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  pdftitle={Free energy of spherical and Ising perceptrons with arbitrary Lipschitz channels}
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\newtheorem{theorem}{Theorem}[section]
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\newcommand{\E}{\mathbb E}
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\title{Parisi formula for Ising and Spherical Perceptrons
}
\author{}
\date{October 7, 2026}

\begin{document}
\maketitle

\begin{abstract}
We prove a Parisi formula for Ising and spherical perceptrons with Gaussian disorder and i.i.d. random potentials having a common Lipschitz bound and integrable values at zero. We assume $M/N\to\alpha\in(0,\infty)$, where $M$ is the number of Gaussian vectors and $N$ is the number of spins. The proof combines cavity in $M$ and $N$ with the Aizenman–Sims–Starr scheme for the lower bound and Guerra’s interpolation for the upper bound. As consequences, we answer several problems posed by Talagrand (2011) and establish a quenched large deviation principle for the empirical distribution of the Gaussian projections, resolving a conjecture of Bolthausen and Kistler (2010). We also resolve the sharp-threshold conjecture of Aubin, Perkins, and Zdeborová (2019) for the $u$-function binary perceptron at every positive width, and establish the predicted variational formulas for the entropy and capacity of the negative spherical perceptron. Finally, the zero-temperature limit yields the scalar projection-pursuit formulas conjectured by Montanari and Zhou (2025, 2026), in both the supervised and unlabelled settings.
\end{abstract}

\setcounter{tocdepth}{1}
\tableofcontents

\subsection*{Disclaimer}
This document and its arguments were generated through discussions with ChatGPT 5.6 Pro. An initial draft for the capacity of the negative spherical perceptron was generated on July 15, 2026, and we extended the argument to i.i.d. lipschitz potentials, both Ising and spherical, on September 17, 2026. While distilling the main ideas, carefully checking the proof, and polishing the draft, we learned on October 6, 2026, that OpenAI had released overlapping results across three manuscripts~\cite{OpenAIIsingPerceptron2026,OpenAISphericalPerceptron2026,OpenAIJamming2026}. Given these circumstances, we decided to post this working draft on Hexagon.

\section{Introduction}\label{sec:introduction}
We study Ising and spherical perceptrons with Gaussian disorder and i.i.d. random Lipschitz potentials. Let $\nu$ be a Borel probability measure on $\mathbb R$, and let $(\theta,x)\mapsto u_\theta(x)$ be Borel measurable on $\mathbb R^2$. We assume that $\Lip(u_\theta)\le L<\infty$  for $\nu$-almost every $\theta$ and that $\int_{\mathbb R}|u_\theta(0)|\nu(\dd\theta)<\infty$.

Let $\Theta_1,\ldots,\Theta_M$ be i.i.d. with law $\nu$, and let $g_1,\ldots,g_M\in\mathbb R^N$ be independent standard Gaussian vectors, independent of $(\Theta_k)_{k\leq M}$. The Hamiltonian of the perceptron model is defined by
\[
H_{N,M}^{\nu}(\sigma)
=\sum_{k=1}^M u_{\Theta_k}(h_k(\sigma)),\quad\textnormal{where}\quad h_k(\sigma)=\frac{g_k\cdot\sigma}{\sqrt N}
\]
is the Gaussian projection. We consider the spherical and Ising configuration spaces
\[
\Sigma_N^{\mathrm S}
=\{\sigma\in\mathbb R^N:\|\sigma\|_2^2=N\},
\qquad
\Sigma_N^{\mathrm I}=\{-1,1\}^N.
\]
Let \(\mu_N^{\mathrm S}\) and \(\mu_N^{\mathrm I}\) be the uniform probability measures on \(\Sigma_N^{\mathrm S}\) and \(\Sigma_N^{\mathrm I}\), respectively. For $\kappa\in\{\mathrm S,\mathrm I\}$, the partition function is
\[
Z_{N,M}^{\kappa,\nu}
=\int_{\Sigma_N^\kappa}
e^{H_{N,M}^{\nu}(\sigma)}\,\mu_N^\kappa(\dd\sigma).
\]
Our main object is the normalized free energy $N^{-1}\log Z_{N,M}^{\kappa,\nu}$. Throughout, we take $M=M_N$ with $M_N/N\to\alpha\in(0,\infty)$.


The study of perceptron model in statistical physics was intiated by Gardner and Gardner--Derrida \cite{Gardner1988,GardnerDerrida1988}.
For constraints \(h_k(\sigma)\ge b\), they studied the volume of
configurations satisfying all \(M\) inequalities and the density
\(M/N\) at which solutions disappear. The entropy records the
logarithmic volume or number of solutions; the storage capacity is
the critical density for their existence. Krauth and M\'ezard
subsequently predicted the capacity for Ising spins
\cite{KrauthMezard1989}.

For spherical spins and nonnegative margin \(b\), Shcherbina and
Tirozzi proved Gardner's replica-symmetric entropy formula
\cite{ShcherbinaTirozziGardner}. For Ising spins, Talagrand developed
cavity arguments, comparing systems with slightly different numbers
of spins or Gaussian vectors, and proved replica-symmetric formulas
under restrictions on the potential and density \cite{TalagrandBookI}.
Bolthausen, Nakajima, Sun, and Xu later obtained small-density formulas
for a broad class of activations \cite{BolthausenNakajimaSunXu}.
Nakajima and Sun proved concentration and universality results,
together with sharp-threshold sequences for broad classes of Ising
constraints \cite{NakajimaSun}. Their results control random
fluctuations and changes in the disorder distribution, and locate
the satisfiability transition around a sequence of densities.
Proving that this sequence converges requires an additional argument.

Beyond replica symmetry, Gy\"orgyi and Reimann developed formulas
involving a full overlap distribution for the spherical perceptron
at positive temperature \cite{GyorgyiReimann2000}. Franz and Parisi
studied the constraints \(h_k(\sigma)\ge b\) with \(b<0\) and their
jamming transition, where feasible configurations disappear as
\(M/N\) increases \cite{FranzParisiJamming}. Their predictions concern
both the entropy and the geometry near this transition. For Ising
spins, Aubin, Perkins, and Zdeborov\'a studied the symmetric
\(U\)-function constraints \(|h_k(\sigma)|\ge K\). They identified
a large-\(K\) regime in which the first-moment estimate, based on the
expected number of solutions, no longer gives the predicted capacity
\cite{AubinPerkinsZdeborova}.

Our main theorem determines the limiting free energy under the
assumptions above. We also prove a quenched large-deviation principle for the empirical
distribution of the Gaussian projections \(h_k(\sigma)\). For a typical
fixed realization of the Gaussian vectors, this describes the
exponential probabilities of atypical projection distributions when
\(\sigma\) is sampled from \(\mu_N^\kappa\). Finally, the
zero-temperature limit determines optimal empirical averages over
one-dimensional projections, in both supervised and unlabelled
settings.

Define the overlap-path space by
\[
 \Q=\left\{q:[0,1]\to[0,1]: q\text{ is right-continuous and nondecreasing},~~~~
 q(1)=1\right\}.
\]
We identify paths that agree almost everywhere on \([0,1)\) and
equip \(\Q\) with the \(L^1\) metric.
The spin cost \(\mathcal A_\kappa\) is defined in
\eqref{eq:spherical-entropy} for spherical spins and
\eqref{eq:intro-ising-cost} for Ising spins. The channel functional
\(\Phi_u\) is defined in \eqref{eq:intro-channel-pde}
through the scalar Parisi PDE \eqref{eq:intro-parisi-pde}.
These definitions follow \cref{thm:iid-channels}.

\begin{theorem}[Free energy for i.i.d.\ Lipschitz channels]
\label{thm:iid-channels}
For either prior,
\begin{equation*}
 \frac1N\log Z_{N,M_N}^{\kappa,\nu}
 \longrightarrow
 \inf_{q\in\Q}\left\{\mathcal A_\kappa(q)
       +\alpha\int_{\mathbb R}\Phi_{u_\theta}(q)\,\nu(\dd\theta)\right\}
 \qquad\text{in }L^1.
\end{equation*}
Convergence holds in \(L^2\) if
\(\int_{\mathbb R}|u_\theta(0)|^2\,\nu(\dd\theta)<\infty\),
and almost surely and in every finite \(L^p\) if there is \(B<\infty\)
such that \(|u_\theta(x)|\le B\) for every \(x\) and
\(\nu\)-almost every \(\theta\).
% For spherical spins, one may again restrict to \(q(1-)<1\).
\end{theorem}


Note that a deterministic potential \(u\) corresponds to a point-mass channel law.
In this case, write \(H_{N,M}^{u}\), \(Z_{N,M}^{\kappa,u}\), and
\(F_{N,\kappa}^{u}=N^{-1}\log Z_{N,M_N}^{\kappa,u}\), and put
\begin{equation*}
 \mathcal P_{\kappa,u}(\alpha)
 =\inf_{q\in\Q}\{\mathcal A_\kappa(q)+\alpha\Phi_u(q)\}.
\end{equation*}
The theorem gives convergence in \(L^2\), since \(u(0)\) is finite.


We next define the functionals \(\Phi_u\) and \(\mathcal A_\kappa\)
on the overlap paths \(q\in\Q\). Recall that these paths are
right-continuous and nondecreasing, take values in \([0,1]\), and
satisfy \(q(1)=1\). Let
\[
 \Pb=\left\{a:[0,1)\to[0,\infty):
 a\text{ is bounded, right-continuous and nondecreasing}\right\}.
\]
These paths will enter the
variational definition of the Ising spin cost. For \(c\in\Pb\),
write \(c_*=c(1-)\) and
\(\gamma_c(t)=\operatorname{Leb}\{r\in[0,1):c(r)\le t\}\).
Thus \(\gamma_c\) is the distribution function of \(c(U)\) for
\(U\) uniform on \([0,1)\). We use the same notation for
\(q\in\Q\), restricting \(q\) to \([0,1)\).

For \(T>0\), a nonnegative Lebesgue-integrable function
\(\gamma:[0,T]\to[0,\infty)\), and a globally Lipschitz
function \(h:\mathbb R\to\mathbb R\), let \(f_{\gamma,h}\) solve
the backward Parisi PDE
\begin{equation}\label{eq:intro-parisi-pde}
 \partial_t f+\frac12\bigl(f_{xx}+\gamma(t)f_x^2\bigr)=0,
 \qquad f(T,x)=h(x).
\end{equation}
We use the stochastic-control solution; its existence and stability
are proved in \cref{lem:mz-control}.  The channel functional is
\begin{equation}\label{eq:intro-channel-pde}
 \Phi_u(q)=f_{\gamma_q,u}(0,0),\qquad T=1.
\end{equation}
The two spin costs are as follows.  For spherical spins,
\begin{equation}\label{eq:spherical-entropy}
 \mathcal A_{\mathrm S}(q)
 =\frac12\int_0^1\left\{
       \frac1{\int_t^1\gamma_q(s)\,\dd s}-\frac1{1-t}\right\}\dd t.
\end{equation}
The outer integrand is nonnegative, and the value may be infinite.
For Ising spins and \(a\in\Pb\), use the same PDE with terminal
function \(\log\cosh\) and terminal time \(a_*\):
\begin{equation}\label{eq:intro-ising-transform}
 \mathcal S_{\mathrm I}(a)
 =f_{\gamma_a,\log\cosh}(0,0)-\frac{a_*}{2},\qquad T=a_*.
\end{equation}
At \(a=0\), define $\mathcal S_{\mathrm I}(0)=0$.  The Ising cost is
\begin{equation}\label{eq:intro-ising-cost}
 \mathcal A_{\mathrm I}(q)
 =\sup_{a\in\Pb}\left\{\mathcal S_{\mathrm I}(a)
                         +\frac12\int_0^1a(r)q(r)\,\dd r\right\}.
\end{equation}
Below, we write \(\langle a,q\rangle=\int_0^1a(r)q(r)\,\dd r\).
% the diagonal value does not enter this pairing.


\subsection{Direct consequences of the free energy formula}
\label{sec:field-bipartite-applications}
In this section, we prove direct consequences of Theorem~\ref{thm:iid-channels}, which may be of independent interest.
\paragraph{Gaussian external fields.}\phantomsection
\label{par:gaussian-field-extension}
For a standard Gaussian vector \(g^0 \in \R^N\), independent of all other randomness, let
\begin{equation}\label{eq:intro-gaussian-field-partition}
 Z_{N,M}^{\kappa,u,b}
 =\int e^{\sum_{k\le M}u(h_k(\sigma))+b\,g^0\cdot\sigma}
        \mu_N^\kappa(\dd\sigma),\qquad
 F_{N,\kappa}^{u,b}=\frac1N\log Z_{N,M_N}^{\kappa,u,b},
\end{equation}
and denote the marked version by \(Z_{N,M}^{\kappa,\nu,b}\).
Write
\begin{equation}\label{eq:intro-field-value}
 \mathsf L(q)=\int_0^1(1-q(r))\,\dd r,\qquad
 \mathcal P_{\kappa,u}^{\mathrm{fld}}(\alpha,b)
 =\inf_{q\in\Q}\left\{\mathcal A_\kappa(q)+\alpha\Phi_u(q)
                         +\frac{b^2}{2}\mathsf L(q)\right\}.
\end{equation}

For a deterministic potential, the Gaussian-field extension of
\cref{thm:iid-channels} gives
\(F_{N,\kappa}^{u,b}\to\mathcal P_{\kappa,u}^{\mathrm{fld}}(\alpha,b)\)
in \(L^2\).  For i.i.d.\ channels, replace \(\Phi_u\) by
\(\E_\nu\Phi_{u_\Theta}\); the same \(L^1\) and \(L^2\) conclusions
as in \cref{thm:iid-channels} hold.  Mixing in linear channels proves
this extension; see \cref{sec:gaussian-field-proof}.

\paragraph{Spherical i.i.d.\ fields.}
Let \(\pi\) be a Borel probability measure on \(\mathbb R\) satisfying
\[
 v_\pi:=\int_{\mathbb R}h^2\,\pi(\dd h)<\infty.
\]
Let \(h_1,h_2,\ldots\) be i.i.d.\ with law \(\pi\), independent
of the Gaussian constraint vectors \((g_k)_{k\le M_N}\).
Define
\[
 \begin{aligned}
 Z_{N,\mathrm S}^{u,\pi}
 &:=
 \int_{S_N}
 \exp\left\{
   \sum_{k=1}^{M_N}u\!\left(\frac{g_k\cdot\sigma}{\sqrt N}\right)
   +\sum_{i=1}^N h_i\sigma_i
 \right\}
 \mu_N^{\mathrm S}(\dd\sigma),\\
 F_{N,\mathrm S}^{u,\pi}
 &:=\frac1N\log Z_{N,\mathrm S}^{u,\pi}.
 \end{aligned}
\]
Here \(\E\) and \(\Pp\) denote expectation and probability
with respect to the joint law of the constraint vectors
\((g_k)_{k\le M_N}\) and external fields \((h_i)_{i\le N}\).

\begin{corollary}[Spherical i.i.d.\ external fields]
\label{cor:spherical-iid-field}
Let $u:\R\to\R$ be real-valued and globally Lipschitz, let
$M_N/N\to\alpha\in(0,\infty)$, and let $\pi$ satisfy
$\int h^2\pi(\dd h)<\infty$ as above.  Then
\begin{equation*}
 F_{N,\mathrm S}^{u,\pi}
 \longrightarrow\mathcal P_{\mathrm S,u}^{\mathrm{fld}}(\alpha,\sqrt{v_\pi})
 \quad\text{in }L^2.
\end{equation*}
% Thus convergence also holds in probability and in mean.
\end{corollary}

\begin{proof}
For constraint vectors
\(\mathbf g=(g_k)_{k=1}^{M_N}\) and a deterministic field
\(\mathbf h\in\mathbb R^N\), define
\[
 \widehat f_N(\mathbf g,\mathbf h)
 :=
 \frac1N\log\int_{S_N}
 \exp\left\{
   \sum_{k=1}^{M_N}
     u\left(\frac{g_k\cdot\sigma}{\sqrt N}\right)
   +\mathbf h\cdot\sigma
 \right\}
 \mu_N^{\mathrm S}(\dd\sigma).
\]
For \(\mathbf h,\widetilde{\mathbf h}\in\mathbb R^N\), Cauchy--Schwarz gives $ |(\mathbf h-\widetilde{\mathbf h})\cdot\sigma|
 \le \sqrt N\,\|\mathbf h-\widetilde{\mathbf h}\|_2$. Comparing the two exponential weights therefore yields
\begin{equation}\label{eq:spherical-field-vector-modulus}
 \left|
   \widehat f_N(\mathbf g,\mathbf h)
   -\widehat f_N(\mathbf g,\widetilde{\mathbf h})
 \right|
 \le \frac{\|\mathbf h-\widetilde{\mathbf h}\|_2}{\sqrt N}.
\end{equation}
For an orthogonal matrix \(O\in\mathbb R^{N\times N}\), write
\(O\mathbf g=(Og_k)_{k=1}^{M_N}\).
The change of variables \(\sigma=O^{\mathsf T}\tau\), together
with rotational invariance of \(\mu_N^{\mathrm S}\), gives $ \widehat f_N(\mathbf g,\mathbf h)
 =
 \widehat f_N(O\mathbf g,O\mathbf h).$ Under the Gaussian row law, \(O\mathbf g\) has the same
distribution as \(\mathbf g\).
Writing \(e_1=(1,0,\ldots,0)\in\mathbb R^N\) and choosing
\(O\) such that \(O\mathbf h=\|\mathbf h\|_2e_1\), we obtain
\[
 \widehat f_N(\mathbf g,\mathbf h)
 \overset{\mathrm d}=
 \widehat f_N(\mathbf g,\|\mathbf h\|_2e_1)
 \qquad(\mathbf h\in\mathbb R^N),
\]
where the equality in distribution is with respect to the
Gaussian constraint vectors.
Let \(z_1,z_2,\ldots\) be independent standard Gaussian
random variables, independent of both the external fields
and the constraint vectors, and put $ \mathbf h^{(N)}=(h_1,\ldots,h_N)$ and $ z^{(N)}=(z_1,\ldots,z_N)$. In the remainder of the proof, \(\E\) also averages over
this auxiliary Gaussian sequence.
Define, using the same constraint vectors in both expressions,
\[
 X_N
 :=
 \widehat f_N\bigl(\mathbf g,\|\mathbf h^{(N)}\|_2e_1\bigr),
 \qquad
 Y_N
 :=
 \widehat f_N\bigl(
   \mathbf g,\sqrt{v_\pi}\,\|z^{(N)}\|_2e_1
 \bigr).
\]
Conditioning on \(\mathbf h^{(N)}\) or \(z^{(N)}\) and applying the
preceding rotational identity gives
\[
 \begin{aligned}
 X_N
 &\overset{\mathrm d}=
 \widehat f_N(\mathbf g,\mathbf h^{(N)})
 =F_{N,\mathrm S}^{u,\pi},\\
 Y_N
 &\overset{\mathrm d}=
 \widehat f_N(\mathbf g,\sqrt{v_\pi}\,z^{(N)})
 \overset{\mathrm d}=
 F_{N,\mathrm S}^{u,\sqrt{v_\pi}},
 \end{aligned}
\]
where \(F_{N,\mathrm S}^{u,\sqrt{v_\pi}}\) is the Gaussian-field
free energy from \eqref{eq:intro-gaussian-field-partition}
with \(b=\sqrt{v_\pi}\). By \eqref{eq:spherical-field-vector-modulus},
\[
 |X_N-Y_N|
 \le
 \left|
   \frac{\|\mathbf h^{(N)}\|_2}{\sqrt N}
   -\sqrt{v_\pi}\frac{\|z^{(N)}\|_2}{\sqrt N}
 \right|.
\]
Using \((\sqrt x-\sqrt y)^2\le |x-y|\) for \(x,y\ge0\),
we conclude that
\[
 \begin{aligned}
 \E|X_N-Y_N|^2
 &\le
 \E\left|
   \frac1N\sum_{i=1}^N h_i^2
   -\frac{v_\pi}{N}\sum_{i=1}^N z_i^2
 \right|\\
 &\le
 \E\left|
   \frac1N\sum_{i=1}^N h_i^2-v_\pi
 \right|
 +v_\pi\,\E\left|
   \frac1N\sum_{i=1}^N z_i^2-1
 \right|
 \longrightarrow0.
 \end{aligned}
\]
Indeed, both terms tend to zero by the law of large
numbers, since \(\E h_1^2=v_\pi<\infty\) and \(\E z_1^2=1\). Set
\[
 P_\pi
 :=
 \mathcal P_{\mathrm S,u}^{\mathrm{fld}}
   (\alpha,\sqrt{v_\pi}).
\]
The Gaussian-field extension of Theorem~\ref{thm:iid-channels}
and the distributional identity for \(Y_N\) imply that
\(\E|Y_N-P_\pi|^2\to0\). Consequently,
\[
 \E\left|F_{N,\mathrm S}^{u,\pi}-P_\pi\right|^2
 =
 \E|X_N-P_\pi|^2
 \le
 2\E|X_N-Y_N|^2+2\E|Y_N-P_\pi|^2
 \longrightarrow0,
\]
which proves the asserted \(L^2\) convergence.
\end{proof}

% \begin{proof}
% For fixed rows \(\mathbf g=(g_k)_{k \le M_N}\) and deterministic
% \(\mathbf h \in\mathbb R^N\), set
% \[
%  \widehat f_N(\mathbf g,\mathbf h)
%  =\frac1N\log\int_{S_N}\exp\left\{
%    \sum_{k \le M_N}u\!\left(\frac{g_k\cdot\sigma}{\sqrt N}\right)
%    +\mathbf h\cdot\sigma\right\}\,\mu_N^{\rm S}(\dd\sigma).
% \]
% Direct comparison on the sphere gives the estimate
% \begin{equation}\label{eq:spherical-field-vector-modulus}
%  \left|\widehat f_N(\mathbf g,\mathbf h)-
%              \widehat f_N(\mathbf g,\widetilde{\mathbf h})\right|
%  \le \frac{\|\mathbf h-\widetilde{\mathbf h}\|_2}{\sqrt N}.
% \end{equation}
% Rotate \(\mathbf h\) to \(\|\mathbf h\|_2e_1\) and rotate the rows by the same
% orthogonal map.  Coupling \(\mathbf h^{(N)}=(h_i)_{i\le N}\) with an independent
% standard Gaussian vector \(z^{(N)}\), the i.i.d.-field and Gaussian-field
% pressures can therefore be coupled within error
% \[
%  \left|\frac{\|\mathbf h^{(N)}\|_2}{\sqrt N}
%        -\sqrt{v_\pi}\frac{\|z^{(N)}\|_2}{\sqrt N}\right|.
% \]
% This tends to zero in \(L^2\) by the \(L^1\) law of large numbers for
% \(h_1^2\) and \((\sqrt x-\sqrt y)^2\le|x-y|\).  Apply
% the \hyperref[par:gaussian-field-extension]{Gaussian-field extension} with \(b=\sqrt{v_\pi}\).
% \end{proof}

\paragraph{Bipartite transposition.}
Summing over one party converts a bipartite Ising model with a homogeneous
deterministic field on only that party into a perceptron.  For $\beta,b\in\R$, define
\begin{equation*}
 Z_{N,M}^{\mathrm{bip}}(\beta,b)
 =2^{-(N+M)}
 \sum_{\substack{\sigma\in\{-1,+1\}^N\\
                  \tau\in\{-1,+1\}^M}}
 \exp\left\{
  \frac\beta{\sqrt N}\sum_{k=1}^M\sum_{i=1}^N
       g_{ki}\tau_k\sigma_i
  +b\sum_{i=1}^N\sigma_i
 \right\},
\end{equation*}
and put
\begin{equation*}
 u_{\alpha,\beta,b}^{\rm bip}(x)
 =\log\cosh\bigl(b+\beta\sqrt\alpha\,x\bigr).
\end{equation*}

\begin{corollary}[Bipartite Ising formula with a field on one party]
\label{cor:bipartite-one-sided-field}
For every $\beta,b\in\R$, if $M_N/N\to\alpha\in(0,\infty)$, then
\begin{equation*}
 \frac1N\log Z_{N,M_N}^{\mathrm{bip}}(\beta,b)
 \longrightarrow
 \alpha\,\mathcal P_{\mathrm I,u_{\alpha,\beta,b}^{\rm bip}}(1/\alpha)
\end{equation*}
in $L^2$ and hence in $L^1$ and probability. Moreover, 
\begin{equation}\label{eq:bipartite-one-sided-field-variance}
 \Var\left(\frac1N\log Z_{N,M_N}^{\mathrm{bip}}(\beta,b)\right)
 \le\frac{\beta^2M_N}{N^2}.
\end{equation}
\end{corollary}

\begin{proof}
Put \(\alpha_N=M_N/N\) and
\(u_N^{\rm bip}(x)=\log\cosh(b+\beta\sqrt{\alpha_N}\,x)\).  Summing first over
\(\sigma\) gives the exact identity
\begin{equation*}
 Z_{N,M_N}^{\mathrm{bip}}(\beta,b)
 =Z_{M_N,N}^{\mathrm I,u_N^{\rm bip}}.
\end{equation*}
Indeed, the columns of the \(M_N\)-by-\(N\) disorder matrix are independent
standard Gaussian rows in \(\mathbb R^{M_N}\).  Write
\(G=(g_{ki})_{k \le M_N,\,i\le N}\) for this matrix.
The one-Lipschitz property of \(\log\cosh\), Cauchy--Schwarz, and the
standard Gaussian operator-norm estimate give
\begin{equation*}
 \left|\frac1{M_N}\log Z_{M_N,N}^{\mathrm I,u_N^{\rm bip}}
       -\frac1{M_N}\log Z_{M_N,N}^{\mathrm I,u_{\alpha,\beta,b}^{\rm bip}}\right|
 \le |\beta|\,|\sqrt{\alpha_N}-\sqrt\alpha|\,
       \frac{\sqrt N\,\|G\|_{\rm op}}{M_N}
 =o_{L^2}(1).
\end{equation*}
Apply \cref{thm:iid-channels} with spin dimension \(M_N\), row count
\(N\), and limiting density \(1/\alpha\), and multiply by
\(M_N/N\to\alpha\).  Finally,
\(\partial_{g_{ki}}[N^{-1}\log Z^{\mathrm{bip}}]
=\beta(N\sqrt N)^{-1}\langle\tau_k\sigma_i\rangle\), so Gaussian
Poincar\'e gives \eqref{eq:bipartite-one-sided-field-variance}.
\end{proof}

\paragraph{Gaussian spins.}
The spherical field formula gives the Euclidean Shcherbina--Tirozzi model
\cite{ShcherbinaTirozziGardner}.  Let \(g^0,g_1,g_2,\ldots\) be independent
standard Gaussian vectors in \(\mathbb R^N\), and define, for \(c>0\),
\begin{equation*}
 \mathcal Z_{N,M}^{\rm ST}(u,h,c)
 =\int_{\mathbb R^N}\exp\left\{
   \sum_{k=1}^M u\left(\frac{g_k\cdot x}{\sqrt N}\right)
   +h\,g^0\cdot x-c\|x\|_2^2
 \right\}\,\dd x.
\end{equation*}
For \(\rho>0\), put
\begin{equation*}
 u_\rho(z)=u(\sqrt\rho\,z)
\end{equation*}
and
\begin{align}
 \mathcal P_{\rm ST}(\alpha;u,h,c)
 :=\frac12\log(2\pi e)
 +\sup_{\rho>0}\Bigg\{
   \frac12\log\rho-c\rho+\inf_{q\in\Q}\left[
      \mathcal A_{\rm S}(q)+\alpha\Phi_{u_\rho}(q)
      +\frac{h^2\rho}{2}\mathsf L(q)
   \right]\Bigg\}.
 \label{eq:st-variational-value}
\end{align}

\begin{theorem}[Gaussian-spin free energy]
\label{thm:st-free energy}
Let \(u:\mathbb R\to\mathbb R\) be real-valued and globally Lipschitz, let
\(c>0\), \(h\in\mathbb R\), and \(M_N/N\to\alpha\in(0,\infty)\).  Then
\begin{equation*}
 \frac1N\log\mathcal Z_{N,M_N}^{\rm ST}(u,h,c)
 \longrightarrow \mathcal P_{\rm ST}(\alpha;u,h,c)
 \qquad\text{in }L^1.
\end{equation*}
% In particular, convergence holds in probability and the expectations
% converge to the same limit; all expectations are over the joint row and
% external-field disorder.
\end{theorem}

In particular, under the hypotheses of Talagrand's Research Problem~3.1.2,
\cref{thm:st-free energy} computes the limit
\cite[Research Problem~3.1.2, p.~193]{TalagrandBookI}.  It does not resolve
the separate Research Problem~3.2.13 on uniqueness of the scalar
replica-symmetric equations
\cite[Research Problem~3.2.13, p.~222]{TalagrandBookI}; the optimizer in
\eqref{eq:st-variational-value} is not assumed to be a constant path.

