on $[0,\max_i a_i^2]$ gives
\[
\prod_i(1+4t^2a_i^2/\sigma^2)^{-1/4}
\le\left(1+\frac{4t^2a_{\max}^2}{\sigma^2}\right)
^{-\sigma^2/(8a_{\max}^2)}.
\]
Since $a_{\max}^2/\sigma^2\le1/16$ for large $n$, the right-hand side is
bounded by $(1+t^2/4)^{-2}$, an integrable function independent of the
parameters.
Hence its density has supremum at
most $C/L$. Its density at the mean is at least $c/L$ by the same
mean-density lemma. Applying these bounds to $f_\lambda$ in the ratio
identity improves the preceding estimate to $|\lambda|L\le C$.
Finally,
\[
|m(0)-x_0|
\le L+|m(0)-m(\lambda)|
\le L+\int_0^{|\lambda|}C L^2\,du\le C_1L.
\]
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-orbits.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-positive.tex ====
\section{Positive-time covariance and macroscopic entropy inputs}
\label{prof:positive-inputs}
We establish the fixed-positive-time estimates used to start the backward
induction, and the estimates for changes of mean needed at macroscopic
entropy. The matrix in a heat-bath conditional mean is
$J^\circ=J-\operatorname{diag}(J_{11},\ldots,J_{nn})$. Thus, in this
section, put
\[
\begin{gathered}
m(y)=\tanh y,\quad q(y)=n^{-1}\|m(y)\|^2,\quad b(y)=1-q(y),\\
V_y=\operatorname{diag}(1-m_i(y)^2),\qquad
F_h(y)=y-h+(b(y)I-J^\circ)m(y).
\end{gathered}
\]
For a nonnegative diagonal matrix $A$ define
\[
\mathcal H(y,A)=I+A^{1/2}
\left(b(y)I-J^\circ-\frac{2m(y)m(y)^{\mathsf T}}n\right)A^{1/2}.
\]
The relevant stability property, with fixed positive constants, is
\begin{equation}\label{prof:eq26}
\left.
\begin{gathered}
\|F_h(y)\|\le\rho_0\sqrt n,\qquad
0\le A\le(1+\eta_0)I,\\
n^{-1/2}\|A-V_y\|_{\mathrm F}\le\epsilon_A
\end{gathered}\right\}
\quad\Longrightarrow\quad \mathcal H(y,A)\succeq c_0 I.
\end{equation}
Only diagonal $A$ are quantified in this implication.
The companion high-temperature article
\cite{AcceptedGap} proves the requisite deterministic residual
identities and posterior consequences. Its probabilistic matrix and
stable-field statements are stated for $j=\beta^2<1$; they do not directly
supply \eqref{prof:eq26} at $j=1$. We give the endpoint replacements below.
In particular, none of the following arguments uses its spectral-gap
theorem at the critical temperature.
\subsection{Inverse words with a strict norm--trace margin}
For a diagonal $A\ge0$ write $\chi_A=n^{-1}\operatorname{Tr}A$ and set
\[
S_A(z;M)=I+z^2\chi_AA-zA^{1/2}MA^{1/2},\qquad
K_A(z;M)=(I-zAM+z^2\chi_AA)^{-1}.
\]
Fix $a_+<\infty$ and $\kappa\in(0,1)$, and define the closed parameter set
\[
\mathcal A_\kappa=
\{A\text{ diagonal}:0\le A\le a_+I,\quad \|A\|\chi_A\le1-\kappa\}.
\]
An ordinary word is a finite product of bounded diagonal matrices and
copies of $M$. An inverse word has, in addition, at most one factor
$K_A(1;M)$. Its diagonal prediction is obtained by noncrossing
contractions $MDM\mapsto n^{-1}\operatorname{Tr}(D)I$. For an inverse
factor use its formal series in $z$ before contracting; the prediction
at $z=1$ is well defined by the polynomial assertion in the next lemma.
The associated normalized trace prediction is cyclic: rotating the
factors of a word does not change
$n^{-1}\operatorname{Tr}\mathsf P(Q)$, coefficient by coefficient
in the formal variable $z$.
\begin{lemma}[Restricted endpoint word estimates]
\label{prof:endpoint-words}
Let $W$ be GOE with off-diagonal variance $1/n$ and diagonal variance
$2/n$, and let $W^\circ$ be its zero-diagonal version. Fix a word length
$L$, a bound $M_0$ on ordinary diagonal factors, and an inverse margin
$c>0$. Except on an event of probability $Ce^{-c_1n}$, simultaneously
for $M=W,W^\circ$ and all such words $Q$,
\[
\|Q\|+
\left(\sum_{i\ne j}|Q_{ij}|^4\right)^{1/4}
+\|\operatorname{diag}(Q-\mathsf P(Q))\|_2\le C.
\]
For inverse words this assertion is restricted to $A\in\mathcal A_\kappa$
and $S_A(1;M)\succeq cI$. All diagonal parameters and permitted
word choices within the fixed bounds may be selected after observing
$M$; the bounds $L,M_0,c,a_+,\kappa$ are fixed beforehand.
Constants depend on the displayed fixed bounds, including $\kappa$.
The prediction for a word with $r$ ordinary matrix letters outside its
inverse is a polynomial in $z$ of degree at most $r$.
For each deterministic $A\in\mathcal A_\kappa$ there is a constant
$c_\kappa>0$ such that
$S_A(z;W)\succeq c_\kappa I$ for all $z\in[0,1]$ except with probability
$Ce^{-c_2n}$. For any fixed finite family of deterministic bounded vectors,
the bilinear deterministic equivalents are, to any fixed positive
accuracy,
\[
G_A\simeq A,\qquad G_AW\simeq\chi_A A,\qquad
WG_AW\simeq\chi_A I+\chi_A^2 A,
\quad G_A=A^{1/2}S_A(1;W)^{-1}A^{1/2},
\]
with the same exponential form of exceptional probability. Here each
comparison concerns only the specified bilinear forms.
\end{lemma}
\begin{proof}
We give the endpoint changes to Lemmas 3.1--3.4 of
\cite{AcceptedGap}, including the uniformity needed below. The proof
separates fixed parameters from adaptive ones. A strict norm--trace bound
gives the deterministic-diagonal path margin and controls expectations.
Chaining makes the word errors simultaneous over the diagonal parameters.
An inverse chosen after observing the matrix is used only where its own
stated positive margin holds.
For $\|p\|=1$ and $w=A^{1/2}p$, Gaussian comparison bounds the expected
supremum of
$zw^{\mathsf T}Ww-z^2\chi_A\|w\|^2$ by that of
\[
2zT_g v-z^2\chi_Av^2,
\qquad 0\le v\le\|A\|^{1/2},\qquad
T_g=n^{-1/2}\|A^{1/2}g\|.
\]
Indeed the increment difference between the comparison process
$2n^{-1/2}\|w\|g^{\mathsf T}w$ and $w^{\mathsf T}Ww$ is
\[
\frac2n\left[(\|w\|^2-\|w'\|^2)^2
+2(\|w\|\|w'\|-w^{\mathsf T}w')^2\right]\ge0.
\]
Also $\mathbb E T_g^2=\chi_A$ and
$\operatorname{Var}(T_g^2)\le2a_+^2/n$, so replacing $T_g$ by
$\sqrt{\chi_A}$ costs $o(1)$ uniformly in $A$.
The supremum then has
integrand $2d-d^2$, where
\[
0\le d=z\sqrt{\chi_A}v\le\sqrt{\|A\|\chi_A}
\le\sqrt{1-\kappa}.
\]
Its distance below one is at least
$\delta_\kappa=(1-\sqrt{1-\kappa})^2$.
The supremum is $a_+$-Lipschitz in $W$ for the Frobenius metric.
Gaussian concentration, the GOE norm tail, and a fixed mesh in $z$
give the asserted path margin, for example with
$c_\kappa=\delta_\kappa/4$.
To prove expectations without conditioning on a matrix event, project
$W$ onto a fixed operator-norm ball and replace the symmetric inverse
by a smooth bounded functional-calculus function agreeing with $x^{-1}$
on an interval containing all spectra with either required margin.
Use
\[
K_A=I+A^{1/2}S_A^{-1}A^{1/2}(zW-z^2\chi_AI),\qquad
K_AA=A^{1/2}S_A^{-1}A^{1/2}.
\]
This defines bounded truncated words. For fixed length their
Frobenius Lipschitz constants in $W$ are bounded, and the increment
between parameter tuples at sup-norm distance $\Delta$ has
Frobenius Lipschitz constant at most $C\sqrt\Delta$.
These follow by the
product rule, $\|A^{1/2}-B^{1/2}\|\le\|A-B\|^{1/2}$, and the
Fourier formula for differentiating a smooth matrix function. Thus
entry variances are $O(n^{-1})$, normalized-trace variances are
$O(n^{-2})$, and fourth centered entry moments are $O(n^{-2})$.
These bounds gain respectively $\Delta,\Delta,\Delta^2$ for increments.
The fixed-parameter agreement event has exponentially small complement.
Gaussian integration by parts therefore gives, with errors $O(n^{-1})$
uniformly in $i,z,A$, the same loop equations as in Lemma 3.3 of the
companion article. Writing $a_i=A_{ii}$,
$k_i=\mathbb E(K_A)_{ii}$, $r=n^{-1}\sum_i a_i k_i$, and $h=r-\chi_A$,
these equations are
\[
\mathbb E(WK_A)_{ii}=zr k_i+O(n^{-1}),\qquad
k_i=1+z^2a_i h k_i+O(n^{-1}).
\]
All expectations here may use the bounded truncations. Their
integration-by-parts error is exponentially small times a fixed
polynomial in $n$, because both true and formal differentials are
bounded and agree on the path event.
If $|h|\le\delta$, the second
equation gives $k_i=1+O(\delta+n^{-1})$. Consequently
\[
\left|z^2n^{-1}\sum_i a_i^2k_i\right|
\le(1+C\delta+C/n)n^{-1}\sum_i a_i^2
\le(1+C\delta+C/n)(1-\kappa)\le1-\kappa/2
\]
for fixed sufficiently small $\delta$. Averaging the loop equation
with weights $a_i$ gives $|h|\le C/n$. Continuity from $h(0)=0$
prevents an exit from $|h|\le\delta$. It follows that
$k_i=1+O(n^{-1})$ and $\mathbb E(WK_A)_{ii}=z\chi_A+O(n^{-1})$.
For completeness, the higher-word recursion has no new stability
requirement. Mark an ordinary matrix letter, integrate it by parts,
and let its differentiated partner split the word as $U\boxed W
V\boxed W Q$. The leading term is
\[
\mathbb E(UQ)_{ii}\,n^{-1}\mathbb E\operatorname{Tr}V;
\]
the reversed contraction is $n^{-1}\mathbb E(UV^{\mathsf T}Q)_{ii}$
and is $O(n^{-1})$. Covariance errors are $O(n^{-1})$ by the preceding
variance bounds. If the derivative meets the inverse, use
$\mathrm dK_A[E]=zK_AAEK_A$.
Splitting then puts one copy of $K_A$
in each smaller word, whose total number of ordinary letters is reduced
by one. Otherwise it is reduced by two. The base formal prediction
is $\mathsf P(K_A)=I$: for each nonempty noncrossing-paired chain,
summing over subsets of its originally adjacent pairs gives
$\sum_T(-1)^{|T|}=0$. Induction proves the polynomial assertion and
entrywise expectation error $O(n^{-1})$. Every inverse in this
calculation has the original coefficient $A$; no enlarged diagonal
parameter set is used.
For a normalized trace, the noncrossing face formula has no marked
starting position on the trace cycle. Rotating the word permutes its
faces and contractions, proving the stated cyclic invariance
coefficient by coefficient, also after the formal inverse expansion.
To make the result simultaneous, take sup-norm meshes whose points lie
in $\mathcal A_\kappa$ and in the bounded boxes for the other diagonal
parameters. At scale $2^{-l}$ their cardinalities are at most
$\exp(C_L(l+1)n)$. The truncated estimates hold in the ambient box,
so mesh increments are legitimate even though no convexity of
$\mathcal A_\kappa$ is assumed. The diagonal $\ell^2$ and off-diagonal
$\ell^4$ seminorms of centered increments have expectations at most
$C2^{-l/2}$ and concentration scale $C\sqrt{2^{-l}/n}$.
A union bound bounds every increment by
$C\sqrt{l+1}\,2^{-l/2}$, with summable failure probabilities
$Ce^{-c(l+1)n}$. Sum the increments and the fixed-parameter biases.
Sign conjugation makes all off-diagonal expectations zero.
This proves the asserted simultaneous estimates for $W$.
Since $\|\operatorname{diag}W\|_{\mathrm F}$ is bounded except with
exponentially small probability, the truncated Frobenius Lipschitz
estimate proves the result for $W^\circ$ as well. Restriction to the
stated positive margin restores the exact inverse factors.
Finally sign conjugation and the loop equations give the diagonal
expectations of $G_A$, $G_AW$ and $WG_AW$. They are respectively $A$,
$\chi_AA$ and $\chi_AI+\chi_A^2A$, to $O(n^{-1})$ in each entry. Their truncated
bilinear forms are uniformly Lipschitz. Gaussian concentration proves
the last assertion for each fixed finite list of vectors.
\end{proof}
\begin{lemma}[Restricted determinant bound]\label{prof:endpoint-logdet}
Fix $\kappa,\varepsilon>0$, a rank bound $r_0$, and a norm bound $M_2$.
There is $\gamma_0>0$ such that for each fixed
$0<\gamma\le\gamma_0$ and each fixed $B>0$, with probability at least
$1-e^{-Bn}$ for sufficiently large $n$, simultaneously for
\[
V\ \text{diagonal},\qquad 0\le V\le I,\qquad
b=n^{-1}\operatorname{Tr}V\le1-\kappa,
\qquad \operatorname{rank}E\le r_0,\quad\|E\|\le M_2,
\]
the matrix $L=I+(bI-W)V+E$ satisfies
\[
n^{-1}\log\det(L^{\mathsf T}L+\gamma I)\le b^2+\varepsilon.
\]
The choices of $\gamma_0$ and the threshold on $n$ may depend on the
fixed margins. The threshold on $n$ may also depend on $B$.
\end{lemma}
\begin{proof}
For deterministic $V$ use Lemma~\ref{prof:endpoint-words} along
$L_z=I+z^2bV-zWV$. On its path event,
\[
\frac{\mathrm d}{\mathrm dz}\frac1n\log\det L_z
=2zb\,\frac{\operatorname{Tr}(K_V(z)V)}n
-\frac{\operatorname{Tr}(WK_V(z)V)}n.
\]
The expectation is $zb^2+O(n^{-1})$, by the loop calculation.
Using a fixed norm cutoff in the path event $\mathcal E_V$, integration
from zero to one gives
$\mathbb E[\mathbf1_{\mathcal E_V}2n^{-1}\log\det L_1]
=b^2+O(n^{-1})$. The inverse norm there is bounded; regularization costs
$C_\kappa\gamma$. Off this event, the regularized logarithm is bounded
in absolute value by $|\log\gamma|+C+2\log(1+\|W\|)$ and its expected
contribution tends exponentially to zero. Hence the regularized mean
is at most $b^2+C_\kappa\gamma+o(1)$ uniformly in $V$.
The map $M\mapsto n^{-1}\log\det(M^{\mathsf T}M+\gamma I)$ has
Frobenius Lipschitz constant $1/\sqrt{n\gamma}$. Gaussian
concentration thus gives a tail $2\exp(-\gamma n^2u^2/4)$ at deviation
$u$. Choose $\gamma_0$ so $C_\kappa\gamma_0<\varepsilon/8$.
A fixed sup-norm mesh of the restricted diagonal set has size
$\exp(O(n))$. On $\|W\|\le R$, a mesh spacing depending on
$R,\gamma,\varepsilon$ controls both the logarithm and $b^2$ to
$\varepsilon/4$. For this last mesh step choose a possibly larger $R$ so that its
exceptional probability is at most $e^{-(B+1)n}$.
This choice does not
change the preceding expectation estimate or its regularizer threshold. The $n^2$ concentration bound
pays for this mesh for sufficiently large $n$. Finally a rank-$r_0$
perturbation changes the logarithm by at most
$M_2\sqrt{r_0/(n\gamma)}$. This proves the statement, including its
arbitrarily large fixed exponential budget. The exponentially small
path exception was used only in the mean estimate; its exponent does
not limit this final budget.
\end{proof}
The diagonality restriction is essential for this simultaneous
statement. If arbitrary positive contractions chosen after \(W\)
were allowed, take \(V=\mathbf1_{\{W\le0\}}\) and \(E=0\).
Then \(b\to1/2\), so the constraint holds, for example, at
\(\kappa=1/4\), while every eigenvalue of \(L\) on the
range of \(V\) is at least \(1+b\), and the other eigenvalues
equal one. For every \(\gamma>0\),
\[
\frac1n\log\det(L^TL+\gamma I)
\ge2b\log(1+b)\longrightarrow\log(3/2)>1/4.
\]
Thus the asserted bound with arbitrarily small \(\varepsilon\)
would fail in that larger class. The application below uses only
the diagonal matrix \(V_y\).
\subsection{Scalar equality and stable fields at positive times}
\begin{lemma}[Endpoint scalar rate]\label{prof:endpoint-scalar}
Let $P$ have finite second moment, put
$q=\mathbb E_P\tanh^2 y$, $b=1-q$, and
$a=\mathbb E_P[(y-d)\tanh y]$. For $\nu>0$ with $\nu\ge q$,
\[
D(P\Vert N(d,\nu))\ge\frac{(a-\nu b)^2}{2\nu(\nu+q)}.
\]
Equality holds exactly when $P=N(d,\nu)$. At every observation time
$s>0$ the consistent equality law, with
$d=s+\mathbb E_P\tanh y$ and $\nu=s+q$, is uniquely
$N(v_*(s),v_*(s))$, where
\[
v_*(s)=s+\mathbb E_{N(v_*(s),v_*(s))}\tanh y.
\]
On compact positive time intervals this curve is continuous and its
$q$ is bounded away from zero. The scalar integral and rate conclusions
of Lemma 5.4(i), (ii), and (iv) of \cite{AcceptedGap} therefore
hold at $j=1$ on such intervals. In particular, outside any fixed
neighborhood of the equality curve, the smoothed scalar integral has
strictly negative rate for sufficiently small smoothing variance.
\end{lemma}
\begin{proof}
Suppose the entropy is finite; then $0<q<1$. Set
$k=q/\nu\in(0,1]$ and $C=a/\nu$. The change of variable
$T_u(y)=y-u\tanh y$, $u<1$, gives
\[
D(P\Vert N(d,\nu))
\ge u(C-b)-\frac{k u^2}{2}-b[-\log(1-u)-u].
\]
This identity is the entropy change-of-variables calculation in
Lemma 4.1 of the companion article and uses no strict restriction on
$j$. Put $h(u)=u^2/2-b[-\log(1-u)-u]$. Since $b=1-q<1$,
\[
h(u)>0\quad(u\le q,\ u\ne0),\qquad h(q)\ge q^3/6.
\]
For negative $u$ use $-\log(1-u)-u\le u^2/2$; for positive $u\le q$
use the increasing series ratio and
$h(q)=\sum_{l\ge3}q^l/[l(l-1)]$.
If $(C-b)/(1+k)\le q$, substitution proves strict inequality unless
$C=b$. Otherwise let $C_0=1+kq$ and
$C_1=C_0+\sqrt{(1+k)/3}\,q^{3/2}$. The choice $u=q$ proves strictness
for $C_0<C<C_1$. For $C\ge C_1$, variance comparison and
Cauchy--Schwarz give
\[
2D(P\Vert N(d,\nu))\ge C^2/k-1-\log(C^2/k).
\]
Subtract the target and write
$f(C)=C^2/k-1-\log(C^2/k)-(C-b)^2/(1+k)$. Direct differentiation gives
\[
f(C_0)\ge-2q^3/3,\qquad
f'(C_0)\ge\frac{2q}{1+q},\qquad f''(C)\ge1.
\]
Consequently
\[
f(C_1)\ge q^3\left[-\frac23+
\frac{2}{\sqrt3(1+q)\sqrt q}+\frac16\right]>0.
\]
Since $f'$ is positive and increasing after $C_0$, this finishes the
strict cases. If $C=b$, the target is zero and equality is equivalent
to vanishing entropy. Gaussian integration by parts verifies equality
for the stated Gaussian law.
For a fixed variance $\nu>0$, the function
$d\mapsto d-\mathbb E_{N(d,\nu)}\tanh y$ is strictly increasing,
onto $\mathbb R$, and zero at $d=0$; its derivative is
$\mathbb E\tanh^2y>0$. Thus its solution at $s\ge0$ is nonnegative.
Let $H_\nu(d)=\mathbb E_{N(d,\nu)}(\tanh y-\tanh^2y)$.
Gaussian reflection gives $H_\nu(\nu)=0$ and
$\mathbb E_{N(d,\nu)}[\tanh y\,\operatorname{sech}^2y]\ge0$ for $d\ge0$.
Hence $|H_\nu'(d)|<1$ for every finite $d\ge0$: the upper bound is
strict since $\mathbb E\operatorname{sech}^2y<1$, and the pointwise
lower bound of $(1-m^2)(1-2m)$ is strictly greater than $-1$.
The strict derivative bound is uniform on the compact segment joining
$d$ and $\nu$. Consistency gives $d-\nu=H_\nu(d)$ and therefore forces
$d=\nu$.
Finally $F(v)=\mathbb E_{N(v,v)}\tanh y$ satisfies $F(0)=0$ and
\[
F'(v)=\mathbb E_{N(v,v)}[(1-m^2)(1-m)]\in[0,1)
\qquad(v>0).
\]
The upper bound uses the same reflection identity. Thus $v-F(v)$
is strictly increasing from zero to infinity, since $0\le F(v)\le1$.
It has a unique inverse at every $s>0$, with $s\le v_*(s)\le s+1$.
This proves continuity and the positive lower bound for $q$ on each
compact positive interval. At $s=0$ there is no equality law with
$q>0$: its mean equation forces $d=0$, while
$\mathbb E_{N(0,q)}\tanh^2 y<q$.
We specify why the integral consequences extend as stated. In the
smoothed integral the parameters are
$d=s+M$, $\nu=s+\sigma^2+q$, and the exponent is
$\psi=(a-\nu b)^2/[2\nu(\nu+q)]$.
With $\varphi_{d,\nu}$ the density of $N(d,\nu)$, the scalar
weight meant here is
\[
G_{s,\sigma}(y)=
\prod_{i=1}^n\varphi_{d,\nu}(y_i)\,\exp\{n\psi\}.
\]
When $\nu\ge\nu_{\min}>0$, Cauchy--Schwarz gives the uniform envelope
$G_{s,\sigma}(y)\le\exp(Cn-c\sum_i y_i^2)$. On a fixed empirical second-moment
ball, $M,q,a,d,\nu,\psi$ are continuous for weak convergence, since
only bounded or at most linearly growing functions are averaged.
The entropy variational formula and a finite covering by fixed
product-Gaussian neighborhoods give
\[
\limsup_{n\to\infty}n^{-1}\log\int_{P_y\in\mathcal C}G_{s,\sigma}(y)\,\mathrm dy
\le\sup_{P\in\mathcal C}\{\psi-D(P\Vert N(d,\nu))\}
\]
for each compact parameter set $\mathcal C$; Gaussian density ratios
on the second-moment ball tend uniformly to one at the exponential
rate when their parameters converge.
The rate is nonpositive, and
its zero set at $\sigma=0$ is exactly the curve just proved.
Upper semicontinuity therefore makes the rate strictly negative on
closed sets outside its prescribed neighborhood. Compactness preserves
this negativity when $\sigma$ is sufficiently small. The exterior of
the ball has any required negative rate by the envelope.
For the tail-square assertion insert
$\exp(\lambda\sum_i y_i^2\mathbf1_{|y_i|>R})$ into each of the finite
Gaussian comparisons, with fixed sufficiently small $\lambda>0$.
Its additional logarithmic moment tends uniformly to zero as
$R\to\infty$. Removing the tilt on a positive tail-square level
produces strict negative rate. These are precisely the three stated
integral conclusions, now with no subcritical assumption.
\end{proof}
\begin{proposition}[Positive-time field inputs]\label{prof:positive-stability}
Fix $0<s_1<T<\infty$. There are positive constants
$\rho_0,\eta_0,\epsilon_A,c_0,c_*,a$, depending on this interval,
such that, at every deterministic $s\in[s_1,T]$, the probability under
the planted law, restricted to the original spectral event, of either
failure of \eqref{prof:eq26} or existence of $y$ with
\[
\|F_{h_s}(y)\|\le\rho_0\sqrt n,\qquad q(y)<c_*
\]
is at most $e^{-an}$ for sufficiently large $n$. The same assertion
holds uniformly on each fixed compact interval after decreasing the
residual threshold.
There are also absolute $c,C>0$ with the following small-time
refinement. For each fixed $A_*>0$, at all sufficiently small
$r=\sqrt s>0$, failure of \eqref{prof:eq26} has probability at most
$e^{-A_*nr^3}$; its margins may depend on the scale. For
$u\ge Cr$ in a sufficiently small fixed range, existence of an
approximate zero with $q(y)>u$ has probability at most $e^{-cnu^3}$.
The approximate-zero tolerance is fixed before taking $n\to\infty$
and may depend on $s,u,A_*$. Uniformity is asserted only on compact
positive bands; finitely many or dyadic values of $u$ may be imposed
together. In the small-time assertions a vanishing maximum diagonal
may be included in the external disorder restriction.
\end{proposition}
\begin{proof}
The scalar calculation separates the overlap from zero and places the
retained parameters near the positive-time equality curve. This gives
a norm--trace margin for nearby diagonals. We next check the conditional
finite-rank matrix correction and make that check simultaneous over
the diagonal neighborhood.
First retain the diagonal and gauge the planted spin to $\mathbf1$.
Then $J=W+\mathbf1\mathbf1^{\mathsf T}/n$ and
$h_s=s\mathbf1+\sqrt s\,g$. Until the final diagonal-removal
paragraph, use the temporary full-matrix residual and Hessian
\[
\widetilde F_h(y)=y-h+(b(y)I-J)m(y),\qquad
\widetilde{\mathcal H}(y,A)=I+A^{1/2}
\left(b(y)I-J-\frac{2m(y)m(y)^T}{n}\right)A^{1/2}.
\]
Let \(E\subseteq\mathbb R^n\times\operatorname{Sym}_n\) be Borel
and independent of the observation \(h_s\). For \(\sigma>0\) set
\[
I_{s,\sigma}(E)=(2\pi\sigma^2)^{-n/2}\int_{\mathbb R^n}
\mathbf1_{\{(y,J)\in E\}}
\exp\left[-\frac{\|\widetilde F_{h_s}(y)\|^2}{2\sigma^2}
+\frac{nb(y)^2}{2}\right]\,\mathrm dy.
\]
The exact Gaussian reduction, Lemma 5.3 of
\cite{AcceptedGap}, is valid for every positive $j$ and
here reads
\[
\mathbb E I_{s,\sigma}(E)
=\int\sqrt{\frac\nu{\nu+q}}\,G_{s,\sigma}(y)
\mathbb P\bigl((y,J)\in E\mid
Wm+\sqrt{s+\sigma^2}\,g'=y-d\mathbf1+bm\bigr)\,\mathrm dy,
\]
where $d=s+M$, $\nu=s+\sigma^2+q$ and $G_{s,\sigma}$ is the
explicit scalar weight in Lemma~\ref{prof:endpoint-scalar}.
Here \(g'\) is a fresh standard normal independent of \(W\).
The identity follows directly by inverting
$\nu I+mm^{\mathsf T}/n$; the determinant factor is
$\sqrt{\nu/(\nu+q)}$.