% \begin{proof}
% Polar disintegration with \(x=\sqrt\rho\,\sigma\) gives
% \begin{equation}\label{eq:st-polar-disintegration}
%  \mathcal Z_{N,M}^{\rm ST}(u,h,c)
%  =C_N\int_0^\infty
%    \rho^{N/2-1}e^{-cN\rho}
%    Z_{N,M}^{\mathrm S,u_\rho,h\sqrt\rho}\,\dd\rho,
%  \qquad C_N=\frac{(\pi N)^{N/2}}{\Gamma(N/2)}.
% \end{equation}
% Stirling's formula yields
% \begin{equation}\label{eq:st-polar-constant}
%  \frac1N\log C_N\longrightarrow\frac12\log(2\pi e).
% \end{equation}
% After the change of variables \(\rho=s^2\),
% \cref{lem:st-radial-laplace} gives \(L^1\) convergence of the
% normalized log radial integral to
% \(\sup_{s>0}\{\mathcal P_{\mathrm S,u_{s^2}}^{\mathrm{fld}}(\alpha,hs)
%                   +\log s-cs^2\}\).
% Together with \eqref{eq:st-polar-constant}, this is
% \eqref{eq:st-variational-value}.  The lemma proves the compact-uniform
% shell convergence, radial localization, and uniform integrability
% needed for this passage.
% \end{proof}

\begin{proof}
Polar disintegration with \(x=\sqrt\rho\,\sigma\),
\(\sigma\in S_N\), gives
\begin{equation}\label{eq:st-polar-disintegration}
 \mathcal Z_{N,M_N}^{\rm ST}(u,h,c)
 =C_N\int_0^\infty
   \rho^{N/2-1}e^{-cN\rho}
   Z_{N,M_N}^{\mathrm S,u_\rho,h\sqrt\rho}\,\dd\rho,
 \qquad
 C_N=\frac{(\pi N)^{N/2}}{\Gamma(N/2)}.
\end{equation}
Stirling's formula yields
\begin{equation}\label{eq:st-polar-constant}
 \frac1N\log C_N
 \longrightarrow\frac12\log(2\pi e).
\end{equation}
For \(s>0\), define, using the same disorder for every \(s\),
\[
 Y_N(s):=\frac1N\log
 Z_{N,M_N}^{\mathrm S,u_{s^2},hs},
 \qquad
 Y(s):=\mathcal P_{\mathrm S,u_{s^2}}^{\mathrm{fld}}(\alpha,hs).
\]
Since \(\rho=s^2\) gives
\(\rho^{N/2-1}\,\dd\rho=2s^{N-1}\,\dd s\), we obtain
\[
 \frac1N\log\mathcal Z_{N,M_N}^{\rm ST}(u,h,c)
 =
 \frac1N\log(2C_N)
 +\frac1N\log\int_0^\infty
   s^{N-1}e^{N[Y_N(s)-cs^2]}\,\dd s.
\]
For these functions \(Y_N\) and \(Y\),
\cref{lem:st-radial-laplace} gives
\[
 \frac1N\log\int_0^\infty
   s^{N-1}e^{N[Y_N(s)-cs^2]}\,\dd s
 \xrightarrow{L^1}
 \sup_{s>0}\{Y(s)+\log s-cs^2\}.
\]
Adding \eqref{eq:st-polar-constant}, with
\(N^{-1}\log2\to0\), proves \(L^1\) convergence of the
full free energy to
\[
 \frac12\log(2\pi e)
 +\sup_{\rho>0}
 \left\{
   \frac12\log\rho-c\rho
   +\mathcal P_{\mathrm S,u_\rho}^{\mathrm{fld}}
      (\alpha,h\sqrt\rho)
 \right\},
\]
where we have returned to \(\rho=s^2\).
Substituting the definition of
\(\mathcal P_{\mathrm S,u_\rho}^{\mathrm{fld}}\)
from \eqref{eq:intro-field-value} identifies this expression
with \(\mathcal P_{\rm ST}(\alpha;u,h,c)\) in
\eqref{eq:st-variational-value}.
\end{proof}

\subsection{Quenched large deviations}
\label{sec:quenched-ldp-statement}

The free energy formula also determines nonlinear weights depending
on the empirical distribution of the constraint fields.  With the
Gaussian rows \((g_k)_k\) held fixed, define
\begin{equation*}
 L_{N,\sigma}=\frac1{M_N}\sum_{k\le M_N}\delta_{h_k(\sigma)},
 \qquad
 \rho_N^\kappa=\mu_N^\kappa\circ L_{N,\cdot}^{-1}.
\end{equation*}
On the space \(\mathcal P(\mathbb R)\) of probability measures, equipped
with its weak topology, put
\begin{equation}\label{eq:quenched-ldp-rate}
 \mathcal J_{\kappa,\alpha}(\mu)
 =\sup_{u\in\mathrm{BL}(\mathbb R)}
  \left\{\int u\,\dd\mu-\frac{\mathcal P_{\kappa,u}(\alpha)}\alpha\right\},
\end{equation}
where \(\mathrm{BL}(\mathbb R)\) denotes bounded Lipschitz functions.

\begin{theorem}[Quenched empirical measure large deviations]
\label{thm:quenched-empirical-ldp}
For either prior, on one event of probability one, \(\rho_N^\kappa\)
satisfies a large-deviation principle with speed \(M_N\) and good
convex rate function \(\mathcal J_{\kappa,\alpha}\).
On the same event, for every bounded weakly continuous
\(\Psi:\mathcal P(\mathbb R)\to\mathbb R\),
\begin{equation}\label{eq:nonlinear-perceptron-free energy}
 \frac1N\log\int e^{M_N\Psi(L_{N,\sigma})}\mu_N^\kappa(\dd\sigma)
 \longrightarrow
 \alpha\sup_\mu\{\Psi(\mu)-\mathcal J_{\kappa,\alpha}(\mu)\}.
\end{equation}
\end{theorem}

Note that no convexity of \(\Psi\) is assumed in Theorem~\ref{thm:quenched-empirical-ldp}.
For unnormalized Ising counting, add \(\log2\) to the limit in Eq.~\eqref{eq:nonlinear-perceptron-free energy}.
At \(M_N=N\), this proves the quenched Laplace principle in
\cite[Definition~2.1 and Conjecture~2.2]{BolthausenKistler} for the
Gaussian Ising perceptron.  The result concerns constraint fields
under the spin prior; it does not assert replica symmetry for a
conditioned spin measure.  The proof in \cref{sec:quenched-ldp-proof}
uses differentiability of the pressure in bounded channel directions.

\subsection{Scalar projection pursuit and supervised learning}
\label{sec:mz-statement}

We next consider the problem of maximizing an empirical objective over a single projection direction.
Let \((X_i)_i\) be independent standard Gaussian vectors in \(\R^d\), where
\(n/d\to\alpha\in(0,\infty)\).  Fix a teacher dimension \(k\ge1\),
deterministic matrices \(W_{*,d}\in\R^{d\times k}\) with orthonormal columns,
and labels
\[
 Y_i=\varphi(W_{*,d}^{\mathsf T}X_i,\varepsilon_i),
\]
where \(\varphi\) is measurable and real-valued and the \(\varepsilon_i\)
are i.i.d.\ with a fixed law, independently of the covariates.  The objective
is \(n^{-1}\sum_i h(Y_i,w\cdot X_i)\), with \(\|w\|_2=1\) and bounded
continuous \(h\).

The limiting value uses the following scalar functional.  Write \(C_b\) for
bounded continuous functions and let \(\mathscr U\) be the nonnegative,
nondecreasing integrable functions on \([0,1)\).  For \(H\in C_b(\R)\) and
\(c>0\), set
\[
 \mathsf H_cH(x)=\sup_z\{H(x+z)-z^2/(2c)\}.
\]
Let \(f_{\mu,c}^{H}\) solve \eqref{eq:intro-parisi-pde} on \([0,1]\)
with coefficient \(\mu\) and terminal function \(h=\mathsf H_cH\).  The function $h$
is bounded and Lipschitz, so the definition applies to every
\(\mu\in\mathscr U\).

\begin{theorem}[Scalar Montanari--Zhou formula]
\label{thm:montanari-zhou}
Under the preceding assumptions, for every fixed \(h\in C_b(\R^2)\),
\begin{equation*}
 \max_{\|w\|_2=1}\frac1n\sum_{i=1}^n h(Y_i,w\cdot X_i)
 \longrightarrow
 \sup_{r\in B_k(1)}\inf_{\mu\in\mathscr U,\ c>0}
                         \mathsf F_{\alpha,h}(\mu,c,r)
\end{equation*}
almost surely and in every \(L^p\) with $p<\infty$, where \(B_k(1)\) is the closed
unit ball and
\begin{equation}\label{eq:mz-supervised-functional}
 \mathsf F_{\alpha,h}(\mu,c,r)
 =\E\left[f_{\mu,c}^{h(Y,\cdot)}(|r|^2,r\cdot G)\right]
   +\frac1{2\alpha}\int_{|r|^2}^1
                 \frac{\dd t}{c+\int_t^1\mu(s)\,\dd s}.
\end{equation}
Here \(G\sim N(0,I_k)\) is independent of \(\varepsilon\), and
\(Y=\varphi(G,\varepsilon)\).  At \(|r|=1\), the inner infimum equals
\(\E h(Y,r\cdot G)\).

Moreover, for every fixed \(H\in C_b(\R)\),
\begin{equation*}
 \max_{\|w\|_2=1}\frac1n\sum_{i=1}^n H(w\cdot X_i)
 \longrightarrow
 \inf_{\mu\in\mathscr U,\ c>0}\left\{
 f_{\mu,c}^{H}(0,0)+\frac1{2\alpha}\int_0^1
                      \frac{\dd t}{c+\int_t^1\mu(s)\,\dd s}\right\}
\end{equation*}
almost surely and in every \(L^p\) with $p<\infty$.
\end{theorem}

These formulas establish the supervised scalar conjecture
\cite[Conjecture~2.2, equations~(14)--(15)]{MZMultiIndex} and the \(m=1\)
projection-pursuit conjecture \cite[Conjecture~2.1 and Remark~1]{MZProjection}.
The first applies, in particular, under their sub-Gaussian label assumption.
The limits also identify their feasible-distribution duals, which equal the
corresponding probability limit inferiors
\cite[equation~(9)]{MZMultiIndex}\cite[equation~(3)]{MZProjection}.

\subsection{Hard constraints and capacities}

We use the free-energy formula for Lipschitz potentials to
derive formulas for the hard-constraint free energy at
margin \(\theta\in\R\).
Let \(C\subsetneq\mathbb R\) be a nonempty closed set containing
either \([\theta,\infty)\) or \((-\infty,\theta]\).  By reflection, we may assume that
\begin{equation*}
 [\theta_C,\infty)\subset C
\end{equation*}
for some finite \(\theta_C\); we call such a set \emph{tail-accessible}.
For \(Z\sim N(0,1)\), put
\begin{equation*}
 d_C(x)=\operatorname{dist}(x,C),\qquad
 u_{\beta,C}(x)=-\beta d_C(x),\qquad \pi_C=\Pp(Z\in C)\in(0,1).
\end{equation*}
The feasible set and its normalized entropy are
\begin{align*}
 \mathcal F_{M,N}^{\kappa}(C)
 &=\{\sigma\in\Sigma_N^\kappa:h_k(\sigma)\in C\text{ for all } k\le M\},
 \\
 \operatorname{Ent}_{N,\kappa}(C)
 &=\frac1N\log\mu_N^\kappa(\mathcal F_{M_N,N}^{\kappa}(C)),
 \qquad\log0=-\infty.
\end{align*}
For Ising spins, the unnormalized counting entropy is
\(\log2+\operatorname{Ent}_{N,\mathrm I}(C)\).
Define the hard channel and variational value by
\begin{align*}
 \mathcal P_{\kappa,\infty,C}(\alpha)
 =\inf_{\substack{q\in\Q:\ \mathcal A_\kappa(q)<\infty}}
 \{\mathcal A_\kappa(q)+\alpha\Phi_{\infty,C}(q)\},\qquad  \Phi_{\infty,C}(q)=\lim_{\beta\to\infty}\Phi_{u_{\beta,C}}(q),
\end{align*}
both taking values in \([-\infty,0]\), with
\(\mathcal P_{\kappa,\infty,C}(0)=0\).

\begin{theorem}[Closed-set hardening]
\label{thm:closed-set-hardening}
Let \(C\) be tail-accessible and \(M_N/N\to\alpha>0\).  For either prior,
\begin{equation}\label{eq:hard-variational-convergence}
 \mathcal P_{\kappa,u_{\beta,C}}(\alpha)
 \downarrow\mathcal P_{\kappa,\infty,C}(\alpha)
 \qquad(\beta\to\infty).
\end{equation}
For spherical spins,
\begin{equation}\label{eq:hard-spherical-volume-limit}
 \operatorname{Ent}_{N,\mathrm S}(C)
 \xrightarrow{\Pp}\mathcal P_{\mathrm S,\infty,C}(\alpha)
\end{equation}
in the extended real line.  For Ising spins, if
\(\mathcal P_{\mathrm I,\infty,C}(\alpha)>-\log2\), then
\begin{align*}
 \operatorname{Ent}_{N,\mathrm I}(C)
 &\xrightarrow{\Pp}\mathcal P_{\mathrm I,\infty,C}(\alpha),
 \\
 \frac1N\log\#\mathcal F_{M_N,N}^{\mathrm I}(C)
 &\xrightarrow{\Pp}\log2+\mathcal P_{\mathrm I,\infty,C}(\alpha)>0.
\end{align*}
If \(\mathcal P_{\mathrm I,\infty,C}(\alpha)<-\log2\), then
\begin{equation}\label{eq:hard-ising-exponential-unsat}
 \limsup_{N\to\infty}\frac1N\log\Pp
 \bigl(\mathcal F_{M_N,N}^{\mathrm I}(C)\ne\varnothing\bigr)<0.
\end{equation}
\end{theorem}

Define the candidate Ising capacity by the variational ratio
\begin{equation*}
 \alpha_c^{\mathrm I}(C)
 =\inf_{\substack{q\in\Q:\ \mathcal A_{\mathrm I}(q)<\infty}}
 \frac{\log2+\mathcal A_{\mathrm I}(q)}{-\Phi_{\infty,C}(q)},
\end{equation*}
where the ratio is zero if \(\Phi_{\infty,C}(q)=-\infty\).
The denominator is positive by \eqref{eq:ising-hard-annealed-bound}.
Under a common infinite sequence of rows, define for either prior
\begin{equation*}
 \mathfrak M_N^\kappa(C)
 =\sup\{M\in\mathbb N_0:\mathcal F_{M,N}^{\kappa}(C)\ne\varnothing\}.
\end{equation*}

\begin{theorem}[Ising capacity]
\label{thm:closed-set-ising-capacity}
For every tail-accessible \(C\), the critical constraint density
satisfies
\begin{equation}\label{eq:closed-set-capacity-upper-bound}
 0<\alpha_c^{\mathrm I}(C)
 \le\frac{\log2}{-\log\pi_C}<\infty.
\end{equation}
It is characterized by positivity of the variational
counting entropy:
\begin{equation}\label{eq:closed-set-ising-capacity-boundary}
 \alpha_c^{\mathrm I}(C)
 =
 \sup\left\{
   \alpha\ge0:
   \log2+\mathcal P_{\mathrm I,\infty,C}(\alpha)>0
 \right\}.
\end{equation}
If \(\alpha\ne\alpha_c^{\mathrm I}(C)\), then the probability that all \(M_N\) constraints admit a
common feasible Ising configuration satisfies
\begin{equation}\label{eq:closed-set-ising-sat-unsat}
 \Pp\bigl(
   \mathcal F_{M_N,N}^{\mathrm I}(C)\ne\varnothing
 \bigr)
 \longrightarrow
 \begin{cases}
  1,&0<\alpha<\alpha_c^{\mathrm I}(C),\\
  0,&\alpha>\alpha_c^{\mathrm I}(C).
 \end{cases}
\end{equation}
If \(0<\alpha<\alpha_c^{\mathrm I}(C)\), then
\[
 \frac1N\log\#\mathcal F_{M_N,N}^{\mathrm I}(C)
 \xrightarrow{\Pp}
 \log2+\mathcal P_{\mathrm I,\infty,C}(\alpha)>0,
\]
with the convention \(\log0=-\infty\).
If \(\alpha>\alpha_c^{\mathrm I}(C)\), then
\[
 \limsup_{N\to\infty}\frac1N
 \log\Pp\bigl(
   \mathcal F_{M_N,N}^{\mathrm I}(C)\ne\varnothing
 \bigr)<0.
\]
Moreover, \(\mathfrak M_N^{\mathrm I}(C)<\infty\)
almost surely for each \(N\), and
\begin{equation}\label{eq:closed-set-random-capacity-limit}
 \frac{\mathfrak M_N^{\mathrm I}(C)}N
 \xrightarrow{\Pp}\alpha_c^{\mathrm I}(C).
\end{equation}
\end{theorem}

% \begin{theorem}[Ising capacity]
% \label{thm:closed-set-ising-capacity}
% For every tail-accessible \(C\),
% \begin{equation}\label{eq:closed-set-capacity-upper-bound}
%  0<\alpha_c^{\mathrm I}(C)\le\frac{\log2}{-\log\pi_C}<\infty.
% \end{equation}
% The variational ratio is the boundary of positive raw entropy:
% \begin{equation}\label{eq:closed-set-ising-capacity-boundary}
%  \alpha_c^{\mathrm I}(C)
%  =\sup\{\alpha\ge0:\mathcal P_{\mathrm I,\infty,C}(\alpha)>-\log2\}.
% \end{equation}
% For every \(M_N/N\to\alpha>0\) away from this threshold,
% \begin{equation}\label{eq:closed-set-ising-sat-unsat}
%  \Pp\bigl(\mathcal F_{M_N,N}^{\mathrm I}(C)
%        \ne\varnothing\bigr)
%  \longrightarrow\mathbf1_{\{\alpha<\alpha_c^{\mathrm I}(C)\}}.
% \end{equation}
% Above threshold the decay is exponential as in
% \eqref{eq:hard-ising-exponential-unsat}; at every positive subcritical load,
% the raw entropy converges as in \eqref{eq:hard-ising-raw-limit}.
% Moreover, \(\mathfrak M_N^{\mathrm I}(C)<\infty\) almost surely and
% \begin{equation}\label{eq:closed-set-random-capacity-limit}
%  \frac{\mathfrak M_N^{\mathrm I}(C)}N
%  \xrightarrow{\Pp}\alpha_c^{\mathrm I}(C).
% \end{equation}
% No assertion is made at the critical load.
% \end{theorem}

\paragraph{Nonmonotone constraints.}
The \(U\)-function constraint corresponds to
\begin{equation*}
 C_K^{\rm U}=(-\infty,-K]\cup[K,\infty),\qquad K>0.
\end{equation*}
Its soft channel is \(-\beta(K-|x|)_+\).  The two theorems give its
spherical hard volume and Ising entropy and capacity, with
\(\pi_{C_K^{\rm U}}=2\Pp(Z\ge K)\).  For \(K>K_*\simeq0.817\), this gives
the full-path variational characterization called for by the wide-regime
analysis of \cite{AubinPerkinsZdeborova}; it does not numerically evaluate
the capacity or identify an optimizing path.  More examples and the limits
of the tail-accessibility assumption are discussed in
\cref{sec:ising-capacities}.  For general \(C\), a spherical
volume formula does not imply an emptiness threshold.

\paragraph{Spherical half-spaces.}
For \(C_\theta=[\theta,\infty)\), geometry turns the hard-volume formula
into a satisfiability threshold.  Define
\begin{equation*}
 \alpha_c^{\rm S}(\theta)
 =\sup\{\alpha\ge0:
       \mathcal P_{\mathrm S,\infty,C_\theta}(\alpha)>-\infty\}.
\end{equation*}

\begin{theorem}[Spherical half-space capacity]
\label{thm:spherical-halfspace-capacity}
For every \(\theta\in\R\) and positive \(\alpha\ne\alpha_c^{\rm S}(\theta)\),
\begin{equation*}
 \Pp\bigl(\mathcal F_{\lfloor\alpha N\rfloor,N}^{\rm S}(C_\theta)
       \ne\varnothing\bigr)
 \longrightarrow\mathbf1_{\{\alpha<\alpha_c^{\rm S}(\theta)\}}.
\end{equation*}
For every $\alpha \in (0, \alpha_c^{\mathrm S})$, the hard-volume limit in
\eqref{eq:hard-spherical-volume-limit} is finite.  Moreover,
\(\mathfrak M_N^{\rm S}(C_\theta)<\infty\) almost surely and
\begin{equation}\label{eq:random-capacity-limit}
 \frac{\mathfrak M_N^{\rm S}(C_\theta)}N
 \xrightarrow{\Pp}\alpha_c^{\rm S}(\theta).
\end{equation}
No assertion is made when
\(M_N/N\to\alpha_c^{\mathrm S}(C_\theta)\).
\end{theorem}
For negative margins, the entropy formula and the capacity
characterization above establish the variational predictions of
Franz and Parisi \cite{FranzParisiJamming}.
Their further predictions concerning the structure of the minimizing
overlap distribution and the jamming critical exponents are not
addressed here.
\paragraph{Relation to three problems of Talagrand.}
At zero Ising margin, \cref{thm:closed-set-hardening,thm:closed-set-ising-capacity}
give a positive deterministic capacity, convergence of the normalized
random capacity, the full subcritical variational entropy, and
exponentially small supercritical satisfiability probability.  These
provide variational formulas for the threshold and entropy requested in
Research Problem~2.1.1
\cite[Research Problem~2.1.1, p.~152]{TalagrandBookI}.
The formulas optimize over all overlap paths; we do not identify a
constant-path optimizer or evaluate the capacity numerically.

The Gaussian-spin formula in \cref{thm:st-free energy} answers
Research Problem~3.1.2.  For every negative margin, with Talagrand's
\(\tau\) equal to our \(\theta\),
\cref{thm:closed-set-hardening,thm:spherical-halfspace-capacity} give the
full-path hard-volume limit at every density and the SAT--UNSAT transition
away from the critical density, answering the free energy and capacity
aspects of Research Problem~8.4.3
\cite[Research Problem~8.4.3, p.~25]{TalagrandBookII}.  We make no claim at
the critical load and do not determine optimizer structure or critical
exponents.

% \paragraph{Proof strategy and further applications.}
% Restoring one and two independent constraints reconstructs the derivative
% correlations.  A one-coordinate calculation for Ising spins and tangent
% integration by parts on the sphere identify the overlap equations.
% Inserting a fixed coordinate block gives the lower bound; the supporting
% equalities supplied by the overlap equations fix the sign of the
% interpolation for the upper bound.  The finite-volume interpolation
% uses finite-depth RPCs; general paths enter through the representation
% of limiting overlap arrays.  Sections~2--7 prove these statements
% and the free energy formula; Section~8 proves the large-deviation
% principle.  Section~9 proves the remaining application results and
% the radial passage used above, including the zero-temperature and
% quadratic-penalty limits, hard constraints, and capacities.

\section{Proof of the free energy formula}
\label{sec:proof-overview}
\label{sec:free energy-proof}

In this section, we outline the proof of the Parisi formula for
the soft free energy, \cref{thm:iid-channels}, and state the main
ingredients. We establish matching lower and upper bounds for
the limiting free energy. The lower bound follows the
Aizenman--Sims--Starr scheme, using increments obtained by
adding finitely many constraints and coordinates. The upper
bound uses a Guerra interpolation.

For both bounds, we add a Gaussian perturbation, independent
of the constraint vectors, that changes the expected normalized
log partition function by \(o(1)\). This perturbation enforces
the joint Ghirlanda--Guerra identities for subsequential limits
of the disorder-averaged replica laws of the spin overlaps and
cascade ancestry labels. Together with synchronization, these
identities yield the limiting-array representation used in the
cavity calculations.

Cavity in \(M\) determines the asymptotic effect of adding a fixed number of constraints.
% The overlap equation follows from a one-coordinate cavity
% calculation for Ising spins and from tangent integration by
% parts for spherical spins. 
Cavity in \(N\) bounds the
increment produced by adding a fixed block of coordinates.
The overlap equation supplies the supporting equality needed
to turn these calculations into the two variational bounds.
We state these results below and deduce
\cref{thm:iid-channels} from them at the end of this section.
Sections~\ref{sec:cavity-M}--\ref{sec:rpc-synchronization}
provide the proofs of these results and verify their
limiting-array hypotheses.

Let us first work with centered smooth channels; this assumption will be removed later.  For the cavity arguments,
assume
\begin{equation}\label{eq:iid-smooth-core-assumption}
 u_\theta(0)=0,\qquad u_\theta\in C^3(\mathbb R),\qquad
 L_j:=\sup_{\theta\in\mathbb R}\|u_\theta^{(j)}\|_\infty<\infty
 \quad(1\le j\le3).
\end{equation}
Put \(L=L_1\), fix \(\kappa\in\{\mathrm I,\mathrm S\}\), and write
\(u_k=u_{\Theta_k}\), where the marks \(\Theta_k\sim\nu\) are i.i.d.\
and independent of the Gaussian vectors and all further randomness
introduced below.
% A mark is held fixed during normalization and
% Gaussian integration, and averaged afterward in \(\E\).
We work at zero external field (meaning $b=0$) and abbreviate
\(\mathcal A=\mathcal A_\kappa\) and \(\mathcal S=\mathcal S_\kappa\).

\subsection{Conditional constraint insertion and reconstruction}
\label{sec:overview-M-cavity}

We first isolate the effect of adding a fixed number of independent constraints $u(h_k)$ to the Hamiltonian.
% We first specify the finite-volume measure and the limiting overlap
% array.  We then define the scalar quantities that describe the effect
% of inserting independent constraints.  
Throughout, the channels and
marks satisfy the centered smooth assumptions stated above, and
\(L=\sup_\theta\|u_\theta'\|_\infty\).

The key result in this section is proposition~\ref{prop:0916-rows}, which has two conclusions. First, adding a fixed number of independent constraints
preserves the limiting joint distribution of the spin overlaps
and ancestry labels
\((R_{\ell m},r_{\ell m})_{\ell,m\ge1}\), after averaging over
the disorder.
Second, under the reconstruction hypotheses (stated in the proposition),
the proposition identifies the limiting distribution of the score-overlap array $Q$~\eqref{eq:overview-composite-score}
jointly with \((R,r)\).

\paragraph{The finite-volume measure and its overlap array.}
Fix \(d\ge2\).  Let \(\mathfrak R_d\) be the RPC with masses
\(j/d\), \(1\le j<d\), viewed as a random probability measure on
the leaf set \(\mathcal L_d=\mathbb N^{d-1}\).
For two leaves \(\eta,\eta'\), let \(|\eta\wedge\eta'|\) be the
number of initial coordinates they have in common.  Define
\[
 r(\eta,\eta'):=
 \begin{cases}
  |\eta\wedge\eta'|/d,&\eta\ne\eta',\\
  1,&\eta=\eta'.
 \end{cases}
\]
We also write \(\eta\cdot\eta'=r(\eta,\eta')\). 
Let \(G_N\) be a random probability measure on
\[
 \mathcal S_N:=\Sigma_N^\kappa\times\mathcal L_d\times\mathsf E_N,
 \qquad s=(\sigma,\eta,\xi),
\]
where \(\mathsf E_N\) is any measurable space, allowing for possibly more integrated coordinates; it is a singleton when none are present.
% No product structure is assumed for \(G_N\).
Let \(\mathscr F_N\) be a sigma-algebra determining \(G_N\).
In the applications, it contains the retained constraint vectors
and channel marks, the cascade weights, and any Gaussian source
fields and perturbations used to define \(G_N\).  It excludes the
marks and Gaussian noises of constraints to be inserted.

Conditional on \(\mathscr F_N\), sample replicas
\(s^\ell=(\sigma^\ell,\eta^\ell,\xi^\ell)\) independently from
\(G_N\).  For a function \(A\) of \(n\) replicas, write
\[
 \langle A\rangle_N
 :=\int A(s^1,\ldots,s^n)\prod_{\ell=1}^nG_N(\dd s^\ell).
\]
% All replica brackets below have this meaning for their specified
% random probability measure: its defining randomness is held fixed
% inside the brackets.  The expectation \(\E\) then averages over
% that randomness, including channel marks.  Any interpolation
% parameters or perturbation amplitudes are kept fixed.
Define
\begin{equation}\label{eq:overview-J-definition}
 R_{\ell m}:=\frac{\sigma^\ell\cdot\sigma^m}{N},
 \qquad
 r_{\ell m}:=r(\eta^\ell,\eta^m).
\end{equation}
Thus \(R_{\ell\ell}=r_{\ell\ell}=1\).
The array \((r_{\ell m})\) records the shared ancestry of the sampled
leaves; this is their genealogy. The pair-ancestry distribution of \(\mathfrak R_d\) has quantile
\begin{equation}\label{eq:finite-ancestry-quantile}
 \rho_d(v)=
 \begin{cases}
  j/d,&v\in[j/d,(j+1)/d),\quad 0\le j\le d-2,\\
  1,&v\in[(d-1)/d,1).
 \end{cases}
\end{equation}
Let \(U=(U_{\ell m})\) be the limiting ancestry array of finite
RPCs with ancestry grids of vanishing mesh, with \(U_{\ell\ell}=1\).
Its law is uniquely determined by
\cite[Theorems~2.13, 2.17 and Remark~2.1]{PanchenkoBook}, and
\(U_{12}\) is uniform on \([0,1)\).
Write \(\E_U\) for expectation under this array law; its entries
are not independent.

We work along a subsequence on which the disorder-averaged replica
laws of \((R,r)\) converge.  Denoting the limiting array by
\((R^\infty,r^\infty)\), the required representation is
\begin{equation}\label{eq:overview-RJ-synchronization}
 (R^\infty_{\ell m},r^\infty_{\ell m})_{\ell<m}
 \stackrel{\mathrm d}=
 (q(U_{\ell m}),\rho_d(U_{\ell m}))_{\ell<m},
 \qquad q\in\Q,
\end{equation}
with diagonal \((1,1)\).
Here \(\Q\) consists of right-continuous nondecreasing paths
\([0,1]\to[0,1]\), with value one at \(1\), identified almost
everywhere on \([0,1)\).
The path \(q\) is deterministic at the fixed parameters.
The array \(U\) represents the limiting law; it is not a coordinate
of \(G_N\) and need not be recoverable from \((R^\infty,r^\infty)\).
Section~\ref{sec:rpc-synchronization} verifies this representation
using joint Ghirlanda--Guerra identities and synchronization.

\paragraph{The scalar response.}
We now define the scalar quantities associated with a covariance
path \(c\in\Q\).  Although \(\rho_d\) is a fixed step function,
\(c\) need not be constant on its intervals.  We therefore use
auxiliary cascades on grids adapted to \(c\).
These cascades are used only for the scalar calculation; they do
not add coordinates to \(G_N\) or change \(\mathfrak R_d\).

First suppose that \(c\) is constant on each interval
\([m_j,m_{j+1})\) of a grid
\[
 \pi=\{0=m_0<\cdots<m_{J+1}=1\}.
\]
Let \(\mathfrak R_\pi\) be the RPC with masses
\(m_1,\ldots,m_J\), with leaves indexed by \(\mathbb N^J\), and put
\[
 r^\pi(\eta,\eta'):=m_{|\eta\wedge\eta'|}.
\]
In particular, \(r^\pi(\eta,\eta)=m_J\).
For distinct replicas set
\(r^\pi_{\ell m}=r^\pi(\eta^\ell,\eta^m)\), and set
\(r^\pi_{\ell\ell}=1\) separately.
Thus, unlike the finite-volume convention, the auxiliary label of
two distinct replicas on the same leaf is \(m_J\), not one.
Write \(\E_\pi\) for the averaged law of these ancestry labels
when the leaves are sampled from the unweighted cascade
\(\mathfrak R_\pi\).

Independently of the cascade weights, take a centered Gaussian
field \((\bar X^c(\eta))_{\eta\in\mathbb N^J}\) with covariance
\[
 \E\bigl[\bar X^c(\eta)\bar X^c(\eta')\bigr]
 =c\bigl(r^\pi(\eta,\eta')\bigr).
\]
Its variance is \(c_*:=c(1-)=c(m_J)\).
Fix a channel \(u\), and write
\[
 \gamma(\dd z):=(2\pi)^{-1/2}e^{-z^2/2}\,\dd z.
\]
Define
\[
 Z_{\pi,c,u}
 :=\int_{\mathbb N^J}\int_{\mathbb R}
 e^{u(\bar X^c(\eta)+\sqrt{1-c_*}\,z)}
 \,\gamma(\dd z)\,\mathfrak R_\pi(\dd\eta).
\]
Set
\begin{equation}\label{eq:overview-completed-row-field}
 \begin{aligned}
 X^c(\eta,z)&:=\bar X^c(\eta)+\sqrt{1-c_*}\,z, \quad
 G_{\pi,c,u}(\dd\eta,\dd z)
 &:=\frac{e^{u(X^c(\eta,z))}}{Z_{\pi,c,u}}
       \mathfrak R_\pi(\dd\eta)\gamma(\dd z).
 \end{aligned}
\end{equation}
Write \(\langle\cdot\rangle_{\pi,c,u}\) for independent-replica
expectation under \(G_{\pi,c,u}\), and put
\(X^{c,\ell}:=X^c(\eta^\ell,z^\ell)\).
The scalar calculation in \cref{lem:rpc-constraint-insertion} gives
\[
 \E\log Z_{\pi,c,u}=\Phi_u(c),
\]
where \(\Phi_u\) is the channel functional from
\eqref{eq:intro-channel-pde}, and defines a step response
\(p_{c,u}\) by
\begin{equation}\label{eq:heuristic-pq-definition}
 \E\left\langle
 f(r^\pi_{12})u'(X^{c,1})u'(X^{c,2})
 \right\rangle_{\pi,c,u}
 =\sum_{j=0}^J(m_{j+1}-m_j)f(m_j)p_{c,u}(m_j)
\end{equation}
for every bounded Borel \(f:[0,1]\to\mathbb R\), with
\(p_{c,u}\) constant on each \([m_j,m_{j+1})\).
The response $p_{c, u}$ is nonnegative, nondecreasing, bounded by
\(\|u'\|_\infty^2\), and constant on every interval on which
\(c\) is constant. For step paths \(c,\widetilde c\) on a common grid, Gaussian
covariance differentiation gives
\[
 \left.\frac{\dd}{\dd s}
 \Phi_u((1-s)c+s\widetilde c)\right|_{s=0+}
 =-\frac12\int_0^1
 p_{c,u}(r)\bigl(\widetilde c(r)-c(r)\bigr)\,\dd r.
\]
Thus the two-replica derivative correlation also determines the
first variation of \(\Phi_u\) in the covariance path.

For general \(c\in\Q\), choose step paths \(c_n\to c\) in
\(L^1([0,1))\) on grids \(\pi_n\) of vanishing mesh.
The averaged finite-replica laws of the ancestry labels and
completed inputs, and the responses \(p_{c_n,u}\), converge
independently of the approximation.
Their limits define \(\E\langle\cdot\rangle_{c,u}\) and
\(p_{c,u}\), respectively; response convergence is in \(L^1\).
For example,
\[
 \E\langle\chi(X^c)\rangle_{c,u}
 :=\lim_{n\to\infty}
 \E\langle\chi(X^{c_n})\rangle_{\pi_n,c_n,u}
\]
for bounded continuous \(\chi:\mathbb R\to\mathbb R\). Section~\ref{sec:cavity-M} proves existence, independence of approximation, and continuity.

For the random channels, define
\begin{equation}\label{eq:iid-response}
 \Phi(c):=\int_{\mathbb R}\Phi_{u_\theta}(c)\,\nu(\dd\theta),
 \qquad
 p_c(r):=\int_{\mathbb R}p_{c,u_\theta}(r)\,\nu(\dd\theta).
\end{equation}
Each channel is normalized separately before averaging over its
mark.  Averaging the fixed-channel derivative identity gives, for
step paths on a common grid,
\begin{equation}\label{eq:0916-channel-derivative}
 \left.\frac{\dd}{\dd s}
 \Phi((1-s)c+s\widetilde c)\right|_{s=0+}
 =-\tfrac12\langle p_c,\widetilde c-c\rangle,
 \qquad
 \langle a,b\rangle:=\int_0^1a(r)b(r)\,\dd r.
\end{equation}
The map \(c\mapsto p_c\) is \(L^1\)-continuous, and for all
\(c,\widetilde c\in\Q\),
\begin{equation}\label{rpc:Phi-L1}
 |\Phi(c)-\Phi(\widetilde c)|
 \le\tfrac12L^2\|c-\widetilde c\|_{L^1}.
\end{equation}

\paragraph{The Gaussian fields to be inserted.}
Return to the finite-volume measure \(G_N\).
Let \(\mathsf C_N\) be an \(\mathscr F_N\)-measurable covariance
kernel on \(\mathcal S_N\), with constant diagonal
\(\mathsf C_N(s,s)=\upsilon_N\), where
\[
 0\le\upsilon_N\le1+\varepsilon_N,
 \qquad \upsilon_N\to\upsilon\in[0,1],
 \qquad \varepsilon_N\to0.
\]
Conditional on \(\mathscr F_N\), construct a centered measurable
Gaussian process \(\bar X_N\) with covariance
\[
 \E[\bar X_N(s)\bar X_N(s')\mid\mathscr F_N]
 =\mathsf C_N(s,s'),
\]
using Gaussian randomness independent of replica sampling and
of the new channel marks.  Define
\[
 X_N(s,z):=\bar X_N(s)
          +\sqrt{1+\varepsilon_N-\upsilon_N}\,z,
 \qquad z\text{ integrated against }\gamma.
\]
% Before insertion, replicas are sampled from \(G_N\otimes\gamma\).
% They use the same realization of \(\bar X_N\) and independent
% standard Gaussian integration coordinates \(z^\ell\).
% Consequently their completed Gaussian inputs have diagonal
% variance \(1+\varepsilon_N\) and off-diagonal covariance
% \(\mathsf C_N(s^\ell,s^m)\), conditional on \(\mathscr F_N\)
% and the sampled old states.
% The usual case is \(\varepsilon_N=0\) and
% \(\upsilon_N=\upsilon\).
For a deterministic \(c\in\Q\) with \(c_*\le\upsilon\), assume
that, for every fixed \(n\ge2\),
\begin{equation}\label{eq:overview-fresh-covariance-limit}
 \bigl(R_{\ell m},r_{\ell m},
       \mathsf C_N(s^\ell,s^m)\bigr)_{1\le\ell<m\le n}
 \Longrightarrow
 \bigl(q(U_{\ell m}),\rho_d(U_{\ell m}),
       c(U_{\ell m})\bigr)_{1\le\ell<m\le n}.
\end{equation}
This convergence averages over \(\mathscr F_N\) and replicas
sampled from \(G_N\), before insertion.
The diagonals of the displayed triples are
\((1,1,\upsilon_N)\to(1,1,\upsilon)\).
% After Gaussian completion, the limiting input variance is one.
Different constraints use conditionally independent copies of
\(\bar X_N\) and separate Gaussian integration coordinates.

For \(n\ge2\), set
\[
 \mathsf D_n:=([-1,1]\times[0,1])^{n\times n},
 \qquad
 A_N^{(n)}(s^1,\ldots,s^n)
 :=\bigl((R_{\ell m},r_{\ell m})\bigr)_{1\le\ell,m\le n}.
\]
Thus \(A_N^{(n)}\) depends only on the spin and leaf coordinates
of the \(n\) states. Define the limiting array block
\[
 A_\infty^{(n)}
 :=
 \bigl(a_{\ell m}\bigr)_{1\le\ell,m\le n},
 \qquad
 a_{\ell m}:=
 \begin{cases}
  (q(U_{\ell m}),\rho_d(U_{\ell m})),&\ell\ne m,\\
  (1,1),&\ell=m,
 \end{cases}
\]
where \(U\) has the array law specified in
\eqref{eq:overview-RJ-synchronization}, and \(\E_U\) denotes
expectation under that law. For a deterministic bounded continuous function
\(F:\mathsf D_n\to\mathbb R\), write
\[
 F_N(s^1,\ldots,s^n):=F(A_N^{(n)}(s^1,\ldots,s^n)).
\]
The function \(F\) is fixed as \(N\) varies and does not depend
on the disorder.
% For a bounded measurable function \(B\) of \(n\ge1\) states,
% define
% \[
%  \langle B\rangle_N
%  :=
%  \int_{\mathcal S_N^n}
%  B(s^1,\ldots,s^n)\prod_{\ell=1}^nG_N(\dd s^\ell).
% \]
The assumed convergence of the replica array under \(G_N\)
means that
\[
 \E\langle F_N\rangle_N\longrightarrow\E_UF(A_\infty^{(n)})
\]
for every \(n\ge2\) and every \(F\) as above.

\begin{proposition}[Independent constraint insertion and reconstruction]
\label{prop:0916-rows}
\label{prop:multirow-restoration}
\label{prop:overview-lower-M-cavity}
Assume the preceding hypotheses on \(G_N\), its limiting array,
and the new Gaussian fields, with deterministic limiting paths
\(q,c\).

\emph{Insertion.}
Fix \(k\ge1\).  Give the new constraints independent marks
\(\Theta_b\sim\nu\), independent of \(\mathscr F_N\) and the
Gaussian noises, and write \(u_b=u_{\Theta_b}\).
Let \(\bar X_{N,b}\), \(1\le b\le k\), be the conditionally
independent Gaussian fields specified above, and put
\[
 X_b(s,z_b^0)
 :=\bar X_{N,b}(s)
   +\sqrt{1+\varepsilon_N-\upsilon_N}\,z_b^0.
\]
Write \(z^0=(z_1^0,\ldots,z_k^0)\), and let \(\gamma\) be
standard Gaussian probability measure on \(\mathbb R\).
With all marks and Gaussian fields held fixed, define
\[
 \begin{aligned}
 Z_{N,k}^+
 &:=
 \int_{\mathcal S_N\times\mathbb R^k}
 \exp\left\{\sum_{b=1}^ku_b(X_b(s,z_b^0))\right\}
 G_N(\dd s)\prod_{b=1}^k\gamma(\dd z_b^0),\\
 G_{N,+k}(\dd s,\dd z^0)
 &:=
 \frac{\exp\{\sum_{b=1}^ku_b(X_b(s,z_b^0))\}}{Z_{N,k}^+}
 G_N(\dd s)\prod_{b=1}^k\gamma(\dd z_b^0).
 \end{aligned}
\]
For a bounded measurable function \(B\) of \(n\ge1\) augmented
states, define
\[
 \langle B\rangle_{+k}
 :=
 \int
 B\bigl((s^\ell,z^{0,\ell})_{\ell=1}^n\bigr)
 \prod_{\ell=1}^n
 G_{N,+k}(\dd s^\ell,\dd z^{0,\ell}),
\]
and write \(X_b^\ell:=X_b(s^\ell,z_b^{0,\ell})\).
% Inside these brackets, \(F_N\) means
% \(F_N(s^1,\ldots,s^n)\): it uses the spin and leaf coordinates
% of replicas sampled from \(G_{N,+k}\) and ignores their added
% coordinates \(z^{0,\ell}\).

For every fixed \(n\ge2\) and every deterministic bounded
continuous \(F:\mathsf D_n\to\mathbb R\),
\[
 \E\langle F_N\rangle_{+k}
 -\E\langle F_N\rangle_N
 \longrightarrow0.
\]
Consequently,
\[
 \E\langle F_N\rangle_{+k}\longrightarrow\E_UF_\infty.
\]
% These are statements about disorder-averaged replica laws;
% no equality of the finite-\(N\) laws is asserted.
Moreover,
\begin{equation}\label{eq:overview-k-row-increment-parameterized}
 \E\log Z_{N,k}^+\longrightarrow k\Phi(c).
\end{equation}
For each \(k\in\{1,2\}\), every \(n\ge2\), and every \(F\)
as above,
\begin{equation}\label{eq:overview-insertion-mixed-moments}
 \E\left\langle
 F_N\prod_{b=1}^k u_b'(X_b^1)u_b'(X_b^2)
 \right\rangle_{+k}
 \longrightarrow
 \E_U\bigl[F_\infty p_c(U_{12})^k\bigr].
\end{equation}

\emph{Reconstruction.}
Consider full measures \(G_{N,M}\), with \(M=M_N\to\infty\),
on states with the same spin and leaf coordinates as above.
Suppose that their disorder-averaged replica laws, including
the channel marks and constraint inputs, are invariant under
permutations of the \(M\) constraints.

For \(j=1,2\), let \(G_{N,M}^{(-j)}\) denote the normalized
measure obtained by deleting constraints \(1,\ldots,j\) and
omitting their added integration coordinates.
Suppose that this measure is independent of the deleted
marks and Gaussian noises, and that restoring these
constraints by the preceding insertion gives \(G_{N,M}\).

Assume that, along a common subsequence, the two deleted
measures satisfy the insertion hypotheses, with deterministic
paths \(q^{(-j)},c^{(-j)}\), \(j=1,2\), not assumed equal.
All limits in this reconstruction statement are taken along
that subsequence.
In each deleted system, suppose that every restored constraint
has the same off-diagonal covariance rule
\[
 \mathsf C_N(s^\ell,s^m)
 =K(R_{\ell m},r_{\ell m}),
 \qquad \ell\ne m,
\]
where \(K:[-1,1]\times[0,1]\to\mathbb R\) is a fixed Borel
function, and that
\[
 c^{(-j)}(v)
 =K(q^{(-j)}(v),\rho_d(v))
 \quad\text{for almost every }v\in[0,1),
 \qquad j=1,2.
\]
% The same conclusion holds with covariance rules \(K_N\to K\)
% uniformly, provided
% \eqref{eq:overview-fresh-covariance-limit} holds for each
% deleted measure.
Then the two deleted measures and the full measure have the
same limiting spin-overlap path:
\[
 q^{(-1)}=q^{(-2)}=:q
 \quad\text{almost everywhere on }[0,1).
\]
Their limiting covariance paths also agree:
\[
 c^{(-1)}=c^{(-2)}=:c,
 \qquad
 c(v)=K(q(v),\rho_d(v))
 \quad\text{almost everywhere on }[0,1).
\]
In particular, the full measure has limiting overlap array
\eqref{eq:overview-RJ-synchronization} with this path \(q\).

% In the remainder of the proposition, reserve the unindexed
% brackets for replicas sampled from the full measure:
% \[
%  \langle B\rangle
%  :=
%  \int B(s^1,\ldots,s^n)
%  \prod_{\ell=1}^nG_{N,M}(\dd s^\ell),
%  \qquad n\ge1.
% \]
% Here each \(s^\ell\) includes all integrated coordinates of
% the full measure. 
Finally, let \(X_b^\ell=\bar X_{N,b}(s_{-b}^\ell)
 +\sqrt{1+\varepsilon_N-\upsilon_N}\,z_b^{0,\ell}\) be the input of constraint
\(b\) at replica \(\ell\), and put \(u_b=u_{\Theta_b}\).
Define
\begin{equation}\label{eq:overview-composite-score}
 Q_{\ell m}
 :=\frac1M\sum_{b=1}^M
 u_b'(X_b^\ell)u_b'(X_b^m).
\end{equation}
Every joint subsequential limit
\((R^\infty,r^\infty,Q^\infty)\) of the disorder-averaged
replica laws under \(G_{N,M}\) satisfies
\begin{equation}\label{eq:score-recovery-L2}
 Q_{12}^\infty
 =\bar p_c(R_{12}^\infty,r_{12}^\infty)
 =p_c(U_{12})
 \quad\text{almost surely},
\end{equation}
where \(\bar p_c:[0,1]^2\to\mathbb R\) is a bounded Borel
function satisfying
\[
 \bar p_c(q(v),\rho_d(v))=p_c(v)
 \quad\text{for almost every }v\in[0,1).
\]
The final equality in \eqref{eq:score-recovery-L2} is understood
in the limiting representation
\eqref{eq:overview-RJ-synchronization}.
For every bounded Borel function
\(\chi:\mathbb R\times\mathbb R\to\mathbb R\) that is continuous
in its second argument,
\begin{equation}\label{eq:empirical-one-row-limit}
 \begin{aligned}
 \E\left\langle\left|
 \frac1M\sum_{b=1}^M\chi(\Theta_b,X_b^1)-m_c(\chi)
 \right|^2\right\rangle
 &\longrightarrow0,\\
 m_c(\chi)
 &:=
 \int_{\mathbb R}
 \E\langle\chi(\theta,X^c)\rangle_{c,u_\theta}
 \,\nu(\dd\theta).
 \end{aligned}
\end{equation}
The scalar expectation is the normalized finite-cascade limit
defined above, with the channel mark averaged afterward.
\end{proposition}

% \begin{proposition}[Independent constraint insertion and reconstruction]
% \label{prop:0916-rows}
% \label{prop:multirow-restoration}
% \label{prop:overview-lower-M-cavity}
% Assume the preceding hypotheses on \(G_N\), its limiting array,
% and the new Gaussian fields.

% \emph{Insertion.}
% Fix \(k\ge1\).  Give the new constraints independent marks
% \(\Theta_b\sim\nu\), independent of \(\mathscr F_N\) and the
% Gaussian noises, and write \(u_b=u_{\Theta_b}\).
% Let \(\bar X_{N,b}\), \(1\le b\le k\), be the conditionally
% independent Gaussian fields just specified, and put
% \[
%  X_b(s,z_b^0)
%  :=\bar X_{N,b}(s)
%    +\sqrt{1+\varepsilon_N-\upsilon_N}\,z_b^0.
% \]
% With all marks and Gaussian fields held fixed, define
% \[
%  \begin{aligned}
%  Z_{N,k}^+
%  &:=\int_{\mathcal S_N\times\mathbb R^k}
%    \exp\left\{\sum_{b=1}^ku_b(X_b(s,z_b^0))\right\}
%    G_N(\dd s)\prod_{b=1}^k\gamma(\dd z_b^0),\\
%  G_{N,+k}(\dd s,\dd z^0)
%  &:=\frac{\exp\{\sum_{b=1}^ku_b(X_b(s,z_b^0))\}}{Z_{N,k}^+}
%    G_N(\dd s)\prod_{b=1}^k\gamma(\dd z_b^0).
%  \end{aligned}
% \]
% Write \(\langle\cdot\rangle_{+k}\) for independent-replica
% expectation under \(G_{N,+k}\), and
% \(X_b^\ell:=X_b(s^\ell,z_b^{0,\ell})\). Write $\langle \cdot \rangle$ for the independent-replica expectation under $G_N$
% The old overlap--ancestry array has the same disorder-averaged
% limit after insertion, and
% \begin{equation}\label{eq:overview-k-row-increment-parameterized}
%  \E\log Z_{N,k}^+\longrightarrow k\Phi(c).
% \end{equation}
% For \(k=1,2\) and every bounded continuous function \(F\) of
% finitely many entries of the old overlap--ancestry array,
% \begin{equation}\label{eq:overview-insertion-mixed-moments}
%  \E\left\langle F\prod_{b=1}^k
%  u_b'(X_b^1)u_b'(X_b^2)\right\rangle_{+k}
%  \longrightarrow
%  \E_U\bigl[F_\infty p_c(U_{12})^k\bigr],
% \end{equation}
% where \(F_\infty\) evaluates \(F\) on
% \((q(U_{\ell m}),\rho_d(U_{\ell m}))\) off the diagonal and
% \((1,1)\) on the diagonal.
% Preservation of the old array means explicitly that
% \(\E\langle F\rangle_{+k}\to\E_UF_\infty\).

% \emph{Reconstruction.}
% Consider full measures \(G_{N,M}\), with \(M=M_N\to\infty\),
% whose disorder-averaged replica laws are invariant under
% permutations of the \(M\) marked constraints and their inputs.
% Suppose that deleting one or two constraints leaves measures
% independent of the deleted marks and Gaussian noises, and that
% restoring them by the preceding insertion gives \(G_{N,M}\).
% Assume that, along a common subsequence, both deleted measures
% satisfy the insertion hypotheses, with paths
% \(q^{(-j)},c^{(-j)}\), \(j=1,2\), not assumed equal.
% Assume also a common off-diagonal covariance rule
% \[
%  \begin{aligned}
%  \mathsf C_N(s^\ell,s^m)
%  &=K(R_{\ell m},r_{\ell m})\quad(\ell\ne m),\\
%  c^{(-j)}(v)&=K(q^{(-j)}(v),\rho_d(v))\quad\text{a.e.}
%  \end{aligned}
% \]
% where \(K:[-1,1]\times[0,1]\to\mathbb R\) is a fixed Borel
% function; the second equality is on \([0,1)\).
% The same conclusion holds with covariance rules \(K_N\to K\)
% uniformly, provided the joint convergence in
% \eqref{eq:overview-fresh-covariance-limit} holds for each deleted
% measure.
% Then the deleted and full measures have the same limiting paths
% \(q,c\).

% Write \(\langle\cdot\rangle\) for independent-replica expectation
% under the full measure \(G_{N,M}\), and let \(X_b^\ell\) denote
% the input of its \(b\)-th constraint at replica \(\ell\).
% Define
% \begin{equation}\label{eq:overview-composite-score}
%  Q_{\ell m}:=\frac1M\sum_{b=1}^M
%  u_b'(X_b^\ell)u_b'(X_b^m).
% \end{equation}
% Every joint subsequential limit \((R^\infty,r^\infty,Q^\infty)\)
% satisfies
% \begin{equation}\label{eq:score-recovery-L2}
%  Q_{12}^\infty
%  =\bar p_c(R_{12}^\infty,r_{12}^\infty)
%  =p_c(U_{12})
%  \quad\text{almost surely},
% \end{equation}
% where \(\bar p_c:[0,1]^2\to\mathbb R\) is a bounded Borel
% function satisfying
% \(\bar p_c(q(v),\rho_d(v))=p_c(v)\) for almost every
% \(v\in[0,1)\).  The last equality in
% \eqref{eq:score-recovery-L2} uses the limiting representation
% \eqref{eq:overview-RJ-synchronization}.