On $[s_1,T]$, discard a large second-moment ball's exterior, a fixed
neighborhood of the equality curve's complement, and the scalar
square-tail event. Lemma~\ref{prof:endpoint-scalar} gives strictly
negative rates on these discarded parts and an arbitrarily small
positive rate for the total integral. On the remaining parameters
$q\ge q_0>0$. Choose $\eta_0$ and the diagonal Frobenius tolerance so
small that
\[
\|A\|\chi_A\le(1+\eta_0)(1-q_0+\epsilon_A)
\le1-q_0/2.
\]
The restricted fixed-parameter and bilinear estimates now apply.
We verify their conditional use and its positive margin.
Set $u=m/\sqrt{nq}$, $Z_1=(y-d\mathbf1)/\sqrt{n\nu}$ and
$P_u=I-uu^{\mathsf T}$. Under the displayed conditioning, using a
fresh GOE $W_0$, put \(\ell_0=u^TW_0u\) and
\[
\begin{split}
P_uWP_u&=P_uW_0P_u,\\
P_uWu&=\bar w+\tau P_uW_0u,\\
\bar w&=\sqrt{q/\nu}\,Z_1-(a/\nu)u,
\qquad \tau^2=(s+\sigma^2)/\nu,\\
u^{\mathsf T}Wu&=\alpha_c
=2(a+bq)/(\nu+q)
+\sqrt{(s+\sigma^2)/(\nu+q)}\,\ell_0.
\end{split}
\]
Writing out the three blocks gives the exact finite-rank difference
\[
\begin{split}
W-W_0={}&u\bar w^T+\bar w u^T
-(1-\tau)\{u(W_0u)^T+(W_0u)u^T\}\\
&+\{\alpha_c+(1-2\tau)\ell_0\}uu^T.
\end{split}
\]
The scalar \(\ell_0\) has variance \(2/n\) and is smaller than
any fixed tolerance outside an exponentially small event.
Write \(S_A=S_A(1;W_0)\) and define
\(\mathcal Y(p)=S_A^{-1/2}A^{1/2}p\). Set
\[
U=\mathcal Y(u),\quad T_1=\mathcal Y(W_0u),\quad
\overline W=\mathcal Y(\bar w),\quad
X_1=\mathcal Y(\mathbf1/\sqrt n).
\]
The exact whitened Hessian is
\[
\begin{split}
S_A^{-1/2}\widetilde{\mathcal H}(y,A)S_A^{-1/2}
={}&I+(b-\chi_A)S_A^{-1/2}AS_A^{-1/2}
-X_1X_1^T-2qUU^T\\
&-\bigl[U\overline W^T+\overline WU^T
-(1-\tau)(UT_1^T+T_1U^T)\\
&\hspace{35mm}
+\{\alpha_c+(1-2\tau)\ell_0\}UU^T\bigr].
\end{split}
\]
For each fixed \(y,A\), the vectors
\(\mathbf1/\sqrt n,u,Z_1\) are deterministic relative to \(W_0\).
The bilinear tests in \Cref{prof:endpoint-words} give
\[
\begin{split}
\mathcal Y(p)^T\mathcal Y(r)&\simeq p^TAr,\\
\mathcal Y(p)^TT_1&\simeq\chi_Ap^TAu,\\
\|T_1\|^2&\simeq\chi_A+\chi_A^2u^TAu
\end{split}
\]
for this fixed finite vector list. In particular
\(\|X_1\|^2\simeq\chi_A\), rather than \(b\) for an arbitrary
\(A\). These are comparisons of actual Gram entries, with a
chosen fixed accuracy.
The comparison Gram for \(T_1\) is represented by
\(\chi_AU+\sqrt{\chi_A}N\), where \(N\) is an abstract unit
direction orthogonal to the other three images. This is a model
of Gram entries, not a coupling with a new random vector.
Set aside the small full-rank term \(b-\chi_A\) and the scalar
\(\ell_0\). Eliminating the \(N\) direction, whose Hessian pivot
is the identity, gives the ideal coefficient
\[
\begin{split}
\beta_U
&=2q-\frac{2a}{\nu}+\frac{2(a+bq)}{\nu+q}
-2(1-\tau)\chi_A+(1-\tau)^2\chi_A\\
&=q\left(2-\frac{\chi_A}{\nu}
-\frac{2(a-\nu b)}{\nu(\nu+q)}\right)
\end{split}
\]
on \(UU^T\) in the form being subtracted. The final term in
the first line is the Schur correction.
At the equality-law limit, \(\sigma=0\), \(d=\nu=v=v_*(s)\),
\(a=vb\), and the limiting diagonal is \(A=V_y\), so
\(\chi_A=b\). Only in this limit is \(\beta_U=q(2-b/v)\).
Write
$b=\mathbb E\operatorname{sech}^2y$ and
$e=\mathbb E[\tanh^2 y\operatorname{sech}^2y]$ under $N(v,v)$.
Use the rescaled triple
\[
M_1=\sqrt q\,\mathcal Y(u),\qquad
G_1=\mathcal Y(Z_1)/\sqrt v.
\]
The ideal remaining matrix is identity minus the form on
$(X_1,M_1,G_1)$ with coefficient matrix \(C\) and limiting
equality-law Gram matrix \(H\):
\[
C=\begin{pmatrix}1&0&0\\0&2-b/v&1\\0&1&0\end{pmatrix},\qquad
H=\begin{pmatrix}
b&e&-2e\\e&e&b-3e\\-2e&b-3e&b/v-2b+6e
\end{pmatrix}.
\]
The Gram matrix here is the limit of the empirical \(A=V_y\)
comparison data, not the actual finite-sample Gram. Its entries
follow by Gaussian integration by parts; for example $\mathbb E[m(1-m^2)]=e$,
$v^{-1}\mathbb E[(y-v)m(1-m^2)]=b-3e$, and the double integration by
parts gives $b/v-2b+6e$. Elementary column operations yield
\[
\det(I-HC)=(q+2e)(q+e)^2>0.
\]
Subtracting the first vector alone is positive because its squared
length is $b<1$.
The remaining two-vector coefficient block has only
one positive eigenvalue. Hence the final matrix can have at most one
nonpositive direction; its positive determinant rules that out.
The preceding orthogonal-direction Schur complement is therefore
positive as well. Continuity on the compact equality curve gives a
fixed positive margin.
The fixed-parameter calculation is complete. To make it simultaneous,
we first quantify its continuity. The margin survives small errors in
all Gram entries, coefficients,
and $\chi_A-b$. To check uniformity over nearby $A$, each tested
coordinate product is bounded by $C(1+y_i^2)/n$. On $|y_i|\le R_1$,
its change under $A-V_y$ is at most
$C_{R_1}\|A-V_y\|_{\mathrm F}/\sqrt n$; the remaining part costs the
prescribed square-tail tolerance. The $V_y$-weighted coordinate
products are bounded continuous functions, so their equality-law
limits follow from weak closeness.
This proves the fixed-$A$
conditional estimate with a uniform positive exponential rate.
The path margin \(S_A\succeq c_\kappa I\) returns the whitened
margin to \(\widetilde{\mathcal H}\), after the small
\((b-\chi_A)\) term is absorbed.
We pay for simultaneity only on the coordinates where a diagonal can
move appreciably. On the bounded conditional-matrix norm event choose
a fixed sup-norm mesh spacing $\delta$ so the $1/2$-H\"older error of
$\widetilde{\mathcal H}(y,A)$ is smaller than one quarter of the margin. Only
$\epsilon_A^2n/\delta^2$ sites can differ from $V_y$ by more than
$\delta$. Mesh those sites and leave the others at $V_y$.
The logarithm of the number of approximants is at most
\[
n\left[\mathfrak h(\alpha)+\alpha\log(C/\delta)+o(1)\right],
\qquad \alpha=\epsilon_A^2/\delta^2,
\]
where $\mathfrak h(\alpha)\to0$ as $\alpha\downarrow0$.
Use twice the Frobenius radius in the fixed-parameter estimate and
then decrease $\epsilon_A$ to pay this mesh. Thus the conditional
matrix failure, integrated against $G$, has strictly negative rate.
This explicitly extends the positive-time part of Lemma 5.5 of the
companion article; its subcritical small-$q$ argument is not used.
The estimates so far control a smoothed integral of bad configurations.
To exclude a bad approximate zero, a witness must force that integral
to be large enough. The deterministic local-volume estimate supplies
this implication, with determinant, smoothing, and residual tolerances
chosen in that order.
Let \(\mathcal D_{\eta_0}\) be the compact set of diagonal
\(0\le A\le(1+\eta_0)I\). Let \(\epsilon_*>0\) be the full
diagonal radius retained by the conditional calculation, and choose
\(c_E\) below half its matrix margin. Use the relaxed Borel set
\[
E^{\rm mat}=\left\{(y,J):\exists A\in\mathcal D_{\eta_0},\
\|A-V_y\|_{\rm F}\le\epsilon_*\sqrt n,\
\lambda_{\min}(\widetilde{\mathcal H}(y,A))\le c_E\right\},
\]
intersected with the external spectral and norm restriction.
Compactness of the diagonal range and continuity make this set
Borel and independent of \(h_s\). The scalar exclusions and the
conditional matrix rate just proved give, for sufficiently small
\(\sigma>0\), uniformly on the fixed positive band,
\[
\E I_{s,\sigma}(E^{\rm mat})\le e^{-a_0n}
\]
for some \(a_0>0\).
The stricter witness event uses
\[
\|\widetilde F_{h_s}(y)\|\le\rho\sqrt n,\qquad
\|A-V_y\|_{\rm F}\le\tfrac12\epsilon_*\sqrt n,\qquad
\lambda_{\min}(\widetilde{\mathcal H}(y,A))\le\tfrac12c_E,
\]
with \(A\in\mathcal D_{\eta_0}\). It depends on the observation,
unlike \(E^{\rm mat}\). Keeping this same \(A\), for
\(d_y=\|y'-y\|/\sqrt n\) we have
\[
\|V_{y'}-V_y\|_{\rm F}\le2\|y'-y\|,\qquad
|b(y')-b(y)|\le2d_y,\qquad
\left\|\frac{m(y')m(y')^T-m(y)m(y)^T}{n}\right\|\le2d_y.
\]
Thus the Hessian changes by at most \(6(1+\eta_0)d_y\).
Every point of the ball
\(\|y'-y\|\le r_{\rm loc}\sqrt n\) lies in the same
\(E^{\rm mat}\) fiber if
\[
r_{\rm loc}\le
\min\{\epsilon_*/4,\ c_E/[12(1+\eta_0)]\}.
\]
We next obtain the overlap lower bound needed for the determinant estimate.
A uniform lower bound on $q$ at any witness is available before
using the determinant estimate. On the external norm event,
\[
\|\tanh h_s\|\le\|m(y)\|+\|h_s-y\|
\le C\sqrt{nq(y)}+\|\widetilde F_{h_s}(y)\|.
\]
Independence of the planted observation coordinates and a fixed
Bernoulli deviation bound give
$\|\tanh h_s\|\ge c\min(\sqrt s,1)\sqrt n$ outside a fixed
exponentially small event, uniformly on compact time intervals.
For instance, at small $s$, a fixed positive fraction of the
standard Gaussian coordinates lie in $[1/2,1]$, on which the
hyperbolic tangent has magnitude at least $c\sqrt s$.
On the fixed band let \(v_->0\) be a common observation lower
bound, and let the external matrix norm be at most \(K\).
Taking \(\rho\le v_-/2\) forces
\[
q(y)\ge q_{\rm vol}:=
\left(\frac{v_-}{2(K+2)}\right)^2>0.
\]
At such a witness the Jacobian is exactly
\[
\widetilde F_{h_s}'(y)
=I+(bI-W)V_y-\frac{\mathbf1\mathbf1^T}{n}V_y
-\frac{2mm^T}{n}V_y.
\]
The last two terms have rank at most two and norm at most three.
Since \(b\le1-q_{\rm vol}\), Lemma~\ref{prof:endpoint-logdet}
with \(\kappa=q_{\rm vol}\) gives its regularized determinant
upper bound with any needed fixed failure budget.
The deterministic local-volume lemma, Lemma 5.6 of
\cite{AcceptedGap}, has hypotheses $j,K<\infty$ and hence
applies at $j=1$ without modification.
With the regularized
Jacobian $Q=\widetilde F_{h_s}'(y)^{\mathsf T}
\widetilde F_{h_s}'(y)+\gamma I$, its change of variable
$y'=y+\sigma Q^{-1/2}g$ proves a lower bound $e^{-\varepsilon n}$
for this bad integral whenever the residual is sufficiently small
and \(2\sigma/\sqrt\gamma\le r_{\rm loc}\).
Its losses per spin are bounded by
\[
\frac{\varepsilon_1}{2}
+C\left[\frac\rho\sigma+\frac\sigma\gamma+
\left(\frac\rho\sigma\right)^2+
\left(\frac\sigma\gamma\right)^2+
\frac\sigma{\sqrt\gamma}\right]+o(1).
\]
Choose the target loss \(\varepsilon<a_0/2\), then the
determinant tolerance $\varepsilon_1$, then the regularizer $\gamma$,
then $\sigma$, and finally the residual tolerance $\rho$.
Markov's inequality now bounds the witness probability by
\(e^{-(a_0-\varepsilon)n}\), together with the assigned determinant
and observation exceptions. For low overlap, choose
\(q_{\rm rel}\) below the equality curve's minimum \(q\), use
the relaxed set \(E^{\rm low}=\{q\le q_{\rm rel}\}\) with the
same external restriction, and take
the witness threshold \(q\le q_{\rm rel}/2\).
Since \(|q(y')-q(y)|\le2d_y\), a radius
\(r_{\rm loc}\le q_{\rm rel}/4\) gives the same inclusion.
The scalar negative rate then replaces the matrix rate in the
same volume argument.
For the stronger small-time budgets use
\Cref{prof:scalar-diagnostics}, without the determinant weight.
Its scalar integral estimates permit any preliminary fixed
exclusion budget \(A_{\rm pre}nr^3\), restrict
$q\asymp_{A_{\rm pre}}r$, and give the outer rate $cnu^3$ for $u\ge Cr$.
Choose \(A_{\rm pre}>A_*\) to reserve rate for smoothing and
local volume. For an upper-tail witness \(q>u\), use the relaxed
set \(E_u^{\rm up}=\{q\ge u/2\}\) and radius
\(r_{\rm loc}\le u/4\). Taking \(u\ge2Lr\), with \(L\) from
the scalar proposition, gives a preliminary rate proportional to
\(n(u/2)^3\).
We compare the actual smoothed scalar weight with the root weight.
On a fixed band \(s\in[s_-,s_+]\), \(s_->0\), put
\(\nu_0=s+q\), \(\nu_\sigma=\nu_0+\sigma^2\), and
\(\mathfrak v=\langle(y-d)^2\rangle\). For fixed \(y\),
\[
\frac1n\log\frac{G_{s,\sigma}(y)}{G_{s,0}(y)}
=-\frac12\log\frac{\nu_\sigma}{\nu_0}
+\frac{\sigma^2\mathfrak v}{2\nu_0\nu_\sigma}
+\psi(\nu_\sigma)-\psi(\nu_0),
\quad
\psi(\nu)=\frac{(a-\nu b)^2}{2\nu(\nu+q)}.
\]
On \(\langle y^2\rangle\le R^2\), all numerators are bounded
and \(\nu\ge s_-\), so the absolute value is at most
\(C_{R,s_-,s_+}\sigma^2\). The uniform Gaussian envelope from
\Cref{prof:endpoint-scalar} makes the exterior integral have any
prescribed fixed negative rate when \(R\) is large.
Choose \(R\) first, then \(\sigma\) so this logarithmic loss is
smaller than one quarter of the reserved gap
\((A_{\rm pre}-A_*)r_-^3\), or than the reserved fixed multiple
of \(u_-^3\) for a finite outer-threshold family. Polynomial
factors are absorbed for large \(n\). This preserves every scalar
rate used here for the smoothed integral.
If a diagnostic itself uses \(\nu_\sigma\), give its thresholds
fixed slack first. On the same ball its scalar parameters change
by \(O_R(\sigma^2)\). On the retained scalar region,
\(q\ge c r_->0\), and the conditional column coefficients have
the same bound. The stricter smoothed bad set is then contained in the relaxed
unsmoothed one. This supplies the required set comparison as well
as the weight comparison.
For the matrix estimate the spectral indicator must remain attached
to the actual conditioned matrix. Denote that matrix by
\(J_\sigma=W_\sigma+\mathbf1\mathbf1^T/n\); the subscript records
the conditioning, not a replacement of the external disorder.
With \(u=m/\sqrt{nq}\), \(E=u^\perp\), and
\(e=P_E\mathbf1/\sqrt n\), its exact regression gives
\[
J_\sigma|_E=W_E+ee^T,\qquad
P_EJ_\sigma u
=\frac q{\nu_\sigma}\frac{P_E(y-d\mathbf1)}{\sqrt{nq}}
+\frac M{\sqrt q}e
+\sqrt{\frac{s+\sigma^2}{\nu_\sigma}}\,w,
\]
where the GOE block \(W_E\) and \(w\sim N(0,I_E/n)\) are
independent. Its scalar block is the regression displayed earlier.
In decreasing eigenvalue order, compression and rank-one interlacing
give, on \(\mathcal G_{\rm spec}(J_\sigma)\),
\[
\lambda_j(J_\sigma)\ge\lambda_j(W_E)
\ge\lambda_{j+2}(J_\sigma),
\qquad 1\le j\le n-2.
\]
Thus every cumulative count changes by at most two, and the top
edge of \(W_E\) is bounded by that of this same full matrix.
Here is the precise spectral information used by the stability
calculation. Put \(\Delta_{\rm sp}=q^{8/5}\) and restrict
\(R=(2+O(q^2)-W_E)^{-1}\) to depths greater than
\(\Delta_{\rm sp}\). The inherited edge counts and bulk
semicircle limit give, uniformly for the \(O(q^2)\) shifts in
the stability Schur calculation,
\[
\frac{\operatorname{Tr}R}{n}
=1+O(\sqrt{\Delta_{\rm sp}})+o_n(1),\qquad
\frac{\operatorname{Tr}R^j}{n}
\asymp_j\Delta_{\rm sp}^{3/2-j}\quad(j\ge2).
\]
The first estimate follows by removing the edge contribution to
the first trace; dyadic counts give the upper estimates for higher
powers, and one band at depth comparable to \(\Delta_{\rm sp}\)
gives their lower estimates. The fixed-rank count error contributes
only \(O((n\Delta_{\rm sp}^j)^{-1})\). Likewise the restriction
to depths at most a fixed \(\zeta\) has first trace divided by
\(n\) at most \(O(\sqrt\zeta)+o_n(1)\).
In particular the first trace is normalized to one, as required
for the cancellation in \eqref{prof:eq11}. The other needed margin
uses the actual full spectrum:
\[
(b+b^{-1})I-J_\sigma\succeq(1-o(1))q^2I,
\qquad b+b^{-1}=2+q^2+O(q^3).
\]
Condition first on the spectrum of \(W_E\). On every spectrum
satisfying these inherited bounds, its Haar eigenframe and the
independent Gaussian column obey the tests in
\eqref{prof:eq11} at any prescribed fixed rate in \(nr^3\),
before imposing \(\mathcal G_{\rm spec}(J_\sigma)\).
The scalar diagnostics bound every regression coefficient uniformly
on the fixed positive band. Choosing their error tolerances first
and then \(\sigma\) small preserves the column tests and their
\(q^2\) Schur margin. Consequently the estimate is for
\[
\int_{\mathcal S}G_{s,\sigma}(y)\,
\mathsf Q_{y,\sigma}
(\mathcal G_{\rm spec}(J_\sigma)\cap\mathcal F_y)\,dy,
\]
with the preliminary rate paying the scalar-weight and mesh losses.
Here \(\mathcal F_y\) is failure of the required Hessian margin.
No event for the fresh comparison matrix \(W_0\) replaces the
displayed indicator, and no division by its conditional probability
is used. This argument transfers the stability calculation, not
the sharper inverse-trace accuracy for \(K_o\), which is not
needed here. All smoothing choices are made on a fixed positive
band before taking \(n\) large.
We spell out the additional diagonal uniformity in this use.
At the given positive scale truncate \(A^{-1}-I\) above at the same level
\(q^{1-\eta}\) used for \(V_y^{-1}-I\), assigning that truncation
value at \(A_{ii}=0\). The truncated scalar function is Lipschitz on $[0,1+\eta_0]$; its
negative part is at most $\eta_0$. Choosing $\eta_0$ small preserves
the norm bound for the truncated perturbation. Shrinking
$\|A-V_y\|_{\mathrm F}/\sqrt n$ makes its normalized first two
moments and all the fixed-axis quadratic and column tests as close
as required to those in \eqref{prof:eq11}.
For axes involving $y$,
first discard their square tails by the scalar diagnostics, then use
bounded site profiles on the remaining sites. For $u=m/\sqrt{nq}$
one can directly use
$\|(B(A)-B(V_y))u\|\le C\epsilon_A/\sqrt q$.
All scales here are fixed before $n\to\infty$. Thus the Schur
calculation retains its margin with the prescribed rate.
The original precision is larger. Its positive lower bound also
gives a positive symmetric-normalization margin: combine it with
the bound $A^{-1}-CI$, then take limits if some entries vanish.
The preceding sparse diagonal mesh costs an arbitrarily small
exponential rate and makes the estimate simultaneous.
The observation lower bound just used has a fixed rate independent
of sufficiently small $r$, so it pays any specified $A_*r^3$ after
shrinking the upper scale. The determinant event can have an
arbitrarily large fixed rate by Lemma~\ref{prof:endpoint-logdet}.
Apply the local-volume argument in the same order, now taking its
loss below the available $A_*r^3$ or $cu^3$ rate. This proves the
small-time assertions, including their order of tolerance choices.
Finally remove the diagonal. On $\max_i|J_{ii}|\le\delta_D$ the
normalized residual changes by at most $\delta_D$, and the
symmetric Hessian by at most $(1+\eta_0)\delta_D$.
On fixed positive bands take a sufficiently small fixed $\delta_D$;
its failure has exponential probability. In the small-scale
assertions keep the vanishing-diagonal event as an external
restriction and then, for each fixed band, take $n$ large enough.
For example, take \(\delta_D\le\rho/2\) and
\((1+\eta_0)\delta_D\le c_E/4\). Then a zero-diagonal
witness with residual at most \(\rho\sqrt n/2\), diagonal radius
\(\epsilon_*/2\), and Hessian threshold \(c_E/4\) implies the
full-matrix witness above.
For the compact path version write \(h_t=t\mathbf1+B_t\) in the
gauged planted process. On a mesh interval of length \(\Delta\),
the coordinate Brownian suprema have a fixed exponential moment
after division by \(\Delta\). Independence across coordinates gives
\[
\Pr\!\left\{\sup_{t_j\le t\le t_{j+1}}
\frac{\|B_t-B_{t_j}\|}{\sqrt n}>\kappa_0\right\}
\le \exp\{-n(c\kappa_0^2/\Delta-C)\}.
\]
Choose \(\kappa_0,\Delta\) so this bound has the required fixed
rate and the Brownian increment plus the deterministic drift is
at most \(\rho/4\) in normalized norm. A finite union covers the
fixed compact interval. A path witness with residual
\(\rho\sqrt n/8\) then has residual below \(\rho\sqrt n/2\)
at the left mesh endpoint. The Hessian, overlap, and diagonal
predicate are independent of the field and persist there.
This proves the compact assertion with its reduced residual
threshold, also for a fixed finite family of scalar thresholds.
All previous exclusions estimated failure intersected with the
original spectral restriction, using interlacing rather than a
new typicality assertion for a conditional largest eigenvalue.
\end{proof}
\subsection{Residual identities and the complete derivative buffer}
\begin{proposition}[Endpoint posterior consequences]
\label{prof:endpoint-posterior}
Fix a matrix norm bound. For the covariance conclusion, assume
positive margins in \eqref{prof:eq26} and suppose that every point
in its small-residual region has \(q(y)\ge c_*>0\).
On the fixed-length ordinary and restricted inverse-word events
specified below,
\[
\|\operatorname{Cov}_{\mu_h}(G,X)\|
\le C\bigl(\operatorname{Var}_{\mu_h}(G)+\mathcal D_h(G)\bigr)^{1/2}
\quad\text{for every }G.
\]
In particular $\|\Sigma_h\|\le C$.
For the following mean conclusion, only the whole-region root
stability implication
\[
\|F_h(y)\|\le\rho_0\sqrt n
\quad\Longrightarrow\quad
\mathcal H(y,V_y)\succeq c_0I
\]
is required, together with ordinary word diagnostics. No lower
bound on \(q(y)\) is assumed in this mean conclusion.
For $N=\mu_h(f^2)>0$ and $\nu_h=f^2\mu_h/N$, and each sufficiently
small fixed $\zeta>0$,
\begin{equation}\label{prof:eq27}
\frac{\|\mathbb E_{\nu_h}X-\mathbb E_{\mu_h}X\|}{\sqrt n}
\le C\zeta+\frac{C_\zeta}{\sqrt n}
\left(1+\frac{\mathcal D_h(f)}N\right).
\end{equation}
The leading constant \(C\) depends on the fixed norm and stability
margins, but not on \(\zeta\).
Without the stability hypothesis, put
\[
Q_\delta(h)=\sup_{\|F_h(y)\|\le\delta\sqrt n}q(y),
\]
with an empty supremum equal to zero. The same ordinary diagnostics give
\begin{equation}\label{prof:eq27-coarse}
\frac{\|\mathbb E_{\nu_h}X-\mathbb E_{\mu_h}X\|}{\sqrt n}
\le 2\sqrt{Q_\delta(h)}+
\frac{C_\delta}{\sqrt n}
\left(1+\frac{\mathcal D_h(f)}N\right).
\end{equation}
For each fixed $\delta>0$ this
assertion holds for sufficiently large $n$ on the corresponding
ordinary diagnostics. All events are independent of $f,G$ and the
unit directions used to test these inequalities.
\end{proposition}
\begin{proof}
The root and inverse statement, Lemma 5.7 of
\cite{AcceptedGap}, is deterministic for every finite positive
$j$ and applies here at $j=1$. Its root conclusion uses only the
whole-region implication at \(A=V_y\) stated above; the overlap
lower bound is used later only for the inverse recipe.
Denote the unique zero by $y_*$,
and set $m_* =\tanh y_*$, $q_*=n^{-1}\|m_*\|^2$.
It gives \(\|y-y_*\|\le C\|F_h(y)\|\) when
\(\|F_h(y)\|<\rho_0\sqrt n\), with
\(C=1+(\|J^\circ\|+3)/c_0\). The strict residual inequality
will be preserved by the inner buffer below.
We verify the precise diagnostic hypotheses of the companion residual
proof. Fix \(J,h\) throughout this deterministic subsection; all
spin expectations below are under \(\mu_h\), unless a displayed
square reweighting specifies otherwise. For a spin function define
\[
d_i g(x)=\frac{g(x)-g(x^{(i)})}{2x_i},\qquad
dg=\nabla g=(d_i g)_i.
\]
For a vector function, \((\nabla g)_{ji}=d_i g_j\).
These are discrete spin-flip derivatives, distinct from the
formal site derivatives in the recipes. Put
\[
m^0=x,\quad b_0=0,\quad h^1=h+J^\circ x,\qquad
m^l=\tanh h^l,\quad b_l=n^{-1}\sum_i(1-(m_i^l)^2),\quad
R_l=m^{l-1}-m^l,
\]
\[
h^{l+1}=h+J^\circ m^l-b_lm^{l-1}.