% For every bounded Borel function
% \(\chi:\mathbb R\times\mathbb R\to\mathbb R\) that is continuous
% in its second argument, one-replica averages satisfy
% \begin{equation}\label{eq:empirical-one-row-limit}
%  \begin{aligned}
%  \E\left\langle\left|
%   \frac1M\sum_{b=1}^M\chi(\Theta_b,X_b^1)-m_c(\chi)
%  \right|^2\right\rangle&\longrightarrow0,\\
%  m_c(\chi)
%  &:=\int_{\mathbb R}
%  \E\langle\chi(\theta,X^c)\rangle_{c,u_\theta}
%  \,\nu(\dd\theta).
%  \end{aligned}
% \end{equation}
% The scalar expectation is the normalized finite-cascade limit
% defined above, with the channel mark averaged afterward.
% \end{proposition}

In particular, define the diagonal derivative correlation by
\begin{equation}\label{eq:overview-pdiag-definition}
 p_{\rm diag}(c)
 :=\int_{\mathbb R}
 \E\langle u_\theta'(X^c)^2\rangle_{c,u_\theta}
 \,\nu(\dd\theta).
\end{equation}
Applying \eqref{eq:empirical-one-row-limit} with
\(\chi(\theta,x)=u_\theta'(x)^2\) gives
\(\E\langle|Q_{11}-p_{\rm diag}(c)|^2\rangle\to0\).
This one-replica quantity need not equal \(p_c(1-)\), which is a
two-replica correlation.  Even when two replicas share a scalar
cascade leaf, their terminal coordinates are sampled separately.
Conditional Jensen in the scalar transition law gives
\[
 0\le p_c(r)\le p_{\rm diag}(c)\le L^2
 \quad\text{for a.e. }r\in[0,1),
 \qquad p_c(1-)\le p_{\rm diag}(c).
\]

To explain the reconstruction, put
\(S_b:=u_b'(X_b^1)u_b'(X_b^2)\), so that
\(Q_{12}=M^{-1}\sum_bS_b\).
Constraint exchangeability gives
\begin{equation}\label{eq:overview-exchangeable-score-moments}
 \begin{aligned}
 \E\langle FQ_{12}\rangle
 &=\E\langle FS_1\rangle,\\
 \E\langle FQ_{12}^2\rangle
 &=\frac{M-1}{M}\E\langle FS_1S_2\rangle
   +\frac1M\E\langle FS_1^2\rangle.
 \end{aligned}
\end{equation}
The last term is bounded in absolute value by
\(\|F\|_\infty L^4/M\).
Deleting and restoring one constraint in the first identity and
two in the second therefore gives
\begin{align}
 \E\langle FQ_{12}\rangle
 &\longrightarrow\E_U[F_\infty p_c(U_{12})],
 \label{eq:overview-Q-first-moment}\\
 \E\langle FQ_{12}^2\rangle
 &\longrightarrow\E_U[F_\infty p_c(U_{12})^2].
 \label{eq:overview-Q-second-moment}
\end{align}
The common covariance rule and the plateau property make
\(p_c(U_{12})\) a Borel function of
\((R_{12}^\infty,r_{12}^\infty)\).
These identities determine the conditional first and second
moments of \(Q_{12}^\infty\) given the limiting overlap--ancestry
array.  Its conditional variance is zero, proving
\eqref{eq:score-recovery-L2}.
When several constraints are inserted together, their weights
share the random normalizer \(Z_{N,k}^+\).  These conclusions
therefore use the normalized RPC calculation, not independence
of the unnormalized constraint weights alone.
Section~\ref{sec:constraint-proof} proves the insertion and
reconstruction statements, including agreement of the deleted
and full limiting paths.

% \subsection{Conditional constraint insertion and reconstruction}
% \label{sec:overview-M-cavity}

% Fix \(d\ge2\).  Let \(\mathfrak R_d\) be the finite-depth RPC with
% masses \(j/d\), \(1\le j<d\), and ancestry labels
% \(0,1/d,\ldots,(d-2)/d,1\); the label is one on a common leaf.
% For replicas \((\sigma^\ell,\eta^\ell)_{\ell \geq 1}\), write
% \begin{equation*}
%  R_{\ell m}=N^{-1}\sigma^\ell\cdot\sigma^m,\qquad
%  r_{\ell m}=\eta^\ell\cdot\eta^m.
% \end{equation*}
% The diagonal is \((R_{\ell\ell},r_{\ell\ell})=(1,1)\).
% The off-diagonal ancestry quantile is
% \begin{equation*}
%  \rho_d(u)=
%  \begin{cases}
%  j/d,&u\in[j/d,(j+1)/d),\quad 0\le j\le d-2,\\
%  1,&u\in[(d-1)/d,1).
%  \end{cases}
% \end{equation*}
% The conditional limiting-array hypothesis is
% \begin{equation}\label{eq:overview-RJ-synchronization}
%  (R_{\ell m},r_{\ell m})_{\ell<m}
%  \stackrel{d}=
%  (q(U_{\ell m}),\rho_d(U_{\ell m}))_{\ell<m},\qquad q\in\Q,
% \end{equation}
% where \(U\) is the limiting genealogy array of finite RPCs whose
% ancestry grids have mesh tending to zero.  Its law is unique by
% \cite[Theorems~2.13, 2.17 and Remark~2.1]{PanchenkoBook}; \(U_{12}\) is
% uniform on \([0,1)\).  The variable \(U\) represents
% the limiting law; it is not part of the finite-volume measure and
% need not be measurable from \((R,r)\).
% Section~\ref{sec:rpc-synchronization} verifies this hypothesis by joint
% GG identities and synchronization.

% % All scalar insertion measures below are defined on finite cascades.
% For a grid \(\pi=\{0=m_0<\cdots<m_{J+1}=1\}\), let
% \(\mathfrak R_\pi\) be the RPC with masses \(m_1,\ldots,m_J\).
% Its off-diagonal genealogy label \(r^\pi\) is the left endpoint of
% the common ancestry interval, including \(m_J\) for a same-leaf pair;
% the true replica diagonal is kept separately at one.  Write
% \(\E_\pi\) for its averaged genealogy law and \(\E_U\) for the
% limiting array law above.

% For a step path \(c\) on \(\pi\), let \(\bar X^c(\eta)\) be an independent Gaussian process with covariance
% \(c(r^\pi_{\ell m})\) between distinct replicas and variance
% \(c_*:=c(1-)\), with Gaussian increments independent of the weights.
% For a fixed channel \(u\), with \(\gamma\) denoting standard Gaussian
% measure, set
% \begin{equation}\label{eq:overview-completed-row-field}
%  X^c(\eta,z)=\bar X^c(\eta)+\sqrt{1-c_*}\,z,\qquad
%  G_{\pi,c,u}(\dd\eta,\dd z)
%  =\frac{e^{u(X^c(\eta,z))}}{Z_{\pi,c,u}}
%        \mathfrak R_\pi(\dd\eta)\gamma(\dd z).
% \end{equation}
% Replicas sampled from this measure define the response \(p_{c,u}\) by
% \begin{equation*}
%  \E\langle f(r^\pi_{12})u'(X^{c,1})u'(X^{c,2})\rangle_{\pi,c,u}
%  =\sum_{j=0}^J(m_{j+1}-m_j)f(m_j)p_{c,u}(m_j).
% \end{equation*}
% Here brackets denote conditional replica expectation under
% \(G_{\pi,c,u}\).  The response $p_{c, u}$ is nonnegative, nondecreasing, bounded
% by \(\|u'\|_\infty^2\), and constant on plateaus of \(c\).
% For arbitrary paths, scalar replica expectations and responses are
% the limits along finite grids of vanishing mesh.  Section~\ref{sec:cavity-M}
% proves that these limits exist and are independent of the approximation. A calculation of the directional derivative gives an equivalent characterization:
% \[
%     \left.\frac{\dd}{\dd s}\Phi_u((1-s)c+s\widetilde c)\right|_{s=0+}
%  =-\tfrac12\langle p_{c, u},\widetilde c-c\rangle.
% \]
% For the random channels, use
% \begin{equation}\label{eq:iid-response}
%  \Phi(c)=\int_{\mathbb R}\Phi_{u_\theta}(c)\,\nu(\dd\theta),\qquad
%  p_c(r)=\int_{\mathbb R}p_{c,u_\theta}(r)\,\nu(\dd\theta).
% \end{equation}
% Thus each channel is normalized separately before averaging over
% \(\theta\).  The same calculation gives, for grid paths,
% \begin{equation}\label{eq:0916-channel-derivative}
%  \left.\frac{\dd}{\dd s}\Phi((1-s)c+s\widetilde c)\right|_{s=0+}
%  =-\tfrac12\langle p_c,\widetilde c-c\rangle.
% \end{equation}
% The map \(c\mapsto p_c\) is \(L^1\)-continuous, and
% \begin{equation}\label{rpc:Phi-L1}
%  |\Phi(c)-\Phi(\widetilde c)|
%  \le\tfrac12L^2\|c-\widetilde c\|_{L^1}.
% \end{equation}
% Let \(\mathcal L_d=\operatorname{supp}\mathfrak R_d\) be the
% leaf set of the finite cascade, with leaves indexed by
% \(\mathbb N^{d-1}\). Given a random probability measure \(G_N\) on
% \[
%  \mathcal S_N
%  :=\Sigma_N^\kappa\times\mathcal L_d\times\mathsf E_N,
%  \qquad s=(\sigma,\eta,\xi),
% \]
% and its genealogy, take an independent
% centered Gaussian process \(\bar X_N(s)\) with conditional covariance
% \(\mathsf C_N(s,s')\) and constant variance \(\upsilon\in[0,1]\).
% Complete it to variance one by
% \(X_N(s,z)=\bar X_N(s)+\sqrt{1-\upsilon}\,z\), with \(z\) integrated
% against \(\gamma\).  Different insertions use independent Gaussian
% noises.  The required joint covariance limit is
% \begin{equation}\label{eq:overview-fresh-covariance-limit}
%  (R_{\ell m},r_{\ell m},\mathsf C_N(s^\ell,s^m))
%  \Longrightarrow(q(U_{\ell m}),\rho_d(U_{\ell m}),c(U_{\ell m}))
%  \quad(\ell\ne m),
% \end{equation}
% for every finite replicated array, with diagonal \((1,1,\upsilon)\) and
% \(c_*\le\upsilon\).  We also allow the variance of \(\bar X_N\) to
% depend on \(N\), with \(\upsilon_N\to\upsilon\), and the completed
% variance to equal \(1+\varepsilon_N\), where \(\varepsilon_N\to0\).
% Provided \(0\le\upsilon_N\le1+\varepsilon_N\), set
% \[
%  X_N(s,z)=\bar X_N(s)+\sqrt{1+\varepsilon_N-\upsilon_N}\,z.
% \]
% The completed variance then converges to one.

% \begin{proposition}[Independent constraint insertion and reconstruction]
% \label{prop:0916-rows}
% \label{prop:multirow-restoration}
% \label{prop:overview-lower-M-cavity}
% Let \(G_N\) be a random probability measure on
% \(\Sigma_N^\kappa\times\operatorname{supp}\mathfrak R_d\), possibly with
% additional integrated variables, whose full replicated \((R,r)\)-array converges to
% \eqref{eq:overview-RJ-synchronization}, with the RPC ancestry law.
% Let the fresh Gaussian constraints satisfy
% \eqref{eq:overview-fresh-covariance-limit} and the preceding convention.

% \emph{Insertion.}
% Give each fresh constraint an independent mark \(\Theta_b\sim\nu\),
% also independent of \(G_N\).  Insert any fixed number \(k\) of them by multiplying
% \(G_N\prod_{b=1}^k\gamma(\dd z_b^0)\) by
% \(\exp\{\sum_{b=1}^ku_b(X_b)\}\) and normalizing with the marks fixed.  Write
% \(Z_{N,k}^+\) for the normalizing integral and
% \(\langle\cdot\rangle_{+k}\) for replica expectation after insertion.
% The old overlap--ancestry array has the same averaged limit, and
% \begin{equation*}
%  \mathbb E\log Z_{N,k}^+\longrightarrow k\Phi(c).
% \end{equation*}
% For \(k=1,2\) and every bounded continuous function \(F\) of finitely
% many entries of that array,
% \begin{equation}\label{eq:overview-insertion-mixed-moments}
%  \mathbb E\left\langle F\prod_{b=1}^k
%  u_b'(X_b^1)u_b'(X_b^2)\right\rangle_{+k}
%  \longrightarrow
%  \mathbb E_U[F_\infty p_c(U_{12})^k],
% \end{equation}
% where \(F_\infty\) evaluates \(F\) on the limiting array.

% \emph{Reconstruction.}
% Consider a family invariant under permutations of its \(M\) marked constraints,
% with \(M\to\infty\).  Suppose that its one- and two-constraint deletions
% satisfy the insertion hypotheses, with a common off-diagonal rule
% \(\mathsf C_N(s,s')=K(R(s,s'),r(s,s'))\) whose limiting path is
% \(c(u)=K(q(u),\rho_d(u))\), for one fixed Borel function \(K\).
% The same conclusion holds for rules \(K_N\) converging uniformly to
% \(K\), provided the joint covariance convergence
% \eqref{eq:overview-fresh-covariance-limit} holds.  Suppose that
% restoring the deleted constraints gives the full measure.  Their
% limiting paths need not be assumed equal.  The deleted and full measures then
% have the same limiting paths \(q,c\).  For
% \begin{equation}\label{eq:overview-composite-score}
%  Q_{\ell m}=\frac1M\sum_{k=1}^M u_k'(X_k^\ell)u_k'(X_k^m),
% \end{equation}
% every joint limiting array satisfies
% \begin{equation*}
%  Q_{12}=\bar p_c(R_{12},r_{12})=p_c(U_{12})\quad\text{almost surely},
% \end{equation*}
% where \(\bar p_c\) is a bounded Borel function satisfying
% \(\bar p_c(q(u),\rho_d(u))=p_c(u)\) almost everywhere.
% The last equality is understood in the limiting representation.
% For every bounded Borel \(\chi(\theta,x)\), continuous in \(x\),
% one-replica averages also satisfy
% \begin{equation}\label{eq:empirical-one-row-limit}
%  \begin{aligned}
%  \mathbb E\left\langle\left|
%  \frac1M\sum_{k=1}^M\chi(\Theta_k,X_k)-m_c(\chi)\right|^2\right\rangle&\longrightarrow0,\\
%  m_c(\chi)&=\int_{\mathbb R}\E\langle\chi(\theta,X^c)\rangle_{c,u_\theta}\,\nu(\dd\theta).
%  \end{aligned}
% \end{equation}
% The scalar expectation includes the Gaussian and cascade randomness,
% with the channel mark averaged after normalization.
% \end{proposition}

% In particular, define the diagonal derivative correlation by
% \begin{equation*}
%  p_{\rm diag}(c)=\int_{\mathbb R}\E\langle u_\theta'(X^c)^2\rangle_{c,u_\theta}\,\nu(\dd\theta).
% \end{equation*}
% It need not equal \(p_c(1-)\).  The latter is an off-diagonal
% two-replica correlation, even at the last ancestry level.  Conditional
% Jensen in the scalar transition law gives
% \(p_c(r)\le p_{\rm diag}(c)\) almost everywhere, hence
% \(p_c(1-)\le p_{\rm diag}(c)\).

% One insertion determines the conditional mean of \(Q_{12}\);
% two independent insertions determine its conditional second moment.
% The second limit is the square of the first, so no conditional
% fluctuation remains.  Restoring deleted constraints also preserves the
% overlap path.  Both conclusions use the normalized RPC calculation:
% the inserted constraints share a random denominator.  Section~\ref{sec:constraint-proof}
% gives the moment calculation.

\subsection{Identifying the overlap equation}
\label{sec:overview-N-cavity}

In this section we identify a self-consistency equation the order parameter $q$, corresponding to spin overlap $R_{12}$, must satisfy. Since this equation is used in the upper bound as well, we specify the interpolation and the perturbation before stating the
overlap equation.  Fix \(d\ge2\), \(t\in[0,1]\), a trial path
\(q^0\in\Q_d\), and a bounded nonnegative nondecreasing path
\(a^0\) on the same grid.  Here \(\Q_d\) consists of paths constant
on \([j/d,(j+1)/d)\).  Write \(b_*=b(1-)\) for a path's last
off-diagonal value and define its lookup on the finite ancestry labels by
\begin{equation*}
 b^\sharp(\rho_d(u))=b(u)\quad\text{for a.e. }u\in[0,1).
\end{equation*}
In particular \(b^\sharp(1)=b_*\).  Let \(\bar Y_k\) be independent
Gaussian fields on the cascade, with covariance
\(q^{0,\sharp}(r_{\eta\eta'})\) and variance \(q_*^0\), and put
\begin{equation*}
 Y_k(\eta,z_k^0)=\bar Y_k(\eta)+\sqrt{1-q_*^0}\,z_k^0,
 \qquad X_{k,t}=\sqrt t\,h_k(\sigma)+\sqrt{1-t}\,Y_k(\eta,z_k^0).
\end{equation*}
The variables \(z_k^0\) are integrated against standard Gaussian measure.
Independently, let \(W^{a^0}=(W_i^{a^0})_{i\le N}\) have independent
spatial coordinates of covariance \(a^{0,\sharp}(r_{\eta\eta'})\)
and variance \(a_*^0\), and set \(W_t=\sqrt{1-t}\,W^{a^0}\).
All Gaussian ancestor marks are independent of the cascade weights.

Use a centered Gaussian perturbation \(V_N(\sigma,\eta)\), independent
of the channel marks, constraint vectors, and source fields, whose law
does not depend on \(M\) or \(t\).  It has amplitudes
\(\mathbf x\in[1,2]^{\mathbb N}\); \(\E_{\mathbf x}\) averages their
independent uniform laws, while \(\E\) holds them fixed.
Use the same realization for every constraint count.
Section~\ref{sec:rpc-synchronization} constructs this perturbation and
proves the properties stated below.

Define
\begin{equation}\label{eq:overview-Guerra-Hamiltonian}
 \mathcal H_{N,M,t}=\sum_{k\le M}u_k(X_{k,t})+W_t\cdot\sigma
       -\frac N2(1-t)a_*^0+V_N,
\end{equation}
and let \(Z_{N,M,t}\) normalize
\begin{equation}\label{eq:overview-model-measure}
 G_{N,M,t}(\dd\sigma,\dd\eta,\dd\boldsymbol z^0)
 =\frac{e^{\mathcal H_{N,M,t}}}{Z_{N,M,t}}
       \mu_N^\kappa(\dd\sigma)\mathfrak R_d(\dd\eta)
       \gamma^{\otimes M}(\dd\boldsymbol z^0).
\end{equation}
Adding \(V_N\) changes normalized expected pressure by \(o(1)\),
uniformly in the amplitudes and time.  At \(t=1\), removing it gives
the original zero-field model.  For every bounded Borel \(f\), the
finite-volume ancestry marginal is exactly
\begin{equation}\label{eq:overview-pair-ancestry}
 \E\langle f(r_{12})\rangle_{N,M,t}
 =\int_0^1f(\rho_d(s))\,\dd s.
\end{equation}
Deleting a constraint leaves its Gaussian noises independent of the
remaining measure.  Its covariance on restoration is
\begin{equation}\label{eq:overview-inserted-constraint-covariance}
 tR_{\ell m}+(1-t)q^{0,\sharp}(r_{\ell m})\quad(\ell\ne m),
 \qquad \operatorname{Var}(X_{k,t}^\ell)=1.
\end{equation}
Thus, if the spin path is \(q\), constraint reconstruction uses
\begin{equation*}
 c=(1-t)q^0+tq.
\end{equation*}

Here are the two scalar responses.  For a step source \(a\), let
\(w^a\) be a Gaussian field on \(\mathfrak R_\pi\) with covariance
\(a(r^\pi)\) and variance \(a_*\).  For Ising spins, set
\begin{equation*}
 \mathbb P_{\pi,a}^{\rm I}(\dd\eta,\dd\tau)
 =\frac{e^{\tau w^a(\eta)-a_*/2}}{Z_{\pi,a}^{\rm I}}
       \mathfrak R_\pi(\dd\eta)\frac{\delta_{-1}+\delta_1}{2}(\dd\tau).
\end{equation*}
Its two-replica correlation defines \(\widehat q_a^{\rm I}\); for
general \(a\), take the finite-cascade limit established in Section~4:
\begin{equation*}
 \mathbb E\langle f(U_{12})\tau^1\tau^2\rangle_a^{\rm I}
 =\int_0^1 f(r)\widehat q_a^{\rm I}(r)\,\dd r.
\end{equation*}
For the sphere, the response has an explicit integral formula.  Put
\(x_a(s)=\inf\{r\in[0,1):a(r)\ge s\}\), with \(\inf\varnothing=1\), and set
\begin{equation*}
 \beta_-(a)=a_*-\int_0^1a(r)\,\dd r,\qquad\mathfrak D_a=(\beta_-(a),\infty).
\end{equation*}
For \(\beta\in\mathfrak D_a\), define
\begin{align}
 \pi_{a,\beta}(s)&=\beta-\int_s^{a_*}x_a(v)\,\dd v,
 \notag\\
 q^{\rm co}_{a,\beta}(r)&=\int_0^{a(r)}\frac{\dd s}{\pi_{a,\beta}(s)^2},
 \qquad V_a(\beta)=\frac1\beta+\int_0^{a_*}\frac{\dd s}{\pi_{a,\beta}(s)^2}.
 \label{eq:general-response-and-variance}
\end{align}
Section~4 proves that \(V_a(\beta)=1\) has a unique solution
\(\bar\beta(a)\in\mathfrak D_a\).  Define
\begin{equation*}
 \widehat q_a^{\rm S}(r)=q^{\rm co}_{a,\bar\beta(a)}(r)\quad(r<1),
 \qquad \widehat q_a^{\rm S}(1)=1.
\end{equation*}
The spherical spin transform is
\[
 \mathcal S_{\rm S}(a)=\frac12\left\{
 \bar\beta(a)-1-a_*-\log\bar\beta(a)
 +\int_0^{a_*}\frac{\dd s}{\pi_{a,\bar\beta(a)}(s)}\right\}.
\]
Both responses are nonnegative, nondecreasing, bounded by one, and
constant on plateaus of \(a\).

\begin{proposition}[Scalar spin identities]
\label{prop:scalar-proof-input}
For either prior, every \(q\in\Q\), and every \(a\in\Pb\),
\begin{align*}
 \mathcal A_\kappa(q)
 &=\sup_{b\in\Pb}\{\mathcal S_\kappa(b)
                         +\tfrac12\langle b,q\rangle\},
 \\
 \mathcal A_\kappa(\widehat q_a^\kappa)
 &=\mathcal S_\kappa(a)
       +\tfrac12\langle a,\widehat q_a^\kappa\rangle<\infty.
\end{align*}
For a fixed step source \(a\) on \(\pi\), let \(W^a\) have
\(N\) independent coordinates distributed as \(w^a\).  Then
\begin{equation}\label{eq:overview-spin-endpoint}
 \frac1N\E\log\int\!\int
 e^{W^a(\eta)\cdot\sigma-Na_*/2}
 \,\mu_N^\kappa(\dd\sigma)\mathfrak R_\pi(\dd\eta)
 \longrightarrow\mathcal S_\kappa(a).
\end{equation}
For Ising spins, equality holds for every \(N\).
\end{proposition}
Section~\ref{sec:spin-laws} proves these identities; the Ising
conjugate formula is already the definition of \(\mathcal A_{\rm I}\).

Call \((q,a)\) a supporting pair when
\(\mathcal A(q)=\mathcal S(a)+\langle a,q\rangle/2\).
Applying the conjugate inequality to each of two supporting pairs
and adding gives
\begin{equation*}
 \langle q_1-q_0,a_1-a_0\rangle\ge0.
\end{equation*}
This is the sign used in the upper bound.  The Ising supporting equality
uses the Chen--Issa--Mourrat concavity theorem~\cite{CIM}; the spherical
one follows from a Gaussian integral.  Neither requires convexity of
the channel functional \(\Phi\).

For the next two propositions, retain fixed amplitudes and a joint
array-convergent subsequence for the full measures and their one- and
two-constraint deletions.  For Ising spins also include the system
obtained by deleting one coordinate from dimension \(D\), retaining
the denominator \(\sqrt D\) and using \(V_{D-1}\), and its constraint
deletions.

Section~\ref{sec:rpc-synchronization} supplies the required limits as
follows.  From any dimension sequence, one may select a common
subsequence for any fixed finite collection of constraint-count
changes and the preceding deletions.  For almost every amplitude
vector at \(t=1\), and for almost every amplitude--time pair in the
interpolation, every further joint array limit has the form
\eqref{eq:overview-RJ-synchronization}.  Compactness supplies such
refinements at each retained parameter.  The paths of different
systems are not assumed to agree.

\begin{proposition}[The overlap equation]
\label{prop:0916-coordinate}
\label{prop:coordinate-cavity-characteristic}
Let \(M/N\to\alpha\in(0,\infty)\), and let \(q\) be the limiting
spin path of \(G_{N,M,t}\).  Set
\begin{equation*}
 c=(1-t)q^0+tq,\qquad a=(1-t)a^0+t\alpha p_c.
\end{equation*}
Then
\begin{equation*}
 q=\widehat q_a\quad\text{almost everywhere},\qquad
 \mathcal A(q)=\mathcal S(a)+\frac12\langle q,a\rangle.
\end{equation*}
For spherical spins, the precision is identified by
\begin{equation*}
 \bar\beta(a)=1+a_*-\langle q,a\rangle=\frac1{1-q(1-)}.
\end{equation*}
\end{proposition}

The two priors lead to different short proofs.  For Ising spins,
retaining \(\sqrt D\) when deleting a coordinate leaves a Gaussian
linear term and a quadratic term independent of that coordinate.
Constraint restoration concentrates this term, and coordinate
exchangeability identifies the response.  On the sphere, integration
by parts in the tangent direction \(\sigma^2-R_{12}\sigma^1\)
cancels the second-derivative and diagonal terms exactly.  The remaining
identity involves only \(Q_{12}\), so constraint reconstruction and
the three-replica RPC law determine \(q\).  Section~5 gives both proofs.

\subsection{The logarithmic increment from a fixed block}

At \(t=1\), with the amplitudes fixed, write
\(A_N(M)=\mathbb E\log Z_{N,M,1}\).  The next proposition computes
the partition-function ratio needed for the lower bound.

\begin{proposition}[Inserting a fixed block of coordinates]
\label{prop:log-block-insertion}
\label{prop:overview-lower-N-cavity}
Fix \(K\ge1\).  If \(M_N/N\to\alpha\) and \(q\) is the limiting
spin path in dimension \(N\) under the preceding hypotheses, then
\begin{equation*}
 \liminf_N\{A_{N+K}(M_N)-A_N(M_N)\}\ge K\mathcal A(q).
\end{equation*}
No array limit at dimension \(N+K\) is required.  For each compact
\(I\subset(0,\infty)\), there is \(C_{K,I}<\infty\) such that,
uniformly in the amplitudes and sufficiently large \(N\),
\begin{equation}\label{eq:log-block-uniform-lower}
 A_{N+K}(M)-A_N(M)\ge-C_{K,I}\qquad(M/N\in I).
\end{equation}
\end{proposition}

All \(K\) coordinates are inserted together.  The expansion uses
\(Q\) and the single coefficient
\(M^{-1}\sum_k\{u_k''(h_k)-h_k u_k'(h_k)\}\), reconstructed before
insertion.  Their qualitative limits suffice because their multipliers
are bounded for a fixed coordinate cutoff.  The overlap equation has
already fixed the scalar precision.  Consequently the spherical proof
passes only expected logarithms to the limit; it does not need a
limit theorem for the inserted coordinates or their second moments.
The lower bound below combines this proposition with constraint
insertion at the same dimension.  The upper bound uses only the
supporting equality in \cref{prop:0916-coordinate}.
\subsection{Proof of the theorem}
\label{sec:completion}
\label{sec:remove-smoothness}

Two elementary bounds complete the stated inputs.  Independent row
insertion and Jensen's inequality give, uniformly in the amplitudes,
\begin{equation}\label{eq:universal-one-row-bound}
 |A_N(M+1)-A_N(M)|\le C_{\rm row},\qquad
 |\Phi(q)|\le C_{\rm row},\qquad
 C_{\rm row}:=\E e^{L|Z|},\quad Z\sim N(0,1).
\end{equation}
The same centered-channel bounds, Gaussian Poincar\'e, and resampling
one mark give, for the original model and \(M/N\) in a fixed compact
subset of \((0,\infty)\),
\begin{equation}\label{eq:unperturbed-iid-variance}
 \Var\!\left(N^{-1}\log Z_{N,M}^{\kappa,\nu}\right)\le C/N.
\end{equation}
The estimates are proved in Sections~\ref{sec:cavity-M} and
\ref{sec:rpc-synchronization}, respectively.

\begin{proof}[Proof of \cref{thm:iid-channels}]
First assume the centered smooth hypotheses
\eqref{eq:iid-smooth-core-assumption}, and abbreviate
\(\Phi=\E_\nu\Phi_{u_\Theta}\), \(\mathcal A=\mathcal A_\kappa\),
and \(\mathcal S=\mathcal S_\kappa\).  Write
\[
 \mathcal P_\kappa(\alpha)
 :=\inf_{q\in\Q}\{\mathcal A(q)+\alpha\Phi(q)\},\qquad
 F_{N,\kappa}:=N^{-1}\log Z_{N,M_N}^{\kappa,\nu}.
\]

\emph{Lower bound.}
For the Aizenman--Sims--Starr lower bound~\cite{ASS}, fix \(d=2\),
use the interpolation at \(t=1\), and set
\begin{equation*}
 m_N=2+\lfloor\alpha N\rfloor,\qquad
 A_N(M)=\E\log Z_{N,M,1},\qquad
 \overline A_N=\E_{\mathbf x}A_N(m_N),
\end{equation*}
where \(A_N(M)\) keeps the amplitudes fixed.  Fix \(K\ge1\).
The number \(k_{N,K}=m_{N+K}-m_N\) satisfies
\begin{equation*}
 |k_{N,K}-\alpha K|\le1.
\end{equation*}
Insert these constraints first and then all \(K\) coordinates:
\begin{equation*}
 \begin{aligned}
 \Delta_{N,K}(\mathbf x)
 &:=A_{N+K}(m_{N+K})-A_N(m_N)\\
 &=A_N(m_{N+K})-A_N(m_N)
   +A_{N+K}(m_{N+K})-A_N(m_{N+K}).
 \end{aligned}
\end{equation*}
From any subsequence, select one on which \(k_{N,K}=k\) is constant
and apply the common selection stated above
to both constraint counts and all deletions required by the cavity
propositions.  At almost every amplitude vector, take a joint
array-convergent refinement.  Proposition~\ref{prop:0916-rows}
identifies the same spin path \(q\) at the two counts and gives
\(A_N(m_{N+K})-A_N(m_N)\to k\Phi(q)\).
Proposition~\ref{prop:log-block-insertion}, applied at the second count,
gives a lower limit \(K\mathcal A(q)\) for the remaining term.  Hence
\[
 \liminf_N\Delta_{N,K}(\mathbf x)
 \ge K\{\mathcal A(q)+\alpha\Phi(q)\}
       +(k-\alpha K)\Phi(q)
 \ge K\mathcal P_\kappa(\alpha)-C_{\rm row}.
\]
The bound is independent of \(q\), so a refinement of any subsequence
realizing a smaller lower limit would give a contradiction.
The row bound \eqref{eq:universal-one-row-bound} and the uniform block
bound \eqref{eq:log-block-uniform-lower} provide a lower envelope for
\(\Delta_{N,K}\) independent of \(N,\mathbf x\).  Fatou's lemma gives
\begin{equation*}
 \liminf_N(\overline A_{N+K}-\overline A_N)
 \ge K\mathcal P_\kappa(\alpha)-C_{\rm row}.
\end{equation*}
Telescope at fixed \(K\) in each residue class modulo \(K\), divide by
the dimension, and let \(K\to\infty\).  This yields
\(\liminf_N\overline A_N/N\ge\mathcal P_\kappa(\alpha)\).
Removing the perturbation costs \(o(1)\) in normalized pressure.
Since \(|M_N-m_N|=o(N)\), the row bound transfers the result to every
\(M_N/N\to\alpha\), proving \(\liminf_N\E F_{N,\kappa}\ge\mathcal P_\kappa(\alpha)\).

\emph{Upper bound.}
Fix \(d\ge2\).  The conjugate formula in
\cref{prop:scalar-proof-input} makes \(\mathcal A\) convex
and lower semicontinuous.  Compactness of \(\Q_d\), continuity of
\(\Phi\), and the finite competitor \(\widehat q_0\in\Q_d\) give a
minimizer \(q^0\) of \(\mathcal A+\alpha\Phi\) on \(\Q_d\).  Put
\begin{equation*}
 a^0=\alpha p_{q^0}.
\end{equation*}
Minimality along \((1-s)q^0+sq\), the channel derivative
\eqref{eq:0916-channel-derivative}, and the convex secant inequality give
\begin{equation}\label{eq:guerra-relative-support}
 \mathcal A(q)-\tfrac12\langle a^0,q\rangle
 \ge\mathcal A(q^0)-\tfrac12\langle a^0,q^0\rangle,
 \qquad q\in\Q_d.
\end{equation}
For infinite \(\mathcal A(q)\) this is automatic.  The plateau properties
put \(\bar q:=\widehat q_{a^0}\) in \(\Q_d\).  The conjugate inequality,
\eqref{eq:guerra-relative-support}, and scalar supporting equality imply
\begin{equation*}
 \mathcal S(a^0)
 \le\mathcal A(q^0)-\tfrac12\langle a^0,q^0\rangle
 \le\mathcal A(\bar q)-\tfrac12\langle a^0,\bar q\rangle
 =\mathcal S(a^0).
\end{equation*}
Thus \((q^0,a^0)\) is a supporting pair, including at boundary
minimizers; no identity \(q^0=\widehat q_{a^0}\) is needed.

Use \eqref{eq:overview-Guerra-Hamiltonian} with these trial paths and set
\(\alpha_N=M_N/N\), \(\varphi_N^{\mathbf x}(t)=N^{-1}\E\log Z_{N,M_N,t}\),
and \(\overline\varphi_N(t)=\E_{\mathbf x}\varphi_N^{\mathbf x}(t)\).
Write \(Q_{t,\ell m}\) for \eqref{eq:overview-composite-score} with inputs
\(X_{k,t}\), and let brackets below denote replicas of \(G_{N,M_N,t}\).
Define
\begin{equation*}
 \mathscr R_N^{\mathbf x}(t)
 :=\E\langle(R_{12}-q^{0,\sharp}(r_{12}))
             (\alpha_NQ_{t,12}-a^{0,\sharp}(r_{12}))\rangle_{t,\mathbf x}.
\end{equation*}
Gaussian integration by parts, conditional on the channel marks,
gives the two contributions
\[
 -\frac{\alpha_N}{2}\E\langle(R_{12}-q^{0,\sharp}(r_{12}))Q_{t,12}\rangle_{t,\mathbf x},
 \qquad
 \frac12\E\langle a^{0,\sharp}(r_{12})R_{12}\rangle_{t,\mathbf x}.
\]
The row diagonals cancel because both fields have variance one; the
source diagonal cancels the derivative of \(-N(1-t)a_*^0/2\).
The exact ancestry marginal \eqref{eq:overview-pair-ancestry} then gives
\begin{equation}\label{eq:guerra-static-sum-rule-pointwise}
 (\varphi_N^{\mathbf x})'(t)
 =\tfrac12\langle a^0,q^0\rangle-\tfrac12\mathscr R_N^{\mathbf x}(t).
\end{equation}
Apply the common selection on the amplitude--time product space.
For almost every \((\mathbf x,t)\), \(0<t<1\), any further subsequence
has a joint array-convergent refinement for the full and required
deleted systems.  If its spin path is \(q_t\), the cavity propositions give
\begin{equation*}
 c_t=(1-t)q^0+tq_t,\qquad
 a_t=(1-t)a^0+t\alpha p_{c_t},\qquad q_t=\widehat q_{a_t}.
\end{equation*}
Joint reconstruction of \((R,r,Q)\) passes the bounded remainder to
\(\mathscr R_t=\langle q_t-q^0,\alpha p_{c_t}-a^0\rangle\);
the trial lookups are continuous on the finite ancestry set.
Both \((q_t,a_t)\) and \((q^0,a^0)\) are supporting pairs, so
\begin{equation*}
 t\mathscr R_t=\langle q_t-q^0,a_t-a^0\rangle\ge0.
\end{equation*}
Every such refinement has a nonnegative limit.  Thus the selected
subsequence has \(\liminf_N\mathscr R_N^{\mathbf x}(t)\ge0\) almost everywhere.
The bound
\begin{equation*}
 |\mathscr R_N^{\mathbf x}(t)|
 \le2\{\alpha_NL^2+\|a^0\|_\infty\}
\end{equation*}
permits Fatou's lemma; since the starting subsequence was arbitrary,
\(\liminf_N\E_{\mathbf x}\int_0^1\mathscr R_N^{\mathbf x}(t)\,\dd t\ge0\).
It remains to evaluate the endpoints.  Removing \(V_N\) costs \(o(1)\)
in normalized expected pressure, uniformly in the amplitudes.
At \(t=0\), the independent row and spin recursions add.  The row
part contributes \(\alpha_N\Phi(q^0)\), and the spin-block limit in
\cref{prop:scalar-proof-input} evaluates the remaining part.  Thus
\begin{equation*}
 \overline\varphi_N(0)
 \longrightarrow\mathcal S(a^0)+\alpha\Phi(q^0).
\end{equation*}
At \(t=1\), the unperturbed model is
the original perceptron, so
\begin{equation*}
 \overline\varphi_N(1)=\E F_{N,\kappa}+o(1).
\end{equation*}
Integrating \eqref{eq:guerra-static-sum-rule-pointwise} now yields
\begin{equation*}
 \limsup_N\E F_{N,\kappa}
 \le\mathcal S(a^0)+\alpha\Phi(q^0)+\tfrac12\langle a^0,q^0\rangle
 =\min_{q\in\Q_d}\{\mathcal A(q)+\alpha\Phi(q)\}.
\end{equation*}
For \(d=2^n\), approximate any \(q\in\Q\) from below by
\(q_d(r)=q(j/d)\) on \([j/d,(j+1)/d)\), with \(q_d(1)=1\).
Then \(q_d\to q\) in \(L^1\) and the conjugate formula gives
\(\mathcal A(q_d)\le\mathcal A(q)\).  Continuity of \(\Phi\) shows that
the grid minima decrease to \(\mathcal P_\kappa(\alpha)\).
Together with the lower bound and the original centered-model variance
estimate \eqref{eq:unperturbed-iid-variance}, this proves convergence in \(L^2\).

\emph{General Lipschitz channels and the convergence assertions.}
For a centered uniformly Lipschitz family, convolve with one common
mollifier and subtract the mollified value at zero.  The resulting
jointly Borel family \(u_{\theta,\delta}\) satisfies
\eqref{eq:iid-smooth-core-assumption} and
\(\varepsilon_\delta:=\sup_\theta\|u_{\theta,\delta}-u_\theta\|_\infty=O(L\delta)\).
Comparison of exponential weights bounds the normalized pressure
difference pointwise by \((M_N/N)\varepsilon_\delta\).
The scalar recursions likewise bound the variational-value difference
by \(\alpha\varepsilon_\delta\).  Send \(N\to\infty\), then
\(\delta\downarrow0\), to obtain the centered formula in \(L^2\).
For general channels set \(\widetilde u_\theta=u_\theta-u_\theta(0)\).  Exactly,
\[
 \log Z^{\kappa,\nu}_{N,M}(u)
 =\sum_{k\le M}u_{\Theta_k}(0)+\log Z^{\kappa,\nu}_{N,M}(\widetilde u),
 \qquad \Phi_{u_\theta}=u_\theta(0)+\Phi_{\widetilde u_\theta}.
\]
The \(L^1\) law of large numbers, or its \(L^2\) version under the stated
second moment, proves the corresponding limits.
Finally, if \(|u_\theta|\le B\) almost surely, resampling one complete
row changes \(\log Z\) by at most \(2B\).  Bounded differences give
summable deviation probabilities for each fixed positive normalized
deviation; Borel--Cantelli and convergence of the means yield almost-sure
convergence.  The deterministic bound
\(|F_{N,\kappa}|\le M_NB/N\) then gives convergence in every finite \(L^p\).
\end{proof}

\paragraph{Organization}
Section~\ref{sec:cavity-M} proves \cref{prop:0916-rows} and the channel
derivative formula.  Section~\ref{sec:spin-laws} proves
\cref{prop:scalar-proof-input}.  Section~\ref{sec:cavity-N} proves
\cref{prop:0916-coordinate,prop:log-block-insertion}.
Section~\ref{sec:rpc-synchronization} constructs the perturbation and
establishes the ancestry identity, common subsequence selection, and
concentration estimates.  Section~\ref{sec:quenched-ldp-proof} proves
the quenched large-deviation principle; Section~\ref{sec:applications-proofs}
proves the Gaussian-field extension and the remaining application results.

\section{Cavity in $M$}
\label{sec:cavity-M}
\label{sec:parameterized-M-cavity}
\label{sec:constraint-insertion}
\label{sec:iid-channel-proof}

One- and two-row insertions determine the conditional first and second
moments of \(Q_{12}\).  We compute these moments on finite cascades
and transfer them to the model.

We use the centered smooth family \eqref{eq:iid-smooth-core-assumption}
and the averaged responses \eqref{eq:iid-response} throughout
Sections~\ref{sec:cavity-M}--\ref{sec:rpc-synchronization}.
Write \(\mathsf N_k=\gamma^{\otimes k}\), where we recall $\gamma$ denotes the standard Gaussian law.  Deletion and restoration
always concern a complete marked row; all estimates below are uniform
in its mark.

\subsection{Independent constraint insertion}
\label{sec:simple-product-row}
\label{sec:rpc-insertion-laws}
Begin with a step path \(c\) on a finite grid \(\pi\).  Take independent
completed fields \(X_b^c\) from \eqref{eq:overview-completed-row-field},
with Gaussian mark trees independent of the entire cascade Poisson tree.
Set
\begin{equation*}
 G_{\pi,c,+k}(\dd\eta,\dd\boldsymbol z)
 =Z_{\pi,c,k}^{-1}e^{\sum_{b\le k}u_b(X_b^c(\eta,z_b))}
       \mathfrak R_\pi(\dd\eta)\mathsf N_k(\dd\boldsymbol z).
\end{equation*}
Here \(Z_{\pi,c,k}\) normalizes the measure, and
\(X_b^{c,\ell}=X_b^c(\eta^\ell,z_b^\ell)\).
For general paths, the brackets \(\E\langle\cdot\rangle_{c,+k}\)
denote the limits of these averaged finite-replica laws, constructed below.
The following conditional specialization of marked-RPC invariance
\cite[Theorem~4.4]{PanchenkoBook} tracks both the inserted inputs and
their genealogy.



\begin{lemma}[Marked-RPC reweighting]
\label{rpc:prop-quasistationarity}
Consider a finite RPC with masses \(0<m_1<\cdots<m_d<1\).
Conditional on root data \(\mathscr F_0\), independent of the Poisson
tree, attach a standard Borel mark tree independent of that entire
tree.  Given ancestral marks, child subtrees are independent with a
common law.  Apply one common positive finite measurable function
\(L\) to each marked branch and the root data, giving weights
\(L_\eta\).  Define
\begin{equation*}
 S_d=L,\qquad S_{j-1}=(\E_j S_j^{m_j})^{1/m_j},
 \qquad j=d,\ldots,1,
\end{equation*}
where \(\E_j\) integrates the child mark given its ancestors.
Assume these conditional moments are finite and positive.  Then
\(\mathcal Z=\sum_\eta v_\eta L_\eta\) is finite and positive.
After a genealogy-preserving relabelling, the weights
\(v_\eta L_\eta/\mathcal Z\) have the original RPC law and are
independent of the mark tree with transitions
\begin{equation}\label{eq:rpc-mark-transition}
 \nu_j^+(\dd y\mid\text{ancestors})
 =(S_j/S_{j-1})^{m_j}\nu_j(\dd y\mid\text{ancestors}).
\end{equation}
In particular, for every bounded finite-genealogy test \(F\),
\begin{equation*}
 \E[\langle F\rangle_+\mid\mathscr F_0]=\E_{\rm RPC}^{\mathbf m}F,
 \qquad \mathbf m=(m_1,\ldots,m_d),
\end{equation*}
and
\begin{equation*}
 \E[\log\mathcal Z-\log S_0\mid\mathscr F_0]=0.
\end{equation*}
Thus \(\E\log\mathcal Z=\E\log S_0\) if \(\E|\log S_0|<\infty\).
All mark and weight assertions are conditional on \(\mathscr F_0\).
\end{lemma}

\begin{proof}
Conditional on the root data, apply \cite[Theorem~4.4]{PanchenkoBook}
with \(X_d=\log L\) and \(X_j=\log S_j\).  Its transitions (4.43)
are \eqref{eq:rpc-mark-transition}; the conditional moment assumption
makes the recursion finite.  The theorem gives independence of the
full reweighted cascade and tilted mark tree, including genealogy.
For unnormalized weights, \cite[Theorem~4.3 and (4.53)]{PanchenkoBook}
also gives
\[
 T':=\sum_\eta w_\eta L_\eta/S_0\stackrel{\rm d}=T:=\sum_\eta w_\eta.
\]
By \cite[Lemmas~2.2 and~2.4]{PanchenkoBook}, \(T\) is a deterministic
scale times a stable Poisson total with integrable logarithm.
Hence \(\log\mathcal Z-\log S_0=\log T'-\log T\) has conditional
mean zero and an absolute first moment bounded independently of the
root data.  This proves the logarithmic assertions and positivity
and finiteness of \(\mathcal Z=S_0T'/T\).  Depth zero is immediate.
\end{proof}

For the finite-step calculation, define
\begin{equation}\label{rpc:channel-semigroup}
 T_{m,s}h(x)=m^{-1}\log\E e^{mh(x+\sqrt s Z)}\quad(m>0),
 \qquad T_{0,s}h(x)=\E h(x+\sqrt s Z),\quad Z\sim N(0,1).
\end{equation}
Write \(c=c_j\) on \([m_j,m_{j+1})\), where
\(0=m_0<\cdots<m_{J+1}=1\), and set \(c_{J+1}=1\).
The Cole--Hopf transform on each interval where \(\gamma_c\) is
constant gives the PDE value \eqref{eq:intro-channel-pde} as
\begin{equation*}
 X_{J+1}=u,\qquad
 X_j=T_{m_{j+1},c_{j+1}-c_j}X_{j+1}\ (j=J,\ldots,0),
 \qquad \Phi_u(c)=[T_{0,c_0}X_0](0).
\end{equation*}
Indeed, \(\gamma_c=m_{j+1}\) on \((c_j,c_{j+1})\), and it is zero
below \(c_0\).  Thus the root is averaged without tilting and the
terminal interval has mass one.  For general paths, control stability
and the quantile identity
\[
 \int_0^1|\gamma_c(t)-\gamma_{\widetilde c}(t)|\,\dd t
 =\|c-\widetilde c\|_{L^1}
\]
identify the PDE value with the continuous extension of this recursion.
The control lemma applies on any finite horizon \(T>0\): under \(t=Ts\),
its diffusion coefficient is \(T\) and its nonlinear coefficient is
\(T\gamma(Ts)\).  Additive constants give
\(\Phi_u=u(0)+\Phi_{u-u(0)}\), while centering leaves normalized laws
unchanged.

\begin{corollary}[Independent constraint insertions]
\label{lem:rpc-constraint-insertion}
For a step path \(c\) on \(\pi\), every fixed \(k\ge1\), and every
bounded Borel test \(F\) of finitely many replica ancestries,
\begin{equation}\label{eq:rpc-insertion-log}
 \E\langle F\rangle_{\pi,c,+k}=\E_\pi F,
 \qquad \E\log Z_{\pi,c,k}=k\Phi(c).
\end{equation}
If these replicas include \(1,2\), then
\begin{equation}\label{eq:rpc-insertion-score}
 \E\left\langle F\prod_{b=1}^k
 u_b'(X_b^{c,1})u_b'(X_b^{c,2})\right\rangle_{\pi,c,+k}
 =\E_\pi[F p_c(r^\pi_{12})^k].
\end{equation}
For bounded Borel measurable \(\chi_b(\theta,x)\), continuous in \(x\),
\begin{equation}\label{eq:rpc-insertion-one-replica}
 \E\left\langle\prod_{b=1}^k\chi_b(\Theta_b,X_b^{c,1})\right\rangle_{\pi,c,+k}
 =\prod_{b=1}^k\E\langle\chi_b(\Theta,X^{c,1})\rangle_{\pi,c}.
\end{equation}
For arbitrary \(c\in\Q\), these identities hold for the limiting
replica laws, with \(r^\pi,\E_\pi\) replaced by \(U,\E_U\).
The limiting expected log normalizer is \(k\Phi(c)\).
The response \(p_c\) is nondecreasing, takes values in \([0,L^2]\),
and is constant on plateaus of \(c\).  Both \(\Phi(c)\) and \(p_c\)
are continuous for \(L^1\) convergence of \(c\).  The derivative formula \eqref{eq:0916-channel-derivative} also holds.
\end{corollary}

\begin{proof}[Finite-step proof]
Condition on the fresh marks.  Represent each row by independent
ancestral Gaussian increments of variances
\(c_0,c_1-c_0,\ldots,c_J-c_{J-1}\), integrate its terminal Gaussian,
and apply \cref{rpc:prop-quasistationarity} conditional on the root
Gaussians.  The bound \(|u_b(x)|\le L|x|\) makes the recursion finite
and its root log value integrable.  Independent rows have factorized
exponential moments, so the recursion adds and the tilted transitions
factor even for different terminals
\cite[Theorem~2.9 and (2.59)--(2.60)]{PanchenkoBook}.
The expected log normalizer is therefore \(\sum_b\Phi_{u_b}(c)\).
Conditional on any sampled genealogy, the entire tilted mark trees
are independent across rows; their one-replica marginals do not depend
on the genealogy.  Averaging the marks proves
\eqref{eq:rpc-insertion-log} and \eqref{eq:rpc-insertion-one-replica}.

For one terminal \(u\), start with \(g_0\sim N(0,c_0)\) and tilt
each transition \(N(g_j,c_{j+1}-c_j)\) by
\(e^{m_{j+1}(X_{j+1}(g_{j+1})-X_j(g_j))}\), including the terminal
step.  Differentiating the recursion shows that
\(M_j=X_j'(g_j)\) is a bounded martingale.  Replicas branching after
level \(j\) have conditionally independent continuations and terminal
scores of conditional mean \(M_j\).  Hence
\begin{equation*}
 p_{c,u}(r)=\E M_j^2\quad\text{on }[m_j,m_{j+1}).
\end{equation*}
Other branches integrate to one, so this identity holds conditional
on the full genealogy.  Independent marks give
\[
 \E_{\Theta_1,\ldots,\Theta_k}\prod_{b=1}^k p_{c,u_{\Theta_b}}(r)
 =p_c(r)^k,
\]
proving \eqref{eq:rpc-insertion-score}.  Conditional Jensen gives
monotonicity and \(0\le p_c(r)\le p_{\rm diag}(c)\le L^2\);
the inequality passes to general paths by their \(L^1\) limits.
Zero-variance transitions
give the plateau property.

Gaussian covariance differentiation, with completed diagonal variance
fixed at one, leaves only the two-replica score term and gives
\eqref{eq:0916-channel-derivative}.  The recursion and its derivatives
are measurable in the marks, and the uniform derivative bounds justify
mark averaging.  Integrating the derivative gives \eqref{rpc:Phi-L1}.
\end{proof}

To pass through the random normalizer, we need positive and inverse
weight moments.  Under any base \(\mu\) independent of fresh row noises,
the completed inputs at one sampled state are independent standard normals.  Thus, for \(W=e^{\sum_{b\le k}u_b(X_b)}\) and every \(p>0\),
\begin{equation}\label{eq:constraint-weight-moments}
 \E\mu(W^p+W^{-p})\le2(\E e^{pL|Z|})^k<\infty.
\end{equation}
For one fresh row under an independent probability measure \(\Gamma\),
Jensen also gives, with \(A=\log\Gamma(e^{L|X|})\ge0\),
\begin{equation}\label{eq:fresh-marked-row-bound}
 |\log\Gamma(e^{u_\Theta(X)})|\le A,\qquad
 \E A^p\le\E e^{pL|Z|}\quad(p\ge1).
\end{equation}
These bounds also control mark fluctuations and row-count changes.


\begin{lemma}[Convergence after changing density]
\label{lem:change-density-limit}
Let \(\mu_n\) be random probability measures, \(W_n>0\) measurable
weights, and \(A_n(s,s')\) maps into a fixed Polish space; single-sample
marks may be included.  For conditional independent samples, put
\begin{equation*}
 Z_n=\mu_n(W_n),\qquad
 \mu_n^+(\dd s)=Z_n^{-1}W_n(s)\mu_n(\dd s).
\end{equation*}
Suppose that every finite sampled array, including its diagonal, converges:
\begin{equation*}
 \left((A_n(s^\ell,s^m))_{\ell,m\le k},
       (W_n(s^\ell))_{\ell\le k}\right)
 \Longrightarrow ((A_{\ell m})_{\ell,m\le k},(W^\ell)_{\ell\le k}),
 \qquad W^\ell>0,
\end{equation*}
and, for every \(p\ge1\),
\begin{equation*}
 \sup_n\E\mu_n(W_n^p+W_n^{-p})<\infty.
\end{equation*}
Then the sampled arrays under \(\mu_n^+\) converge to a unique
consistent replica law determined by the array on the right.  There is
also a positive random variable \(Z\), determined by that array, such that
\begin{equation*}
 Z_n\Longrightarrow Z,\qquad \E\log Z_n\longrightarrow\E\log Z.
\end{equation*}
No random-measure representation of the limiting array is required.
If one is given, the limits agree with its ordinary change of density.
\end{lemma}

\begin{proof}
Fix a bounded continuous test \(F\) of a \(k\)-replica array, with
\(|F|\le1\), and take \(P\) conditional independent samples
\(s^1,\ldots,s^P\) from \(\mu_n\).  Set
\[
 \begin{split}
 H_n(s^1,\ldots,s^k)
 &=F\bigl((A_n(s^\ell,s^m))_{\ell,m\le k}\bigr)
       \prod_{\ell=1}^kW_n(s^\ell),\\
 B_n&=\mu_n^{\otimes k}(H_n),\qquad
 B_{n,P}=P^{-k}\sum_{i_1,\ldots,i_k\le P}
              H_n(s^{i_1},\ldots,s^{i_k}),\\
 Z_{n,P}&=P^{-1}\sum_{i\le P}W_n(s^i),\qquad
 M_p=1+\sup_n\E\mu_n(W_n^p+W_n^{-p}).
 \end{split}
\]
Single-sample marks are included in the argument of \(F\) when present.
Repeated sample indices are allowed in \(B_{n,P}\); this is why the
array-convergence hypothesis includes the diagonal.

Conditional independence gives
\(\E|Z_{n,P}-Z_n|^2\le M_2/P\).
To estimate the numerator, expand \(\E|B_{n,P}-B_n|^2\) as a sum
over two ordered \(k\)-tuples of sample indices.  If all \(2k\) indices
are different, the corresponding centered product has conditional
expectation zero.  There are at most
\(\binom{2k}{2}P^{2k-1}\) remaining pairs of tuples.
For a tuple with distinct indices of multiplicities
\(m_1,\ldots,m_j\), where \(\sum_i m_i=k\), H\"older with exponents
\(k/m_i\) and \(k/(k-m_i)\) (or equality when \(m_i=k\)) gives
\[
 \E\bigl[|H_n(s^{i_1},\ldots,s^{i_k})|^2\mid\mu_n,W_n,A_n\bigr]
 \le\prod_{i=1}^j\mu_n(W_n^{2m_i})
 \le\mu_n(W_n^{2k}).
\]
Also \(|B_n|^2\le\mu_n(W_n^{2k})\).  Cauchy--Schwarz therefore
bounds each remaining centered product in absolute expectation by
\(4M_{2k}\), and hence
\[
 \E|B_{n,P}-B_n|^2
 \le4\binom{2k}{2}M_{2k}/P.
\]
Jensen bounds all positive and inverse moments of \(Z_n,Z_{n,P}\).
Since \(|B_n|\le Z_n^k\) and \(|B_{n,P}|\le Z_{n,P}^k\), expanding
the difference using the larger denominator gives
\[
 \left|\frac{B_{n,P}}{Z_{n,P}^k}-\frac{B_n}{Z_n^k}\right|
 \le\frac{|B_{n,P}-B_n|}{\max(Z_n,Z_{n,P})^k}
       +k\frac{|Z_{n,P}-Z_n|}{\max(Z_n,Z_{n,P})}.
\]
Cauchy--Schwarz, with exponents \(2,2\), now yields
\[
 \sup_n\E\left|B_{n,P}/Z_{n,P}^k-B_n/Z_n^k\right|
 \le C_kP^{-1/2},
\]
uniformly over \(|F|\le1\).