\]
For each fixed depth the primary differentiation estimates of
Lemma 6.1 of the companion article need only ordinary words, with
contraction coefficient one. Explicitly their formal derivatives are
\[
H_1=J^\circ,\quad M_0=I,\quad
M_l=\operatorname{diag}(1-(m^l)^2)H_l,\quad
H_{l+1}=J^\circ M_l-b_lM_{l-1}.
\]
The diagonal predictions of $H_l$ cancel by inclusion--exclusion over
originally adjacent pairs of each noncrossing pairing.
The word
estimates give bounded operator, diagonal $\ell^2$, and off-diagonal
entrywise $\ell^4$ norms. Taylor's error in a coordinate difference
is at most $C|d_i h_j^l|^2$; summing its square uses that $\ell^4$
bound. Differentiating $b_lm^{l-1}$ adds a rank-one term of bounded
Frobenius norm, since $\|db_l\|\le C_l/\sqrt n$.
Thus actual derivatives differ from $H_l,M_l$ by bounded Frobenius
errors. Every single flip changes a fixed primary vector by at most
$C_l$ in Euclidean norm. None of these finite recurrences uses
$j<1$ or convergence as $l\to\infty$.
We use the following finite recipe class, as in Definition~6.2 of
\cite{AcceptedGap}. Fix the primary depth and finitely many seeds
$s^1,\ldots,s^v\in\mathbb R^n$ with bounded Euclidean norms.
The seeds may depend on the fixed $(J,h)$ but are constant as functions
of the spin configuration $x$. Let $\theta(x)$ be a finite tuple of
scalar parameters with
$\sup_x(|\theta_\alpha(x)|+\|d\theta_\alpha(x)\|)\le C$
for every component; the averages $b_l$ may be among them.
Given auxiliary fields $y^1,\ldots,y^{a-1}$, a source has the form
\[
w_i^a=\sum_{j=1}^v
\phi_i^{a,j}((h_i^l)_{l\in I_a},\theta)s_i^j
+\sum_{b<a}\psi_i^{a,b}((h_i^l)_{l\in I_a},\theta)y_i^b.
\]
All index sets are finite. The site coefficients and every fixed
derivative needed by the recipe are uniformly bounded on their argument
ranges and on the line segments joining evaluations. Deterministic
site labels are allowed with the same bounds. Define
\[
y^a=J^\circ w^a
-\sum_l\left(n^{-1}\sum_i\partial_lw_i^a\right)m^{l-1}
-\sum_{b<a}\left(n^{-1}\sum_i\partial_{y^b}w_i^a\right)w^b.
\]
The partial $\partial_l$ differentiates only the explicit $h_i^l$
argument, keeping $\theta$ and all auxiliary arguments fixed;
$\partial_{y^b}w_i^a=\psi_i^{a,b}$, and an absent partial is zero.
A terminal vector has the same source form. When a source is reused
in a later source or terminal vector, substitute its source formula
without expanding the auxiliary fields.
A recipe is admissible from level $p$ when every explicit primary
argument in these source and terminal formulas has index at least $p$.
This restriction does not apply to scalar parameters or to the
correcting vectors $m^{l-1}$ in the auxiliary formula. All constants
may depend on the fixed recipe and its coefficient bounds. In particular,
the finite residual inductions below may enlarge the recipe before its
diagnostic length and the eventual dimension threshold are fixed.
Only the directional covariance argument uses an inverse recipe.
Fix $k\ge2$ and
$\Omega_{a\rho}=\{\max(\|R_k\|,\|R_{k-1}\|)\le a\rho\sqrt n\}$.
When several indices occur we write this set as
\(\Omega_{a\rho}^{(k)}\); the superscript is suppressed while
\(k\) is fixed.
The exact recurrence gives
\[
F_h(h^k)=J^\circ R_k-b_{k-1}(R_{k-1}+R_k)
+(b_k-b_{k-1})m^k,
\qquad |b_k-b_{k-1}|\le2\|R_k\|/\sqrt n.
\]
Choose \(\rho\) so the displayed recurrence puts
\(\Omega_{2\rho}\) strictly inside the root lemma's residual
region. Consequently on $\Omega_{2\rho}$,
\[
\|h^k-y_*\|+\|m^{k-1}-m_*\|\le C\rho\sqrt n,
\qquad |1-b_{k-1}-q_*|\le C\rho,
\qquad q(h^k)\ge c_*.
\]
These constants are independent of $k$. Define
\[
\Phi(z;r,a)=\int_0^1e^{a(1-u^2)/2}
\frac{\cosh(u(z-r))}{\cosh z\cosh r}\,\mathrm du,
\qquad a=1-b_{k-1}-q_*,\qquad A_i=\Phi(h_i^k;(y_*)_i,a).
\]
Differentiation in $u$ proves the exact identity
\[
[(z-r-a\tanh z)-a\partial_z]\Phi(z;r,a)=\tanh z-\tanh r.
\]
At $a=0$, $\Phi$ is the secant slope of $\tanh$.
Its positive integral
and the bounded derivatives of its logarithmic factors give
$0<\Phi\le e^{|a|/2}$ and bounded ratios of every fixed derivative to
$\Phi$. The derivative recipes evaluate their coefficients at nearby spin
configurations as well as at the selected residual point. We therefore
establish the diagonal estimates on the whole outer buffer. Once the
finite recipe and its number of flips are fixed, the threshold in the
dimension will ensure that every evaluation stays in that buffer.
The preceding root-distance estimates imply, on the whole outer buffer,
\begin{equation}\label{prof:implicit-diagonal-buffer}
\|A-V_{h^k}\|_{\mathrm F}\le C\rho\sqrt n,
\quad |\chi_A-b_k|\le C\rho,
\quad \|A\|\le e^{C\rho},
\quad
\|A\|\chi_A\le e^{C\rho}(1-c_*+C\rho)\le1-c_*/2.
\end{equation}
Choose $\rho$ small enough for the last inequality and for all stable
field tolerances. This choice precedes the depth and recipe choices.
The inverse diagonal therefore belongs to $\mathcal A_\kappa$ with
$\kappa=c_*/2$ everywhere it will be evaluated.
For either $w_{0,i}=A_i e_i$, $\|e\|\le1$, or
$w_{0,i}=n^{-1/2}\partial_r\Phi(h_i^k;(y_*)_i,a)$, the implicit recipe is
\[
w=w_0+A(y^{\mathrm{imp}}-c_1m_*),\qquad
y^{\mathrm{imp}}=J^\circ w-\chi_A w-c_1m^{k-1},\qquad
c_1=n^{-1}\sum_i\partial_k w_i.
\]
In this site partial, $y^{\mathrm{imp}},c_1$ and $a$ are held fixed.
After this one implicit solve, $y^{\mathrm{imp}}$ may be used as the
first auxiliary in an ordinary recipe, with $\sqrt n c_1$ as an
additional scalar parameter and $m_*/\sqrt n$ as an additional seed.
The corresponding coefficient and parameter bounds are required on
the flip buffers below; no such bounds are imposed on the arbitrary
exterior extension of the implicit solution.
Put $\ell_i=\partial_z\Phi/\Phi$, $u_0=w_0/A$ and
$p_0=A\partial_k u_0$. Then $\partial_k w=\operatorname{diag}(\ell)w+p_0$
and elimination gives the actual finite-dimensional solve
\[
\left[I-A(J^\circ-\chi_AI)
+\frac1n A(m^{k-1}+m_*)\ell^{\mathsf T}\right]w
=A\left[u_0-(m^{k-1}+m_*)n^{-1}\sum_i(p_0)_i\right].
\]
Put
\[
\mathcal L=\chi_AI-J^\circ+
(m^{k-1}+m_*)\ell^{\mathsf T}/n.
\]
Because $\|\ell+m_*\|\le C\rho\sqrt n$, the two-sided normalized
coefficient \(I+A^{1/2}\mathcal LA^{1/2}\), where the displayed solve is
\((I+A\mathcal L)w=\cdots\), differs in operator norm by \(O(\rho)\)
from \(\mathcal H(h^k,A)\). The normalized coefficient need not
be symmetric. The latter matrix has a fixed positive margin by
\eqref{prof:eq26}, so a perturbation smaller than half that margin
still has smallest singular value bounded below and a bounded
inverse.
The unnormalized inverse is bounded by
\[
(I+A\mathcal L)^{-1}
=I-A^{1/2}(I+A^{1/2}\mathcal LA^{1/2})^{-1}
A^{1/2}\mathcal L.
\]
This proves
$\|w\|+\|y^{\mathrm{imp}}\|+\sqrt n|c_1|\le C$, uniformly in
$k$ and the unit seed. Removing the negative rank term in
$\mathcal H(h^k,A)$ and replacing $b_k$ by $\chi_A$ also gives
\[
S_A(1;J^\circ)\succeq cI.
\]
This separate adaptive stability condition is essential; the norm--trace
restriction alone was only used for fixed-parameter expectations.
Every coordinate-difference solve uses the same displayed coefficient
matrix evaluated at the flipped configuration. All coefficient
differences multiply vectors at the common starting configuration.
Their squared column norms sum to a constant: for a coefficient $\psi$
and a fixed bounded-norm vector $v$,
\[
\sum_{i,j}v_i^2|d_j\psi_i|^2
\le\|v\|^2\max_i\sum_j|d_j\psi_i|^2\le C.
\]
The averaged coefficients have differences of order $n^{-1/2}$;
the factors $m^{k-1}+m_*$ have norm at most $2\sqrt n$.
Columnwise inversion therefore gives
$\|\nabla w\|_{\mathrm F}+\|\nabla y^{\mathrm{imp}}\|_{\mathrm F}
+\sqrt n\|dc_1\|\le C_k$, as in Lemma 6.3 of the companion article.
If the finished finite recipe uses at most $N_f$ successive flips,
the primary bounds put every required flip-neighborhood of
$\Omega_{3\rho/2}$ inside $\Omega_{2\rho}$ once
$n>(2N_f\max_l C_l/\rho)^2$. Thus all four bounds in
\eqref{prof:implicit-diagonal-buffer} and the adaptive inverse margin
hold at every derivative evaluation. The recipe and its depth are
fixed before this threshold on $n$ is chosen.
We also fix the whole-cube convention for these local recipes.
Keep the actual implicit solution \(y^{\rm imp},c_1\) on
\(\Omega_{2\rho}^{(k)}\), assign arbitrary finite values to them
outside that set, and define \(w\) and all later ordinary
auxiliaries by their stated formulas. The implicit equation is
required only on the outer buffer. Every scalar test in a local
residual statement is supported on \(\Omega_\rho^{(k)}\).
Its discrete differences use only the fixed finite flip
neighborhoods already placed inside the buffers, so the exterior
values never enter a tested estimate.
We record the remaining closure of the residual proof to ensure that
no other inverse diagonal is being assumed. Freezing parameters in
formal differentiation yields
\[
\nabla w=A\nabla y^{\mathrm{imp}}+
\operatorname{diag}(\partial_k w)H_k+E_1,
\quad
\nabla y^{\mathrm{imp}}=(J^\circ-\chi_AI)\nabla w-c_1M_{k-1}+E_2.
\]
Solving this equation introduces exactly the one factor
$K=(I-AJ^\circ+\chi_AA)^{-1}$ multiplying
$\operatorname{diag}(\partial_k w)H_k-c_1AM_{k-1}+E_1+AE_2$.
Later auxiliaries are ordinary, so error propagation multiplies by
ordinary words only.
The Taylor errors have bounded Frobenius norm
and bounded trace against the required words. For the only
non-nuclear product error, the decisive bound is
\[
\sum_{i\ne j}|P_{ji}|\,|d_js_i|\,Q_{ij}
\le\|P_{\mathrm{off}}\|_{\ell^4}
\|\nabla s\|_{\mathrm F}\|Q\|_{\ell^4}\le C,
\qquad Q_{ij}=\sum_l|d_jh_i^l|.
\]
The diagonal part uses the primary diagonal $\ell^2$ estimate.
All other dropped terms are bounded in nuclear norm, using
$\|BC\|_{\mathrm{nuc}}\le\|B\|_{\mathrm F}\|C\|_{\mathrm F}$.
The formal trace cancellation in Lemma 6.4 of the companion article
is algebraic. Its only cyclic node closes as
\[
\mathfrak S_{\mathrm{imp}}
=K_A(z;J^\circ)
\{\operatorname{diag}(\partial_k w)\mathfrak H_{h^k}
-z^2c_1A\mathfrak S_{h^k}\}.
\]
Both terms, including the averaged correction, must be retained.
For each dependency path, inclusion--exclusion over its originally
adjacent contracted pairs cancels every complete noncrossing pairing.
There is no empty derivative path since seeds do not depend on $x$.
The contracted trace prediction vanishes as a formal series and
is a polynomial by Lemma~\ref{prof:endpoint-words}. Its evaluation
at \(z=1\) is therefore zero; this does not require convergence
of the inverse's formal series. Each resulting trace has either a bounded-norm small
vector as a diagonal factor or a scalar $O(n^{-1/2})$.
In the first case rotate the small diagonal to the beginning of
the trace. Cyclic invariance of the normalized prediction preserves
the canceled value, and Cauchy--Schwarz gives
\[
\left|\operatorname{Tr}\bigl(\operatorname{diag}(u)
(Q-\mathsf P(Q))\bigr)\right|
\le \|u\|_2\,
\|\operatorname{diag}(Q-\mathsf P(Q))\|_2\le C.
\]
In the scalar case the diagonal word error has trace at most
\(C\sqrt n\), canceled by its \(O(n^{-1/2})\) coefficient.
These bounds remain uniform after a norm-one diagonal test.
Thus every small vector \(U\) in an ordinary recipe obeys the
following bound on the whole cube, and every local recipe vector
obeys it on \(\Omega_\rho^{(k)}\) with the stated flip buffers,
including the vector \(\partial_k w\):
\[
\|U\|+\|\nabla U\|_{\mathrm F}
+\|\operatorname{diag}\nabla U\|_1\le C.
\]
This proves the derivative assertion with only the single inverse
whose full buffer has already been checked.
We state the residual interfaces that now apply, specializing
Lemmas 6.5--6.6 of \cite{AcceptedGap} to coefficient one. Put
\[
\mathcal E_h(G)=
\bigl(\mu_h(G^2)+\mathcal D_h(G)\bigr)^{1/2},
\qquad
\mathcal D_h(G)=\mu_h\sum_i(1-(m_i^1)^2)(d_iG)^2.
\]
If \(U\) is a terminal vector of an ordinary recipe admissible
from level \(p\), then
\[
|\mu_h[G R_p^TU]|\le C\mathcal E_h(G)
\]
for every scalar \(G\). For a terminal vector of a local recipe
admissible from level \(p\), with implicit starting index \(k\),
the same inequality holds when \(p\le k\) and \(G\) is supported
on \(\Omega_\rho^{(k)}\), with the buffers active.
For an ordinary recipe admissible from level \(p\) and a multiplier satisfying
\(\sup_x(|H_0(x)|+\|dH_0(x)\|)\le C_0\), the square-density
version is
\[
|\mu_h[f^2H_0R_p^TU]|
\le C\bigl(\mu_h(f^2)+\mathcal D_h(f)\bigr).
\]
Only ordinary recipes are used in this last version.
The induction can be checked directly. Its base identity is
\[
\mu_h\sum_i(x_i-m_i^1)GU_i
=\mu_h\sum_i(1-(m_i^1)^2)d_i(GU_i).
\]
The preceding diagonal $\ell^1$ bound and Cauchy--Schwarz control it
by \(C\mathcal E_h(G)\).
For the induction from $p$ to $p+1$ multiply the terminal vector by
$\Phi(h_i^p;h_i^{p+1},b_p-b_{p-1})$, add one ordinary auxiliary,
and use the displayed scalar identity for $\Phi$.
Here $|b_p-b_{p-1}|\le1$, so all required coefficient bounds hold.
The term $n(b_p-b_{p-1})\operatorname{av}(\partial_pU')$ cancels by
\[
R_p^{\mathsf T}m^{p-1}
=n(b_p-b_{p-1})-R_p^{\mathsf T}m^p.
\]
Only smaller residual indices remain; no further inverse is introduced.
The square-density version follows from
$d_i(f^2)=2f\,d_if-2x_i(d_if)^2$ and has right side
$C(\mu_h(f^2)+\mathcal D_h(f))$. It uses ordinary recipes only.
All these operations are finite, so one fixed diagnostic length
covers the whole argument.
For the implicit recipe define the terminal residual
\[
\mathcal T_k=
\{h^k-h-(J^\circ-b_{k-1}I)m^k\}^Tw
-(b_k-b_{k-1})nc_1.
\]
For every \(G\) supported on \(\Omega_\rho^{(k)}\),
\[
|\mu_h[G\mathcal T_k]|\le C\mathcal E_h(G).
\]
Indeed, on the buffer the exact expansion is
\[
\mathcal T_k=
R_k^Ty^{\rm imp}+(\chi_A-b_{k-1})R_k^Tw
-b_{k-1}R_{k-1}^Tw-c_1R_k^Tm^k.
\]
The local weak rule applies to the first three terms. In the last
one use target \(m^k/\sqrt n\) and multiplier \(\sqrt n\,c_1\),
whose size and discrete differences are bounded on the buffer.
We finish by spelling out selection and the uniformity in the tests.
Put $T_l=R_l^{\mathsf T}m^l/\sqrt n$. The ordinary residual rule gives
$\mu_h(T_l^2)\le C_l$ and
\[
\mathbb E_{\nu_h}(T_l/\sqrt n)^2
\le\frac{C_l}{\sqrt n}\left(1+\frac{\mathcal D_h(f)}N\right).
\]
The exact telescoping identity
$\|R_l\|^2/n=b_l-b_{l-1}-2T_l/\sqrt n$
supplies a pair of consecutive small residuals. More precisely,
choose a fixed depth \(d_{\rm sel}\) with
\(\lfloor d_{\rm sel}/2\rfloor\rho^2/4>2\), and then a scalar
tolerance with \(2d_{\rm sel}\varepsilon<1\). If
\(\max_{l\le d_{\rm sel}}|T_l|/\sqrt n\le\varepsilon\), the
telescoping sum is less than \(2\), forcing a consecutive pair
whose two norms are at most \(\rho\sqrt n/2\).
Take a smooth \(\vartheta:[0,\infty)\to[0,1]\), equal to one
on \([0,1/4]\) and zero on \([1,\infty)\), and set
\[
C_k=\vartheta\!\left(\frac{\|R_k\|^2}{\rho^2n}\right)
\vartheta\!\left(\frac{\|R_{k-1}\|^2}{\rho^2n}\right),
\qquad
\widehat C_k=C_k\prod_{i=2}^{k-1}(1-C_i),
\quad 2\le k\le d_{\rm sel}.
\]
Pointwise,
\[
0\le\widehat C_k\le C_k\le1,\qquad
\operatorname{supp}C_k\subseteq\Omega_\rho^{(k)},\qquad
\|dC_k\|+\|d\widehat C_k\|\le C_\rho/\sqrt n.
\]
The derivative bounds follow from the primary differences and the
finite product rule. Also \(C_k=1\) when both residual norms are
at most \(\rho\sqrt n/2\), and
\[
D:=1-\sum_{k=2}^{d_{\rm sel}}\widehat C_k
=\prod_{k=2}^{d_{\rm sel}}(1-C_k)\in[0,1].
\]
The scalar moment bounds and a finite union bound give
\[
\mu_h(D>0)\le C_\rho/n,
\qquad
\nu_h(D>0)\le\frac{C_\rho}{\sqrt n}
\left(1+\frac{\mathcal D_h(f)}N\right).
\]
These are the conclusions of the companion Lemma 7.1, with the
preceding derivation of its endpoint hypotheses.
Residual selection now turns the finite recipe estimates into posterior
bounds. The covariance argument uses its one controlled inverse; the
square-reweighted mean argument below uses only ordinary recipes.
This distinction also separates their diagnostic requirements.
We make the terminal implication explicit. On the selected buffer
put \(p_k=m^k-m_*\) and
\(L_*=\sqrt n(\chi_A-(1-q_*))\). Subtracting the root equation gives
\[
h^k-h-(J^\circ-b_{k-1}I)m^k
=h^k-y_*-a m^k-J^\circ p_k+(1-q_*)p_k.
\]
Using the implicit equations and the scalar identity for \(\Phi\)
then yields the exact identity
\[
\begin{split}
\mathcal T_k={}&
\sum_i\bigl[(h_i^k-(y_*)_i-a m_i^k)w_{0,i}
-a\,\partial_k w_{0,i}\bigr]\\
&+(1-q_*-\chi_A)p_k^Tw-c_1R_k^Tp_k.
\end{split}
\]
The final term is controlled by the local weak rule with target
\(p_k/\sqrt n\) and multiplier \(\sqrt n\,c_1\). The remaining
scalar error satisfies pointwise
\[
|(1-q_*-\chi_A)p_k^Tw|
=\frac{|L_*|}{\sqrt n}|p_k^Tw|
\le C_0\rho |L_*|.
\]
Here the root-distance and source-size bounds give \(C_0\)
depending only on the fixed norm and stability margins, not on
\(k\) or the selection depth. The earlier choice of \(\rho\)
includes \(C_0\rho<1/2\).
For the source \(w_{0,i}=n^{-1/2}\partial_r\Phi\), differentiation
of the scalar identity in \(r\), with \(a\) fixed, gives
\[
[(z-r-a\tanh z)-a\partial_z]\partial_r\Phi
=\Phi-\operatorname{sech}^2r.
\]
Thus the sum in the terminal identity is \(L_*\). Globally,
\(|L_*|\le C\sqrt n\) and \(\|dL_*\|\le C_k\).
Use the local test \(G=C_k^2L_*\), whose support lies in
\(\Omega_\rho^{(k)}\). The cutoff difference bound gives
\[
\mathcal E_h(C_k^2L_*)\le C_{\rho,k}
+\|C_kL_*\|_{L^2(\mu_h)}.
\]
The tested scalar error is at most
\(C_0\rho\,\mu_h(C_k^2L_*^2)\). Absorbing it into the left side
of the terminal inequality proves
\[
\|C_kL_*\|_{L^2(\mu_h)}\le C_{\rho,k}.
\]
For the source \(w_0=Ae\), \(\|e\|\le1\), the same sum is
\(p_k^Te\). Use the local test \(\widehat C_kG\).
Its support and difference bounds control its test norm by
\(C_\rho\mathcal E_h(G)\). For the scalar error use
\(\widehat C_k\le C_k\), the last \(L^2\) bound, and
Cauchy--Schwarz. The terminal and weak residual inequalities give
\[
|\mu_h[\widehat C_kG(m^k-m_*)^Te]|
\le C_{\rho,k}\mathcal E_h(G).
\]
Summing the ordinary weak rule with target \(e\) from level one
through \(k\) also controls the tested \((X-m^k)^Te\).
The deficit costs at most
$2\sqrt n\|G\|_2\mu_h(D>0)^{1/2}$.
This is the deterministic argument of Proposition 7.2, applied only
after checking its hypotheses on the selected regions and verifying
both of its actual source diagonals above. It yields the asserted covariance inequality after
replacing $G$ by $G-\mu_h(G)$ and taking the supremum over $e$.
For $G=e^{\mathsf T}X$, $\mathcal D_h(G)\le1$; taking a top covariance
eigenvector gives $\|\Sigma_h\|\le C(\|\Sigma_h\|+1)^{1/2}$ and hence
a constant bound.
For the mean estimate, the ordinary residual rule suffices; no implicit
source is needed. On each selected
region $m^k$ is within $C\zeta\sqrt n$ of $m_*$ if the residual
threshold was chosen below a fixed multiple of $\zeta$.
The ordinary square-density rule, summed with the constant target $e$
through $k$, bounds the tested difference between $X$ and $m^k$ by
$C_\zeta(1+\mathcal D_h(f)/N)$. The deficit has the same bound.
This proves the conclusion of the companion Proposition 7.3 relative
to $m_*$.
Apply it also with $f=1$ and use the triangle inequality
to obtain \eqref{prof:eq27}.
If stability is absent, choose the iteration threshold so the exact
formula for $F_h(h^k)$ is at most $\delta\sqrt n$ on each selected
region. Then $\|m^k\|\le\sqrt{nQ_\delta(h)}$, and the same argument,
once for $\nu_h$ and once for $\mu_h$, proves the coarser assertion.
This also covers an empty approximate-zero set, since then every
selected cutoff vanishes and the displayed deficit estimate controls
the entire measure.
All bounds use only norm bounds on seeds, so they hold for the whole
Euclidean unit ball simultaneously. The finite recipe choices may
depend on the requested scalar tolerance, but the diagnostic event
quantifies over every bounded diagonal parameter after $J$ is observed.
It therefore does not depend on $f,G$, or the directional seed.
For covariance choose the stability margins and $c_*$, then $\rho$
and $\kappa$, then the selection depth and finite recipes, then their
word length, coefficient bounds and truncations, and finally the
size threshold guaranteeing every flip buffer. For
\eqref{prof:eq27} and its coarser form, only the ordinary diagnostics
are required throughout.
\end{proof}
\subsection{Large observation time and the product endpoint}
\begin{proposition}[Terminal extension, uniformly in the initial spin law]
\label{prof:terminal-extension}
There are fixed $T,C,c>0$ and disorder events of ordinary probability
at least $1-e^{-cn}$ with the following property. For every matrix in
these events, let $X$ have any probability law on the cube, possibly
depending on the matrix, and let the observation noises be independent
of $X$. Set $h_T=TX+\sqrt T\,g$. The reference Gibbs law $\mu_{h_T}$
has unscaled gap at least $1/2$ except with probability $e^{-cn}$.
On the original spectral event choose a fixed $c_+$ such that
$R=J+c_+I\succeq0$. For one further time unit add observations with
drift $RX$ and covariance rate $R$, obtaining $h_{T+\lambda}$.
Define the reference laws
\[
\mu_{h,\lambda}(x)\ \propto\
\exp\left(\frac{1-\lambda}{2}x^{\mathsf T}J^\circ x
+h^{\mathsf T}x\right),
\qquad 0\le\lambda\le1.
\]
The covariance and inverse unscaled gap of
$\mu_{h_{T+\lambda},\lambda}$ are bounded by $C$, simultaneously in
$\lambda$, outside an event of conditional probability $e^{-cn}$,
uniformly over the law generating $X$. At $\lambda=1$ the reference
law is product. These reference laws are the posteriors of the original
Gibbs prior evaluated at the observations; no assertion about the
posterior of the arbitrary law generating $X$ is made.
\end{proposition}
\begin{proof}
The signed curvature inequality, Lemma 8.1 of
\cite{AcceptedGap}, is deterministic for arbitrary symmetric
zero-diagonal interaction matrices with sufficiently small entries.
For interaction $B$, write
$U_{ij}=\tanh B_{ij}$, $Q_{ij}=U_{ij}^2$ (entrywise),
$r_4(U)=\max_i\sum_jU_{ij}^4$, and
$a_i=v_i d_i f$, where $v_i$ is the conditional spin variance.
It states
\[
\mu_h[(-Lf)^2]\ge(1-Cr_4(U))\mathcal D_h(f)
-\mu_h[a^{\mathsf T}Ua+C|a|^{\mathsf T}Q|a|].
\]
Its proof has no inverse-temperature parameter, so this is an exact
companion input.
Here is the endpoint matrix event, with exponential rather than only
asymptotic probability. Fix a curvature tolerance
\(\varepsilon>0\) so small that \((1+C)\varepsilon\le1/8\).
Choose a fixed global bound \(M\) for the row-square sums and
\(\|J^\circ\|\), with exponentially small exception. Next choose
an entry tolerance \(\delta>0\) small enough for the curvature
lemma and for \(M\delta/3\le\varepsilon/2\) and
\(CM\delta^2\le1/8\). Gaussian tails give
\(\max|J^\circ_{ij}|\le\delta\) outside probability
\(e^{-c_\delta n}\). For each fixed unit vector \(z\),
\[
\mathbb P(|z^{\mathsf T}J^\circ z|>t)\le2e^{-nt^2/4},\qquad
\mathbb P\left(\sum_{l\in S}(J^\circ_{il})^2>t\right)
\le e^{-nt/4+|S|\log2/2}.