Place the consistent limiting input arrays on one probability space.
Given their first \(P\) samples, choose indices \(I_1,\ldots,I_k\)
independently with probabilities \(W^i/\sum_{j\le P}W^j\), and let
\(\Lambda_{P,k}\) be the averaged law of
\((A_{I_\ell I_m})_{\ell,m\le k}\), together with any sampled marks.
For fixed \(P\), its expectation of \(F\) is the limit of
\(\E B_{n,P}/Z_{n,P}^k\): the normalized empirical integral is a
bounded continuous function of the finite array and its positive weights.
The preceding estimate shows, for all bounded continuous \(|F|\le1\),
\[
 |\Lambda_{P,k}(F)-\Lambda_{Q,k}(F)|
 \le C_k(P^{-1/2}+Q^{-1/2}).
\]
For probability measures on a Polish space, the supremum over these
tests equals the supremum over bounded measurable tests.  Thus
\(\Lambda_{P,k}\) is Cauchy in total variation and has a probability
limit \(\Lambda_k\).  Consistency in \(k\) holds for each \(P\) and
passes to the limit.  Comparing the exact tilted law with its empirical
law, then taking \(n\to\infty\) and \(P\to\infty\), proves convergence
to \(\Lambda_k\) and its uniqueness.

On the limiting input array put \(Z_P=P^{-1}\sum_{i\le P}W^i\).
Finite-array convergence and the Portmanteau theorem transfer the bound
\[
 \E|Z_{n,P}-Z_{n,Q}|^2\le2M_2(P^{-1}+Q^{-1})
\]
to \(Z_P,Z_Q\).  Consequently \(Z_P\to Z\) in \(L^2\).
Jensen and Fatou give \(\E(Z^p+Z^{-p})\le M_p\), so \(Z>0\).
For any bounded Lipschitz function, comparison successively through
\(Z_{n,P}\) and \(Z_P\), first sending \(n\to\infty\) and then
\(P\to\infty\), proves \(Z_n\Rightarrow Z\).
Finally \(|\log z|^2\le C(z+z^{-1})\) makes \(\log Z_n\) uniformly
integrable; continuous mapping gives convergence of its expectation.
If a random-measure representation of the limiting array is supplied,
the same empirical estimates identify these limits with its ordinary
change of density and normalizer.
\end{proof}

\begin{proof}[General paths in \cref{lem:rpc-constraint-insertion}]
\emph{Covariance convergence.}
Choose step paths \(c_n\to c\) in \(L^1\) on grids \(\pi_n\) of mesh
\(h_n\to0\), adding zero-variance levels if necessary.  Couple their
finite genealogy arrays by
\(r^n_{\ell m}=\kappa_{\pi_n}(U_{\ell m})\) off the diagonal, where
\(\kappa_\pi\) rounds down to the left grid endpoint.  This exact
finite-RPC law follows from monotone rounding
\cite[(2.118)--(2.122)]{PanchenkoBook} and the GG characterization
\cite[Theorems~2.10 and~2.13]{PanchenkoBook}.
Since \(U_{12}\) is uniform and \(c_n\) is constant on grid cells,
\[
 \E|c_n(r^n_{12})-c(U_{12})|=\|c_n-c\|_1.
\]
Every completed Gaussian covariance matrix therefore converges in
probability, with diagonal fixed at one, regardless of \(c_n(1-)\).
When true diagonals differ from
same-leaf pair marks, adjoin an independent uniform label to each
sampled state and assign diagonal values only to identical complete
states.  Distinct replicas then have distinct complete states almost
surely; all weights and Gaussian fields ignore the extra labels.

\emph{Normalized limits.}
Condition on the fresh marks and couple finite Gaussian vectors by
their matrix square roots, using independent copies across rows.
This gives joint convergence of genealogy, completed inputs, and
weights.  The bound \eqref{eq:constraint-weight-moments} and
\cref{lem:change-density-limit} define their unique normalized replica
limits and expected log normalizers.  They depend only on \(c\), not
its approximation.  Finite-RPC invariance passes to continuous tests
and then to Borel tests by equality of finite measures, so the limiting
genealogy still has law \(U\).  Uniform moments permit mark averaging.

\emph{Response continuity.}
For continuous \(f\), define
\[
 \nu_c(f)=\E\langle f(U_{12})u_\Theta'(X^{c,1})u_\Theta'(X^{c,2})\rangle_c.
\]
Helly selection and \(0\le p_{c_n}\le L^2\) give an
\(L^1\)-convergent further subsequence of every subsequence.  The
finite-step score identity identifies every such limit \(p\) by
\[
 \int_0^1 f(u)p(u)\,\dd u=\nu_c(f),
\]
since replacing \(f(r^n_{12})\) by \(f(U_{12})\) costs at most
\(L^2\omega_f(h_n)\).  Continuous tests determine finite measures,
so \(p\) is unique almost everywhere.  This defines \(p_c\) and proves
full-sequence \(L^1\) convergence.  Boundedness passes \(p_{c_n}^k\)
to \(p_c^k\), giving all three insertion identities.  The logarithmic
bound \eqref{rpc:Phi-L1} passes to the limit as well.

For an arbitrary sequence \(c_n\to c\), choose successively finer
step approximations to each \(c_n\).  Their completed arrays converge
to the same array with covariance \(c(U)\); the uniform empirical
bounds in \cref{lem:change-density-limit} give
\(\nu_{c_n}(f)\to\nu_c(f)\).  The same Helly argument proves
\(p_{c_n}\to p_c\) in \(L^1\).  To preserve a plateau of \(c\), use
\(2^{-n}\lfloor2^nc\rfloor\) off the diagonal and refine its grid.
These step approximations show that \(p_c\) is constant there.
One-replica convergence gives \(p_{\rm diag}(c_n)\to p_{\rm diag}(c)\).
Passing the finite-grid inequality \(p_{c_n}\le p_{\rm diag}(c_n)\)
almost everywhere, then taking the essential supremum, gives
\begin{equation*}
 p_{\rm diag}(c)\ge p_c(1-).
\end{equation*}
Convergence of \(p_{c_n}(1-)\) is neither needed nor asserted.
\end{proof}


\subsection{Array limits and reconstruction}
\label{sec:constraint-proof}

We transfer the insertion identities to the finite-volume measures
of Section~2.  Their limiting observed array is
\((q(U_{\ell m}),\rho_d(U_{\ell m}))\) off the diagonal, with
diagonal \((1,1)\).

\begin{proof}[Proof of \cref{prop:0916-rows}]
\emph{Insertion.}
For every fixed \(k\), apply \cref{lem:change-density-limit} with
\[
 \mu_N=G_N\otimes\mathsf N_k,\qquad
 W_N=\exp\Bigl\{\sum_{b=1}^ku_b(X_b)\Bigr\}.
\]
Conditional on the old environment, sampled old states, and fresh
marks, the completed Gaussian input vectors are centered and independent
across rows, with off-diagonal covariance \(\mathsf C_N\) and true
diagonal \(1+\varepsilon_N\to1\), including \(\varepsilon_N=0\).  Thus
\eqref{eq:overview-fresh-covariance-limit} gives joint convergence
of the old array, completed inputs, and weights.
The bound \eqref{eq:constraint-weight-moments}, with its Gaussian
variance replaced by the bounded \(1+\varepsilon_N\), supplies all positive
and inverse weight moments, uniformly in the marks.  The lemma gives
convergence of the normalized laws and expected logarithms; these
conclusions pass through the mark average.  Explicitly,
\begin{align*}
 \E\log Z_{N,k}^+&\longrightarrow k\Phi(c),
 \\
 \E\langle F\Psi\rangle_{+k}
 &\longrightarrow\E\langle F_\infty\Psi\rangle_{c,+k},
\end{align*}
for bounded continuous old-array tests \(F\) and bounded \(\Psi\)
measurable in the fresh marks and continuous in the inserted inputs.
Here \(F_\infty\) evaluates \(F\) on the observed limiting array.
Corollary~\ref{lem:rpc-constraint-insertion} preserves that array's
law and, with \(\Psi=\prod_bu_b'(X_b^1)u_b'(X_b^2)\), gives
\eqref{eq:overview-insertion-mixed-moments}.

Restoring deleted rows gives the full measure exactly and preserves
the limiting observed array.  Its spin marginal, and hence its
nondecreasing quantile \(q\), agree across the full and deleted
systems.  The common limiting rule \(c=K(q,\rho_d)\) identifies
\(c\); a uniform error \(\|K_N-K\|_\infty\to0\) does not change
the covariance limit.
The same argument applies to any fixed collection of row counts linked
by finitely many insertions.

\emph{Reconstruction.}
Put \(S_k=u_k'(X_k^1)u_k'(X_k^2)\), so \(|S_k|\le L^2\) and
\(Q_{12}=M^{-1}\sum_kS_k\).  For a bounded continuous old-array test
\(F\), exchangeability gives
\begin{equation*}
 \E\langle FQ_{12}\rangle=\E\langle FS_1\rangle,\qquad
 \E\langle FQ_{12}^2\rangle
 =\frac{M-1}{M}\E\langle FS_1S_2\rangle+O(M^{-1}).
\end{equation*}
The repeated-index error is at most \(\|F\|_\infty L^4/M\).
One- and two-row restoration therefore give
\begin{align*}
 \E\langle FQ_{12}\rangle
 &\longrightarrow\E_U[F_\infty p_c(U_{12})],
 \\
 \E\langle FQ_{12}^2\rangle
 &\longrightarrow\E_U[F_\infty p_c(U_{12})^2].
\end{align*}
The common covariance rule \(c=K(q,\rho_d)\) and the plateau property
give \(p_c(u)=\bar p_c(q(u),\rho_d(u))\) almost everywhere, with
\(\bar p_c\) bounded Borel; see \cref{lem:finite-rpc-representation}.
For any joint subsequential limit \((A,Q)\), where \(A\) retains the
finite observed overlap--ancestry array containing \((R_{12},r_{12})\),
the first- and second-moment identities against continuous tests
identify finite signed Borel measures.  Hence
\[
 \E[Q\mid A]=\bar p_c(R_{12},r_{12}),\qquad
 \E[Q^2\mid A]=\bar p_c(R_{12},r_{12})^2.
\]
The conditional variance vanishes, so
\(Q=\bar p_c(R_{12},r_{12})\) almost surely; in the canonical representation
this equals \(p_c(U_{12})\).  Boundedness gives compactness, and every
joint limit has this form.

For \(T_N=M^{-1}\sum_k\chi(\Theta_k,X_k^1)\), restore one row to
compute \(\mathbb E\langle T_N\rangle\) and two rows to compute
\(\mathbb E\langle T_N^2\rangle\).  The product identity
\eqref{eq:rpc-insertion-one-replica} gives the limits \(m_c(\chi)\)
and \(m_c(\chi)^2\); repeated indices contribute \(O(M^{-1})\).
This proves \eqref{eq:empirical-one-row-limit}.  The scalar limits
exist by the same covariance and weight-moment passage, so no separate
approximation argument is needed.

For the family of Section~\ref{sec:overview-N-cavity}, deletion leaves
the row noises independent, and their covariance is
\eqref{eq:overview-inserted-constraint-covariance}.  The lookup
\(q^{0,\sharp}\) is a function on the fixed finite ancestry alphabet,
so convergence of the observed \((R,r)\)-array passes its covariance
to the limit.  Since
\(q^{0,\sharp}(\rho_d(u))=q^0(u)\), this verifies
\eqref{eq:overview-fresh-covariance-limit} with
\(c=(1-t)q^0+tq\).
\end{proof}

The following comparison will control changes in the insertion field
during coordinate lifting.

\begin{lemma}[Comparison of tilted averages]
\label{lem:tilted-comparison}
Let \(\mu\) be a random probability measure on a measurable space.  For
\(Z_H=\mu(e^H)\in(0,\infty)\), write
\(\langle F\rangle_H=(e^H\mu/Z_H)^{\otimes n}(F)\).
Let \(F\) be a bounded \(n\)-replica test, with deterministic bound
\(\|F\|_\infty\).
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item \emph{General comparison.}  For real measurable fields \(X,Y\), put
\[
 d^2=\E\mu((Y-X)^2),\qquad
 B=\max_{\substack{H\in\{X,Y\}\\\epsilon=\pm1}}
                  \E\mu(e^{4\epsilon H}).
\]
If \(d,B<\infty\), then
\begin{equation*}
 \begin{aligned}
 |\E\log Z_Y-\E\log Z_X|&\le\sqrt B\,d,\\
 |\E\langle F\rangle_Y-\E\langle F\rangle_X|
 &\le2n\|F\|_\infty\sqrt B\,d.
 \end{aligned}
\end{equation*}
No independence among \(\mu,X,Y,F\) is required.
If \(|Y-X|\le\varepsilon\), both bounds hold before averaging with
\(\sqrt B\,d\) replaced by \(\varepsilon\), requiring only finite
positive normalizers; the averaged log bounds then hold whenever the
logarithms are integrable.
\item \emph{Gaussian comparison.}  Condition on an environment determining
\(\mu\) and \(F\).  Let \(X,Y\) be conditionally centered Gaussian
fields with covariance kernels \(C_X,C_Y\), whose diagonal values are
bounded by a finite deterministic constant.  If
\(\|C_Y-C_X\|_\infty\le\delta\), then
\begin{equation*}
 \begin{aligned}
 |\E\log Z_Y-\E\log Z_X|&\le\delta,\\
 |\E\langle F\rangle_Y-\E\langle F\rangle_X|
 &\le n(2n+1)\|F\|_\infty\delta.
 \end{aligned}
\end{equation*}
\end{enumerate}
\end{lemma}

\begin{proof}
For (i), set \(D=Y-X\), \(H_t=X+tD\), and \(Z_t=\mu(e^{H_t})\).
Holding \(F\) fixed, differentiation gives
\begin{equation*}
 \frac{\dd}{\dd t}\log Z_t=\langle D\rangle_t,\qquad
 \frac{\dd}{\dd t}\langle F\rangle_t
 =\left\langle F\left(\sum_{i=1}^nD(s^i)-nD(s^{n+1})\right)\right\rangle_t.
\end{equation*}
Convexity bounds \(\E\mu(e^{\pm4H_t})\) by \(B\).  Jensen gives
\(Z_t^{-4}\le\mu(e^{-4H_t})\), so H\"older yields
\[
 \E\langle|D|\rangle_t
 \le d\,(\E\mu(e^{4H_t}))^{1/4}(\E\mu(e^{-4H_t}))^{1/4}
 \le\sqrt B\,d.
\]
Integration proves (i); truncation and this bound justify differentiation.
For \(|D|\le\varepsilon\), use
\(\langle|D|\rangle_t\le\varepsilon\) directly, before averaging.

For (ii), take conditionally independent endpoint copies and set
\(V_t=\sqrt{1-t}X+\sqrt tY\), \(Z_t=\mu(e^{V_t})\), and
\(\Delta_{ij}=(C_Y-C_X)(s^i,s^j)\).
Gaussian integration by parts gives
\begin{equation}\label{eq:gaussian-log-interpolation}
 \frac{\dd}{\dd t}\E\log Z_t
 =\frac12\E\langle\Delta_{11}-\Delta_{12}\rangle_t
\end{equation}
and
\begin{equation*}
 \begin{split}
 \frac{\dd}{\dd t}\E\langle F\rangle_t
 =\frac12\E\Big\langle F\Big[&\sum_{i,j\le n}\Delta_{ij}
 -2n\sum_{i\le n}\Delta_{i,n+1}\\
 &-n\Delta_{n+1,n+1}
 +n(n+1)\Delta_{n+1,n+2}\Big]\Big\rangle_t.
 \end{split}
\end{equation*}
These identities hold first for finite state spaces.  If \(B_0\) bounds
the field variances, then
\(\E\mu(e^{pV_t}+e^{-pV_t})\le2e^{p^2B_0/2}\).
The empirical measure argument of \cref{lem:change-density-limit},
using conditional samples independent of the Gaussian noises, passes
the integrated identities to general \(\mu\).
Bounding each covariance difference by \(\delta\) and summing the
absolute coefficients proves (ii).
\end{proof}

\subsection{The quadratic coefficient for block insertion}
\label{sec:simple-empirical-row-recovery}

The block calculation at \(t=1\) uses the coefficient
\begin{equation*}
 B_N=\frac1M\sum_{k=1}^M\{u_k''(h_k)-h_k u_k'(h_k)\}.
\end{equation*}
The diagonal correlation \(Q_{11}\) is already covered by
\eqref{eq:empirical-one-row-limit}.  We recover \(B_N\) by Gaussian
integration by parts and the limiting GG identity.  The spectral bound
below controls its unbounded factor \(h_k\).

\begin{lemma}[Uniform constraint bounds]
\label{lem:spectral-moments}
Let \(G\) be the \(M\times N\) Gaussian constraint matrix and put
\(\Lambda_N=\|G\|_{\rm op}^2/M\).  If \(M/N\) stays in a compact
subset of \((0,\infty)\), then, for every \(r,c\ge0\),
\[
 \sup_N\E\Lambda_N^r<\infty,\qquad
 \sup_N\E e^{c\sqrt{\Lambda_N}}<\infty.
\]
On either spin space,
\begin{equation}\label{eq:simple-spectral-empirical}
 \frac1M\sum_k h_k(\sigma)^2\le\Lambda_N,\qquad
 \frac1M\sum_k|h_k(\sigma)|\le\sqrt{\Lambda_N}.
\end{equation}
\end{lemma}

\begin{proof}
Use \(\|\sigma\|^2=N\), Cauchy--Schwarz, and the Gaussian matrix
tail bound
\(\mathbb P\{\|G\|_{\rm op}\ge\sqrt M+\sqrt N+s\}\le e^{-s^2/2}\).
\end{proof}

In particular, the common bounds on \(u_k',u_k''\) give the uniform envelope
\begin{equation}\label{eq:quadratic-coefficient-exponential-envelope}
 |B_N|\le C_u(1+\sqrt{\Lambda_N}).
\end{equation}

\begin{lemma}[The quadratic coefficient]
\label{lem:empirical-row-coefficients}
At \(t=1\), assume the full measure and its one- and two-constraint
deletions satisfy the array-limit hypotheses of \cref{prop:0916-rows},
with common spin path \(q\).  Put
\begin{equation*}
 \overline B=\langle q,p_q\rangle-p_{\rm diag}(q).
\end{equation*}
Then
\begin{align*}
 \E\langle|Q_{11}-p_{\rm diag}(q)|^2\rangle&\longrightarrow0,
 \\
 \E\langle|B_N-\overline B|^2\rangle&\longrightarrow0.
\end{align*}
\end{lemma}

\begin{proof}
The first limit is \eqref{eq:empirical-one-row-limit} with
\(\chi(\theta,x)=u_\theta'(x)^2\).

To recover \(B_N\), first add back its diagonal derivative term.  Put
\[
 C_N=B_N+Q_{11},\qquad K_{\ell m}=R_{\ell m}Q_{\ell m},\qquad
 \overline K_N=\E\langle K_{12}\rangle.
\]
For an \(n\)-replica test \(F\), let
\[
 D_n(F)=\frac1{M\sqrt N}\sum_{k,i}
 \E\langle\sigma_i^1u_k'(h_k^1)\partial_{g_{ki}}F\rangle,
\]
where the derivative holds the replica states and marks fixed.  Gaussian
integration by parts conditional on the marks, using independence of the perturbation from
the constraint vectors, gives
\begin{equation}\label{eq:reaction-replica-ibp}
 \E\langle F C_N^1\rangle
 =\E\left\langle F\left(nK_{1,n+1}
                     -\sum_{\ell=2}^nK_{1\ell}\right)\right\rangle-D_n(F).
\end{equation}
The derivatives of \(u_k'(h_k^1)\) and of the first replica's density
produce \(u_k''(h_k^1)\) and \(u_k'(h_k^1)^2\); the other density
derivatives produce the displayed overlaps.  With \(n=1,F=1\),
this yields \(\E\langle C_N\rangle=\overline K_N\).

The uniform derivative bounds and \(\|\sigma^\ell\|=\sqrt N\) give
\[
 \|\nabla_{g_k}C_N^1\|\le\frac{C_u}{M}(1+|h_k^1|),\qquad
 \|\nabla_{g_k}K_{12}\|\le\frac{C_u}{M}.
\]
Contracting with \(\sigma^1\) and using
\eqref{eq:simple-spectral-empirical} yields
\begin{equation*}
 |D_1(C_N^1)|+|D_2(K_{12})|
 \le\frac{C_u}{M}\bigl(1+\E\sqrt{\Lambda_N}\bigr)=O(M^{-1}).
\end{equation*}
The same bound and \eqref{eq:quadratic-coefficient-exponential-envelope}
justify these integrations by parts by truncation.  Applying
\eqref{eq:reaction-replica-ibp} to these two tests gives
\[
 \E\langle(C_N-\overline K_N)^2\rangle
 =\{2\E\langle K_{12}K_{13}\rangle
        -\E\langle K_{12}^2\rangle-\overline K_N^2\}
       -D_2(K_{12})-D_1(C_N^1).
\]
Row reconstruction identifies the limiting
\(K_{12}=q(U_{12})p_q(U_{12})\).  The GG identity for \(U\),
applied to the bounded Borel function \(u\mapsto q(u)p_q(u)\),
makes the expression in braces vanish.  This is the identity of
\cite[Theorem~2.10]{PanchenkoBook}, extended by
\cite[Theorem~2.17]{PanchenkoBook} and equality of the resulting
finite dimensional laws to bounded Borel tests.  Thus
\(C_N\to\langle q,p_q\rangle\) in averaged \(L^2\).
Subtract the established limit of \(Q_{11}\) to obtain the result.
\end{proof}
\section{Scalar transforms and supporting inequalities}
\label{sec:spin-laws}

The Ising response is a two-spin correlation.  The spherical response
is the explicit function obtained from a Gaussian quadratic integral.
We prove that both responses give supporting pairs for the spin cost.
The same Gaussian integral supplies the spherical block increment.
Write \(\mathcal S_{\rm I},\mathcal S_{\rm S}\),
\(\mathcal A_{\rm I},\mathcal A_{\rm S}\), and
\(\widehat q_a^{\rm I},\widehat q_a^{\rm S}\) for the two priors.
For a partition \(\pi=(x_0,\ldots,x_n)\), write a step source as
\begin{equation}\label{eq:spin-step-source}
 0=x_0<x_1<\cdots<x_n=1,\qquad
 a(r)=a_j\quad(x_{j-1}\le r<x_j),\qquad
 0\le a_1\le\cdots\le a_n,
\end{equation}
and put \(a_*=a_n\).  The cascade \(\mathfrak R_\pi\) has masses
\(x_1,\ldots,x_{n-1}\).  Its off-diagonal ancestry is the left
endpoint of the common ancestry interval, including \(x_{n-1}\) for
two replicas on the same leaf; the true replica diagonal is one.
Write \(\E_\pi\) for its averaged genealogy law.  Gaussian ancestor
marks are independent of the entire Poisson tree; root increments form
the common environment in \cref{rpc:prop-quasistationarity}.
Limiting Ising replica laws use the uniform genealogy \(U\), with
expectation \(\E_U\), and are defined by finite partitions of vanishing
mesh.  Zero-variance levels may always be added.

\subsection{The spherical Gaussian integral}
\label{sec:spherical-insertion-proof}

Use the domain \(\mathfrak D_a\), precision \(\pi_{a,\beta}\), and
functions \(V_a,q^{\rm co}_{a,\beta}\) defined in Section~2.  Put
\begin{equation}\label{eq:general-scalar-normalizer}
 K_a(\beta)=-\frac12\log\beta
       +\frac12\int_0^{a_*}\frac{\dd t}{\pi_{a,\beta}(t)}.
\end{equation}
For a step source,
let \(w^a\) have covariance path \(a\) and variance \(a_*\), and set
\begin{equation*}
 K_{a,R}(\beta)=\mathbb E\log\int\mathfrak R_\pi(\dd\eta)
       \int_{-R}^R\frac{\dd\tau}{\sqrt{2\pi}}
       e^{\tau w^a(\eta)-\beta\tau^2/2}.
\end{equation*}
The next lemma also defines this quantity for general sources.

\begin{lemma}[Gaussian recursion, precision, and logarithmic cutoff]
\label{prop:rpc-spherical-insertion}
For a step source and \(\beta\in\mathfrak D_a\),
\begin{equation*}
 \mathbb E\log\int\mathfrak R_\pi(\dd\eta)
       \int_{\mathbb R}\frac{\dd\tau}{\sqrt{2\pi}}
       e^{\tau w^a(\eta)-\beta\tau^2/2}=K_a(\beta).
\end{equation*}
The function \(V_a\) is continuous and strictly decreasing from infinity
to zero on \(\mathfrak D_a\).  Its unique solution
\(V_a(\bar\beta(a))=1\) satisfies
\begin{equation*}
 1+\beta_-(a)\le\bar\beta(a)\le1+a_*,\qquad
 \inf_t\pi_{a,\bar\beta(a)}(t)\ge1.
\end{equation*}
If step paths \(a_n\) on refining partitions satisfy
\(a_n(1-)=a_*\), \(\|a_n-a\|_{L^1}\to0\),
\(\beta_n\to\beta\), and \(\inf_{n,t}\pi_{a_n,\beta_n}(t)>0\), then
\begin{equation*}
 K_{a_n}(\beta_n)\to K_a(\beta),\qquad
 \|q^{\rm co}_{a_n,\beta_n}-q^{\rm co}_{a,\beta}\|_{L^1}\to0,
 \qquad V_{a_n}(\beta_n)\to V_a(\beta).
\end{equation*}
For each fixed \(R>0\), the quantities \(K_{a_n,R}(\beta_n)\)
also converge to a limit \(K_{a,R}(\beta)\) independent of the
approximation, and
\begin{equation}\label{eq:scalar-log-cutoff}
 K_{a,R}(\beta)\uparrow K_a(\beta)\qquad(R\to\infty).
\end{equation}
\end{lemma}

\begin{proof}
\emph{Gaussian recursion.}
For a step source, set
\begin{equation*}
 d_n=\beta,\qquad
 d_j=d_{j+1}-x_j(a_{j+1}-a_j),\qquad 1\le j<n.
\end{equation*}
These numbers are positive precisely when \(\beta>\beta_-(a)\).
The leaf integral is \(\beta^{-1/2}e^{w^2/(2\beta)}\).
Square completion at level \(j\), with
\(\Delta_j=a_{j+1}-a_j\), gives
\[
 \frac1{x_j}\log\E_Z
 e^{x_j[C+(w+\sqrt{\Delta_j}Z)^2/(2d_{j+1})]}
 =C+\frac1{2x_j}\log\frac{d_{j+1}}{d_j}
      +\frac{w^2}{2d_j}.
\]
This is the Gaussian recursion of
\cite[Lemma~3.5]{TalagrandSpherical}; positivity of \(d_j\)
checks exactly the fractional moments in
\cref{rpc:prop-quasistationarity}.  Averaging the unchanged root
Gaussian of variance \(a_1\) gives
\begin{equation}\label{eq:scalar-log-normalizer}
 K_a(\beta)=-\frac12\log\beta+\frac{a_1}{2d_1}
 +\frac12\sum_{j=1}^{n-1}\frac1{x_j}\log\frac{d_{j+1}}{d_j}.
\end{equation}

We need only the one-coordinate marginal of the normalized scalar
integral, to control the logarithmic cutoff below.  Under the tilted
transition \eqref{eq:rpc-mark-transition}, the next field, given its
current value \(w\), is Gaussian with mean
\(d_{j+1}w/d_j\) and variance \(\Delta_jd_{j+1}/d_j\).
At the leaf, the coordinate has conditional law
\(\mathcal N(w/\beta,\beta^{-1})\).  Consequently its averaged law is
\begin{equation}\label{eq:scalar-gaussian-representation}
 \tau\stackrel{\rm d}=
 \frac{Z_n}{\sqrt\beta}+\frac{\sqrt{a_1}}{d_1}Z_0
 +\sum_{j=1}^{n-1}\sqrt{\frac{a_{j+1}-a_j}{d_jd_{j+1}}}\,Z_j,
\end{equation}
with independent standard Gaussians.  Its variance is
\begin{equation*}
 V_a(\beta)=\frac1\beta+\frac{a_1}{d_1^2}
 +\sum_{j=1}^{n-1}\frac{a_{j+1}-a_j}{d_jd_{j+1}}.
\end{equation*}
On \((0,a_1]\), \(\pi_{a,\beta}=d_1\); on
\((a_j,a_{j+1}]\),
\(\pi_{a,\beta}(t)=d_j+x_j(t-a_j)\).
Integrating \(\pi^{-1}\) and \(\pi^{-2}\) gives
\eqref{eq:general-scalar-normalizer} and
\eqref{eq:general-response-and-variance}.  In particular,
\begin{equation}\label{eq:scalar-coordinate-response}
 q^{\rm co}_{a,\beta}(r)=\frac{a_1}{d_1^2}
 +\sum_{i=1}^{j-1}\frac{a_{i+1}-a_i}{d_id_{i+1}},
 \qquad x_{j-1}\le r<x_j.
\end{equation}

\emph{Precision and continuity.}
For general sources the explicit integrals give
\begin{equation*}
 V_a'(\beta)=-\frac1{\beta^2}
 -2\int_0^{a_*}\frac{\dd t}{\pi_{a,\beta}(t)^3}<0.
\end{equation*}
With \(\delta=\beta-\beta_-(a)\), layer cake gives
\(\pi_{a,\beta}(t)=\delta+\int_0^t x_a(s)\dd s\le\delta+t\)
and \(\beta\le\delta+a_*\).  Thus \(V_a(\beta)\ge\delta^{-1}\);
also \(V_a(\beta)\to0\) as \(\beta\to\infty\).
At \(\beta=1+a_*\), \(\pi_{a,\beta}(t)\ge1+t\), so
\(V_a(1+a_*)\le1\).  These facts prove the existence, uniqueness,
and bounds for \(\bar\beta(a)\).
Endpoint-preserving step approximations exist by rounding source values
and setting a shrinking terminal interval equal to \(a_*\).
The quantile identities
\begin{equation*}
 \int_0^{a_*}|x_{a_n}-x_a|=\|a_n-a\|_{L^1},\qquad
 \int_0^{a_*}x_a(t)\,\dd t=a_*-\int_0^1a(r)\,\dd r
\end{equation*}
imply
\(\|\pi_{a_n,\beta_n}-\pi_{a,\beta}\|_\infty
\le|\beta_n-\beta|+\|a_n-a\|_{L^1}\).
The positive lower bound on precision permits dominated convergence
in \(K_a,V_a\).  If that lower bound is \(\underline\pi\), then
\[
 \|q^{\rm co}_{a_n,\beta_n}-q^{\rm co}_{a,\beta}\|_{L^1}
 \le\underline\pi^{-2}\|a_n-a\|_{L^1}
 +a_*\|\pi_{a_n,\beta_n}^{-2}-\pi_{a,\beta}^{-2}\|_\infty\to0.
\]

\emph{Logarithmic cutoff.}
At fixed \(R\), use the probability base
\(\mathfrak R_{\pi_n}\otimes\gamma(\,\cdot\mid[-R,R])\) and weight
\(e^{\tau w^{a_n}-(\beta_n-1)\tau^2/2}\).
Its positive and inverse moments of order \(p\) are bounded by
\(\exp\{pR^2\sup_n|\beta_n-1|/2+p^2R^2a_*/2\}\).
The sampled covariance arrays converge, including their common
diagonal \(a_*\).  Lemma~\ref{lem:change-density-limit} therefore
passes the expected logarithmic normalizer to the limit, including
the factor \(\gamma([-R,R])\), and defines \(K_{a,R}(\beta)\).

For a finite step approximation, let \(L_R(w)\) denote the scalar
coordinate integral and put
\[
 P_{\beta,R}(w)=\frac{L_R(w)}{L_\infty(w)}
 =\mathbb P\{|w/\beta+Z/\sqrt\beta|\le R\}.
\]
Jensen's inequality under the normalized full scalar integral gives
\[
 0\le K_{a_n}(\beta_n)-K_{a_n,R}(\beta_n)
 \le\E\langle-\log P_{\beta_n,R}(w^{a_n})\rangle.
\]
By \eqref{eq:scalar-gaussian-representation}, the averaged coordinate
is \(\mathcal N(0,V_{a_n}(\beta_n))\).  Its conditional distribution
given the field is \(\mathcal N(w/\beta_n,\beta_n^{-1})\), so
characteristic functions show that the averaged field under this
normalized integral is
\(\mathcal N(0,\beta_n^2V_{a_n}(\beta_n)-\beta_n)\).
These variances are uniformly bounded.  For \(R\ge1\), integration
over \([-1,1]\) gives
\(0\le-\log P_{\beta_n,R}(w)\le C(1+w^2)\), uniformly in \(n\).
The left side tends to zero uniformly on bounded field intervals;
Gaussian tails therefore make the logarithmic loss tend to zero
uniformly in \(n\).  Passing to the limit at fixed \(R\), and then
letting \(R\to\infty\), proves \eqref{eq:scalar-log-cutoff}.
\end{proof}

\subsection{The scalar spin transforms}

\paragraph{Block transforms and terminal completion.}
For a finite-step source as in \eqref{eq:spin-step-source}, let
\((v_\eta)\) be the weights of \(\mathfrak R_\pi\), and let
\(W^a=(W_1^a,\ldots,W_K^a)\) consist of independent copies of the
Gaussian field \(w^a\), with variance \(a_*\).
Use normalized counting measure \(\mu_K^{\rm I}\) on
\(\Sigma_K^{\rm I}=\{-1,1\}^K\) and normalized surface measure
\(\mu_K^{\rm S}\) on \(\Sigma_K^{\rm S}=S^{K-1}(\sqrt K)\).  Set
\begin{equation}\label{eq:spin-finite-block}
 \mathcal S_{\kappa,K}(a)
 =\frac1K\mathbb E\log\sum_\eta v_\eta
 \int_{\Sigma_K^\kappa}
 e^{W^a(\eta)\cdot\tau-Ka_*/2}\,\mu_K^\kappa(\dd\tau).
\end{equation}
Adding zero-variance levels leaves this finite recursion unchanged.
If the Gaussian diagonal is \(d\ge a_*\), complete the field by
\(\sqrt{d-a_*}\,z\), integrate \(z\) inside the spin integral,
and replace the compensation by \(-Kd/2\).  The resulting transform
satisfies
\begin{equation*}
 \mathcal S_{\kappa,K}(a;d)=\mathcal S_{\kappa,K}(a),
\end{equation*}
because \(\|\tau\|^2=K\) and
\begin{equation*}
 \int e^{\sqrt c\,z\cdot\tau-c\|\tau\|^2/2}\,
       \mathsf N_K(\dd z)=1\qquad(c\ge0).
\end{equation*}

These transforms satisfy the uniform continuity bound
\begin{equation}\label{eq:spin-transform-lipschitz}
 |\mathcal S_{\kappa,K}(a)-\mathcal S_{\kappa,K}(\widetilde a)|
 \le\tfrac12\|a-\widetilde a\|_{L^1}.
\end{equation}
Indeed, for \(a_t=(1-t)a+t\widetilde a\) on a common grid, the leaf
factor lies between
\(e^{-Ka_{t,*}/2-\sqrt K\|W^{a_t}\|}\) and
\(e^{-Ka_{t,*}/2+\sqrt K\|W^{a_t}\|}\).
Its recursive conditional moments are finite and positive, so
\cref{rpc:prop-quasistationarity} preserves ancestry.  Apply
\eqref{eq:gaussian-log-interpolation} to \(W^{a_t}(\eta)\cdot\tau\).
Its diagonal derivative cancels the compensation \(-Ka_{t,*}/2\), giving
\[
 \left|\frac{\dd}{\dd t}\mathcal S_{\kappa,K}(a_t)\right|
 \le\frac12\mathbb E_t\left\langle
 \frac{|\tau^1\cdot\tau^2|}{K}
 |\widetilde a(r_{12})-a(r_{12})|\right\rangle
 \le\tfrac12\|\widetilde a-a\|_{L^1}.
\]
Integration proves the bound; its unique \(L^1\)-continuous extension
defines \(\mathcal S_{\kappa,K}\) on \(\Pb\).

\paragraph{The spherical transform.}
The variance constraint enters the block transform through the multiplier
functional
\begin{equation*}
 F_a(\beta)=K_a(\beta)+\tfrac12(\beta-1-a_*),
 \qquad \beta\in\mathfrak D_a.
\end{equation*}
The explicit formula for \(K_a\) gives
\begin{equation*}
 F_a'(\beta)=\tfrac12\{1-V_a(\beta)\}.
\end{equation*}
Thus \(F_a\) is strictly convex, with unique minimizer
\(\bar\beta(a)\).  The spherical block transform is
\begin{equation}\label{eq:general-scalar-spherical-variational}
 \mathcal S_{\rm S}(a)=\min_{\beta\in\mathfrak D_a}F_a(\beta)
 =F_a(\bar\beta(a)),
\end{equation}
and
\begin{equation}\label{eq:spherical-large-block}
 \lim_{K\to\infty}\mathcal S_{{\rm S},K}(a)=\mathcal S_{\rm S}(a).
\end{equation}
For a finite-step source with \(a_*>0\), apply
\cite[Proposition~3.1 and (3.34)]{TalagrandSpherical} with
\[
 \xi(s)=a_*s^2/2,\qquad h=0,\qquad b=\beta,\qquad
 x(s)=\operatorname{Leb}\{r\in[0,1):a(r)\le a_*s\}.
\]
The overlap levels are \(a_j/a_*\), with masses \(x_j\);
coinciding terminal levels are allowed in \cite[(3.2)]{TalagrandSpherical}.
Both surface measures are normalized, so his spin pressure is
\(\varphi_K(0)=\mathcal S_{{\rm S},K}(a)+a_*/2\).
His \(d(s)=a_*\int_s^1x(u)\dd u\) satisfies
\(d(0)=\beta_-(a)\) and
\(\beta-d(s)=\pi_{a,\beta}(a_*s)\).  Changing variables in (3.34) gives
\[
 W(x,\beta)-a_*/2
 =K_a(\beta)+(\beta-1-a_*)/2=F_a(\beta).
\]
Proposition~3.1 therefore proves \eqref{eq:spherical-large-block}.
Its restriction \(b>1\) loses no minimizer, since
\(V_a(\bar\beta(a))=1\) implies \(\bar\beta(a)>1\) for \(a_*>0\).
For \(a=0\), both sides are zero and \(\bar\beta(0)=1\).

For general \(a\), choose endpoint-preserving step approximations
\(a_m\to a\).  Their minimizing precisions lie in \([1,1+a_*]\)
and satisfy \(\pi\ge1\).  The stability in
\cref{prop:rpc-spherical-insertion} shows that every subsequential limit
solves \(V_a(\beta)=1\), hence
\(\bar\beta(a_m)\to\bar\beta(a)\) and
\(F_{a_m}(\bar\beta(a_m))\to F_a(\bar\beta(a))\).
The uniform bound \eqref{eq:spin-transform-lipschitz} then passes both
\eqref{eq:general-scalar-spherical-variational} and
\eqref{eq:spherical-large-block} to \(a\).

For \(q_*=q(1-)<1\), \(\gamma_q=1\) above \(q_*\), so
\eqref{eq:spherical-entropy} becomes
\begin{equation}\label{eq:spherical-entropy-classical}
 \mathcal A_{\rm S}(q)
 =\frac12\int_0^{q_*}\frac{\dd t}{\int_t^1\gamma_q(s)\,\dd s}
   +\frac12\log(1-q_*).
\end{equation}
For its finite-grid expression, put
\(\mathscr Q_{<1}=\{q\in\mathscr Q:q(1-)<1\}\) and
\(\lambda_{\gamma_q}(t)=\int_t^1\gamma_q(s)\dd s\).
If \(q=q_j\) on the grid intervals and \(q_n<1\), its susceptibilities are
\begin{equation*}
 \lambda_n=1-q_n,\qquad
 \lambda_j=\lambda_{\gamma_q}(q_j)
 =\lambda_{j+1}+x_j(q_{j+1}-q_j),\quad j<n.
\end{equation*}
The Crisanti--Sommers entropy \cite[(4.4)--(4.6)]{TalagrandSpherical}
has the following expression in these quantile coordinates:
\begin{equation}\label{eq:spherical-step-entropy}
 2\mathcal A_{\rm S}(q)
 =\frac{q_1}{\lambda_1}
 +\sum_{j=1}^{n-1}\frac1{x_j}\log\frac{\lambda_j}{\lambda_{j+1}}
 +\log\lambda_n.
\end{equation}
The quantile identity and dominated convergence also give
\begin{equation}\label{eq:spherical-entropy-continuity}
 \mathcal A_{\rm S}(q_m)\to\mathcal A_{\rm S}(q)
\end{equation}
whenever \(q_m\to q\) in \(L^1\) with a common endpoint bound below one.

\paragraph{Ising spins as constraint insertions.}
\label{par:ising-coordinate-ancestry}
For a step source \(a\) on \(\pi\), choose
\(A>\max\{1,\|a\|_\infty\}\), and put
\[
 u_A(x)=\log\cosh(\sqrt A\,x)-A/2,\qquad
 c_a(r)=a(r)/A\ (r<1),\quad c_a(1)=1.
\]
For \(X=(w^a+\sqrt{A-a_*}\,z)/\sqrt A\), terminal Gaussian
integration gives
\begin{align*}
 \int e^{u_A(X)}\,\gamma(\dd z)
 &=e^{-a_*/2}\cosh w^a,\\
 \frac{\int u_A'(X)e^{u_A(X)}\,\gamma(\dd z)}
      {\int e^{u_A(X)}\,\gamma(\dd z)}
 &=\sqrt A\,\tanh w^a.
\end{align*}
The first identity identifies \(K\) spin coordinates with \(K\)
independent constraint insertions; the second identifies their response.
The recursion for this spin insertion is the step-coefficient
solution of \eqref{eq:intro-parisi-pde} with terminal \(\log\cosh\)
and horizon \(a_*\), followed by subtraction of \(a_*/2\).
Thus it is exactly \eqref{eq:intro-ising-transform}.
For general sources, use endpoint-preserving step approximations
\(a_n\) on partitions \(\pi_n\) of vanishing mesh.  The finite bases
\(\mathfrak R_{\pi_n}\otimes(\delta_{-1}+\delta_1)/2\) and weights
\(e^{\tau w^{a_n}-a_*/2}\) have uniformly bounded positive and inverse
moments of every order.  Their sampled Gaussian arrays converge, with
off-diagonal path \(a\) and fixed diagonal \(a_*\).
Lemma~\ref{lem:change-density-limit} therefore defines the limiting
averaged Ising replica laws \(\E\langle\cdot\rangle_a^{\rm I}\).
Control stability and the quantile identity pass the transform
identification to the limit.  The constraint identities in
\cref{lem:rpc-constraint-insertion} give ancestry preservation and
\begin{equation}\label{eq:ising-tensorization}
 \mathcal S_{{\rm I},K}(a)=\mathcal S_{{\rm I},1}(a)
 =\mathcal S_{\rm I}(a),\qquad K\ge1,
\end{equation}
\begin{equation*}
 \mathcal S_{\rm I}(a)=\Phi_{u_A}(c_a),\qquad
 \widehat q_a^{\rm I}=A^{-1}p_{c_a,u_A}.
\end{equation*}
In particular, \(\widehat q_a^{\rm I}\) is nonnegative,
nondecreasing, bounded by one, and constant on the plateaus of \(a\).
Choose one \(A\) for both endpoints of a segment
\(a_s=(1-s)a+s\widetilde a\).  The constraint derivative and response
continuity give
\begin{equation}\label{eq:ising-derivative-from-constraint}
 \left.\frac{\dd}{\dd s}\right|_{s=0+}\mathcal S_{\rm I}(a_s)
 =-\tfrac12\langle\widehat q_a^{\rm I},\widetilde a-a\rangle.
\end{equation}
For general sources, apply the integrated finite-grid identity and pass
to the limit using the \(L^1\)-continuity of the response and
\(0\le\widehat q^{\rm I}\le1\).  The fixed \(A\) keeps every completed
constraint variance equal to one.

\paragraph{The Chen--Issa--Mourrat concavity input.}
\label{par:ising-cim-normalization}
\label{prop:centered-Ising-response-input}
Concavity follows by identifying the spin transform with their
symmetric-Ising functional:
\begin{equation*}
 \mathcal S_{\rm I}(a)=-\psi^\circ(a/2).
\end{equation*}
Indeed, \(\sqrt2\,w^{a/2}\) has covariance \(a\), the diagonal
subtraction is \(a_*/2\), and the Ising prior is normalized as in
\eqref{eq:spin-finite-block}.  The identity extends from finite grids by
\eqref{eq:spin-transform-lipschitz}.  Convexity of \(\psi^\circ\) in
\cite[Proposition~2.2]{CIM} gives concavity of \(\mathcal S_{\rm I}\).
% This is the only Chen--Issa--Mourrat input and supplies the supporting
% inequality; constraint insertion gives the response and derivative.

\begin{corollary}[Responses and supporting equalities]
\label{prop:scalar-coordinate-laws}
For either prior, \(\widehat q_a^\kappa\), with true diagonal one,
belongs to \(\mathscr Q\) and is constant on every plateau of \(a\).
The finite and limiting Ising scalar laws preserve genealogy:
\begin{equation*}
 \E\langle F\rangle_{\pi;a}^{\rm I}=\E_\pi F,
 \qquad \E\langle F\rangle_a^{\rm I}=\E_U F
\end{equation*}
for every bounded Borel test of finitely many replica ancestries.
Their supporting equality is
\begin{equation}\label{eq:ising-exposure-identity}
 \mathcal A_{\rm I}(\widehat q_a^{\rm I})
 =\mathcal S_{\rm I}(a)+\tfrac12\langle a,\widehat q_a^{\rm I}\rangle
 <\infty.
\end{equation}
For the spherical response,
\(\widehat q_a^{\rm S}(1-)=1-\bar\beta(a)^{-1}\le a_*/(1+a_*)<1\),
and
\begin{align*}
 K_a(\bar\beta(a))
 &=\mathcal S_{\rm S}(a)+\tfrac12\{1+a_*-\bar\beta(a)\},
 \\
 \mathcal A_{\rm S}(\widehat q_a^{\rm S})
 &=\mathcal S_{\rm S}(a)+\tfrac12\langle a,\widehat q_a^{\rm S}\rangle.
\end{align*}
\end{corollary}

\begin{proof}[Proof of the Ising assertions]
The constraint specialization gives genealogy preservation and the
response properties.  Concavity and
\eqref{eq:ising-derivative-from-constraint} give
\[
 \mathcal S_{\rm I}(\widetilde a)
 \le\mathcal S_{\rm I}(a)
       -\tfrac12\langle\widehat q_a^{\rm I},\widetilde a-a\rangle.
\]
The supremum over \(\widetilde a\in\Pb\), after adding
\(\langle\widehat q_a^{\rm I},\widetilde a\rangle/2\), is attained
at \(a\), proving \eqref{eq:ising-exposure-identity}.
\end{proof}

\subsection{The spherical supporting equalities}
\label{sec:spherical-supporting-equalities}

\begin{proof}[Proof of the spherical assertions of \cref{prop:scalar-coordinate-laws}]
The definition \eqref{eq:general-scalar-spherical-variational} gives
the normalizer identity.  Also
\(\widehat q_a^{\rm S}(1-)=1-\bar\beta(a)^{-1}\le a_*/(1+a_*)\);
monotonicity and the plateau property follow from
\eqref{eq:general-response-and-variance}.

One finite-grid calculation proves the supporting equality and the
Fenchel inequality needed below.  Given \(q,a\) on a common grid with
\(q_n<1\), set \(d_1=\lambda_1^{-1}\) and
\(d_{j+1}=d_j+x_j(a_{j+1}-a_j)\).  Define
\(h(z)=z-1-\log z\ge0\), and recall \(x_n=1\).  Substituting
\eqref{eq:scalar-log-normalizer} and
\eqref{eq:spherical-step-entropy}, then collecting terms, gives
\begin{equation}\label{eq:spherical-fenchel-gap}
 2F_a(d_n)+\langle q,a\rangle
 =2\mathcal A_{\rm S}(q)
 -\sum_{j=2}^n\left(\frac1{x_{j-1}}-\frac1{x_j}\right)h(d_j\lambda_j).
\end{equation}
Here \(\lambda_1=1-\int_0^1q\) cancels the root term.  For the remaining
terms, \(\lambda_{j-1}=\lambda_j+x_{j-1}(q_j-q_{j-1})\) makes the
coefficient of \(d_j\) equal to
\(-\lambda_j(1/x_{j-1}-1/x_j)\), including \(j=n\).
Since \(\mathcal S_{\rm S}(a)\le F_a(d_n)\), this yields
\begin{equation}\label{eq:spherical-fenchel}
 \mathcal S_{\rm S}(a)+\tfrac12\langle q,a\rangle
 \le\mathcal A_{\rm S}(q).
\end{equation}

For \(q=\widehat q_a^{\rm S}\), the explicit response formula
\eqref{eq:scalar-coordinate-response} and \(V_a(\bar\beta)=1\)
give, with the minimizing precisions,
\begin{equation*}
 q_1=a_1d_1^{-2},\qquad q_n=1-d_n^{-1},\qquad
 q_{j+1}-q_j=\frac{d_j^{-1}-d_{j+1}^{-1}}{x_j}.
\end{equation*}
Thus \(\lambda_j=d_j^{-1}\), the matching relation in
\cite[Lemma~4.1, (4.8)--(4.11)]{TalagrandSpherical}; all gaps vanish
and \(d_n\) is the minimizer.  Consequently
\begin{equation*}
 \mathcal S_{\rm S}(a)
 =\mathcal A_{\rm S}(q)-\tfrac12\langle q,a\rangle.
\end{equation*}
Endpoint-preserving step approximations of \(a\), the stability in
\cref{prop:rpc-spherical-insertion}, and the common response bound
\(q(1-)\le a_*/(1+a_*)\) pass this equality to general sources by
\eqref{eq:spherical-entropy-continuity}.  The same continuity and
\eqref{eq:spin-transform-lipschitz} extend
\eqref{eq:spherical-fenchel} to all \(q\in\mathscr Q_{<1}\), \(a\in\Pb\).
\end{proof}

For later use, let \(\overline q_x\) be the constant off-diagonal path:
\begin{equation*}
 \overline q_x(r)=x\quad(r<1),\qquad \overline q_x(1)=1,
 \qquad 0\le x<1.
\end{equation*}

\begin{theorem}[Spherical conjugate formula]
\label{thm:spherical-entropy-identification}
For every \(q\in\mathscr Q\), the cost defined in
\eqref{eq:spherical-entropy} satisfies
\begin{equation}\label{eq:spherical-conjugate-identification}
 \mathcal A_{\rm S}(q)
 =\sup_{a\in\Pb}\{\mathcal S_{\rm S}(a)+\tfrac12\langle a,q\rangle\}.
\end{equation}
For every \(L^1\)-continuous \(\Psi:\mathscr Q\to\mathbb R\),
\begin{equation}\label{eq:spherical-endpoint-density}
 \inf_{q\in\mathscr Q}\{\mathcal A_{\rm S}(q)+\Psi(q)\}
 =\inf_{q\in\mathscr Q_{<1}}\{\mathcal A_{\rm S}(q)+\Psi(q)\}.
\end{equation}
\end{theorem}

\begin{proof}
Denote the supremum in \eqref{eq:spherical-conjugate-identification} by
\(C(q)\).  The Fenchel inequality gives \(C(q)\le\mathcal A_{\rm S}(q)\)
when \(q(1-)<1\).  For step \(q\), construct its inverse source by
\[
 a_1=\frac{q_1}{\lambda_1^2},\qquad
 a_{j+1}-a_j=\frac1{x_j}
       \left(\frac1{\lambda_{j+1}}-\frac1{\lambda_j}\right).
\]
This is a bounded nonnegative nondecreasing source.  With
\(d_j=\lambda_j^{-1}\), its scalar response is \(q\) and
\(V_a(d_n)=q_n+d_n^{-1}=1\), so the trial precision is the minimizer.
All gaps in \eqref{eq:spherical-fenchel-gap} vanish, giving equality,
also when \(n=1\) and the sum is empty.
For general \(q(1-)\le\bar q<1\), take step approximations \(q_m\)
with this same endpoint bound.  Their inverse sources obey
\(\|a_m\|_\infty\le\bar q/(1-\bar q)^2\), and hence
\[
 C(q)\ge\mathcal S_{\rm S}(a_m)+\tfrac12\langle q,a_m\rangle
 =\mathcal A_{\rm S}(q_m)+\tfrac12\langle q-q_m,a_m\rangle
 \longrightarrow\mathcal A_{\rm S}(q).
\]

For any \(q\in\mathscr Q\), put \(q_\delta=(1-\delta)q\) off the
diagonal and \(q_\delta(1)=1\).  Nonnegativity of the sources and
interchange of the two suprema give
\[
 C(q_\delta)
 =\sup_{a\in\Pb}\{\mathcal S_{\rm S}(a)
                   +\tfrac12(1-\delta)\langle a,q\rangle\}
 \uparrow C(q).
\]
Since \(\gamma_{q_\delta}\downarrow\gamma_q\), monotone convergence
in \eqref{eq:spherical-entropy} gives
\(\mathcal A_{\rm S}(q_\delta)\uparrow\mathcal A_{\rm S}(q)\).
The identity \(C(q_\delta)=\mathcal A_{\rm S}(q_\delta)\) therefore
proves \(C(q)=\mathcal A_{\rm S}(q)\), including infinite values.  In particular, the spherical supporting equality is
\begin{equation*}
 \mathcal A_{\rm S}(\widehat q_a^{\rm S})
 =\mathcal S_{\rm S}(a)+\tfrac12\langle a,\widehat q_a^{\rm S}\rangle.
\end{equation*}
Finally, \(q_\delta\to q\) in \(L^1\) and
\(\mathcal A_{\rm S}(q_\delta)\to\mathcal A_{\rm S}(q)\).
Applying this approximation to every finite-cost competitor proves
\eqref{eq:spherical-endpoint-density}; the reverse inequality follows
from inclusion of domains.
\end{proof}


\begin{proof}[Proof of \cref{prop:scalar-proof-input}]
The Ising conjugate formula is \eqref{eq:intro-ising-cost}; the spherical
formula is \cref{thm:spherical-entropy-identification}.
Corollary~\ref{prop:scalar-coordinate-laws} gives the supporting
equalities.  The left-hand side of \eqref{eq:overview-spin-endpoint} is
\(\mathcal S_{\kappa,N}(a)\) from \eqref{eq:spin-finite-block}.
Its asserted value and limit follow from \eqref{eq:ising-tensorization}
and \eqref{eq:spherical-large-block}, respectively.
\end{proof}

\section{Cavity in $N$}
\label{sec:cavity-N}

We first prove the overlap equation in \cref{prop:0916-coordinate}.
For Ising spins, separating one coordinate determines its normalized
law.  On the sphere, integration by parts gives the equation directly.
We then compute the logarithmic increment from inserting a fixed block
of coordinates at \(t=1\).  This last calculation is the only one that
uses the quadratic coefficient \(B_N\).