\]
A $1/4$-net of a $k$-dimensional unit sphere has size at most $9^k$.
Union these bounds over supports of size at most $\alpha n$ and the
$9^{\alpha n}$ nets. Since
$n^{-1}\log\binom n{\lfloor\alpha n\rfloor}\to\mathfrak h(\alpha)$,
choosing $\alpha$ sufficiently small gives, with exponential probability,
\[
\|(J^\circ)_{SS}\|\le\varepsilon/2,
\qquad \max_i\sum_{l\in S}(J^\circ_{il})^2\le\varepsilon
\quad(|S|\le\alpha n).
\]
These are the net and row arguments of Lemma 8.2 of the companion
article, now evaluated at variance $1/n$.
They work simultaneously for all $B=\theta J^\circ$, $0\le\theta\le1$.
Indeed $|\tanh z-z|\le|z|^3/3$ gives
$\|\tanh(\theta J^\circ)-\theta J^\circ\|\le M\delta/3$ by row sums;
$Q_{ij}\le(J^\circ_{ij})^2$ and $r_4(U)\le M\delta^2$.
By the preceding choice of \(\delta\), these errors fit the
\(\varepsilon\) and curvature budgets. We obtain uniformly in
$\theta$ bounded global norms of $B,U,Q$, small principal submatrix
norms of $U,Q$ on $|S|\le\alpha n$, and $Cr_4(U)\le1/8$.
The bounds hold pointwise over spin configurations and require no
union over configurations or initial laws.
The observation source law may differ from the Gibbs law whose gap is
being estimated. Throughout this argument the posterior is formed from
the Gibbs reference law. A norm estimate makes most coordinates strongly
biased for every source configuration; the small-submatrix bounds
control the remaining coordinates uniformly.
At time $T$ the observation field is $TX+\sqrt T\,g$.
During the extension it is
\[
h_{T+\lambda}=TX+\sqrt T\,g+\lambda RX+R^{1/2}B_\lambda,
\]
where $B$ is a fresh standard Brownian motion independent of $X,g$.
Updating the original Gibbs prior by this likelihood subtracts
$\lambda x^{\mathsf T}Rx/2$ from its interaction; the diagonal is
constant on the cube. This identifies its posterior with the stated
reference law. The probability estimates below are uniform over the
possibly different law generating the observation source $X$.
On the norm event $\|R\|\le C$, the perturbation to $TX$ obeys
\[
\sup_{0\le\lambda\le1}\|h_{T+\lambda}-TX\|
\le C(1+\sqrt T)\sqrt n
\]
outside an event of probability $e^{-cn}$, independently of the law of
$X$. The Gaussian norm tail controls $g$, and exponential Markov on
the sum of the coordinate Brownian suprema squared controls $B$.
Thus, for sufficiently large fixed $T$, fewer than $\alpha n/2$
coordinates have $|h_{T+\lambda,i}|<T/2$, uniformly in $\lambda$.
For every configuration $x$ simultaneously,
$\|(1-\lambda)J^\circ x\|\le M\sqrt n$, so at most
$16M^2n/T^2\le\alpha n/2$ coordinates have interaction field
larger than $T/4$. Consequently the set where the full conditional
field has magnitude below $T/4$ has size at most $\alpha n$,
for every $x$ and $\lambda$.
Put $q_f(x)=\sum_i v_i(x)(d_i f(x))^2$ and
$\eta=\operatorname{sech}^2(T/4)$. On the complement of that small
set, $\|a_{S^c}\|^2\le\eta q_f$, while $\|a\|^2\le q_f$.
The small-submatrix and global bounds therefore give pointwise
\[
a^{\mathsf T}Ua+C|a|^{\mathsf T}Q|a|
\le(1+C)\{\varepsilon+M(2\sqrt\eta+\eta)\}q_f.
\]
With \(\varepsilon\) already fixed, choose \(T\) large enough
both for the sparse-set bounds above and for
\((1+C)M(2\sqrt\eta+\eta)\le1/8\). The displayed expression
is then at most \(\mathcal D_h(f)/4\) after averaging.
The curvature inequality yields
$\mu_h[(-Lf)^2]\ge\mathcal D_h(f)/2$.
We finish by converting curvature to the gap.
By the finite-dimensional
spectral theorem each nonzero eigenvalue
of $-L$ is at least $1/2$. The kernel consists of constants because
the Ising law is positive on every cube edge. Thus
$\operatorname{Var}_{\mu_h}f\le2\mathcal D_h(f)$, uniformly in
$\lambda$ and $f$. Applying this to unit linear tests also gives
$\|\Sigma_h\|\le2$. This proves the proposition.
\end{proof}
We conclude by making explicit the probability interfaces used elsewhere.
For each fixed positive compact interval, combine
Propositions~\ref{prof:positive-stability} and
\ref{prof:endpoint-posterior} with the finite diagnostic events of
Lemma~\ref{prof:endpoint-words}. Their ordinary exponentially small
exceptions remain exponentially small after planting: if
$L=Z(0)/\mathbb E Z(0)$, the critical overlap-binomial identity gives
$\mathbb E L^2\le C(n+1)$, for example by the pointwise Chernoff bound
$\mathbb P(\sum_i\varepsilon_i=k)\le2e^{-k^2/(2n)}$ and summing its
$n+1$ possible values.
Cauchy--Schwarz therefore pays only a polynomial
factor in transferring any disorder event. It follows that, at each
deterministic positive time in that interval, $\|\Sigma_s\|$ is bounded
except with probability $e^{-cn}$ integrated under planting and the
original spectral restriction. The terminal proposition gives the
corresponding terminal gap input.
For the macroscopic entropy argument, only the ordinary diagnostics in
\eqref{prof:eq27} and its $Q_\delta$ version are needed. They are
field-independent and may be imposed externally on typical ordinary
disorder for each fixed tolerance. Their exponent need not be compared
with the arbitrarily large multiplier $A_*$ in
Proposition~\ref{prof:positive-stability}. On any compact positive
band with finitely many outer thresholds, use one residual tolerance
smaller than every threshold's permitted tolerance.
Existence of a
large-$q$ point at this tolerance implies existence at each
corresponding looser tolerance, so the outer rates remain valid.
Here is how the moving thresholds in the entropy argument fit this
finite construction. Let \(u_*>0\) be a fixed upper endpoint for
the small-range outer estimate, and on a fixed compact band put
\(R_b=C'A_*^{1/3}\sqrt b\), with \(A_*\ge1\) and \(C'\ge2C\).
Restrict the initial interval so that \(R_b\le u_*\).
For every moving threshold \(u=2^jR_b\le u_*\), choose a lower
point \(v\) in a fixed dyadic grid on that band with
\(u/2\le v\le u\). Then \(v\ge C\sqrt b\), and
\[
\{Q_\delta(h_b)>u\}\subseteq\{Q_\delta(h_b)>v\},\qquad
cv^3\ge(c/8)u^3.
\]
Only finitely many grid points occur on a fixed positive band.
For each point use the compact time subband on which
\(v\ge C\sqrt b\), and choose one residual tolerance for this
finite collection. Thresholds above \(u_*\) are grouped into the
single event \(Q_\delta>u_*\), using its fixed rate
\(e^{-cnu_*^3}\) and the bound \(Q_\delta\le1\).
Countably many bands and tolerances are handled by the diagonal
subsequence procedure in the entropy argument; no estimate is claimed
uniformly as positive time or a chosen residual tolerance tends to zero.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-positive.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-roots.tex ====
\section{Counting exact field roots}\label{prof:roots}
We express expected counts of field roots as integrals. Conditioning at
a root leaves a Gaussian matrix on its orthogonal complement, while the
scalar part of the integral determines the root's empirical moments.
An excluded root count will later bound the probability of an excluded
field. All integrated estimates below use the planted law, with the
original spectral restriction imposed when indicated.
The organization by a Gaussian root density and a conditional determinant
follows the Kac--Rice analysis of TAP equations in
\citet[Sections~3--4]{FanMeiMontanari2021} and its local-convexity refinement
in \citet[Lemma~4.8 and Appendix~A.5]{CelentanoFanMei2023}.
The critical shrinking scale, polynomial determinant bound, and
$nq^3$ exclusion rates are proved here.
At observation precision $s=r^2$, the planted experiment may be
gauged so that its spin is $\mathbf1$. If $S_X=\operatorname{diag}X$,
this gauge sends $J$ to $S_XJS_X$ and sends $h,y,m$ to
$S_Xh,S_Xy,S_Xm$. In the gauged coordinates
$J=W+\mathbf1\mathbf1^{\mathsf T}/n$ and
$h_s=s\mathbf1+\sqrt s\,g$, with $W$ a GOE matrix and $g$ an
independent standard Gaussian. We study the roots of
\[
\Phi_J(y):=y+(1-q)m-Jm=h_s,\qquad
m=\tanh y,\qquad q=\langle m^2\rangle.
\]
Write
\[
b=1-q,\quad v=1-m^2,\quad V=\operatorname{diag}v,\quad
M=\langle m\rangle,
\]
and, for the scalar Gaussian regression, put
\[
\nu=q+s,\quad d=M+s,\quad a=\langle(y-d\mathbf1)m\rangle,\quad
C_a=a/\nu,\quad j_0=q/\nu,\quad
\alpha=(C_a-b)/(1+j_0).
\]
For fixed $y$, let $\mathsf Q_y$ be the conditional law
of $J=W+\mathbf1\mathbf1^{\mathsf T}/n$ given
\[
Wm+\sqrt s\,g=y-d\mathbf1+bm.
\]
The field represented by $(y,J)$ is $\Phi_J(y)$. All root-counting
arguments in this section use $k=nr^3\ge\log^{10}n$ and small $r$.
\begin{lemma}[The exact counting density]\label{prof:root-count}
For every nonnegative Borel test $T(y,J)$,
\[
\E_{\rm pl}\sum_{\substack{y:\,\Phi_J(y)=h_s\\
\det D\Phi_J(y)\ne0}}T(y,J)
=\int (1+j_0)^{-1/2}G(y)\,
\E_{\mathsf Q_y}\!\left[T(y,J)\mathcal W(y,J)\right]\,dy,
\]
where
\[
G(y)=\prod_i \varphi_{d,\nu}(y_i)\exp\{n(1+j_0)\alpha^2/2\}
\]
and
\begin{equation}\label{prof:eq8}
\mathcal W(y,J)
=e^{-nb^2/2}|\det(I+(bI-J-2mm^{\mathsf T}/n)V)|.
\end{equation}
Here $\varphi_{d,\nu}$ is the density of a real Gaussian with mean
$d$ and variance $\nu$. Matrix restrictions are included in $T$.
\end{lemma}
\begin{proof}
In the conditioning above, $y$ is fixed and $m=\tanh y$ is its
deterministic profile. The root map is proper,
since $\Phi_J(y)-y$ is bounded for fixed $J,n$. Its critical target values
have Lebesgue measure zero, and the Gaussian field has an everywhere
positive smooth density. The area formula therefore expresses the expected
root sum as the integral of the observation density at $\Phi_J(y)$ times
$|\det D\Phi_J(y)|$; exhaustion by compact balls and monotone convergence
handle arbitrary nonnegative tests.
The derivative is
\[
D\Phi_J(y)=I+(bI-J-2mm^T/n)V.
\]
The conditioned observation has covariance $\nu I+mm^T/n$, determinant
$\nu^n(1+j_0)$, and inverse
$\nu^{-1}I-mm^T/[n\nu^2(1+j_0)]$.
After dividing its density by $\prod_i\varphi_{d,\nu}(y_i)$, the log ratio
per site, apart from the determinant prefactor, is
\[
-bC_a-\frac{b^2j_0}{2}
+\frac{(C_a+bj_0)^2}{2(1+j_0)}
=\frac{(1+j_0)\alpha^2}{2}-\frac{b^2}{2}.
\]
This proves the formula, including the conditional matrix law and all
nonnegative restrictions.
\end{proof}
\begin{lemma}[A determinant square bound]\label{prof:determinant-square}
For each fixed $D$, the conditional second moment
$\E_{\mathsf Q_y}\mathcal W(y,J)^2$ is at most $n^{C_D}$ on
$\|y\|\le n^D$, uniformly over the parameters under consideration.
Outside these balls the bound has a fixed polynomial envelope in
$\|y\|$.
\end{lemma}
\begin{proof}
The identity $\det(I+AB)=\det(I+BA)$ writes the determinant in
$\mathcal W$ as the symmetric determinant
$\det(I+\sqrt V(bI-J-2mm^{\mathsf T}/n)\sqrt V)$.
First compare the conditional matrix with an unconditioned GOE matrix
$W_0$. For $q>0$, put $u=m/\sqrt{nq}$ and $E=u^\perp$. The conditional
law leaves $P_EWP_E$ as a fresh GOE block. Its remaining row and scalar
are Gaussian; with $e=P_E\mathbf1/\sqrt n$ they have the representation
\[
\begin{split}
P_EJu={}&\frac{M}{\sqrt q}e+
\frac{\sqrt q}{\sqrt n\,\nu}P_E(y-d\mathbf1)
+\sqrt{\frac{s}{n\nu}}\,g_E,\\
u^{\mathsf T}Ju={}&\frac{M^2}{q}
+\frac{2(a+bq)}{2q+s}
+\sqrt{\frac{2s}{n(2q+s)}}\,\xi,
\end{split}
\]
where $g_E,\xi$ are independent standard Gaussians, also independent
of the $E$ block. Share that block with $W_0$. The row change, the
plant $ee^{\mathsf T}$, and the TAP spike are a perturbation of fixed
rank. Its norm is polynomial in $n,\|y\|$ and the fresh Gaussian
norms: $|M|/\sqrt q\le1$, $a^2\le q\langle(y-d)^2\rangle$, and
$\nu,2q+s\ge s$ with $s^{-1}$ polynomial on the present range.
When $q=0$, $m=0$ and the conditioning is independent of $W$, so the
same conclusion is immediate.
Here is a useful bound for this reduction. For symmetric $H$ and
rank-$j$ symmetric $N$,
\[
|\det(H+N)|\le(1+\|N\|/\varepsilon)^j
|\det(H+\mathrm i\varepsilon I)|.
\]
It follows from singular-value interlacing after factoring
$H+\mathrm i\varepsilon I$: the factor
$H(H+\mathrm i\varepsilon I)^{-1}$ is a contraction and the remaining
change has rank $j$. Take $\varepsilon=n^{-5}$. Markov's polynomial
derivative inequality on $[-1/n,1/n]$, followed by integration on a
small interval around a maximizer, gives for every real polynomial
$P$ of degree at most $n$
\[
|P(\mathrm i\varepsilon)|^2
\le n^C\int_{-1/n}^{1/n}|P(z)|^2\,dz.
\]
Indeed $\|P'\|_\infty\le Cn^3\|P\|_\infty$ on this interval; iterating
this bound controls the imaginary displacement, and an interval of
length $cn^{-3}$ controls the maximum by the integral.
Truncate enormously large fresh entries at negligible cost by crude determinant bounds.
It remains to bound the unconditioned square. For real $z$, put
$x_i=1+z+bv_i$. The exact identity is
\[
\E_{W_0}\det\!\left((1+z)I+bV-\sqrt V W_0\sqrt V\right)^2
=\mathbb E_H\prod_i
\left[(x_i+\mathrm i v_i H_1)(x_i+\mathrm i v_i H_2)
+v_i^2(|h_1|^2+|h_2|^2)\right],
\]
where $H_1,H_2$ are independent real normals of variance $1/n$ and
$h_1,h_2$ are independent circular normals with
$\mathbb E|h_j|^2=1/n$. For completeness, represent each determinant
as the top coefficient of the exponential in one set of exterior
variables $\bar\psi,\psi$. Averaging $W_0$ adds
$-[X_{11}^2+X_{22}^2+2X_{12}X_{21}+2PQ]/(2n)$ to the exponent, where
$X_{ab}=\sum_i v_i\bar\psi_i^a\psi_i^b$,
$P=\sum_i v_i\bar\psi_i^1\bar\psi_i^2$, and
$Q=\sum_i v_i\psi_i^1\psi_i^2$. Decoupling these four quadratic terms
with the displayed Gaussian variables and extracting the site
coefficients gives the identity.
Move each $H_j$ contour to $H_j+\mathrm i b$. The factors become
$1+z+\mathrm i v_iH_j$, while the two Gaussian densities cost
$e^{nb^2}$ in modulus. This exactly cancels the factor $e^{-nb^2}$
in $\mathcal W^2$. The contour move is valid because the integrand is
an entire polynomial times a Gaussian. If $e_1,e_2$ are the eigenvalues
of the two-by-two Hermitian matrix with diagonals $H_1,H_2$ and
squared off-diagonal modulus $|h_1|^2+|h_2|^2$, the remaining modulus
is at most $\prod_{j=1}^2[(1+z)^2+e_j^2]^{n/2}$, since $0\le v_i\le1$.
More explicitly, write
$R_H^2=H_1^2+H_2^2+2(|h_1|^2+|h_2|^2)=e_1^2+e_2^2$.
The Gaussian exponent is $-nR_H^2/2$. For $R_H\le1$, the inequality
$\log(1+x)\le x-cx^2$ on a fixed bounded interval bounds the remaining
integrand by $C\exp(-cnR_H^4)$; for $R_H>1$, split off a fixed compact
annulus and then use $\log(1+x)\le x-cx$ for $x$ sufficiently large,
obtaining an integrable $Ce^{-cn\min(R_H^4,R_H^2)}$ bound.
There are six real Gaussian variables, so their normalized integral is
at most $Cn^3\int_{\mathbb R^6}e^{-cn\min(|u|^4,|u|^2)}du
\le Cn^{3/2}$. The bounded-rank and real-shift reductions multiply this
by another fixed polynomial. This proves the second-moment bound.
\end{proof}
\begin{proposition}[Scalar root diagnostics]\label{prof:scalar-diagnostics}
Fix an exclusion budget $A>0$ and an accuracy for the little-oh terms.
After choosing a sufficiently small upper bound on $r$ and then a
sufficiently large $n$, the following statements hold for
$k=nr^3\ge\log^{10}n$.
The total integral of $G(y)\,dy$ is at most polynomial in $n$. Outside
mass $n^Ce^{-Ak}$ under this scalar integral,
\[
q\asymp_A r,\qquad |M|=O_A(q),\qquad |\alpha|=O_A(q).
\]
There are absolute $L,c>0$ such that the part with
$q\ge q_{\min}\ge Lr$ has mass at most
$n^Ce^{-cnq_{\min}^3}$. For a sufficiently small fixed exclusion
budget, the retained parameters instead satisfy
$q,M=(1\pm\epsilon)r$ and $|\alpha|\le\epsilon q$, for any requested
fixed $\epsilon>0$.
Under either budget the retained empirical moments satisfy
\begin{equation}\label{prof:eq9}
\langle m^4\rangle=3q^2-8q^3+o(q^3),\qquad
\langle m^6\rangle=15q^3+o(q^3),\qquad
\langle m^3\rangle=3qM+O_A(q^3).
\end{equation}
For $l=\langle ym\rangle/q$,
$|\langle y-lm\rangle|\le C_Aq^3$, with an arbitrarily small leading
constant under a sufficiently small budget. For any fixed sufficiently
small $\eta>0$, one may also retain
\[
\langle y^2\mathbf1_{\{|y|>q^{1/2-\eta}\}}\rangle\le C_Aq^4,
\qquad
\frac1n\|P_{\operatorname{span}(m,\mathbf1)^\perp}y\|^2\asymp q^3.
\]
Finally, fix $\tau>0$ and $D<\infty$. There are constants $c,C>0$
such that, for every $(t_1,t_3)$ with
$\tau\le\|(t_1,t_3)\|\le n^D$ and every $\theta\in\mathbb R$, at
least $cn$ indices satisfy
\[
|m_i|\le C\sqrt q,\qquad |y_i-lm_i|\le Cq^{3/2},\qquad
\operatorname{dist}\left(
t_1\frac{m_i}{\sqrt q}+t_3\frac{y_i-lm_i}{q^{3/2}}+\theta,
\pi\mathbb Z\right)\ge c.
\]
The determinant-weighted exclusions follow from
Lemma~\ref{prof:determinant-square} after increasing the preliminary
budget. Through Lemma~\ref{prof:root-count}, they bound the expected
number of excluded regular roots. After removing the null set of
critical target fields, the probability that any root is excluded is
at most that expected count. This is the simultaneous-root meaning of
the diagnostics.
\end{proposition}
\begin{proof}
\par\smallskip\noindent\textit{Scalar comparison and exclusion of remote parameters.}\enspace
First restrict to a fixed polynomial ball in $y$. The bound
$a^2\le q\langle(y-d)^2\rangle$ and $j_0/(1+j_0)\le1/2$ give a
Gaussian tail envelope outside it. On the ball, the logarithmic
likelihood and its parameter derivatives are polynomially bounded,
since $\nu\ge s$ and $s^{-1}$ is polynomial. Divide $q,M,a$ into a
polynomial number of bins with sufficiently fine inverse-polynomial
mesh. Replacing their empirical values by a bin representative changes
the log likelihood by $O(1)$. In the transport and variance comparisons
through the retained-range bound below, the symbols
$q,M,a,b,\nu,d,C_a,j_0,\alpha$ denote these frozen representatives;
$y$ and $m=\tanh y$ remain the variables being integrated. In the
variance comparison, $j_0^{-1}$ is used only where $C_a>1$; Cauchy--Schwarz
and the polynomial ball then also bound $q^{-1}$ polynomially.
Suppose $\alpha<1$ stays a fixed distance from one. The map
$z'_i=y_i-\alpha\tanh y_i$ has Jacobian
$\prod_i(1-\alpha v_i)$; here $\alpha$ is the frozen bin value.
Direct expansion of the two Gaussian exponents gives
\[
\frac{G(y)\,dy}{\prod_i\varphi_{d,\nu}(z'_i)\,dz'}
=e^{O(1)-n(b\alpha+\alpha^2/2)}
\prod_i(1-\alpha v_i)^{-1}
\le e^{O(1)-nH(\alpha)},
\]
where
\[
H(u)=\frac{u^2}{2}-b[-\log(1-u)-u].
\]
The inequality uses $\log(1-uv)\ge v\log(1-u)$ for $0\le v\le1$.
In particular $H(u)\ge qu^2/6$ for $u\le q$, with a cubic or stronger
penalty on the negative side when $q$ is small.
The remaining branch is $C_a>C_0:=1+j_0q$, equivalently $\alpha>q$.
Compare there with the product Gaussian on $y$ having variance
$\nu C_a^2/j_0$. Cauchy--Schwarz on $a$ gives the remaining rate
$f(C_a)/2$, where
\[
f(C_a)=C_a^2/j_0-1-\log(C_a^2/j_0)
-(C_a-b)^2/(1+j_0).
\]
The bounds
$f(C_0)\ge-2q^3/3$, $f'(C_0)\ge2q/(1+q)$, and $f''\ge1$
exclude $\alpha>q+C_AR_*^{3/2}$ at budget $AnR_*^3$, where
$R_*:=\max(r,q)$. The $H$ comparison excludes
$\alpha<-C_AR_*$. On the retained range
\[
-C_AR_*\le\alpha\le q+C_AR_*^{3/2},\qquad
H(\alpha)\ge-\delta R_*^3,
\]
where $\delta$ can be made arbitrarily small by decreasing the upper
scale.
The frozen-symbol convention for those two comparisons ends here.
We also need a fixed negative rate away from $q=0$ as $s\downarrow0$.
Fix $q_0>0$. For a probability law $P$ with finite second moment and $\E_P\tanh^2Y\ge q_0$, define
\[
q=\E_P\tanh^2Y,\quad M=\E_P\tanh Y,\quad d=M+s,\quad \nu=q+s,
\]
\[
a=\E_P[(Y-d)\tanh Y],\quad b=1-q,\quad j_0=q/\nu,\quad
\alpha=(a/\nu-b)/(1+j_0).
\]
Put $\psi_s(P)=(1+j_0)\alpha^2/2$ and
\[
\mathscr R_s(P)=\psi_s(P)
-\operatorname{KL}\bigl(P\Vert N(d,\nu)\bigr).
\]
The endpoint scalar inequality of Lemma~\ref{prof:endpoint-scalar},
proved independently below, gives $\mathscr R_s\le0$, with equality
only for the indicated Gaussian. At $s=0$, a consistent equality law
with $q>0$ would be $N(M,q)$. Its mean equation
$M=\E_{N(M,q)}\tanh Y$ forces $M=0$, since the difference of the two
sides is strictly increasing and vanishes at zero. But then
$\E_{N(0,q)}\tanh^2Y<\E Y^2=q$, a contradiction.
For completeness, this pointwise exclusion has the uniformity needed
by the integral. Fix a bound on $\E_PY^2$. The resulting set of laws
with $q\ge q_0$ is weakly compact. Its first moments are
uniformly integrable, so the average of $y\tanh y$ and hence $\psi_s$
are continuous there; $\nu\ge q_0$ keeps the denominators separated
from zero. Relative entropy is lower semicontinuous, making
$\mathscr R_s$ upper semicontinuous. Its maximum at $s=0,q\ge q_0$
is therefore strictly negative, and remains so for sufficiently small
$s$. The second-moment bound also makes replacement of nearby Gaussian
parameters uniformly $e^{o(n)}$ in the product density. A finite cover
by neighborhoods on which one bounded continuous entropy test has
this slack bounds the scalar integral by $e^{-cn}$. This is the
compact-set integral bound in the proof of
Lemma~\ref{prof:endpoint-scalar}, with its smoothing variance set to zero
and $\nu=q+s\ge q_0$; that proof records the finite-cover argument of
\cite[Lemma~5.4]{AcceptedGap}.
The Gaussian envelope $G(y)\le e^{Cn-c\sum_i y_i^2}$ on $q\ge q_0$
removes the second-moment restriction. This proves the fixed-$q_0$
exclusion; the cubic rate for shrinking $q$ is derived next.
\par\smallskip\noindent\textit{Empirical moments at the cubic scale.}\enspace
The transport comparison handles the negative and moderate scalar
parameters, and variance inflation excludes the remaining positive
range. On the retained range we can now impose product-Gaussian
moment diagnostics and use root consistency to locate the overlap scale.
Return to the root integral. For the shifted-coordinate reference of
the bin containing $y$, write $\alpha_{\rm bin},d_{\rm bin},\nu_{\rm bin}$
for its frozen values and put
\[
z'_{{\rm bin},i}=y_i-\alpha_{\rm bin}\tanh y_i,\qquad
Z_i=(z'_{{\rm bin},i}-d_{\rm bin})/\sqrt{\nu_{\rm bin}}.
\]
The rate parameters in this product-reference estimate also use the
frozen bin values. For any fixed $A'>0$, exponential Markov
bounds give the following outside product probability
$e^{-A'nR_*^3}$: the second moment on $|Z_i|>R_*^{-\eta}$ is at most
$C A'R_*^3$, and clipped moments through any prescribed fixed degree
differ from their Gaussian values by $O(\sqrt{A'}R_*^{3/2})$.
Choose $\eta$ sufficiently small for that degree. These estimates use
a fixed exponential tilt for the tail square and a tilt of order
$\sqrt{A'}R_*^{3/2}$ for each clipped polynomial; their constants stay
bounded as $A'\downarrow0$.
The scalar change of measure costs at most $e^{\delta nR_*^3}$ on a
retained bin. Hence a reference exception of size $e^{-A'nR_*^3}$
contributes at most $e^{-(A'-\delta)nR_*^3}$ to the scalar integral,
up to the polynomial bin factor. For a small target budget choose
$A'$ proportional to that budget and then make $\delta<A'/2$ by
shrinking the upper scale. This preserves the small
$O(\sqrt{A'})$ moment errors needed below.
For the consistency and residual conclusions below, the symbols
$q,M,a,b,\nu,d,C_a,j_0,\alpha$ again denote the exact empirical quantities
defined at the start of the section. Expand the exact identity
$m=\tanh(z'_{\rm bin}+\alpha_{\rm bin}m)$ on the clipped region and use
the tail second moment on its complement. Replacing the bin parameters
in the resulting expressions by the empirical quantities is included
in the displayed errors, using the initially chosen sufficiently fine
mesh. The first consistency estimate is
\[
q=\nu+d^2+O_A(R_*)(\nu+d^2).
\]
It implies $s+d^2\le C_AR_*q$. If $q\le r$, the inequality
$r^2=s\le C_Arq$ forces $q\ge c_Ar$; otherwise this is automatic.
Thus $R_*=O_A(q)$ and $|d|=O_A(q)$. On this narrower range the
retained $H$ is nonnegative. Keeping the next terms gives
\[
0=s+d^2+2\alpha q-2q^2+o_A(q^2),\qquad
M=d+\alpha d-qd+O_A(q^2).
\]
Apart from an $o(q^2)$ error, the last constant tends to zero with the
small budget. The $H$ and $f$ penalties then force $|\alpha|\ll q$.
Since $d=M+s$, division of the two consistency equations by $q^2$
gives
\[
\frac{s}{q^2}=\frac dq+o_{\rm small}(1),\qquad
\frac{s}{q^2}+\left(\frac dq\right)^2=2+o_{\rm small}(1).
\]
The first ratio is nonnegative. These two relations therefore force
$s/q^2$ and $d/q$ close to one, proving the small-budget location.
In particular, bins with $q\gg r$ have a fixed negative rate
$cnq^3$. This tail also pays a prescribed budget $Ak$ in the
large-budget calculation: choose $q_{\min}/r$ so that
$c(q_{\min}/r)^3>A$, with slack for the polynomial bins.
Finally the total scalar integral is polynomial. The exceptional bins
just considered have strict negative rates. On the retained bins the
reference product integral is one and $H\ge0$; indeed
$H(q)\ge q^3/6$ and this sign persists up to
$q+O_A(q^{3/2})$. Summing over the polynomial grid proves the claim.
\par\smallskip\noindent\textit{Fourth and sixth moments and the cubic residual.}\enspace
The scale comparison is not yet enough for the matrix estimates: they
also use the first correction to the fourth moment. We therefore keep
the cubic-order terms in the same expansion. The cubic residual will
require a separate cancellation at small exclusion budget.
The same expansion has linear coefficient $1+\alpha+O(q^2)$ and
cubic coefficient $-1/3+O(q)$. The consistency equation for $q$ gives
\[
3q^2=3\nu^2+6\nu d^2+12\alpha\nu^2-12\nu^3+o(q^3),
\]
whereas direct expansion of the fourth moment gives
\[
\langle m^4\rangle
=3\nu^2+6\nu d^2+12\alpha\nu^2-20\nu^3+o(q^3).
\]
Since $\nu=q+O(q^2)$, their difference is $-8q^3+o(q^3)$.
Likewise $\langle m^6\rangle=15\nu^3+o(q^3)$ and
$\langle m^3\rangle=3\nu d+O_A(q^3)=3qM+O_A(q^3)$.
The clipped empirical errors are $o(q^3)$ at this scale even for a
fixed large budget. On the complement, boundedness of $\tanh$ and the
original-scale tail second moment $O_A(q^4)$ control the remainders.
For the residual, use the identity
\[
\langle y-lm\rangle
=\langle y-m\rangle-
\frac{\langle(y-m)m\rangle}{q}M.
\]
Expand it in $z'_{\rm bin}$ through degree six, including the occurrences of
$M$ and $q$ in this expression. Replacing empirical moments by their
Gaussian values costs $O(\sqrt{A'}q^3)+o(q^3)$. The order-$q^2$
terms cancel: the cubic Gaussian moment starts with $3\nu d$ and the
fourth with $3\nu^2$. After division by $q^3$, the remaining leading
coefficient is continuous in the bounded ratios
$\nu/q,d/q,\alpha/q$, with $\nu/q$ bounded below.
At the small-budget limit these ratios approach $1,1,0$. The leading
coefficient vanishes there by an exact reflection identity. If
$Y\sim N(\nu,\nu)$, its density obeys
$\varphi(-y)=e^{-2y}\varphi(y)$, so pairing $y$ and $-y$ gives
\[
\E F(Y)=\E[F(Y)\tanh Y]
\quad\text{for every integrable odd }F.
\]
With $F(y)=y$ and $F(y)=\tanh y$, this gives
$\E Y=\E[Y\tanh Y]$ and $\E\tanh Y=\E\tanh^2Y$.
The integrated residual is therefore exactly zero at
$\alpha=0,d=\nu$. Continuity and the empirical error prove the stated
small leading constant.
\par\smallskip\noindent\textit{Nonlattice and spanning diagnostics.}\enspace
On bounded standardized inputs, the normalized profiles
$m/\sqrt q$ and $(y-lm)/q^{3/2}$ converge to the nonsingular pair
$z$ and $z^3/3-z$. Their Gram matrix on a fixed bounded input set is
consequently bounded above and below. Put $w=y-lm$.
Then $\langle wm\rangle=0$, $\langle w^2\rangle\asymp q^3$, and
$1-M^2/q=1-O_A(q)$ is bounded below. The normalized squared norm of
the projection of $w$ onto $\operatorname{span}(m,\mathbf1)$ is
$\langle w\rangle^2/(1-M^2/q)=O_A(q^6)$. Removing both directions
therefore leaves squared norm comparable to $nq^3$, as asserted.
For phase dispersion we need derivative control of the actual profiles,
because the frequencies may grow with $n$. In a frozen reference bin,
write $Y(z)$ for the inverse of
\[
Y-\alpha\tanh Y=d+\sqrt\nu\,z,
\qquad
Y'(z)=\frac{\sqrt\nu}{1-\alpha\operatorname{sech}^2Y(z)}.
\]
The retained parameters satisfy $d=O_A(q)$, $\nu=q+O_A(q^2)$,
$\alpha=O_A(q)$, and
$l=(a+dM)/q=1+q+O_A(q^2)$. The last expansion follows by expanding
$ym=m^2+m^4/3+m^6/5+\cdots$ on the clipped region and using
\eqref{prof:eq9} and the tail-square bound. On every fixed bounded
$z$ interval, the functions
\[
f_q(z)=\frac{\tanh Y(z)}{\sqrt q},\qquad
g_q(z)=\frac{Y(z)-l\tanh Y(z)}{q^{3/2}}
\]
therefore satisfy
$f_q(z)=z+O_A(\sqrt q)$ and
$g_q(z)=z^3/3-z+O_A(\sqrt q)$ in $C^1$. For example,
\[
g_q'(z)=
\frac{(1-l\operatorname{sech}^2Y(z))\sqrt\nu}
{q^{3/2}(1-\alpha\operatorname{sech}^2Y(z))},
\]
which converges uniformly to $z^2-1$ on that interval.
Put $R=\|(t_1,t_3)\|$ and $v=(t_1,t_3)/R$. Each polynomial
$v_1z+v_3(z^3/3-z)$, with $\|v\|=1$, has an interval of fixed
positive length on which its derivative is bounded above and away
from zero in absolute value. Compactness in $v$ permits a finite
choice of such intervals and uniform constants. The $C^1$ bounds
give the same derivative bounds for the actual function
$v_1f_q+v_3g_q$ when the upper scale is small. On its chosen interval
this function is monotone. For all sufficiently large $R$, counting
periods of $R(v_1f_q+v_3g_q)+\theta$ shows that a fixed positive
fraction of the interval stays a fixed distance from $\pi\mathbb Z$,
uniformly in $\theta$. For $\tau\le R$ in a fixed bounded range,
compactness in $R,v,\theta\bmod\pi$ gives the same conclusion with
possibly smaller constants. Gaussian density is bounded below on
the bounded intervals used. No approximation error is multiplied
by the growing frequency in this argument.
Use a smooth periodic test supported a little inside this separated
phase set and a smooth cutoff inside the bounded input interval.
For each fixed choice, product concentration gives failure $e^{-cn}$.
All parameter derivatives on the retained bins are polynomially
bounded, since $q^{-1}$ is polynomially bounded. The parameter $l$
is determined by the existing bins through $(a+dM)/q$. Refine their
mesh, and mesh frequencies and phase, at inverse-polynomial accuracy
chosen after $D$, so that every test changes by less than the fixed
separation slack even for $R\le n^D$. There are only polynomially
many tests. Their union retains failure $e^{-c'n}$, and the scalar
change of measure preserves that rate after the upper scale is
decreased. The same bounded-input concentration gives the Gram
estimates used above.
\end{proof}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-roots.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-spectral.tex ====
\section{Spectral and spherical estimates for the support-profile argument}
\label{prof:sec:spectral}
The support-profile argument begins with estimates that involve only the
spectrum and unrestricted rotation orbits. These estimates have two uses.
At positive observation times they provide fixed-rate overlap concentration
and exponential-moment bounds with arbitrarily prescribed exceptional-set
rates. At the
smallest times a more accurate comparison with the spherical model gives
covariance estimates and an independent lower-bound argument.
\subsection{Conventions and the observation law}
\label{prof:subsec:conventions}
The matrix $J$ has the GOE normalization: its independent off-diagonal
entries have variance $1/n$, and its diagonal entries have variance $2/n$.
The diagonal does not affect the spin law. Write $\pi_{\mathrm c}$ for
uniform probability measure on $\{-1,1\}^n$, and $\pi_{\mathrm o}$ for
uniform probability measure on the sphere of radius $\sqrt n$. Define
\[
Z(h)=\int \exp\{x^TJx/2+h^Tx\}\,\dd\pi_{\mathrm c}(x),
\qquad
Z_{\mathrm o}(h)=\int \exp\{x^TJx/2+h^Tx\}\,\dd\pi_{\mathrm o}(x).
\]
Thus the partition functions use probability, rather than counting or
surface-area, normalization. The spin law at field $h$ is $\mu_h$.
Empirical averages are denoted by $\av{a}=n^{-1}\sum_i a_i$; vector norms
are Euclidean, matrix norms without a subscript are operator norms, and
$\|\cdot\|_{\mathrm{HS}}$ is the Hilbert--Schmidt norm.
Given $J$, sample $X$ from $\mu_0$ and observe
\[
h_s=sX+B_s,\qquad r=\sqrt s,\qquad k=nr^3,
\]
where $B$ is an independent standard Brownian motion in $\R^n$.
Completing the square in the Gaussian likelihood shows that the
conditional spin law is $\mu_s:=\mu_{h_s}$. Its mean and covariance will
be denoted by $m_s^{\mathrm{post}}$ and $\Sigma_s$, respectively.
There are two probability laws for the disorder. The ordinary law is
the centered GOE law. The planted law is its size bias by
$Z(0)/\E Z(0)$, together with the Gibbs spin and observation. Under the
present normalization $\E Z(0)=e^{n/4}$. Under the
planted law, $X$ is uniform on the cube and
\[
J=W+XX^T/n,
\]
where $W$ is an independent centered GOE matrix. This follows directly
by completing squares in its Gaussian density. A second exact identity
will be used repeatedly. Fix the spectrum and write $J=U\Lambda U^T$,
randomizing the finite eigenframe choices so that $U$ belongs to the
proper rotation group.
In eigen-coordinates $c=U^Th_s$, the planted joint
law of $(U,c)$ has density
\[
\frac{Z(Uc)}{Z_{\mathrm o}(Uc)}
\]
relative to Haar measure in $U$ and the spherical-model observation law
in $c$. In this ratio both partition functions use the co-rotated
matrix. Indeed the conditional density of $U$ is
$Z_U(0)/Z_{\mathrm o}(0)$, whereas the observation density contributes
$Z_U(Uc)/Z_U(0)$ times a rotation-invariant Gaussian factor.
Averaging $Z_U(Uc)$ over $U$ gives $Z_{\mathrm o}(Uc)$. In particular,
the marginal law of $c$ is exactly the spherical observation law.
Constants may change from line to line. All rate budgets in this
section are fixed before $n$ tends to infinity. At a small scale, an
$o(1)$ error may also be made small by decreasing a fixed upper bound
on $r$; the time itself need not tend to zero. Thus a statement with
exceptional mass $e^{-Bk}$, for every fixed $B$, means that the allowed
upper scale and the size threshold may depend on $B$ and on the desired
fixed accuracy. Polynomial factors are absorbed by rate slack when
$k/\log n$ is sufficiently large.
\subsection{Resolvents at the upper edge}
\label{prof:subsec:spectrum}
Put
\[
\ell=\log\log(n+1000),\qquad d_0=n^{-1/3}\ell,
\qquad b_{\mathrm{sc}}(z)=\frac{z-\sqrt{z^2-4}}2.
\]
Let $\rho_{\mathrm{sc}}$ be the semicircle probability law on $[-2,2]$.
The next estimates are imposed on the \emph{ordinary} disorder. Later
planted probabilities will be intersected with this spectral event;
the event need not have large planted probability.
\begin{lemma}[Spectral estimates]\label{prof:lem:spectrum}
There are events of ordinary probability tending to one on which,
uniformly for $d_0\le d\le D_*$ with any fixed $D_*<\infty$,
\begin{equation}\label{prof:eq1}
\lambda_{\max}(J)\le 2+o(d_0^2),\qquad
\frac1n\Tr(2+d^2-J)^{-1}
=b_{\mathrm{sc}}(2+d^2)
+o\!\left(\frac{d}{\sqrt{nd^3}}\right).
\end{equation}
For every fixed integer $p\ge2$, the trace of the $p$th resolvent
power is asymptotic to its semicircle value and is $O_p(nd^{3-2p})$.
There are order $nd^3$ eigenvalues at depth comparable to $d^2$ below
$2$, and at most $Cnd^3$ within an upper-edge interval of that width.
Moreover $\|J\|\le3$ and the empirical spectral measure converges to
$\rho_{\mathrm{sc}}$. Fixed-rank interlacings preserve the resolvent
estimates when the evaluation point stays at least a fixed positive
multiple of $d^2$ from every pole in both spectra.
\end{lemma}
\begin{proof}
The Gaussian--chi representation is the classical tridiagonal
construction; see \cite[Theorem~2.1 and Section~2.3]{DumitriuEdelman2002}
for the general beta-ensemble model. In the present normalization,
successive orthogonal Gaussian elimination gives a tridiagonal matrix
with the same spectrum as $J$. Its independent diagonal entries are
$N(0,2/n)$ and its adjacent entries are $\chi_{n-i}/\sqrt n$,
$1\le i<n$. To see the independence, rotate the first column below
the diagonal onto the second coordinate. Orthogonal invariance leaves
a fresh independent GOE matrix on the remaining coordinates, and the
procedure can then be iterated.
Let $A$ be the deterministic Jacobi matrix with zero diagonal and
adjacent entries $\sqrt{1-i/n}$, and write the tridiagonal matrix as
$A+\Delta$. Centered entries of $\Delta$ are sub-Gaussian at scale
$n^{-1/2}$. Gaussian norm concentration and the second moment of a
chi variable give adjacent-entry means
$O((n(n-i))^{-1/2})$. For
\[
R=(2+d^2-A)^{-1},\qquad a_i=(d^2+i/n)^{1/2},
\]
the positive walk expansion gives
\begin{equation}\label{prof:eq:jacobi-diagonals}
(R^p)_{ii}\le C_p a_i^{1-2p},\qquad p\ge1.
\end{equation}
Indeed the positive series
\[
(R^p)_{ii}
=\sum_{j\ge0}\binom{p+j-1}{j}
(2+d^2)^{-p-j}(A^j)_{ii}
\]
is a sum over returning walks. Their count divided by $2^j$ is
$O((j+1)^{-1/2})$. A walk staying above $i/2$ gains
$e^{-cij/n}$ from the adjacent weights, so its contribution is bounded by
\[
C_p\sum_{j\ge0}(j+1)^{p-3/2}
e^{-c(d^2+i/n)j}\le C_p a_i^{1-2p}.
\]
By reflection and the binomial tail bound, the relative count of walks
that visit below $i/2$ is at most $Ce^{-ci^2/(j+1)}$. Their sum is at
most $C_p d^{1-2p}e^{-cdi}$. Since $nd^3\ge1$,
$di\ge i/(nd^2)$; exponential decay in this last quantity bounds the
sum by $C_p d^{1-2p}(1+i/(nd^2))^{(1-2p)/2}$, which is again
$C_pa_i^{1-2p}$.
The perturbation is small in the resolvent metric already at
$d=n^{-1/3}\sqrt\ell$:
\begin{equation}\label{prof:eq:relative-noise}
\|R^{1/2}\Delta R^{1/2}\|=o(1)
\quad\hbox{with probability tending to one}.
\end{equation}
The block-energy estimate below follows the partial-sum and
summation-by-parts method used in
\cite[Proposition~7 and Lemmas~8--9]{LedouxRider2010}; see also
\cite[Lemma~5.6 and Section~6]{RamirezRiderVirag2011}.
The resolvent precision needed here is verified explicitly.
With $b_i=\sqrt{1-i/n}$ and $b_0=b_n=0$, the Jacobi energy is
\[
u^{\mathsf T}(2+d^2-A)u
=\sum_{i<n}b_i(u_{i+1}-u_i)^2+
\sum_i(2+d^2-b_i-b_{i-1})u_i^2.
\]
It controls
\[
\sum_i a_i^2 u_i^2+
\sum_{i<n/2}(u_{i+1}-u_i)^2.
\]
The identity
$2\Delta_{i,i+1}u_iu_{i+1}
=\Delta_{i,i+1}(u_i^2+u_{i+1}^2)
-\Delta_{i,i+1}(u_{i+1}-u_i)^2$
separates each noisy adjacent term into a diagonal contribution and a
squared difference.
The maximum entry of $\Delta$ is $o(1)$, so the
difference terms and the terms on the bulk half are harmless. Divide
the first half into consecutive blocks of length comparable to $a_i^{-1}$,
merging a short last block. The values of $a_i$ are comparable within
each block. Sub-Gaussian maximal bounds, applied separately to each
shifted entry sequence, show that centered diagonal partial sums from
the beginning of a block are at most $\epsilon a_i$, except with
probability $Ce^{-c\epsilon^2na_i^3}$ per block.
For a block $B$ of length comparable to $a_B^{-1}$, summation by parts
against a centered diagonal sequence $\zeta_i$ with these partial sums
gives
\[
\left|\sum_{i\in B}\zeta_i u_i^2\right|
\le C\epsilon\left(
a_B^2\sum_{i\in B}u_i^2+
\sum_{i,i+1\in B}(u_{i+1}-u_i)^2\right).
\]
To see the scale, average the endpoint value of $u^2$ over the block
and bound the variation of $u^2$ by Cauchy--Schwarz. Multiplication by
the partial-sum bound $\epsilon a_B$ then gives the two terms displayed.
The first-half means are $O(n^{-1})$, a relative
$O((na_B^2)^{-1})=o(1)$ contribution to the potential term.
A dyadic shell contains $O(na_B^3)$ blocks. At
$d=n^{-1/3}\sqrt\ell$, the smallest value of $na_B^3$ is
$\ell^{3/2}$. Choose $\epsilon\downarrow0$ with
$\epsilon^2\ell^{3/2}\gg\log\ell$. The block counts and the exponents
then grow geometrically on successive shells, so the exceptions are
summable. Summing the block inequalities proves
\eqref{prof:eq:relative-noise}. Positivity of $R^{-1}-\Delta$ at this
value of $d$ also places the upper edge below
$2+n^{-2/3}\ell=2+o(d_0^2)$.
The relative perturbation estimate has placed the spectrum below the
resolvent pole. The trace estimate requires more precision: its error
must be small on the unnormalized trace scale. We first estimate the
linear and quadratic perturbation terms, then identify the deterministic
Jacobi trace through its Hermite equation.
Expanding the resolvent and using \eqref{prof:eq:relative-noise} gives
\[
\Tr(2+d^2-J)^{-1}
=\Tr R+\Tr R^2\Delta
+O\bigl(\Tr R\Delta R^2\Delta\bigr).
\]
The quadratic remainder is nonnegative, so its expectation controls it
by Markov's inequality. Its centered-noise expectation
is $O(d^{-2})$, by \eqref{prof:eq:jacobi-diagonals}, Cauchy--Schwarz
for neighboring entries, and
$n^{-1}\sum_i a_i^{-4}=O(d^{-2})$. For the mean contribution, first
use $R\preceq Cd^{-2}I$. On the bulk half the adjacent-entry means
are $O(n^{-1/2})$ and the diagonals of $R^2$ are bounded; on the other
half the means are $O(n^{-1})$ and the trace bound applies.
The linear
term has standard deviation $O(d^{-2})$ and mean $O(d^{-1}+1)$.
These estimates extend to each fixed-ratio interval in $d$. In an
eigenbasis of $A$, with $\rho_i=(2+d^2-\lambda_i(A))^{-1}$, the quadratic
term is
\[
Q(d)=\sum_{i,j}\rho_i\rho_j^2|\widetilde\Delta_{ij}|^2.
\]
It is nonnegative and decreases with $d$, regardless of the signs in
$\Delta$. For the linear term $L(d)=\Tr R^2\Delta$,
\[
d\,\partial_d L(d)=-4d^2\Tr R^3\Delta.
\]
The diagonal bounds give this logarithmic derivative the same
$O(d^{-2})$ fluctuation scale as $L(d)$ on a fixed-ratio interval.
Integrating it controls the supremum there. Markov's inequality for
$Q$ and the corresponding second-moment bound for $L$, at threshold
$(nd^3)^{1/4}d^{-2}$, have summable failures over the dyadic intervals
because $nd^3$ starts at $\ell^3$ and grows geometrically. Finally
\[
\frac{(nd^3)^{1/4}d^{-2}}{\sqrt{n/d}}
=(nd^3)^{-1/4}\longrightarrow0,
\]
which gives the stated uniform trace error.
It remains to identify the deterministic trace. Reversing the Jacobi
indices in its principal-determinant recurrence gives monic Hermite
polynomials. The characteristic polynomial $P_n$ satisfies
\[
n^{-1}P_n''-zP_n'+nP_n=0.
\]
Therefore $b_A=P_n'/(nP_n)$ obeys
\[
b_A^2-zb_A+1=-b_A'/n.
\]
The right side is $O((nd)^{-1})$ by
\eqref{prof:eq:jacobi-diagonals}. Continuation from large $z$ selects
the smaller branch and gives
$b_A(2+d^2)=b_{\mathrm{sc}}(2+d^2)+O((nd^2)^{-1})$.
Together with the perturbation estimate this proves
\eqref{prof:eq1}.
For a fixed integer $p\ge2$, take the $(p-1)$st finite difference at
spacing $hd^2$, with $\Delta_a f(u)=f(u+a)-f(u)$. The scalar identity is
\[
\frac{(-1)^{p-1}\Delta_{hd^2}^{p-1}(t+d^2)^{-1}}
{(p-1)!(hd^2)^{p-1}}
=\prod_{j=0}^{p-1}(t+(1+jh)d^2)^{-1}
\]
For an eigenvalue $v$ of $J$, take $t=2-v$. The upper-edge bound
already proved in \eqref{prof:eq1} gives
$t+d^2\ge d^2-o(d_0^2)\ge d^2/2$ for all sufficiently large $n$,
uniformly for $d\ge d_0$. Hence
\[
(1+2(p-1)h)^{-(p-1)}(t+d^2)^{-p}
\le \prod_{j=0}^{p-1}(t+(1+jh)d^2)^{-1}
\le (t+d^2)^{-p}.
\]
The same comparison holds for semicircle depths $t\ge0$. For fixed
$h>0$, \eqref{prof:eq1} gives the semicircle limit for the finite
difference with a relative error tending to zero. Letting $h$ decrease
after $n$ tends to infinity now squeezes the $p$th power to its
semicircle value.
For the edge counts at a shrinking scale, put
$t_i=(2-\lambda_i(J))/d^2$ and consider the finite measure
\[
\nu_{n,d}=\frac1{nd^3}\sum_i(1+t_i)^{-2}\delta_{t_i}.
\]
To identify its limit, use the coordinate $x=(1+t)^{-1}$. The upper-edge
bound places the pushed-forward measures on $[0,1+o(1)]$, and for every
fixed integer $j\ge0$ their moments are
\[
\int (1+t)^{-j}\,d\nu_{n,d}(t)
=\frac{d^{2j+1}}n\Tr(2+d^2-J)^{-j-2}.
\]
The trace-power estimates include the total mass, corresponding to
$j=0$. Polynomial approximation on a fixed compact interval therefore
determines the weak limit of the pushed-forward finite measures. For
$d\to0$, the semicircle change of variables identifies its pullback as
\[
\pi^{-1}\sqrt t(1+t)^{-2}\mathbf1_{\{t>0\}}\,dt.
\]
The interval $[1,2]$ has positive limiting mass and boundary of mass
zero. Since its weight $(1+t)^{-2}$ is bounded above and below, it
contains order $nd^3$ eigenvalues. This conclusion is uniform over
shrinking scales: any violating sequence would have the same moment
limits and hence the same interval-mass limit. The second-power trace bound
gives at most $Cnd^3$ eigenvalues in any fixed multiple of the
$d^2$ upper-edge window. For $d$ in a fixed interval bounded away
from zero, ordinary semicircle convergence gives the same conclusions
with constants depending on that interval. Applying the same
argument to $-J$ gives the lower-edge bound.
For the interlacing assertion, the two eigenvalue counting functions
differ by at most a fixed integer. Write $z$ for the evaluation point
and let $(z-cd^2,z+cd^2)$ be free of both spectra. The function
$(z-v)^{-p}$, defined outside this gap, has an extension across the
gap of total variation $O_p(d^{-2p})$. Integration by parts against
the difference of the counting functions therefore bounds the raw
trace difference by $O_p(d^{-2p})$. This argument also applies when
$z$ is interior to the full spectral range. For $p\ge2$ the error
is relative $O((nd^3)^{-1})$; for $p=1$ it is
$o(\sqrt{n/d})$, the unnormalized error scale in \eqref{prof:eq1}.
\end{proof}
We will use these estimates under conditional matrix laws through a
deterministic interlacing implication. If $E$ has codimension one,
$e\in E$, and
\[
M=(P_EJP_E)|_E-ee^{\mathsf T},
\]
then the ordered eigenvalues satisfy
\[
\lambda_j(J)\ge\lambda_j(M)\ge\lambda_{j+2}(J)
\qquad(1\le j\le n-2).
\]
In particular $\lambda_{\max}(M)\le\lambda_{\max}(J)$, cumulative
upper-edge counts differ by at most two, and band counts differ by at
most four. Thus the retained spectrum of the actual full matrix supplies
the edge spread and trace estimates for its compressed block. The
first-trace error $O(d^{-2})$ is also $o(\sqrt{n/d})$ because
$nd^3\to\infty$. When a conditional block has a Haar eigenframe, its
concentration estimates can be proved uniformly on these inherited
spectra before intersecting with the original full-matrix spectral
event; this does not require conditioning on that event's probability.
\subsection{Conditioning a Gaussian on the sphere}
\label{prof:subsec:spherical}
For a fixed spectrum and field, choose the unique $\lambda>\lambda_{\max}(J)$
such that
\begin{equation}\label{prof:eq:spherical-root}
D=(\lambda-J)^{-1},\qquad m_{\mathrm o}=Dh,
\qquad \Tr D+\|m_{\mathrm o}\|^2=n.
\end{equation}
The left side is strictly decreasing from infinity to zero, so the
root is well defined. The spherical Gibbs law is precisely
$N(m_{\mathrm o},D)$ conditioned on $\|x\|^2=n$, in the sense of
disintegration by its length density. Superscript $0$ will denote
the zero-field quantities.
Lemma~\ref{prof:lem:spectrum} implies
\begin{equation}\label{prof:eq:zero-shell}
|\lambda^0-2|\le Cd_0^2,\qquad \|D^0\|\le Cnd_0.
\end{equation}
Indeed at $2+Cd_0^2$ the normalized trace deficit is $O(d_0)$.
At distance $c/(nd_0)$ above the last pole, that pole contributes
enough to close this deficit if $c$ is small; monotonicity gives the
second bound as well as the first.
The density estimates used in this conditioning are recorded with
their proofs, since polynomial conditioning losses must be tracked
at the lower edge.
\begin{lemma}[Density at the mean]\label{prof:quadratic-mean-density}
Let $g$ be a finite-dimensional standard Gaussian vector and let
$Q=g^TAg+b^Tg$ with $A\succeq0$ and $\sigma^2=\Var Q>0$.
The density of $Q$ at $\E Q$ is at least $c/\sigma$, for an absolute
$c>0$. The same lower bound, with a smaller absolute constant,
holds at a displacement $o(\sigma)$ along any sequence of such forms.
\end{lemma}
\begin{proof}
After diagonalization and normalization it is enough to consider
\[
\sum_i a_i(g_i^2-1)+2\sum_i b_i g_i,\qquad a_i\ge0,
\qquad 2\sum_i a_i^2+4\sum_i b_i^2=1.
\]
Argue by subsequences. If an $a_i$ stays bounded below, write the sum
as $a_i(g_i^2-1)+2b_ig_i+W$. Cantelli's inequality gives a positive
lower bound for $\Pprob(W\le a_i/2)$, and Chebyshev's inequality removes
$W<-C$ after a fixed large $C$ is chosen. On the remaining event the
one-coordinate target lies in $[-a_i/2,C]$, a compact interval strictly
above the minimum $-a_i-b_i^2/a_i$ of the quadratic. Its density there
has a positive lower bound. This proves the claim along the subsequence.
If $\max_i a_i\to0$ but $\|b\|$ stays bounded below, write
$g=z\,b/\|b\|+g_\perp$. The coefficient of $z^2$ tends to zero, and
the random correction to its linear coefficient has variance
$O((\max_i a_i)^2)$. Thus that linear coefficient stays bounded away
from zero with high probability. The remaining term has bounded second
moment. Restrict it to a fixed compact interval and the cross coefficient
to a small interval; the change of variables in $z$ then has a bounded
Jacobian and a preimage in a fixed compact interval, where the Gaussian
density is bounded below. In the remaining case Fourier inversion gives
a local Gaussian limit. The quadratic-product formula supplies the
integrable bound
\[
\bigl(1+4(\max_i a_i)^2t^2\bigr)^{-c/(\max_i a_i)^2}.
\]
The same proof permits an $o(1)$ displacement of the target in
standard-deviation units.
\end{proof}
Suppose $\lambda-2\asymp d^2$, where $d_0\le d\lesssim1$.
The shell variable has standard deviation comparable to $\sqrt{n/d}$.
The lower bound follows from $\Tr D^2\asymp n/d$, and its upper bound
uses \eqref{prof:eq:spherical-root} to give
$\|m_{\mathrm o}\|^2=O(nd)$ and
$m_{\mathrm o}^TDm_{\mathrm o}=O(n/d)$.
The mean-density lower bound controls the denominator of shell
conditioning. For a projection test we also need an upper bound for the
remaining length density. The spectral spread survives a compression
of fixed codimension.
More explicitly, write $x=m_{\mathrm o}+D^{1/2}g$, let
$S=\|x\|^2$, and let $Z=V^{\mathsf T}g$ for a matrix $V$ with a fixed
number of orthonormal columns. Then
\[
p_{Z\mid S=n}(z)=p_Z(z)\,
\frac{f_{S\mid Z=z}(n)}{f_S(n)}.
\]
Let $P=I-VV^{\mathsf T}$. After fixing $Z$, the quadratic matrix
for the remaining white coordinates is the compression
$\left.PDP\right|_{\operatorname{ran}P}$. The conditional covariance of
$x$ is $D^{1/2}PD^{1/2}$, whose nonzero eigenvalues agree with those of
this compression. The surviving spectral block of the compression and
the quadratic-product formula therefore give
$f_{S\mid Z=z}(n)\le C\sqrt{d/n}$ uniformly in the translated mean,
hence uniformly in $z$. The mean-density lemma gives the matching
lower scale for $f_S(n)$. This ratio supplies Gaussian moment and tail
bounds for every fixed collection of white projections. In particular,
\begin{equation}\label{prof:eq2}
\left\|
\E_{\mathrm{shell}}
\left[D^{-1/2}(x-m_{\mathrm o})(x-m_{\mathrm o})^TD^{-1/2}\right]
\right\|\le1+o(1).
\end{equation}
The density ratio gives uniform moment control; the sharp coefficient
requires the limiting conditional law.
To obtain the coefficient $1$,
take any sequence of unit projection directions. The standardized shell variable and the projection have
joint Gaussian subsequential limits, since
\[
\frac{\|D\|}{\sqrt{\Tr D^2}}
=O((nd^3)^{-1/2})=o(1).
\]
Their possible limiting correlation comes only from the linear part
of the shell variable. Fourier inversion conditional on the
projection, with the integrable bound just obtained, gives local
convergence at shell value zero against bounded projection tests.
The uniform projection tails make the squared projections uniformly
integrable, so this local convergence also gives their second moments. Conditioning a
joint Gaussian at its mean, or at an $o(1)$ displacement, can only
decrease the second moment, which proves \eqref{prof:eq2} uniformly
over directions.
When $\lambda\le2+Cd_0^2$, the same density-ratio argument costs at
most $\ell^C$. Indeed $D\preceq D^0$,
$\|m_{\mathrm o}\|^2=O(nd_0)$, and the edge levels and remaining
dyadic shells give
\[
\Tr D^2\le
C\bigl[(nd_0^3)(nd_0)^2+n/d_0\bigr],
\qquad m_{\mathrm o}^TDm_{\mathrm o}=O((nd_0)^2).
\]
The spread at depth comparable to $d_0^2$ supplies the required
density upper bound after fixed-dimensional compression. Thus the
ratio to the lower bound at the mean is bounded by a power of $\ell$.
We will also condition coarse tail events on exact spheres. The rule
used below is the following radial comparison. For a fixed number of
Gaussian vectors, let $I_E(n,\ldots,n)$ be the unnormalized joint
density of their squared lengths at $(n,\ldots,n)$, restricted to an
angular event $E$. Suppose the Gaussian precision and its product with
the mean have polynomial norms. If $E_{\rm buf}$ contains the radial
images of $E$ for $|s_j-n|\le n^{-C_0}$, then
\[
I_E(n,\ldots,n)\le n^C\,\Pprob_{\rm G}(E_{\rm buf})
\]
for a fixed $C$ after $C_0$ is sufficiently large. Indeed, in polar
coordinates the Gaussian log density and the radial log Jacobian vary
by $O(1)$ on this interval, uniformly in the angular variables.
Integrating along those rays proves the inequality. Dividing by a
reference shell density bounded below by an inverse power of $n$
gives the corresponding normalized bound. At each use below, $E$ is
specified by quadratic or fixed-degree Gram tests on a polynomial
ball. Their polynomial derivative and inverse-tolerance bounds ensure
that a slight relaxation of the tests contains this radial buffer.
\begin{lemma}[Zero-field spherical normalization]\label{prof:lem:spherical-Z}
With ordinary probability tending to one,
\begin{equation}\label{prof:eq3}
e^{n/4}n^{-C}\le Z_{\mathrm o}(0)\le e^{n/4+o(n)}.
\end{equation}
\end{lemma}
\begin{proof}
The Gaussian shell formula gives
\[
\log Z_{\mathrm o}(0)
=\frac{n(\lambda^0-1)}2
-\frac12\log\det(\lambda^0-J)
+\log\frac{\text{length density for }N(0,D^0)\text{ at }n}
{\text{length density for }N(0,I)\text{ at }n}.
\]
The last term is $O(\log n)$ in the direction needed for the lower
bound. Replacing $\lambda^0$ by $2+Cd_0^2$ in the first two terms
increases them by only $O(nd_0^3)$, by convexity and
\eqref{prof:eq1}. At this new argument, concavity of log determinant
gives the upper bound
\[
\log\det(z-J)\le\log\det(z-A)-\Tr(z-A)^{-1}\Delta.
\]
The linear correction has mean $O(1)$ and standard deviation
$O(\sqrt{\log n})$, by \eqref{prof:eq:jacobi-diagonals}.
Integrating the deterministic Hermite transform estimate shows that
the deterministic log determinant differs from the semicircle value
by $O(\log n)$. Beyond a fixed large $z$, the same Hermite equation
and $\|A\|\le2$ give normalized-transform error
$O(n^{-1}(z-2)^{-3})$, so the integral to infinity is harmless.
Finally,
\[
\frac{z-1}{2}-\frac12\int\log(z-v)\,\dd\rho_{\mathrm{sc}}(v)
\]
has value $1/4$ at $z=2$, as follows by integrating
$b_{\mathrm{sc}}$, and is increasing thereafter. This proves the
lower bound. The upper bound follows from the same Gaussian
representation at a fixed positive shift, using the length-density
upper bound and then sending the shift to zero.
\end{proof}
\begin{lemma}[Spherical observations]\label{prof:lem:spherical-observation}
Assume the spectral event in Lemma~\ref{prof:lem:spectrum}.
For every fixed tolerance $\epsilon>0$, when $k\gg\log n$ and the
allowed upper bound on $r$ is sufficiently small, the spherical
observation satisfies
\[
\lambda=2+(1\pm\epsilon)s,
\qquad \|m_{\mathrm o}\|^2=(1\pm2\epsilon)nr,
\]
except with probability $e^{-c_\epsilon k}$. For every fixed $B<\infty$ there
are constants $c_B,C_B>0$ such that
\[
c_Bs\le\lambda-2\le C_Bs,
\qquad \|h_s\|\le C_Br\sqrt n,
\]
except with probability $e^{-Bk}$. Polynomial conditioning factors
can be accommodated already at $k>K\log n$, with $K$ chosen after the
tolerance and rate budget. For every deterministic actual time
$0\le t\le s=r^2$, the same upper test gives
\[
\lambda(h_t)-2\le C_Bs,\qquad \|h_t\|\le C_Br\sqrt n
\]
outside probability $e^{-Bk}$, where $k=nr^3$ is the nominal scale.
The lower and sharp width bounds and the sharp mean estimate above
concern the actual time $s$.
\end{lemma}
\begin{proof}
Test \eqref{prof:eq:spherical-root} at $\lambda=2+\gamma s$.
Before conditioning, $X\sim N(0,D^0)$ and $h_s=rg+sX$ with an
independent white Gaussian $g$. Put $\delta=\lambda-\lambda^0$.
Since $k\gg\log n$, we have $r/d_0\to\infty$ and
$\delta=\gamma r^2(1+o_n(1))$ for fixed $\gamma>0$.
The partial fraction identity
\[
D^2D^0=\delta^{-2}(D^0-D)-\delta^{-1}D^2
\]
together with \eqref{prof:eq1} and $\Tr D^0=n$ gives
\[
\frac{\E\|D(rg+sX)\|^2}{nr}
=\frac1{2\sqrt\gamma}+\frac1{2\gamma^{3/2}}
+O_\gamma(r)+o_n(1),
\qquad
\frac{n-\Tr D}{nr}=\sqrt\gamma+O_\gamma(r)+o_n(1).
\]
The leading expressions agree at $\gamma=1$; their errors are small
in the stated order of upper scale and dimension.
The signal square is $s^2X^{\mathsf T}D^2X$. Subtract
$(s^2/\delta^2)(\|X\|^2-n)$, which has mean zero and vanishes on the
exact sphere. In a white eigen-coordinate with
$t=\lambda^0-\lambda_i(J)>0$, the remaining quadratic coefficient is
\[
\frac{s^2}{t}\left(\frac1{(\delta+t)^2}-\frac1{\delta^2}\right)
=-\frac{s^2(2\delta+t)}{\delta^2(\delta+t)^2}.
\]
The pole in $D^0$ has disappeared. The spectral trace bounds give
operator norm $O_\gamma(r^{-2})$ and squared Hilbert--Schmidt norm
$O_\gamma(n/r)$ for this quadratic matrix. The noise square has the
same bounds. Gaussian exponentiation at a deviation of order $nr$
therefore gives exponent
$\min\{(nr)^2/(n/r),(nr)/r^{-2}\}\asymp k$.
Conditional on $X$, the cross term is Gaussian with variance
$O(n/r)$ when the signal square is $O(nr)$. The radial conditioning
comparison preserves these rates after polynomial loss.
Testing on either side of $\gamma=1$ proves the sharp assertions.
For a prescribed rate, first choose a large fixed $L$. At a lower
trial point $2+\gamma r^2$, the $L^3k$ modes at depth comparable to
$L^2r^2$ contribute noise-square mean comparable to $nr/L$, whereas
the trace deficit is comparable to $n\sqrt\gamma r$. Choose
$\sqrt\gamma$ smaller than a fixed multiple of $L^{-1}$. A
fixed-fraction Gaussian small-ball bound on these modes then has
failure $e^{-cL^3k}$, uniformly in $X$ because translation decreases
the probability of a centered ball.
For the upper width, test at $2+L^2r^2$. The trace deficit is of order
$nLr$, while the noise and signal means are $O(nr/L)$ and
$O(nr/L^3)$. The noise quadratic matrix has norm
$O(L^{-4}r^{-2})$ and squared Hilbert--Schmidt norm $O(n/(L^5r))$;
the canceled signal matrix has the corresponding bounds
$O(L^{-6}r^{-2})$ and $O(n/(L^9r))$. Their Gaussian tail exponents at
the deficit scale are at least $cL^3k$. For an actual time $t\le s$,
write $h_t=\sqrt t\,g+tX$ and use this same trial point. Both noise
and signal prefactors decrease, so the upper-width estimate retains
its nominal $k$-scale rate. Choose $L$ from the requested rate, then
the lower trial constant $\gamma$, then the upper scale so that these
spectral windows are available. Finally choose the logarithmic
threshold to absorb the polynomial shell cost.
On the exact sphere $\|X\|=\sqrt n$; Gaussian norm
concentration applied to $h_t=\sqrt t\,g+tX$ gives the stated field
bound uniformly over the deterministic choices $0\le t\le s$.
\end{proof}
\section{Comparison on unrestricted rotation orbits}
\label{prof:sec:unconstrained}
In this section the matrix, field, and spherical mean rotate together.
The spectral data and the field in eigen-coordinates are fixed.
The comparison first controls unnormalized replica integrals.
A controlled Brownian argument then supplies the lower partition-function
tail needed to normalize those integrals.
\begin{proposition}[Unrestricted orbit comparison]
\label{prof:prop:unconstrained}
Assume Lemma~\ref{prof:lem:spectrum}, let $d=\sqrt{\lambda-2}$ be
comparable to $r>0$, and suppose $\|h\|\le Cr\sqrt n$ and
$k=nr^3\gg\log n$. Fix the width-comparison factors. For every fixed
$B$ and every requested fixed relative accuracy, the following holds
outside Haar mass $e^{-Bk}$ once the allowed upper bound on $r$ is
sufficiently small:
\begin{equation}\label{prof:eq4}
\left|\log\frac{Z(h)}{Z_{\mathrm o}(h)}\right|\le o(k).
\end{equation}
For each fixed replica number $p$ and fixed symmetric zero-diagonal
$p\times p$ matrix $P$ of sufficiently small norm, chosen before the
upper scale and exceptional set, the same Haar bound holds for
\begin{equation}\label{prof:eq5}
\log\E_{\mu_h^{\otimes p}}
\exp\!\left\{
\frac{s}{2}\sum_{a,b}P_{ab}(X^a-m_{\mathrm o})^T(X^b-m_{\mathrm o})
\right\}
\le C\Tr(P^2)k+o(k).
\end{equation}
The leading constant and the norm threshold depend only on the
width-comparison factors; the allowed upper scale may also depend on
the fixed replica data. The joint Haar--posterior probability that
a pair overlap differs from $\|m_{\mathrm o}\|^2/n$ by more than any
prescribed fixed tolerance times $r$ is at most $e^{-ck}$, with $c>0$
depending on that tolerance.
The same conclusions are available at $k>K\log n$ with explicit
polynomial losses. For any prescribed $\eta>0$, the errors $o(k)$ in
\eqref{prof:eq4} and \eqref{prof:eq5} may be replaced by
$\eta k+C_*\log n$. The unabsorbed exceptional masses have the form
$n^{C_*}e^{-B'k}$ for the partition and moment bounds and
$n^{C_*}e^{-c'k}$ for the joint pair bound. Here $C_*$ may depend on
the fixed replica data, accuracy, and preliminary rate budget.
Choose $B'>B$ and then $K$ to absorb these losses; for the pair bound
retain a smaller positive rate than $c'$. At fixed $K$ the logarithmic
loss is a chosen small fraction of $k$, rather than an error vanishing
with $n$. The exceptional-mass conclusions
also hold after weighting Haar measure by $Z(h)/Z_{\mathrm o}(h)$,
with an arbitrarily small decrease of the rate exponents.
\end{proposition}
\subsection{Replica integrals and their tails}
\label{prof:subsec:replica-integrals}
For independent uniform signs, write
$R_{ab}=(X^a)^TX^b/n$. On polynomially small boxes near the identity
Gram matrix, their off-diagonal overlaps have probability at most
the corresponding spherical probability times
\begin{equation}\label{prof:eq6}
n^{C_p}\exp\!\left\{C_pn\max_{a\ne b}|R_{ab}|^4\right\}.
\end{equation}
The spherical density is proportional to
$(\det R)^{(n-p-1)/2}$, with polynomial normalization obtained by
successive projections. For signs, the sum Chernoff bound has an
analytic finite-dimensional Legendre expansion near zero, with
identity covariance.
Its quadratic and cubic terms agree with
$-\tfrac12\log\det R$: the only nonzero third products are triangles,
and each has expectation one. The difference thus begins at fourth
order, which proves \eqref{prof:eq6}. Conditional on the Gram matrix,
Haar rotation supplies exactly the same angular integral for both
priors. The boxes can have any sufficiently small inverse-polynomial
side length. Their Gram eigenvalues remain bounded below, and a
smooth Gram factor changes the energy and the additional tilts in
\eqref{prof:eq5} by a polynomially bounded amount times the box width.
Under the spherical Gibbs energies, pair overlaps are $O(r)$.
For $a\gg r$ up to a small fixed constant, the event
$\max_{i\ne j}|R_{ij}|\ge a$ costs at most a polynomial times
$e^{-cna^3}$, for every fixed number of replicas. To prove this,
one of $(X^i+X^j)/\sqrt2$ and $(X^i-X^j)/\sqrt2$ has squared length
at most $n(1-a)$. Translations decrease the relevant Gaussian
small-ball probability. Add a negative square tilt of size
$\eta a^2$, with $\eta>0$ small. By \eqref{prof:eq1}, the normalized
trace lost under this tilt is $O(r+\sqrt\eta a)$. The logarithmic
Gaussian integral then gives the asserted $na^3$ exponent; exact
length conditioning costs only a polynomial.
For the tilted calculation write $\xi^a=X^a-m_{\mathrm o}$.
Before shell conditioning these are independent Gaussians of
precision $D^{-1}$. Choose the norm threshold on $P$ so that
$s\|P\|\le\theta(\lambda-\lambda_{\max}(J))$ for a fixed $\theta<1$.
This is possible uniformly under the width comparison. Factoring the
Gaussian integral leaves the determinant
$\det(I-sP\otimes D)^{-1/2}$ and a normalized centered Gaussian law
with covariance
\[
\bigl(I_p\otimes D^{-1}-sP\otimes I\bigr)^{-1}.
\]
The precision remains positive. The determinant series has no linear
term because $\Tr P=0$, and the remaining terms cost at most
$Cs^2\Tr(P^2)\Tr(D^2)\le Ck\Tr(P^2)$.
The new covariance is bounded between fixed shifted resolvents.
Under this law $X^a=m_{\mathrm o}+\xi^a$; the preceding small-ball
argument for their sums and differences still applies because it is
uniform in translations. The joint length numerator is estimated
under this normalized correlated law, while the original single-shell
densities remain the denominators. The radial comparison gives the
stated polynomial conditioning cost. Without a tilt, Gaussian
quadratic and linear tails show concentration about
$\|m_{\mathrm o}\|^2/n$ within any fixed tolerance times $r$.
Consequently the annealed unnormalized products, divided by the
corresponding powers of $Z_{\mathrm o}(h)$ and restricted to all
overlaps in a sufficiently small fixed interval, have logarithmic
cost at most $Ck\Tr P^2+o(k)$. On overlaps of order $r$, the cost
$nr^4=rk=o(k)$ in \eqref{prof:eq6} is negligible. For a fixed replica
number, choose the overlap cutoff $\eta$ so that
$C_pa^4\le(c/2)a^3$ for $0<a\le\eta$. On the rest of the small interval,
the negative $na^3$ exponent under the tilted Gaussian law then
absorbs the quartic prior-comparison error. The argument applies to repetitions in disjoint
replica groups. For the proposition's joint Haar--posterior pair
conclusion, take $P=0$. For any prescribed fixed tolerance, restrict
the pair to the complement of the interval centered at
$\|m_{\mathrm o}\|^2/n$ with that tolerance times $r$ as its radius.
The no-tilt concentration estimate above, combined with the same
prior comparison, retains a strict negative rate of order $k$ for
this restricted pair integral.
Only the pair calculation requires a completely unrestricted
unnormalized overlap tail. For every fixed $\eta>0$,
$|R_{12}|>\eta$ contributes at most $e^{-c_\eta n}$ relative to
$Z_{\mathrm o}(h)^2$. The field costs $O(nr)$; we will absorb it by
choosing the upper scale after $\eta$.
After rotating the orthogonal sum and difference, their angular
energy is at most
\[
\exp\!\left\{
\frac n4\bigl[(1+R_{12})^2+(1-R_{12})^2\bigr]+o(n)
\right\}.
\]
At a parallel endpoint only the nonzero direction is needed.
Here the elementary one-direction bound is that a uniform direction
of squared radius $n$, tilted by strength $u\ge0$, has logarithmic
integral per coordinate at most $u^2/4+o(1)$, uniformly on compact
$u$-ranges, also in a fixed-codimension compression.
Indeed use a
centered Gaussian with precision $z-uJ$, $z>2u$, to obtain the upper
bound
\[
\frac{z-1}{2}-\frac12\int\log(z-uv)\,\dd\rho_{\mathrm{sc}}(v).
\]
At a fixed shift the shell-density loss is subexponential. For
$u\le1$, choose $z=1+u^2$; for $u\ge1$, send $z\downarrow2u$.
The derivative of the latter continuation is
$1-1/(2u)\le u/2$, proving the claimed quadratic upper bound.
Iterating over the two directions proves the display. Its exponent is
$n/2+nR_{12}^2/2+o(n)$. For the denominator we need only
$\log Z_{\mathrm o}(0)\ge n/4-o(n)$, which follows deterministically
from the retained spectral event. Indeed \eqref{prof:eq:zero-shell}
gives $\lambda^0\to2$ and a polynomial bound on $\|D^0\|$.
The zero-field shell formula and the density-at-mean lower bound give
\[
\log Z_{\mathrm o}(0)\ge
\frac{n(\lambda^0-1)}2
-\frac12\log\det(\lambda^0-J)-O(\log n).
\]
For every fixed $\delta>0$, eventually $\lambda^0\le2+\delta$,
so the log determinant is at most $\log\det((2+\delta)I-J)$.
Semicircle convergence identifies its normalized limit. Sending
$\delta\downarrow0$ gives the claimed lower bound. Thus no additional
normalization event is needed here.
By symmetry $Z_{\mathrm o}(h)\ge Z_{\mathrm o}(0)$, so the denominator
removes the $n/2$ term. The Cram\'er rate of the overlap of independent
signs exceeds $R_{12}^2/2$ by a fixed positive amount on
$|R_{12}|\ge\eta$. Choosing the upper bound on $r$ so that the field
cost is smaller than this gap proves the global pair estimate.
\subsection{The lower partition-function tail}
\label{prof:subsec:controlled-rotation}
Set $F(U)=\log(Z(h)/Z_{\mathrm o}(h))$ on the rotation orbit.
The Haar averaging behind this first-moment identity follows
Comets \cite[Equations~(2.1)--(2.2)]{Comets1996Spherical}; the present
field-dependent identity follows from the rotation convention above.
Haar averaging and the pair bounds give
\[
\E_Ue^F=1,\qquad \E_Ue^{2F}\le e^{o(k)}.
\]
Paley--Zygmund therefore supplies a set of high anchors of mass
$e^{-o(k)}$ on which $F\ge-o(k)$. These anchors may also be
required to satisfy
\[
\left\|\E_{\mu_h}XX^T\right\|_{\mathrm{HS}}=o(n).
\]
Indeed the square of this Hilbert--Schmidt norm is
$\E_{\mu_h^{\otimes2}}(X^1\cdot X^2)^2$, and the global pair
estimate removes the contrary event at exponentially small
unnormalized mass.
For every fixed $B'$ and $\epsilon>0$,
\[
\Pprob_U(F>\epsilon k)\le e^{-B'k}
\]
once the upper bound on $r$ is small enough. To see this, let
$q_\eta(U)=\mu_h^{\otimes2}(|R_{12}|>\eta)$. The global pair estimate is
$\E_U[e^{2F}q_\eta]\le e^{-c_\eta n}$. Hence
\[
\Pprob_U(F>\epsilon k,\ q_\eta>\delta)
\le\delta^{-1}e^{-2\epsilon k}\E_U[e^{2F}q_\eta].
\]
Fix a replica number $p$ and choose
$\delta<(2\binom p2)^{-1}$. On the remaining orientations at least
half of the posterior $p$-replica mass has all overlaps below $\eta$.
The annealed bound for that truncated integral then gives upper-event
mass at most $2e^{-(p\epsilon-o_p(1))k}$. Choose $p$ from $B'$, then
the truncation tolerance and the upper scale. This obtains the strong
upper tail from truncated moments.
We now turn the upper-tail bound into a lower-tail bound. The tangent
inequality at a high anchor compares nearby endpoints. A reflected third
rotation will cancel their first-order displacement, while its entropy
cost allows us to control its endpoint by the upper tail just proved.
At a high anchor, Jensen's inequality gives the deterministic lower
tangent estimate
\begin{equation}\label{prof:eq7}
F(UR)\ge F(U)+\langle H_U,R-I\rangle_{\mathrm{HS}}
-o(n)\|R-I\|_{\mathrm{HS}}^2,
\qquad \|H_U\|_{\mathrm{HS}}=o(n).
\end{equation}
To verify it, expand the rotated quadratic and linear energies under
the Gibbs law at $U$. The linear coefficient is bounded by a
constant times
$\|J\|\|\E XX^T\|_{\mathrm{HS}}+\sqrt n\|h\|=o(n)$.
The quadratic remainder is bounded by
$C\|J\|\|\E XX^T\|\|R-I\|_{\mathrm{HS}}^2$.
The denominator is rotation invariant. This proves
\eqref{prof:eq7}, with arbitrarily small fixed leading errors at
sufficiently small upper scale.
Suppose, toward a contradiction, that the low set
$\{F<-\epsilon k\}$ has Haar mass at least $e^{-Bk}$.
The following controlled reflection produces three nearby rotations:
one at a high anchor, one in the low set, and one whose upper tail
can be bounded by entropy.
Use rotational Brownian motion started at the identity, with
right-translated antisymmetric noise $\dd W$ whose upper entries
have variance $\dd t/n$. Its It\^o drift is
$-(1-1/n)I/2$. Choose a deterministic terminal time sufficiently
large that its endpoint law approximates Haar as accurately as
needed. The heat kernel on the compact connected group permits
this even for the exponentially small target sets at each fixed
$n$.
The control cost is the entropy--drift representation associated
with F\"ollmer's process; the Euclidean form and its converse are
given in \cite[Proposition~1 and Theorem~2]{Lehec2013Entropy}.
The group normalization needed here follows directly from the
likelihood martingale. For a positive smooth terminal likelihood of free mean
one, its backward heat extension is the density-process function.
It\^o's formula under the Doob transform shows that the expected
integrated squared drift in standard noise coordinates is twice
the endpoint relative entropy.
An antisymmetric matrix velocity
$E$ corresponds to standard-coordinate drift with squared norm
$n\|E\|_{\mathrm{HS}}^2/2$. Thus its entropy cost is exactly
\begin{equation}\label{prof:eq:control-entropy}
\frac n4\E\int\|E_t\|_{\mathrm{HS}}^2\,\dd t.
\end{equation}
Bounded positive smooth approximations to normalized target
indicators attain endpoint hit probabilities arbitrarily close to
one and entropies within an additive constant of the indicator
entropy. Conversely Girsanov's formula and data processing bound
the endpoint entropy of any adapted control above by
\eqref{prof:eq:control-entropy}; stopping at finite energy justifies
this statement without further integrability assumptions.
Choose feedback controls for two motions $U,V$ so that their
endpoints hit the high-anchor and low sets, respectively, with
probability at least $0.95$. Their expected joint matrix energy is
$O_B(k/n)$. Stop all controls when the joint energy exceeds a
sufficiently large fixed multiple of $k/n$. Markov's inequality
makes this a small probability loss, so the two endpoint requirements
still hold jointly with probability greater than $0.7$.
Couple their noises as follows, and introduce a third motion $V'$.
If $D=U^TV$, conjugate the noise for $V$ to
$D^{-1/2}\dd W D^{1/2}$; use the analogous conjugation for $V'$.
These are Brownian marginal noises because conjugation is
orthogonal on antisymmetric matrices. In base coordinates the
three controls are
\[
E_U,\qquad E_V,\qquad 2E_U-E_V.
\]
Here $E_V$ is obtained by undoing the conjugation on the prescribed
feedback drift for $V$, which preserves its norm. The first two
motions therefore retain their prescribed marginals up to the
energy stopping. Initially use the local square roots near the
identity; the estimates below show that their chart is never left.
The identity
\[
\dd W\,S\,\dd W=(S^T-\Tr(S)I)\,\dd t/n
\]
gives the relative equation
\begin{align*}
\dd D={}&-\dd W D+\sqrt D\,\dd W\sqrt D\\
&+\left[-(1-1/n)D+
\frac{\Tr\sqrt D}{n}\sqrt D-\frac In\right]\dd t
+(\sqrt D E_V\sqrt D-E_UD)\dd t.
\end{align*}
The same equation holds for $D'=U^TV'$ with the reflected control.
The noise has trace zero, so $q_D$ below has finite variation. For
$q_D=\|D-I\|_{\mathrm{HS}}^2
=2(n-\Tr D)$, expansion in rotation angles gives
\[
-(1-1/n)\Tr D+\frac{(\Tr\sqrt D)^2}{n}-1
=\left(\frac14-\frac1{2n}
+O(\|D-I\|^2)\right)q_D.
\]
The control contribution to $q_D$ is bounded by
$C\sqrt{q_D}(\|E_U\|_{\mathrm{HS}}+\|E_V\|_{\mathrm{HS}})$.
Hence, while in the chart,
\[
q_D'\le-cq_D+C\sqrt{q_D}\,e_t
\le-\tfrac c2q_D+Ce_t^2,
\qquad e_t=\|E_U\|_{\mathrm{HS}}+\|E_V\|_{\mathrm{HS}}.
\]
The reflected motion obeys the same inequality with a constant
multiple of $e_t$. The energy cutoff therefore gives
\begin{equation}\label{prof:eq:relative-radius}
\sup_t\bigl(\|D_t-I\|_{\mathrm{HS}}
+\|D'_t-I\|_{\mathrm{HS}}\bigr)
\le C_B\sqrt{k/n}.
\end{equation}
Taking $r$ sufficiently small in terms of $B$ prevents the first
exit and verifies the use of the local square roots.
The two relative rotations have opposite first-order displacements.
Their sum therefore begins with the quadratic error:
\begin{equation}\label{prof:eq:reflection-error}
\E\|D_t+D'_t-2I\|_{\mathrm{HS}}^2\le C_B(k/n)^2,
\end{equation}
uniformly in the deterministic time $t$. This uniformity is needed
when the terminal Brownian time is taken large to approximate Haar
measure. We retain the damping term in the error equation to obtain it. Put $A=D-I$, $A'=D'-I$, and
$H=A+A'$. The linear noise in $A$ is $[A,\dd W]/2$ and its linear
drift is $-(1/2-1/n)A$.
The term $(\Tr A)I/(2n)$ is quadratic
because $\Tr A=-\|A\|_{\mathrm{HS}}^2/2$. The constant control
terms cancel upon adding the two equations. If
$a^2=C_Bk/n$, the resulting equation is
\[
\dd H=\tfrac12[H,\dd W]-(1/2-1/n)H\,\dd t
+\dd M_R+R_t\,\dd t,
\]
where the Taylor remainders satisfy
\[
\frac{\dd\langle M_R\rangle_{\mathrm{HS}}}{\dd t}\le Ca^4,
\qquad \|R_t\|_{\mathrm{HS}}\le Ca^2+Ca e_t.
\]
These follow from \eqref{prof:eq:relative-radius} and the convergent
matrix power series for the square root.
For every real matrix $H$
(the sum here need not be antisymmetric), the conditional noise
variance is
\[
\frac{\E\|[H,\dd W]/2\|_{\mathrm{HS}}^2}{\dd t}
=\frac{1-1/n}{2}\|H\|_{\mathrm{HS}}^2
+\frac{\Tr(H^2)-(\Tr H)^2}{2n}
\le\frac12\|H\|_{\mathrm{HS}}^2.
\]
It\^o's formula and Young's inequality, with small fixed constants
to absorb the quadratic-covariation cross term, yield
\[
\frac{\dd}{\dd t}\E\|H\|_{\mathrm{HS}}^2
\le-c\E\|H\|_{\mathrm{HS}}^2
+Ca^4+Ca^2\E e_t^2.
\]
Integrating against the exponential damping and using the pathwise
bound $\int e_t^2\,\dd t\le Ca^2$ proves
\eqref{prof:eq:reflection-error}.
It remains to put the endpoint events and the matrix error on one
positive-probability event. Let $P_T$ be the free endpoint law and
$Q_T$ the reflected endpoint law. By \eqref{prof:eq:control-entropy},
$\operatorname{KL}(Q_T\Vert P_T)\le C_Bk$. For an event $E$ with
$0<P_T(E)<1$, binary relative entropy gives
\[
Q_T(E)\le
\frac{\operatorname{KL}(Q_T\Vert P_T)+\log2}
{\log(1/P_T(E))}.
\]
If $P_T(E)=0$, finite relative entropy gives $Q_T(E)=0$.
Choose the upper-tail rate $B'$ large relative to $C_B$, and choose
the terminal Brownian time so that its Haar approximation error is
smaller than $e^{-B'k}$. The upper-tail estimate then gives
$P_T(F>\epsilon k/2)\le2e^{-B'k}$, making the reflected failure
probability as small as required. Markov's inequality in
\eqref{prof:eq:reflection-error} likewise gives its stated order
with any desired fixed high probability.
There is consequently
positive probability of the two endpoint hits, this bound for $V'$,
and both relative-matrix bounds, simultaneously. On that event,
sum \eqref{prof:eq7} from the high anchor $U$ to $V$ and $V'$.
The linear error is at most
$o(n)\|D+D'-2I\|_{\mathrm{HS}}=o(k)$, and the two quadratic
errors are $o(k)$ by \eqref{prof:eq:relative-radius}.
Thus $F(V)+F(V')\ge-o(k)$. This contradicts
$F(V)<-\epsilon k$ and $F(V')\le\epsilon k/2$ after fixing the
little-oh accuracies. The lower tail, and hence
\eqref{prof:eq4}, follows.
\begin{proof}[Completion of Proposition~\ref{prof:prop:unconstrained}]
Normalize the replica integral bounds from
Section~\ref{prof:subsec:replica-integrals} using \eqref{prof:eq4}.
To handle large overlaps between replica groups after normalization,
put $q_\eta(U)=\mu_h^{\otimes2}(|R_{12}|>\eta)$. The global pair
bound gives
\[
\E_U[e^{2F}q_\eta]\le e^{-c_\eta n}.
\]
On the retained lower-partition event $F\ge-\epsilon k$, Markov's
inequality shows that $q_\eta\le e^{-c_\eta n/2}$ outside Haar mass
$e^{-c_\eta n/2+2\epsilon k}$. A union bound applies to any fixed
number of replica pairs. Their fixed tilts cost at most $C_pnr^2$,
since $\|X^a\|=\sqrt n$ and $\|m_{\mathrm o}\|\le\sqrt n$.
Choosing the upper scale after $\eta$ therefore makes the tilted
exterior exponentially small in $n$ as well.
On the remaining replica configurations the truncated integral
estimates apply. Markov's inequality with repeated tilted replica
groups gives \eqref{prof:eq5}, choosing the denominator accuracy
arbitrarily small first. The pair concentration estimate transfers
in the same way, taking the denominator error below its strict
rate. The polynomial Gram-box and shell-density factors cost
$C_*\log n$ in these logarithmic estimates and in the exponents
of the raw failure bounds. Choosing $K$ after the rate
and accuracy absorbs them when $k>K\log n$, as stated.
Finally, for an event with an arbitrarily prescribed raw rate,
\[
\E_U[e^F\mathbf1_E]\le
(\E_Ue^{2F})^{1/2}\Pprob_U(E)^{1/2}
\]
gives any desired weighted rate by starting with more than twice that
rate. For a bounded posterior failure probability $0\le q(U)\le1$
having only a fixed strict rate, split instead at $F=\delta k$:
\[
\E_U[e^Fq]\le e^{\delta k}\E_Uq+
(\E_Ue^{2F})^{1/2}\Pprob_U(F>\delta k)^{1/2}.
\]
The second term has any fixed rate by the strong upper tail, and the
first loses only $\delta$ from its rate. Taking $\delta$ small proves
the stated weighted conclusions.
\end{proof}
\section{The polylogarithmic edge window}
\label{prof:sec:polylog}
The unrestricted comparison above has exponential accuracy on the
$k$-scale. In the lowest spectral window a different second-moment
calculation gives convergence of the partition-function ratio to
one and an absolute error smaller than the critical covariance
scale. This calculation also supplies an independent, supplementary
lower-bound argument. At zero field the partition-function
comparison follows the quantitative critical comparison of
\cite[Theorem~1.6 and Section~3.3]{DuHuang2026CriticalFluctuations}.
The mixed cube--sphere cancellation for matrix moments is the
tensor-barycenter method of
\cite[Section~3.2, Lemma~3.5]{DuHuang2026CriticalOverlap}.
The field range and the weighted moment errors required here are
proved below.
\begin{proposition}[Edge comparison]\label{prof:prop:polylog}
Assume the spectral event of Lemma~\ref{prof:lem:spectrum}. Suppose
\[
r\le n^{-1/3}\log^{C_1}n,\qquad
\lambda-\lambda^0\le n^{-2/3}\log^{C_2}n,\qquad
\|h\|\le\sqrt n\,r\log n,
\]
where $C_1,C_2$ are fixed. Set $L=n^{2/3}$ and $A_0=2I-J$.
Outside Haar mass $o(1)$,
\[
\frac{Z(h)}{Z_{\mathrm o}(h)}=1+o(1).
\]
The cube and spherical raw second moments about $m_{\mathrm o}$
differ in operator norm by at most $n^{1/2+o(1)}$. The cube raw
second moment, sandwiched by $A_0$, has operator norm at most
$n^{1/6+o(1)}$. Each exponent bound can be read with any fixed
positive exponent slack. Unspecified powers of logarithms may
depend on $C_1,C_2$.
\end{proposition}
\begin{proof}
For two independent prior vectors, the sphere--sphere and
cube--sphere overlap distributions agree exactly. The cube--cube
overlap has lattice spacing $2/n$. Stirling's formula and the
spherical density $c_n(1-R_{12}^2)^{(n-3)/2}$ show that its mass,
divided by the lattice spacing and the spherical density, equals
\begin{equation}\label{prof:eq:edge-Stirling}
1+O(n^{-1/3}\log^C n)
\qquad\text{for }|R_{12}|\le L\log^{C'}n/n.
\end{equation}
Choose $C'$ sufficiently large. Outside this window the second
integrals, including any fixed bounded-degree polynomial insertions,
are negligible.
Indeed the negative-square-tilt argument of
Section~\ref{prof:subsec:replica-integrals} applies at depths much
larger than $(\lambda-2)_+^{1/2}+d_0$; combine it with
\eqref{prof:eq6}, then with the global pair estimate. A cube-only
first moment averages exactly to the spherical first moment.
The following cancellation is needed to turn
\eqref{prof:eq:edge-Stirling} into an operator estimate. Before
length conditioning, write independent Gaussian replicas as
\[
x_i=m_{\mathrm o}+D^{1/2}\eta_i,\qquad i=1,2,
\]
with white $\eta_i$, and use Fourier inversion for
\begin{equation}\label{prof:eq:edge-statistics}
\frac{\|x_1\|^2-n}{L},\qquad
\frac{\|x_2\|^2-n}{L},\qquad
\frac{x_1^Tx_2}{L}.
\end{equation}
The reciprocals of the two length densities at zero, in these
units, are at most $\log^C n$.
The Fourier inversions, including any fixed number of derivatives, cost
only powers of $\log n$ apart from the insertion size. Choose an edge
width $d_*=n^{-1/3}\log^M n$ larger than the polylogarithmic saddle
width. The spectral count supplies $N\asymp\log^{3M}n$ modes on which
$D_i/L\ge\log^{-C}n$. A frequency vector for the three statistics is
a symmetric $2\times2$ matrix $T$. At least one eigenvalue of $T$ has
size comparable to $\|T\|$, so the quadratic-product formula on those
modes gives
\[
|\varphi(T)|\le
\bigl(1+c\|T\|^2\log^{-C}n\bigr)^{-cN}.
\]
The noncentral factor does not increase the modulus, because the real
part of the inverse complex precision is positive. This bound is
integrable with any prescribed fixed frequency polynomial, with a
polylogarithmic integral. It remains so after deleting finitely many
modes, as required below.
We now assemble the signed comparison before estimating it. Let
$\pi_{\mathrm c}$ and $\pi_{\mathrm o}$ be the uniform cube and sphere
priors, put $E_U(x)=x^{\mathsf T}Jx/2+h^{\mathsf T}x$, and write
$H_U(x)=(x-m_{\mathrm o})(x-m_{\mathrm o})^{\mathsf T}-D$.
Set $Z_{\mathrm c}=Z$. For $i\in\{\mathrm c,\mathrm o\}$ define
\[
z_i=\frac{Z_i}{Z_{\mathrm o}},\qquad
\widehat M_i(U)=\frac1{Z_{\mathrm o}}
\int H_U(x)e^{E_U(x)}\,d\pi_i(x),\qquad z_{\mathrm o}=1.
\]
All matrix data rotate with $U$. For a symmetric $A$ commuting with
$D$, expansion of the square gives
\[
\mathbb E_U\|A(\widehat M_{\mathrm c}-\widehat M_{\mathrm o})A\|_{\rm HS}^2
=\mathcal I_{\mathrm{cc}}-\mathcal I_{\mathrm{co}}
-\mathcal I_{\mathrm{oc}}+\mathcal I_{\mathrm{oo}},
\]
where
\[
\mathcal I_{ij}=\frac1{Z_{\mathrm o}^2}
\iint\mathbb E_U\!\left[e^{E_U(x)+E_U(y)}
\operatorname{Tr}\bigl(H_U(x)A^2H_U(y)A^2\bigr)\right]
d\pi_i(x)d\pi_j(y).
\]
Conditional on the prior overlap, Haar integration gives the same
angular integral in all four terms. The three pair laws containing a
sphere use the same continuous overlap density; only the cube pair
uses the lattice and its multiplier in \eqref{prof:eq:edge-Stirling}.
After whitening, the common signed angular insertion is
\begin{equation}\label{prof:eq:edge-signed-kernel}
\operatorname{Tr}\!\left[
(\eta_1\eta_1^{\mathsf T}-I)K
(\eta_2\eta_2^{\mathsf T}-I)K\right],
\qquad K=D^{1/2}A^2D^{1/2}.
\end{equation}
Under the complex Fourier tilt its covariance differs from identity by
\[
\Delta_T=(I-2\mathrm iT\otimes D/L)^{-1}-I.
\]
Its Hilbert--Schmidt norm, and the tilted mean using
$\|D^{1/2}m_{\mathrm o}\|/L\le\log^C n$, are bounded by frequency
polynomials times powers of $\log n$. Centering in
\eqref{prof:eq:edge-signed-kernel} cancels the baseline and the terms
belonging to only one replica. With zero tilted mean the remaining Wick
terms have the forms
\[
\operatorname{Tr}(K\Delta_{22}K\Delta_{11}),\qquad
\operatorname{Tr}(K\Delta_{21}K\Delta_{21}),\qquad
(\operatorname{Tr}K\Delta_{12})^2.
\]
The first two are bounded using $\|K\|^2$ and the Hilbert--Schmidt
bounds on $\Delta_T$. For the last,
$|\operatorname{Tr}K\Delta_{12}|\le
C\|T\|\operatorname{Tr}(|K|D)/L$. Terms containing the tilted mean
obey the same envelope. Fourier inversion therefore bounds the signed
density of the insertion and its overlap derivative by a power of
$\log n$ times
\begin{equation}\label{prof:eq:edge-kernel-size}
\|K\|^2+\left(\frac{\operatorname{Tr}(|K|D)}L\right)^2.
\end{equation}
The frequency decay above absorbs all fixed derivative polynomials.
The spectral estimates give
\[
\|D\|\le L\log^C n,\quad \operatorname{Tr}D^2\le L^2\log^C n,
\quad \|A_0^2D\|\le\log^C n,\quad
\operatorname{Tr}(A_0^2D^2)\le n\log^C n.
\]
The last two follow from $A_0D=I-(\lambda-2)D$ and $\|A_0\|\le5$.
Thus \eqref{prof:eq:edge-kernel-size} is at most $L^2\log^C n$ for
$A=I$ and $n^{2/3}\log^C n$ for $A=A_0$. Insert a smooth cutoff in
the central overlap window. The lattice spacing in the last statistic
of \eqref{prof:eq:edge-statistics} is $2/L$, so its Riemann-sum error
is the same envelope times $L^{-1}\log^C n$. The leading continuous
integrals cancel in the four-term sum, and
\eqref{prof:eq:edge-Stirling} leaves a squared error bounded by
$n^{-1/3}$ times the envelope, up to logarithms. The mass calculation
has envelope one.
Consequently the root-mean-square errors are
\[
\begin{aligned}
(\E_U|z_{\mathrm c}-1|^2)^{1/2}
&\le n^{-1/6}\log^C n,\\
(\E_U\|\widehat M_{\mathrm c}-\widehat M_{\mathrm o}\|_{\rm HS}^2)^{1/2}
&\le n^{1/2}\log^C n,\\
(\E_U\|A_0(\widehat M_{\mathrm c}-\widehat M_{\mathrm o})A_0\|_{\rm HS}^2)^{1/2}
&\le n^{1/6}\log^C n.
\end{aligned}
\]
Here $n^{-1/3}L^2=n$ and $n^{-1/3}n^{2/3}=n^{1/3}$ are the two
squared matrix scales. Markov's inequality gives the stated bounds
outside Haar mass $o(1)$, with any fixed exponent slack.
Finally let $Q_i$ be the normalized raw second moment about
$m_{\mathrm o}$. Then
\[
Q_{\mathrm c}=D+\widehat M_{\mathrm c}/z_{\mathrm c},\qquad
Q_{\mathrm o}=D+\widehat M_{\mathrm o},
\]
so the common $D$ cancels and
\[
Q_{\mathrm c}-Q_{\mathrm o}
=z_{\mathrm c}^{-1}(\widehat M_{\mathrm c}-\widehat M_{\mathrm o})
+(z_{\mathrm c}^{-1}-1)\widehat M_{\mathrm o}.
\]
The shell density-ratio bound controls $\widehat M_{\mathrm o}$ in
operator norm by $L\log^C n$, and its $A_0$ sandwich by $\log^C n$.
The scalar mass error therefore costs at most $n^{1/2+o(1)}$ in the
unweighted comparison and less in the weighted one. The spherical
weighted raw moment itself is polylogarithmically bounded. This proves
all claims.
\end{proof}
\subsection{An independent covariance and slab lower bound}
\label{prof:subsec:supplementary-lower}
The realized-equilibrium-start and linear-test results in
\cite[Theorems~1.2--1.3]{AcceptedCritical} have stronger quantifiers:
they apply to every deterministic time sequence below the critical
scale and every deterministic diverging multiplicative slack,
respectively. This supplementary subsection gives an independent
covariance obstruction and a worst-start slab proof with fixed
exponent slack.
At zero field, Proposition~\ref{prof:prop:polylog} and
\eqref{prof:eq3} give, with ordinary probability tending to one,
\begin{equation}\label{prof:eq:cube-Z-lower}
Z(0)\ge n^{-C}e^{n/4}.
\end{equation}
This is also the lower partition-function event used later to
undo the planted size bias at polynomial cost.
On the same type of event the zero-field Gibbs covariance has an
eigenvalue at least $n^{2/3-o(1)}$. Choose a spherical spectral
mode at depth of order $n^{-2/3}\ell^{10}$ below $2$, available by
Lemma~\ref{prof:lem:spectrum}. Its unconditioned variance is
$n^{2/3}\ell^{-O(1)}$. More precisely, the chosen mode has variance
$O(n^{2/3}\ell^{-10})$, whereas the other modes at depth comparable
to $d_0^2$ give remaining length standard deviation at least
$c n^{2/3}\ell^{-1/2}$. Conditioning the chosen standardized
projection to lie between $1$ and $2$ therefore shifts the remaining
length target by $O(\ell^{-19/2})$ of that standard deviation.
Lemma~\ref{prof:quadratic-mean-density}
and the length-density upper bound show that this mode retains its
variance within an $\ell^{O(1)}$ factor on the sphere.
Symmetry at
zero field and the $n^{1/2+o(1)}$ cube--sphere error then give
\begin{equation}\label{prof:eq:supplementary-covariance}
\|\Cov_{\mu_0}X\|\ge n^{2/3-o(1)}.
\end{equation}
For a unit linear test $f_v(x)=v^Tx$, the unscaled heat-bath form
satisfies
\[
\D(f_v)=\sum_i\E_{\mu_0}
\Var(f_v\mid X_{[n]\setminus\{i\}})
\le\sum_i v_i^2=1.
\]
Thus the inverse unscaled spectral gap is at least
$n^{2/3-o(1)}$. A top nonconstant eigenfunction gives the same
exponent lower bound for fixed mixing accuracies less than $1/2$.
The comparison also proves that worst-start distance tends to one
below this exponent. Let $v$ be the same unit spectral mode, and
put
\[
Y=\frac{v^TX}{\sigma_Y},\qquad
\sigma_Y=n^{1/3}\ell^{-O(1)},
\]
where $\sigma_Y$ is its unconditioned spherical Gaussian standard
deviation. For every fixed $K>0$ and $w=\ell^{-K}$, the spherical
probabilities of both $|Y|\le w$ and $|Y|\le2w$ lie between
$cw\ell^{-C}$ and $Cw\ell^C$. The density estimates just used are
uniform when this standardized projection lies in the indicated
range, which proves these bounds.
For these events, the unnormalized cube probabilities, measured
in units of $Z_{\mathrm o}(0)$, differ from their spherical
counterparts by at most $n^{-c}$, outside ordinary Haar mass
$o(1)$, for an absolute small $c>0$. This follows from the same
mass second-moment comparison, with the projection indicator on
each copy. The angular integrals at fixed Gram matrix continue
to agree because $v$ co-rotates with the matrix. In the central
overlap window, the bounded Stirling error is sufficient.
For
the lattice quadrature, condition first on the two tested white
Gaussian coordinates, each of absolute value at most one.
Deleting this one spectral mode leaves the eigenvalue spread for
the remaining two lengths and overlap. Their joint density and
its overlap derivative are therefore bounded by powers of
$\log n$ in the units of \eqref{prof:eq:edge-statistics}, uniformly
in the conditioned coordinates. The quadrature error is again
at most a power of $\log n$ divided by $L$. The exterior of the
window has the same absolute tail bound as before. Markov's
inequality proves the asserted $n^{-c}$ comparison.
Since the total mass ratio tends to one, choose $K>C+1$.
With ordinary probability tending to one, the narrow slab
$A=\{|Y|\le w\}$ then has Gibbs mass at least $n^{-o(1)}$, while
the wider slab $A^+=\{|Y|\le2w\}$ has mass tending to zero.
For the stationary rate-one-per-site chain, the spectral theorem
and $\D(f_v)\le1$ give
\[
\E_{\mu_0}(f_v(X_t)-f_v(X_0))^2\le2t.
\]
For $m$ single-site attempts the same bound is $2m/n$, since
the attempt kernel is an average of conditional-expectation
projections and hence has spectrum in $[0,1]$.
Conditioning the stationary chain at its initial point to lie
in $A$ increases the bound by at most $1/\mu_0(A)$.
Consequently, for every fixed $\delta>0$ and
$t\le n^{2/3-\delta}$ in rate-one-per-site time,
\[
\Pprob_{\mu_0(\,\cdot\mid A)}(X_t\notin A^+)
\le\frac{2t}{\mu_0(A)\,w^2\sigma_Y^2}=o(1).
\]
The stationary mass of $A^+$ is $o(1)$, so this conditional-start
law has total variation distance $1-o(1)$ from equilibrium.
Averaging over its starting points, after the disorder and time have
been fixed, yields a starting configuration with the same conclusion.
In attempts, the corresponding time
range is $m\le n^{5/3-\delta}$.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//profile-spectral.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//proof-overview.tex ====
\section{Two upper-bound mechanisms}\label{sec:proof-overview}
Both proofs use the geometry of the upper spectral edge, but transport
different estimates. The limiting GOE spectral density decays as the
square root of the distance to its upper edge $2$, making $nu^{3/2}$ the
characteristic dimension of an edge window of width $u$. The spectral
estimates below make this count precise at the scales where it is used.
A spherical spin law can be represented as a
Gaussian conditioned on $\|X\|^2=n$, so its resolvent turns this edge
geometry into explicit covariance estimates. The first route observes
all directions at the same precision. The second observes only the soft
edge directions and enlarges that set one spectral cap at a time.
All changes of law below use finite-dimensional likelihood identities
with explicit partition normalizers. The planted law weights the disorder
by its partition function and introduces an auxiliary spin. Estimates
under that law, or under a further tilt, are returned to ordinary disorder
with the stated probability cost. No critical contiguity assertion is
used; the distinction at the spiked-Wigner threshold is discussed in
\cite[Appendix~A]{PerryWeinBandeiraMoitra2018}.
\subsection{Isotropic observations: from covariance to a support profile}
The endpoint of the first route is \Cref{prof:support-profile}: for every
fixed $\omega>0$, on one disorder event of probability tending to one,
\[
\D_J(f)\ge c_\omega n^{-2/3-\omega}\log(1/p)\,\mu_J(f^2),
\qquad 0<p:=\mu_J(f\ne0)\le\tfrac12,
\]
simultaneously for every nonzero real $f$. The factor $\log(1/p)$ says
that a function concentrated on a smaller set must pay more heat-bath
energy relative to its squared norm. For an indicator,
$\D_J(\mathbf1_A)=\sum_{x\in A}\mu_J(x)\sum_iQ_i(x,A^c)$ is the
stationary probability current out of $A$ per unit of per-site time.
The profile controls the early evolution of a point-start density;
the spectral gap controls the final decay.
\paragraph{A local covariance comparison.}
At isotropic precision $s$, write $r=\sqrt s$ and $k=nr^3=ns^{3/2}$.
The spectral and spherical estimates in \Cref{prof:sec:spectral} locate
the Gaussian saddle and control its length densities, projections, and
replica overlaps. The unrestricted rotation comparison in
\Cref{prof:sec:unconstrained} then compares the spin and spherical
partition functions and their replica moments. These estimates give the
continuous reference and the probability budgets for the root analysis.
The root analysis studies solutions of
\[
y+(1-q)m-Jm=h_s,\qquad m=\tanh y,\qquad q=n^{-1}\|m\|^2.
\]
The Kac--Rice formula in \Cref{prof:roots} expresses planted expectations
of sums over regular solutions as an integral against the observation
density and the absolute Jacobian determinant. After gauging the planted
spin to $\mathbf1$, the
conditional matrix calculation in \Cref{prof:matrices} isolates three
directions: $m$; the component of $\mathbf1$ orthogonal to $m$; and the
component of the root field $y$ orthogonal to both. Their span is
$\mathcal P$. The last direction records the remaining cubic part of the
root profile. Conditioning on these directions leaves Haar rotations on
$\mathcal P^\perp$. \Cref{prof:orbits} compares their Haar weights with
the determinant weights in the root count and locates the Gaussian trace.
The exclusions are integrated over the root and conditional matrix data,
with the imposed spectral restriction. The degree-one argument controls
multiplicity when this root count is returned to the planted field law.
Here is the cancellation at the center of the comparison. Put $z=X-m$.
For each replica, fix its projection on $\mathcal P$ and its translated
length; for a pair, also record the overlap of their residual components
in $\mathcal P^\perp$. The two replica types are a product spin prior
with means $m$ and a variable-variance Gaussian prior. Multiplication
by the common quadratic weight
$\exp\{z^{\mathsf T}(J-(1-q)I)z/2\}$ turns the product spin prior
into the posterior $\mu_s$ in the translated coordinates: the root
equation cancels its linear term. The quantities
$\mathcal R_{ij}$ in \eqref{prof:eq16} are their conditional saddle-density
ratios for this residual overlap, relative to a common continuous
reference, for the four ordered type pairs. A projection multiplier first
matches the one-replica saddle densities. The signed combination
$\mathcal R_{pp}-\mathcal R_{pg_0}-\mathcal R_{g_0p}
+\mathcal R_{g_0g_0}$, with the common projection weight in
\eqref{prof:eq16}, cancels common errors and errors depending on only one
replica type. The actual spin statistics retain their lattice measures:
\Cref{prof:common-angular-kernel,prof:mixed-coordinate-transfer} transfer
only the smooth leading terms to common volume, after bounding each
remainder on its own measure.
This signed estimate, including the centered matrix kernel of
\Cref{prof:gaussian-signed-kernel}, yields
\Cref{prof:median-covariance}. For $k\ge\log^{10}n$ and sufficiently small
$r$, most Haar orientations on each retained root orbit satisfy
$\|\Sigma_s\|\le(1+\epsilon)/s$,
together with the weighted covariance and mean bounds in
\eqref{prof:eq13}. Those additional bounds control derivatives along
rotations in the next stage.
\paragraph{A probability bound along the observation path.}
\Cref{prof:prop-amplified-covariance} combines short backward variance
transfers with concentration around the local majority just obtained.
Starting from fixed positive observation times, it bounds, at each
deterministic time with $0<s\le s_0$ and $k\ge K_{\log}\log n$, the
integrated planted mass of
$\mathcal G\cap\{\|\Sigma_s\|>(1+\epsilon)/s\}$ by $e^{-ak}$.
Here $\mathcal G$ is the rotation-invariant good-spectrum event in the
proposition. The positive-time
covariance and terminal gap inputs are proved at the critical parameter
in \Cref{prof:positive-inputs}; the subcritical spectral-gap theorem is
not applied at its endpoint. In the intervening polylogarithmic window,
the induction uses the unrestricted comparison of
\Cref{prof:sec:polylog} in place of root-preserving rotations.
For the bottom range, put $\ell=\log\log(n+1000)$. At each deterministic
time with $k\le K_{\log}\log n$, \Cref{prof:prop-bottom-covariance} bounds
the integrated planted mass of
$\mathcal G\cap\{\|\Sigma_s\|>Cn^{2/3}\ell^C\}$ by $n^{-B}$ for any
fixed $B>0$. Its
finite-replica expansion retains the central prefactors and lattice
covolumes. After conversion to common volume, the alternating sum cancels
every term that omits a replica label; increasing the fixed moment order
then supplies the required probability exponent. These are estimates
along the observation law at deterministic times, with their stated
integrated exceptions.
\paragraph{Entropy and the uniform event.}
\Cref{prof:entropy} first transfers the planted bounds to one typical
ordinary-disorder event, before choosing $f$. After normalizing
$\mu_J(f^2)=1$, the restricted step assumes
$0<p:=\mu_J(f\ne0)\le1/2$ and $\|f\|_\infty^2\le100p^{-2}$.
It treats small, moderate, and order-$n$ values of $\log(1/p)$ separately.
Variance transport uses the bottom-scale estimate in the first range;
replica moments and the sharp covariance bound retain entropy in the
second; square-reweighted mean estimates control the last range. The
product endpoint converts retained entropy to heat-bath energy, giving
\Cref{prof:restricted-profile}. Dyadic layers remove the bounded-test
restriction and prove the support profile stated above.
\subsection{Spectral caps: from a radial deficit to log-Sobolev}
The second route proves \Cref{cap:thm-lsi}: for every fixed $\rho>0$,
with probability tending to one,
\[
\Ent_{\mu_J}(f^2)\le n^{2/3+\rho}\D_J(f)
\qquad\text{for every real }f.
\]
This classical logarithmic Sobolev inequality controls the smoothing of
arbitrary densities through hypercontractivity and also implies the
Poincar\'e inequality.
\paragraph{The cap and the radial deficit.}
Let $d=2I-J$. At scale $p$, observe with precision
$B_p=(p^2-d)_+$ and write $J_p=J-B_p$. On the spectral event used below,
the active projection $P_p=\mathbf1_{\{d\le p^2\}}$ has rank of order
$k_p=np^3$; the cap makes $J_p$ scalar on its range. For $h$ in this range, the
spherical comparison uses
$R_p=(\max(d,p^2)+\alpha)^{-1}$, where $\alpha$ is chosen so that
$\operatorname{Tr}R_p+\|R_ph\|^2=n$. \Cref{cap:static-comparison}
controls partition values and projected means uniformly on each prescribed
ball $\|h\|\le R p^2M_p$, where $M_p=\sqrt{np}$ and $R$ is fixed.
Changing the field radially changes the length of the Gaussian mean, so
the sphere constraint also changes $\alpha$. This reduces the radial
response relative to the frozen resolvent $R_p$. The induction records
that reduction explicitly. The true potential is
$\phi_p(h)=\log Z_{J_p}(h)$, whose Hessian is the capped Gibbs covariance.
For this potential,
$K_p=(P_p\nabla^2\phi_p(h)P_p)|_{\operatorname{ran}P_p}$.
During initialization, $K_p$ instead denotes the Hessian of an auxiliary
potential on the same space. Restrict $R_p$ there as well. With
$e=h/\|h\|$, on the prescribed annulus
the matrix $D=p^2(K_p-R_p)$ has a radial entry at most $-0.30$, small
transverse and mixed blocks, and a bounded lower error; see
\eqref{cap:invariant}. This negative radial correction is the
\emph{radial deficit}. It provides the gain needed to propagate the
covariance estimate to a smaller cap.
\paragraph{Transport and initialization.}
The curvature induction proceeds from larger caps to smaller caps.
After the estimates have been proved, the entropy argument uses them
along the forward observation process, in which $p$ and $B_p$ increase.
For a child scale $c=\gamma p$, the exact Gaussian convolution
\eqref{cap:heat-convolution} expresses the child potential through the
parent. The exact increment formula conjugates the covariance of the
parent field under the transition law by the inverse precision increment,
then subtracts that inverse increment, to obtain the child's Hessian.
The Brascamp--Lieb upper bound retains
the radial correction through the inverse in \eqref{cap:bl-recursion};
the variational Fisher lower bound controls the opposite side and the
mixed block. Averaging the rank-one correction regenerates the radial
deficit by \eqref{cap:radial-regeneration}. Meanwhile the still hidden
eigenframe inside the flat parent space is Haar. Its compression makes
the transverse error small enough at the child scale. This conditional
rotation is used under the canonical Gaussian bridge before the change
to the actual tilted transition. \Cref{cap:curvature-induction} is the
resulting one-step statement.
The initial radial deficit comes from the small-field TAP expansion in
\Cref{cap:classical}, specifically \Cref{cap:tap-expansion}.
\Cref{cap:auxiliary-initialization} uses an independent conditional
completion of the first cap to construct an auxiliary potential with the
curvature invariant and a bounded Euclidean gradient error from the true
potential. The companion estimate used for that error is at one fixed
$0<\beta<1$. \Cref{cap:gradient-contraction} contracts the error over a
finite number of cap steps, and \Cref{cap:remove-auxiliary} then transfers
the invariant to the true potential. The induction continues down to
$np^3\asymp\sqrt{\log n}$. For fields $h\in\operatorname{ran}P_p$, its
final output, \Cref{cap:true-curvature}, gives a sharp
$(1+\epsilon)/p^2$ upper bound for the covariance projected onto
$\operatorname{ran}P_p$ on a thin saddle annulus, and a $C/p^2$ bound for
the same projected covariance on any prescribed bounded field ball, at
every scale in the stated interval.
\paragraph{Entropy retention and layers.}
Write $N=\mu_J(f^2)=e^{-L}>0$ and $\nu=f^2\mu_J/N$. The bounded-test
reduction assumes $0\le f\le1$, $\mu_J(f>0)\le1/2$,
$\operatorname{KL}(\nu\Vert\mu_J)\ge L/2$, and $L\le Cn$. The disorder
event is chosen before $f$. For $L\le\lambda n$, where the small fixed
$\lambda>0$ is chosen after the curvature and ordinary-path tolerances,
\Cref{cap:entropy-retention} chooses a first cap with dimension comparable
to $\max\{\sqrt{\log n},c_0L\}$. The curvature bounds control the
information revealed as the cap grows, retaining enough entropy for the
product estimate in \Cref{cap:product-estimate}. For
$\lambda n\le L\le Cn$, \Cref{cap:large-entropy} argues by contradiction
under $\D_J(f)/N\le n^{-2/3}L$. Square-reweighted mean bounds and relative
entropy of stopped observation paths retain order-$n$ entropy until the
product estimate contradicts that assumption. The noise diagnostics and
stability tolerances are chosen before the test function. Finally,
\Cref{cap:layer-lemma} passes from the bounded-test energy estimate to
set capacities, then to arbitrary half-supported functions, and then by
a median decomposition and the entropy shift \eqref{cap:entropy-shift}
to the classical logarithmic Sobolev inequality.
\paragraph{The final dynamical transfer.}
\Cref{sec:dynamics} converts either functional endpoint to worst-start
mixing: the support profile truncates large densities, while logarithmic
Sobolev hypercontractivity supplies the second smoothing argument. The
identity $\mathcal L_J=n(P_J-I)$ and the nonnegative spectrum of $P_J$
keep the factor $n$ between per-site time and attempted updates explicit.
The linear lower tests and \Cref{thm:accepted-critical} supply the
opposite exponents for mixing and the functional constants. The
companion's realized-start and arbitrary-diverging-factor quantifiers
remain those stated in \Cref{thm:accepted-critical}.
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//proof-overview.tex ====
==== BEGIN Critical-mixing-in-the-Sherrington-Kirkpatrick-model//references.bib ====
@misc{Wang2026OptimalMixing,
author = {Wang, Sihan},
title = {Optimal Mixing of {Glauber} Dynamics for the {Sherrington--Kirkpatrick} Model at $\beta<1/2$},
year = {2026},
eprint = {2608.22159},
archivePrefix = {arXiv},
primaryClass = {math.PR},
note = {Version 2, August 28, 2026},
url = {https://arxiv.org/abs/2608.22159v2}
}
@misc{BobanLiOveisGharan2026,
author = {Boban, Mathews and Li, Anqi and Oveis Gharan, Shayan},
title = {Rank-1-perturbed trickledown theorems: Mixing time of {Glauber} dynamics for the {Sherrington--Kirkpatrick} model up to $\beta\leq\frac{1}{2}+\varepsilon$},
year = {2026},
eprint = {2609.13138},
archivePrefix = {arXiv},
primaryClass = {cs.DS},
note = {Version 1, September 11, 2026},
url = {https://arxiv.org/abs/2609.13138v1}
}
@article{EldanKoehlerZeitouni2022,
author = {Eldan, Ronen and Koehler, Frederic and Zeitouni, Ofer},
title = {A spectral condition for spectral gap: fast mixing in high-temperature {Ising} models},
journal = {Probability Theory and Related Fields},
volume = {182}, number = {3--4}, pages = {1035--1051}, year = {2022},
doi = {10.1007/s00440-021-01085-x},
eprint = {2007.08200}, archivePrefix = {arXiv},
note = {Preprint version 2, August 6, 2021},
url = {https://arxiv.org/abs/2007.08200v2}
}
@inproceedings{AnariKoehlerVuong2024,
author = {Anari, Nima and Koehler, Frederic and Vuong, Thuy-Duong},
title = {Trickle-Down in Localization Schemes and Applications},
booktitle = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing},
pages = {1094--1105}, year = {2024},
doi = {10.1145/3618260.3649622},
eprint = {2407.16104}, archivePrefix = {arXiv},
note = {Full preprint version 1, July 23, 2024},
url = {https://arxiv.org/abs/2407.16104v1}
}
@article{BilloireCampbell2011CriticalDynamics,
author = {Billoire, Alain and Campbell, I. A.},
title = {Dynamics in the {Sherrington--Kirkpatrick Ising} spin glass at and above {$T_g$}},
journal = {Physical Review B},
volume = {84},
pages = {054442},
year = {2011},
doi = {10.1103/PhysRevB.84.054442},
eprint = {1105.1902},
archivePrefix = {arXiv},
note = {arXiv version 3, 28 July 2011},
url = {https://arxiv.org/abs/1105.1902v3}
}
@misc{DuHuang2026CriticalFluctuations,
author = {Du, Hang and Huang, Brice},
title = {Fluctuations of the {Sherrington--Kirkpatrick} free energy at critical temperature},
year = {2026},
eprint = {2607.02172},
archivePrefix = {arXiv},
note = {Version 2, 8 July 2026},
url = {https://arxiv.org/abs/2607.02172v2}
}
@misc{DuHuang2026CriticalOverlap,
author = {Du, Hang and Huang, Brice},
title = {Overlap distribution of the critical {Sherrington--Kirkpatrick} model},
year = {2026},
eprint = {2608.08752},
archivePrefix = {arXiv},
note = {Version 1, 9 August 2026},
url = {https://arxiv.org/abs/2608.08752v1}
}
@article{GMT2006,
author = {Goel, Sharad and Montenegro, Ravi and Tetali, Prasad},
title = {Mixing Time Bounds via the Spectral Profile},
journal = {Electronic Journal of Probability},
volume = {11},
number = {1},
pages = {1--26},
year = {2006},
doi = {10.1214/EJP.v11-300},
url = {https://emis.de/ft/42421},
eprint = {math/0505690},
archivePrefix = {arXiv}
}
@article{DSC1996,
author = {Diaconis, Persi and Saloff-Coste, Laurent},
title = {Logarithmic {Sobolev} inequalities for finite {Markov} chains},
journal = {The Annals of Applied Probability},
volume = {6},
number = {3},
pages = {695--750},
year = {1996},
doi = {10.1214/aoap/1034968224},
url = {https://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/ConvergeRates/DiaconisSaloffCosteSobolev1996.pdf}
}
@article{BR2003,
author = {Barthe, Franck and Roberto, Cyril},
title = {{Sobolev} inequalities for probability measures on the real line},
journal = {Studia Mathematica},
volume = {159},
number = {3},
pages = {481--497},
year = {2003},
doi = {10.4064/sm159-3-9},
url = {https://www.impan.pl/shop/publication/transaction/download/product/89465}
}
@article{BobkovTetali2006Modified,
author = {Bobkov, Sergey G. and Tetali, Prasad},
title = {Modified Logarithmic {Sobolev} Inequalities in Discrete Settings},
journal = {Journal of Theoretical Probability},
volume = {19},
number = {2},
pages = {289--336},
year = {2006},
doi = {10.1007/s10959-006-0016-3},
url = {https://www-users.cse.umn.edu/~bobko001/papers/2006_JTP_BT_Modified.pdf}
}
@article{Comets1996Spherical,
author = {Comets, Francis},
title = {A spherical bound for the {Sherrington--Kirkpatrick} model},
journal = {Ast{\'e}risque},
volume = {236},
pages = {103--108},
year = {1996},
url = {https://www.numdam.org/item/AST_1996__236__103_0/},
note = {In Hommage {\`a} P. A. Meyer et J. Neveu}
}
@article{DumitriuEdelman2002,
author = {Dumitriu, Ioana and Edelman, Alan},
title = {Matrix models for beta ensembles},
journal = {Journal of Mathematical Physics},
volume = {43},
number = {11},
pages = {5830--5847},
year = {2002},
doi = {10.1063/1.1507823},
eprint = {math-ph/0206043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math-ph/0206043v1}
}
@article{LedouxRider2010,
author = {Ledoux, Michel and Rider, Brian},
title = {Small deviations for beta ensembles},
journal = {Electronic Journal of Probability},
volume = {15},
number = {41},
pages = {1319--1343},
year = {2010},
doi = {10.1214/EJP.v15-798},
eprint = {0912.5040},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/0912.5040v1}
}
@article{RamirezRiderVirag2011,
author = {Ram{\'i}rez, Jos{\'e} A. and Rider, Brian and Vir{\'a}g, B{\'a}lint},
title = {Beta ensembles, stochastic {Airy} spectrum, and a diffusion},
journal = {Journal of the American Mathematical Society},
volume = {24},
number = {4},
pages = {919--944},
year = {2011},
doi = {10.1090/S0894-0347-2011-00703-0},
eprint = {math/0607331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/0607331v4}
}
@article{BrascampLieb1976,
author = {Brascamp, Herm Jan and Lieb, Elliott H.},
title = {On extensions of the {Brunn--Minkowski} and {Pr{\'e}kopa--Leindler} theorems, including inequalities for log concave functions, and with an application to the diffusion equation},
journal = {Journal of Functional Analysis},
volume = {22},
number = {4},
pages = {366--389},
year = {1976},
doi = {10.1016/0022-1236(76)90004-5},
url = {https://doi.org/10.1016/0022-1236(76)90004-5}
}
@article{BobkovLedoux2000,
author = {Bobkov, Sergey G. and Ledoux, Michel},
title = {From {Brunn--Minkowski} to {Brascamp--Lieb} and to logarithmic {Sobolev} inequalities},
journal = {Geometric and Functional Analysis},
volume = {10},
number = {5},
pages = {1028--1052},
year = {2000},
doi = {10.1007/PL00001645},
url = {https://www.esi.ac.at/preprints/esi819.pdf}
}
@article{PerryWeinBandeiraMoitra2018,
author = {Perry, Amelia and Wein, Alexander S. and Bandeira, Afonso S. and Moitra, Ankur},
title = {Optimality and sub-optimality of {PCA} {I}: Spiked random matrix models},
journal = {The Annals of Statistics},
volume = {46},
number = {5},
pages = {2416--2451},
year = {2018},
doi = {10.1214/17-AOS1625},
eprint = {1807.00891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1807.00891v2},
note = {The cited arXiv version includes the supplement as appendices}
}
@article{Eldan2013StochasticLocalization,
author = {Eldan, Ronen},
title = {Thin shell implies spectral gap up to polylog via a stochastic localization scheme},
journal = {Geometric and Functional Analysis},
volume = {23},
number = {2},
pages = {532--569},
year = {2013},
doi = {10.1007/s00039-013-0214-y},
eprint = {1203.0893},
archivePrefix = {arXiv},
primaryClass = {math.MG},
url = {https://arxiv.org/abs/1203.0893v5},
note = {Proof locators refer to arXiv:1203.0893v5, 16 December 2012}
}
@article{ChenEldan2025Localization,
author = {Chen, Yuansi and Eldan, Ronen},
title = {Localization schemes: A framework for proving mixing bounds for {Markov} chains},
journal = {Duke Mathematical Journal},
volume = {174},
number = {8},
pages = {1431--1510},
year = {2025},
doi = {10.1215/00127094-2024-0063},
eprint = {2203.04163},
archivePrefix = {arXiv},
primaryClass = {math.PR},
url = {https://arxiv.org/abs/2203.04163v2},
note = {Proof locators refer to arXiv:2203.04163v2, 6 June 2022}
}
@article{ChewiPooladian2023Entropic,
author = {Chewi, Sinho and Pooladian, Aram-Alexandre},
title = {An entropic generalization of {Caffarelli}'s contraction theorem via covariance inequalities},
journal = {Comptes Rendus. Math{\'e}matique},
volume = {361},
number = {G9},
pages = {1471--1482},
year = {2023},
doi = {10.5802/crmath.486},
url = {https://doi.org/10.5802/crmath.486},
}
@article{Lehec2013Entropy,
author = {Lehec, Joseph},
title = {Representation formula for the entropy and functional inequalities},
journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
volume = {49},
number = {3},
pages = {885--899},
year = {2013},
doi = {10.1214/11-AIHP464},
url = {https://www.numdam.org/articles/10.1214/11-AIHP464/}
}
@article{TAP1977,
author = {Thouless, David J. and Anderson, Philip W. and Palmer, Richard G.},
title = {Solution of {'Solvable model of a spin glass'}},
journal = {Philosophical Magazine},
volume = {35}, number = {3}, pages = {593--601}, year = {1977},
doi = {10.1080/14786437708235992},
url = {https://doi.org/10.1080/14786437708235992}
}
@article{FanMeiMontanari2021,
author = {Fan, Zhou and Mei, Song and Montanari, Andrea},
title = {{TAP} free energy, spin glasses, and variational inference},
journal = {The Annals of Probability},
volume = {49}, number = {1}, pages = {1--45}, year = {2021},
doi = {10.1214/20-AOP1443},
url = {https://arxiv.org/abs/1808.07890v2},
note = {Proof locators refer to arXiv:1808.07890v2, 26 March 2020}
}
@article{CelentanoFanMei2023,
author = {Celentano, Michael and Fan, Zhou and Mei, Song},
title = {Local convexity of the {TAP} free energy and {AMP} convergence for {$\mathbb Z_2$}-synchronization},
journal = {The Annals of Statistics},
volume = {51}, number = {2}, pages = {519--546}, year = {2023},
doi = {10.1214/23-AOS2257},
url = {https://arxiv.org/abs/2106.11428v3},
note = {Proof locators refer to arXiv:2106.11428v3, 15 April 2023}
}
@article{SherringtonKirkpatrick1975,
author = {Sherrington, David and Kirkpatrick, Scott},
title = {Solvable Model of a Spin-Glass},
journal = {Physical Review Letters},
volume = {35}, number = {26}, pages = {1792--1796}, year = {1975},
doi = {10.1103/PhysRevLett.35.1792},
url = {https://doi.org/10.1103/PhysRevLett.35.1792}
}
@article{KirkpatrickSherrington1978,
author = {Kirkpatrick, Scott and Sherrington, David},
title = {Infinite-ranged models of spin-glasses},
journal = {Physical Review B},
volume = {17}, number = {11}, pages = {4384--4403}, year = {1978},
doi = {10.1103/PhysRevB.17.4384},
url = {https://doi.org/10.1103/PhysRevB.17.4384}
}
@article{SompolinskyZippelius1982,
author = {Sompolinsky, Haim and Zippelius, Annette},
title = {Relaxational dynamics of the {Edwards--Anderson} model and the mean-field theory of spin-glasses},
journal = {Physical Review B},
volume = {25}, number = {11}, pages = {6860--6875}, year = {1982},
doi = {10.1103/PhysRevB.25.6860},
url = {https://doi.org/10.1103/PhysRevB.25.6860}
}
@article{Billoire2011SlowSamples,
author = {Billoire, Alain},
title = {What makes slow samples slow in the {Sherrington--Kirkpatrick} model},
journal = {Journal of Physics A: Mathematical and Theoretical},
volume = {44}, pages = {075001}, year = {2011},
eprint = {1011.3420}, archivePrefix = {arXiv},
note = {arXiv version 2; equation (1), pages 2--3},
url = {https://arxiv.org/abs/1011.3420v2}
}
@article{BauerschmidtBodineau2019,
author = {Bauerschmidt, Roland and Bodineau, Thierry},
title = {A very simple proof of the {LSI} for high temperature spin systems},
journal = {Journal of Functional Analysis},
volume = {276}, number = {8}, pages = {2582--2588}, year = {2019},
doi = {10.1016/j.jfa.2019.01.007},
eprint = {1712.03676}, archivePrefix = {arXiv}, primaryClass = {math.PR},
note = {Proof locators refer to arXiv:1712.03676v2, 31 January 2018},
url = {https://arxiv.org/abs/1712.03676v2}
}
@misc{AnariJainKoehlerPhamVuong2021,
author = {Anari, Nima and Jain, Vishesh and Koehler, Frederic and Pham, Huy Tuan and Vuong, Thuy-Duong},
title = {Entropic Independence {I}: Modified Log-{Sobolev} Inequalities for Fractionally Log-Concave Distributions and High-Temperature {Ising} Models},
year = {2021},
eprint = {2106.04105}, archivePrefix = {arXiv}, primaryClass = {cs.DS},
note = {Version 2, 5 November 2021; Theorem 12 and Section 5},
url = {https://arxiv.org/abs/2106.04105v2}
}
@misc{AcceptedCritical,
author = {{OpenAI}},
title = {{Critical slowing down in the Sherrington--Kirkpatrick model}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf}{OAI:Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026}},
year = {2026},
note = {Theorems 1.2 and 1.3}
}
@misc{AcceptedGap,
author = {{OpenAI}},
title = {{A spectral gap throughout the high-temperature Sherrington--Kirkpatrick phase}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf}{OAI:A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026}},
year = {2026},
note = {Lemma 5.7 and Propositions 7.2--7.3; see the hypotheses of Sections 3 and 8}
}
@book{Meckes2019ClassicalGroups,
author = {Meckes, Elizabeth S.},
title = {The Random Matrix Theory of the Classical Compact Groups},
series = {Cambridge Tracts in Mathematics},
volume = {218},
publisher = {Cambridge University Press},
year = {2019},
doi = {10.1017/9781108303453},
isbn = {9781108419529},
url = {https://case.edu/artsci/math/esmeckes/Haar_book.pdf},
note = {Proof locators refer to the author's final pre-copy-editing draft}
}
==== END Critical-mixing-in-the-Sherrington-Kirkpatrick-model//references.bib ====