\subsection{The Ising overlap equation}
\label{sec:simple-coordinate-equation}

\begin{proof}[Proof of the Ising assertion of \cref{prop:0916-coordinate}]
Put \(D=N+1\), write \(\sigma^D=(\sigma,\tau)\), and separate
the last entries \(z_k\) of the constraint vectors.  In the measure
with the last coordinate deleted, retain the denominator \(\sqrt D\):
\[
 X_k^- =\sqrt{t/D}\,g_k\cdot\sigma+\sqrt{1-t}\,Y_k,
 \qquad X_k^D=X_k^-+\sqrt{t/D}\,z_k\tau.
\]
Use \(V_N\) in this deleted measure.  Its fresh constraint variance is
\(1-t/D\), and its off-diagonal covariance is
\(t(N/D)R+(1-t)q^{0,\sharp}(r)\).  The variable-diagonal version of
\cref{prop:0916-rows} therefore applies.  Select a common refinement
for this measure and its one- and two-constraint deletions, and write
\(q^-\) for their common limiting spin path.  Set
\[
 c^-=(1-t)q^0+tq^-,\qquad
 a^-=(1-t)a^0+t\alpha p_{c^-}.
\]
No comparison with the usual \(N\)-dimensional denominator is made.

The lifted and full overlaps differ by at most \(2/D\), while their
ancestries agree.  The covariance of \(V_N\) is \(N^{3/4}\) times
a uniformly bounded Lipschitz kernel.  Consequently replacing
\(V_D(\sigma,\tau,\eta)\) by \(V_N(\sigma,\eta)\) changes its
covariance by \(O(N^{-1/4})\), uniformly in the states and amplitudes.
Apply \cref{lem:tilted-comparison}(ii) with the perturbations removed
from the common density.  The replacement changes bounded replicated
tests by \(o(1)\).

Since \(\tau^2=1\), Taylor expansion gives
\begin{equation*}
 \begin{gathered}
 \sum_{k\le M}\{u_k(X_k^D)-u_k(X_k^-)\}
 =\tau Z_N+\frac{tM}{2D}C_N+\varepsilon_N,\\
 Z_N=\sqrt{t/D}\sum_{k\le M}z_k u_k'(X_k^-),\quad
 C_N=M^{-1}\sum_{k\le M}u_k''(X_k^-).
 \end{gathered}
\end{equation*}
Before insertion, the centered quadratic term has conditional second
moment at most \(2t^2ML_2^2/D^2\).  The cubic remainder has averaged
second moment \(O(N^{-1})\), by the uniform third-derivative bound
and Gaussian moments.  Thus \(\E\mu_N(\varepsilon_N^2)=O(N^{-1})\),
where \(\mu_N\) is the deleted measure times the symmetric Ising
prior of \(\tau\).
The exact increment is uniformly Lipschitz in \((z_k)\), with squared
constant at most \(tML^2/D\), and its conditional Gaussian mean has
absolute value at most \(tML_2/(2D)\).  The approximate increment
has the same positive and inverse exponential moment bounds, since
\(C_N\) is bounded and \(Z_N\) has bounded conditional variance.
The fresh source term \(\tau W_{t,D}\) also has bounded Gaussian
variance.  Lemma~\ref{lem:tilted-comparison}(i) therefore transfers
the Taylor estimate through the normalizer.

Apply \eqref{eq:empirical-one-row-limit} with
\(\chi(\theta,x)=u_\theta''(x)\) and with
\(\chi(\theta,x)=u_\theta'(x)^2\).  It gives
\[
 C_N\longrightarrow c'',\qquad
 Q_{11}^-\longrightarrow p_{\rm diag}(c^-)
 \quad\text{in averaged }L^2,
\]
where \(c''\) is deterministic on the selected refinement.
Row reconstruction also gives
\(Q_{12}^-\Rightarrow p_{c^-}(U_{12})\), jointly with the retained
array.  Conditional on the old replicas, \(Z_N+W_{t,D}\) is Gaussian,
with off-diagonal covariance converging to \(a^-(U_{12})\) and
diagonal converging to
\((1-t)a_*^0+t\alpha p_{\rm diag}(c^-)\).

The covariance and coefficient limits, together with the preceding
moment bounds, verify \cref{lem:change-density-limit}.  Thus the
normalized insertion law passes to its scalar limit.  Represent the
excess diagonal variance by an independent terminal Gaussian of
variance
\(\delta=t\alpha\{p_{\rm diag}(c^-)-p_{c^-}(1-)\}\).
Its integral is \(e^{\delta\tau^2/2}=e^{\delta/2}\).
The factor from \(C_N\) tends to \(e^{t\alpha c''/2}\).
Both factors are constants and cancel from normalization.  The
remaining law is exactly the scalar Ising law with weight
\(e^{\tau w^{a^-}-a_*^-/2}\).  This argument uses concentration of
\(C_N\); the equality \(\tau^2=1\) alone would not remove a
coefficient depending on the old configuration.

The scalar Ising law preserves the genealogy, by
\cref{prop:scalar-coordinate-laws}.  The old array after insertion
therefore still has law \((q^-(U),\rho_d(U))\).  Since full and old
overlaps differ by \(O(N^{-1})\), the limiting path \(q\) of the
full system equals \(q^-\).  Coordinate exchangeability gives
\begin{equation*}
 \E\langle\tau^1\tau^2f(R_{12}^D,r_{12})\rangle_D
 =\E\langle R_{12}^Df(R_{12}^D,r_{12})\rangle_D.
\end{equation*}
Passing this bounded identity to the scalar law yields
\[
 \int_0^1 f(q(r),\rho_d(r))
       \{\widehat q_a^{\rm I}(r)-q(r)\}\,\dd r=0,
 \qquad a=(1-t)a^0+t\alpha p_{(1-t)q^0+tq}.
\]
Both paths factor through \((q,\rho_d)\), by their plateau
properties.  Continuous tests determine the resulting signed measure,
so \(q=\widehat q_a^{\rm I}\) almost everywhere.  The supporting
equality follows from \cref{prop:scalar-coordinate-laws}.
\end{proof}

\subsection{The spherical overlap equation}

\begin{proof}[Proof of the spherical assertion of \cref{prop:0916-coordinate}]
Let \(F=f(R_{12},r_{12})\), with \(f\) continuously differentiable
in its first variable; the second variable lies in the finite ancestry
alphabet.  First retain only the first \(J\) terms of \(V_N\), and
write \(H^{[J]}\) and \(\langle\cdot\rangle_J\) for this Hamiltonian
and its replica averages.  Put
\[
 v=\sigma^2-R_{12}\sigma^1,\qquad d_\ell=N^{-1}v\cdot\sigma^\ell;
 \qquad d_1=0,\quad d_2=1-R_{12}^2,\quad
 d_3=R_{23}-R_{12}R_{13}.
\]
The vector \(v\) is tangent to the sphere at \(\sigma^1\).  Its
spherical divergence and its action on the test are
\[
 \operatorname{div}_{\mathbb S}(v/N)=-(1-N^{-1})R_{12},\qquad
 (v/N)\cdot\nabla F=N^{-1}d_2\partial_Rf.
\]
Integrating the divergence of \((v/N)F e^{H^{[J]}(\sigma^1)}\)
in the first replica therefore gives
\begin{equation*}
 (1-N^{-1})\E\langle R_{12}F\rangle_J
 =N^{-1}\E\langle d_2\partial_Rf\rangle_J
 +N^{-1}\E\langle Fv\cdot\nabla H^{[J]}(\sigma^1)\rangle_J.
\end{equation*}

For the Gaussian integrations, condition on the underlying cascade
weights and channel marks, and hold each replica state fixed inside
its integral.  Differentiating a two-replica average with respect to
a Gaussian coordinate \(g\) gives
\[
 \partial_g\langle A\rangle_J
 =\left\langle\partial_gA+
 A\{\partial_gH^{[J],1}+\partial_gH^{[J],2}
                         -2\partial_gH^{[J],3}\}\right\rangle_J.
\]
The two numerator densities give the coefficients \(1,1\), and the
square of the normalizer gives \(-2\).  Neither \(F\) nor \(v\)
has an explicit Gaussian derivative; their dependence through the
sampling measure is included in these density terms.

Write \(u_k^{\prime\ell}=u_k'(X_k^\ell)\) and
\(H_{\rm con}=\sum_k u_k(X_k)\).  Since
\(\partial_{g_{ki}}X_k^\ell=\sqrt{t/N}\,\sigma_i^\ell\),
Gaussian integration by parts gives the full constraint contribution:
\[
 \begin{aligned}
 &N^{-1}\E\langle Fv\cdot\nabla H_{\rm con}(\sigma^1)\rangle_J\\
 &\quad=\frac tN\sum_k\E\left\langle F\left[
 d_1u_k''(X_k^1)+u_k^{\prime1}
 (d_1u_k^{\prime1}+d_2u_k^{\prime2}-2d_3u_k^{\prime3})
 \right]\right\rangle_J\\
 &\quad=\frac{tM}{N}\E\langle
 F(d_2Q_{12}-2d_3Q_{13})\rangle_J.
 \end{aligned}
\]
The \(u''\) term and the diagonal \((u')^2\) term both multiply
\(d_1=0\), which is the required cancellation.

For the source, set \(b_{11}=(1-t)a_*^0\) and
\(b_{1\ell}=(1-t)a^{0,\sharp}(r_{1\ell})\) for \(\ell=2,3\).
Its covariance and the same density derivative give
\[
 N^{-1}\E\langle Fv\cdot W_t(\eta^1)\rangle_J
 =\E\langle F(d_1b_{11}+d_2b_{12}-2d_3b_{13})\rangle_J.
\]
Again the diagonal term vanishes.  Thus the constraints and source
together contribute \(\E\langle F(d_2A_{12}^N-2d_3A_{13}^N)\rangle_J\),
where
\[
 A_{\ell m}^N=t(M/N)Q_{\ell m}+(1-t)a^{0,\sharp}(r_{\ell m}).
\]

The truncated perturbation has covariance \(N^{3/4}k_J(R,r)\), where
\(k_J(R,r)=\sum_{j\le J}4^{-j}x_j^2K_j((1+R)/2,r)\).
Differentiating this covariance in its first spin argument shows that
its contribution is exactly
\[
 N^{-1/4}\E\left\langle F\left[
 d_2\partial_Rk_J(R_{12},r_{12})
 -2d_3\partial_Rk_J(R_{13},r_{13})\right]\right\rangle_J.
\]
Here the self term also vanishes because \(d_1=0\).
Since \(|d_2|,|d_3|\le1\) and
\(\|\partial_Rk_J\|_\infty\le\frac12\sum_j4^{-j}x_j^2\le2/3\),
this is bounded by \(2\|F\|_\infty N^{-1/4}\), uniformly in \(J\).

Now keep \(N\) fixed and let \(J\to\infty\).
The covariance tail is \(O(N^{3/4}4^{-J})\), so
\cref{lem:tilted-comparison}(ii) passes all the bounded replica
averages in the combined identity to the full perturbation.
The covariance-derivative tail is \(O(4^{-J})\), which also passes
the displayed perturbation term to the limit.  No convergence of
sample gradients is needed.  Dropping the truncation subscript, we obtain
\begin{equation}\label{eq:spherical-replicated-equation}
 \begin{split}
 (1-N^{-1})\E\langle R_{12}F\rangle
 ={}&N^{-1}\E\langle d_2\partial_Rf\rangle
 +\E\langle F(d_2A_{12}^N-2d_3A_{13}^N)\rangle+\varepsilon_N,\\
 &|\varepsilon_N|\le2\|F\|_\infty N^{-1/4}.
 \end{split}
\end{equation}

Row reconstruction gives
\(A_{\ell m}^N\Rightarrow a(U_{\ell m})\), jointly with the
overlap--ancestry array.  We may now let \(N\to\infty\) in
\eqref{eq:spherical-replicated-equation}: the test-derivative term
and \(\varepsilon_N\) vanish.  To evaluate the limit, recall the
conditional triple law of the uniform RPC: for bounded \(H\),
\begin{equation}\label{eq:uniform-rpc-triple-law}
 \begin{split}
 2\E[H(U_{13},U_{23})\mid U_{12}=r]
 ={}&\int_0^r H(s,s)\,\dd s\\
 &+\int_r^1\{H(s,r)+H(r,s)\}\,\dd s+rH(r,r).
 \end{split}
\end{equation}
The GG identity gives the joint law of \((U_{12},U_{13})\) as
one half product measure plus one half diagonal measure;
ultrametricity and replica symmetry then give this formula.

Set
\[
 \begin{gathered}
 B=\int_0^1qa,\qquad C(r)=1-B+\int_r^1a,\qquad
 I(r)=\int_0^rqa,\\
 D(r)=1-\int_r^1q,\qquad \Delta(r)=D(r)-rq(r).
 \end{gathered}
\]
Apply \eqref{eq:uniform-rpc-triple-law} to
\(H(s,v)=\{q(v)-q(r)q(s)\}a(s)\).  Collecting terms yields
\[
 \int_0^1 f(q(r),\rho_d(r))
       \{q(r)C(r)+I(r)-a(r)\Delta(r)\}\,\dd r=0.
\]
On every common fiber of \(q\) and \(\rho_d\), the path \(a\)
is constant.  The expression in braces is also constant there:
the derivatives of its integral terms cancel.  Thus it factors
through \((q,\rho_d)\), by the factorization argument in
\cref{lem:finite-rpc-representation}.  Smooth tests in the first
variable and arbitrary tests on the finite ancestry alphabet are
dense in continuous tests.  Equality of the resulting finite measures
therefore implies
\begin{equation}\label{eq:spherical-path-equation}
 q(r)C(r)+I(r)=a(r)\Delta(r)\quad\text{for a.e. }r.
\end{equation}

This equation can be solved without assuming differentiability of
the paths.  The function \(CD+rI\) is absolutely continuous, and
\eqref{eq:spherical-path-equation} gives
\[
 (CD+rI)'=-aD+Cq+I+rqa=0\quad\text{a.e.}
\]
Its value at \(r=1\) is one.  Expanding the product and using
\eqref{eq:spherical-path-equation} proves
\begin{equation}\label{eq:spherical-precision-matching}
 P(r)\Delta(r)=1,\qquad P(r):=C(r)+ra(r).
\end{equation}
Right-continuous versions give this identity at every \(r<1\).
At the terminal point it yields
\[
 (1+a_*-B)(1-q_*)=1.
\]
Thus \(q_*<1\), and \(\beta:=1+a_*-B=(1-q_*)^{-1}\).
Layer cake gives
\(P(r)=\pi_{a,\beta}(a(r))\), using the precision function defined
in Section~2.  Moreover \(P(0)=1/\Delta(0)>0\), so
\(\beta\in\mathfrak D_a\).

In the sense of Stieltjes measures, \(\dd P=r\,\dd a\) and
\(\dd\Delta=-r\,\dd q\).  The continuous parts of
\eqref{eq:spherical-precision-matching} give
\(\dd q=\dd a/P^2\) away from zero.  At a jump at \(r>0\),
the same identity gives exactly
\[
 q(r)-q(r-)=\frac{a(r)-a(r-)}{P(r-)P(r)}
 =\int_{a(r-)}^{a(r)}
   \frac{\dd v}{[P(r-)+r\{v-a(r-)\}]^2}.
\]
At zero, \eqref{eq:spherical-path-equation} gives
\(q(0)=a(0)/P(0)^2\).  Together these identities prove
\[
 q(r)=\int_0^{a(r)}\pi_{a,\beta}(v)^{-2}\,\dd v
      =q^{\rm co}_{a,\beta}(r).
\]
The terminal identity says \(V_a(\beta)=q_*+\beta^{-1}=1\).
Uniqueness of this precision gives \(\beta=\bar\beta(a)\), hence
\(q=\widehat q_a^{\rm S}\).  The supporting equality follows from
\cref{prop:scalar-coordinate-laws}.
\end{proof}

\subsection{The logarithmic increment of a fixed block}
\label{sec:simple-small-coordinate}

\begin{proof}[Proof of \cref{prop:log-block-insertion}]
Fix \(K\ge1\), put \(D=N+K\), and use \(t=1\).
The old measure has \(M\) constraints in dimension \(N\).
For \(\tau\in\mathbb R^K\), use
\[
 \iota_\tau^{\rm I}\sigma=(\sigma,\tau),\qquad
 \iota_\tau^{\rm S}\sigma=
       \left(\sqrt{\frac{D-\|\tau\|^2}{N}}\,\sigma,\tau\right).
\]
In the Ising case \(\tau\in\{-1,1\}^K\).
On every fixed cube \([-R,R]^K\), the old and full overlaps satisfy
\begin{equation*}
 \max_{\ell,m}|R_{\ell m}^D-R_{\ell m}^N|
 \le C_{K,R}/N,
\end{equation*}
and their ancestry labels agree.  The covariance comparison used above
therefore replaces \(V_D\circ\iota\) by \(V_N\) with error
\(O_{K,R}(N^{-1/4})\) in expected logarithms.  In this comparison
the common density contains neither perturbation; any old partition
function in the insertion ratio cancels from the difference.

For spherical spins, the density of the last \(K\) coordinates is
\begin{equation*}
 f_{D,K}(\tau)=
 \frac{\Gamma(D/2)}{(\pi D)^{K/2}\Gamma((D-K)/2)}
 \left(1-\frac{\|\tau\|^2}{D}\right)_+^{(D-K-2)/2}.
\end{equation*}
Conditionally on \(\tau\), the first coordinates have the spherical
law specified by \(\iota_\tau^{\rm S}\).
Stirling's formula gives
\begin{equation*}
 f_{D,K}(\tau)=(2\pi)^{-K/2}e^{-\|\tau\|^2/2}
                      \{1+O_{K,R}(N^{-1})\}
 \quad\text{uniformly on }[-R,R]^K.
\end{equation*}
Let \(\nu_{N,R}\) be this prior conditioned on the cube, and
\(\zeta_{N,R}\) its mass.  For Ising spins take \(R\ge1\),
\(\nu_{N,R}\) the product Ising prior and \(\zeta_{N,R}=1\).
Write \(\mu_N=G_{N,M,1}\otimes\nu_{N,R}\).

Separate \(g_k^D=(g_k,z_k)\), where \(z_k\in\mathbb R^K\), and put
\[
 c_D(\tau)=\sqrt{1-\|\tau\|^2/D},\qquad
 \xi_k=D^{-1/2}z_k\cdot\tau,\qquad
 Z_i=D^{-1/2}\sum_k z_{ki}u_k'(h_k).
\]
The exact new constraint input is \(c_D(\tau)h_k+\xi_k\), for
both priors.  Taylor expansion gives
\begin{equation*}
 \begin{split}
 U_N&:=\sum_k\{u_k(c_Dh_k+\xi_k)-u_k(h_k)\}
       =E_N+\tfrac12T_N+\mathcal R_N,\\
 E_N&=\tau\cdot Z+\frac{M\|\tau\|^2}{2D}B_N,\qquad
 T_N=\sum_k u_k''(h_k)
                  \{\xi_k^2-\|\tau\|^2/D\}.
 \end{split}
\end{equation*}
Here \(B_N=M^{-1}\sum_k\{u_k''(h_k)-h_ku_k'(h_k)\}\).
Before insertion the \(z_k\)'s are independent of \(\mu_N\), so
\[
 \E_zT_N^2=\frac{2\|\tau\|^4}{D^2}\sum_k u_k''(h_k)^2
                     \le C_{K,R}/N.
\]
Taylor's theorem also gives
\[
 |\mathcal R_N|\le C_{K,R}\left\{
 D^{-2}\sum_k(|h_k|+h_k^2)
 +D^{-1}\sum_k|h_k\xi_k|+\sum_k|\xi_k|^3
 +D^{-3}\sum_k|h_k|^3\right\}.
\]
Using \(\sum_k h_k^2\le M\Lambda_N\),
\(\sum_k|h_k|^3\le(M\Lambda_N)^{3/2}\), and Gaussian moments,
\cref{lem:spectral-moments} gives
\begin{equation*}
 \E\mu_N(T_N^2+|\mathcal R_N|^2)\le C_{K,R}/N.
\end{equation*}

For fixed old state and \(\tau\), the squared Lipschitz constant
of \(U_N\) in \((z_k)\) is at most \(ML^2\|\tau\|^2/D\),
and
\[
 |\E_zU_N|\le L|c_D-1|\sum_k|h_k|
                   +\frac{ML_2\|\tau\|^2}{2D}
          \le C_{K,R}(1+\sqrt{\Lambda_N}).
\]
Gaussian concentration bounds its positive and negative exponential
moments.  The same bound holds for \(E_N\), since its conditional
Gaussian variance is bounded and
\(|B_N|\le C_u(1+\sqrt{\Lambda_N})\).
H\"older and the spectral exponential bound therefore give, for each
fixed \(p<\infty\),
\begin{equation}\label{eq:coordinate-exponent-moment-envelope}
 \sup_{0\le s\le1}\E\mu_N
 \left(e^{p\{(1-s)E_N+sU_N\}}+
       e^{-p\{(1-s)E_N+sU_N\}}\right)\le C_{p,K,R}.
\end{equation}
Only the exact and quadratic endpoint exponents are used here; no
exponential moment of the cubic Taylor bound is required.
Lemma~\ref{lem:tilted-comparison}(i) now changes the expected log
normalizer by \(O_{K,R}(N^{-1/2})\) when \(U_N\) is replaced by
\(E_N\).

On the selected refinement, row reconstruction and
\cref{lem:empirical-row-coefficients} give
\[
 Q_{12}\Rightarrow p_q(U_{12}),\qquad
 Q_{11}\longrightarrow p_{\rm diag}(q),\qquad
 B_N\longrightarrow\langle q,p_q\rangle-p_{\rm diag}(q),
\]
jointly with the old array; the last two limits hold in averaged
\(L^2\).  Conditional on the old measure and sampled states, the
inserted Gaussian vector has
covariance
\[
 \E_z Z_i^\ell Z_j^m=\delta_{ij}(M/D)Q_{\ell m}.
\]
Put \(a=\alpha p_q\).  Its limit consists of \(K\) independent
Gaussian trees of covariance path \(a\), each completed by independent
terminal variance \(\alpha\{p_{\rm diag}(q)-p_q(1-)\}\).
The covariance limits and
\eqref{eq:coordinate-exponent-moment-envelope} verify
\cref{lem:change-density-limit} at fixed \(K,R\).
Equivalently, approximate the paths by finite cascades, preserving
\(a_*\); sampled covariances, coefficients, and logarithmic
normalizers have the same limit.  The prior mass
\(\zeta_{N,R}\) is restored after using the probability base
\(\mu_N\).

Integrating the terminal Gaussians cancels the
\(p_{\rm diag}(q)\) term.  For Ising spins the resulting block
weight is
\[
 e^{K\langle q,a\rangle/2}
 \prod_{i=1}^K e^{\tau_iw_i^a-a_*/2}
       \left(\frac{\delta_{-1}+\delta_1}{2}\right)^{\!\otimes K}
                   (\dd\tau).
\]
The independent Gaussian trees have product normalized transitions;
their logarithmic RPC recursion is additive.  Its expected logarithm
is \(K\{\mathcal S_{\rm I}(a)+\langle q,a\rangle/2\}\).
The Ising overlap equation already proved identifies this as
\(K\mathcal A_{\rm I}(q)\).

For spherical spins the limiting cutoff weight is
\[
 \mathbf1_{[-R,R]^K}(\tau)
 \exp\left\{\sum_{i=1}^K\tau_iw_i^a
                         -\frac\beta2\|\tau\|^2\right\}
 \frac{\dd\tau}{(2\pi)^{K/2}},\qquad
 \beta=1+a_*-\langle q,a\rangle.
\]
The same product recursion gives expected logarithm
\(K K_{a,R}(\beta)\).  Restriction to the cube can only decrease
the finite dimensional normalizer.  The spherical overlap equation
gives \(\beta=\bar\beta(a)\), and the logarithmic cutoff limit
\eqref{eq:scalar-log-cutoff} consequently yields
\[
 \liminf_N\{A_{N+K}(M)-A_N(M)\}
 \ge K K_a(\bar\beta(a))=K\mathcal A_{\rm S}(q).
\]
The last equality uses the supporting identity and
\(1+a_*-\bar\beta(a)=\langle q,a\rangle\).
This proves the claimed increment for both priors.  Only the old
array was used; no limiting array in dimension \(N+K\) is needed.

Finally, take \(R=1\) to obtain the uniform lower bound, without
any array-limit hypothesis.  The preceding lifting and Taylor
comparisons are uniform for \(M/N\) in a compact subset of
\((0,\infty)\) and for all amplitudes.  The Gaussian part of \(E_N\)
has mean zero under \(\mu_N\).  Gaussian integration by parts gives
\[
 \E\langle B_N\rangle
 =\E\langle R_{12}Q_{12}-Q_{11}\rangle\ge-2L^2.
\]
The coordinate prior is independent of the old measure and
\(\nu_{N,1}(\|\tau\|^2)\le K\).  Jensen therefore gives
\[
 A_{N+K}(M)-A_N(M)
 \ge\log\zeta_{N,1}+\E\log\mu_N(e^{E_N})+o(1)
 \ge\log\zeta_{N,1}-MKL^2/D+o(1)\ge-C_K.
\]
Here \(\zeta_{N,1}=1\) for Ising spins and
\(\zeta_{N,1}\to\gamma([-1,1])^K>0\) on the sphere.
\end{proof}

\section{Perturbation, positivity, and synchronization}
\label{sec:rpc-synchronization}

The free energy bounds require the conditional cavity hypotheses to
hold simultaneously for the interpolating measures and their fixed
row and dimension shifts.  We construct a perturbation that enforces
the joint GG identities at vanishing pressure cost.  Pressure
concentration permits a common subsequence selection; synchronization
then represents the limiting joint array through a limiting genealogy array.
Throughout, assume \eqref{eq:iid-smooth-core-assumption} and use the common bound \(L=L_1\).

\subsection{The perturbation and its covariance}
\label{sec:overview-enriched-free energy}
\label{sec:overview-RJ-perturbation}

Fix \(\kappa\in\{\mathrm I,\mathrm S\}\), \(d\ge2\), and the
trial paths \(q^0\in\Q_d\) and \(a^0\) used in
Section~\ref{sec:overview-N-cavity}.  Use the finite RPC
\(\mathfrak R_d\) with masses \(j/d\), \(1\le j<d\), and ancestry
quantile \(\rho_d\), together with the independent constraint vectors
and Gaussian fields specified there.  Trial covariances use the lookup
functions \(q^{0,\sharp},a^{0,\sharp}\); in particular their values at
the terminal label \(r=1\) are \(q_*^0,a_*^0\).  We keep \(d\) fixed
while taking the dimension limit.

For the interpolation
\eqref{eq:overview-Guerra-Hamiltonian}--\eqref{eq:overview-model-measure},
define the perturbation as follows.

Put \(R^+=(1+R)/2\), enumerate
\((\theta_j,k_j)\in(\mathbb Q\cap[0,1])\times\mathbb N_+\), and define
\begin{equation*}
 K_j(x,r)=\frac{[\theta_jx+(1-\theta_j)r]^{k_j}}{k_j}.
\end{equation*}
Let \(G_{j,N}\) have independent centered Gaussian noises, independent
of the constraint vectors, source fields, and cascade weights, with
\begin{equation*}
 \mathbb E G_{j,N}(s^1)G_{j,N}(s^2)=N K_j(R_{12}^+,r_{12}),
 \qquad s^\ell=(\sigma^\ell,\eta^\ell).
\end{equation*}
For \(\mathbf x=(x_j)_{j\ge1}\in[1,2]^{\mathbb N}\), set
\begin{equation}\label{eq:RJ-perturbation}
 V_N=N^{-1/8}\sum_{j\ge1}2^{-j}x_jG_{j,N}.
\end{equation}
Write \(b_h=h/d\) for \(0\le h\le d-2\) and \(b_{d-1}=1\).
The finite ancestral construction is
\begin{equation*}
 K_j(x,r)=K_j(x,0)+\sum_{h=1}^{d-1}\mathbf1_{\{r\ge b_h\}}
       \{K_j(x,b_h)-K_j(x,b_{h-1})\}.
\end{equation*}
Every polynomial in braces, and \(K_j(x,0)\), has nonnegative
coefficients in \(x\).  Since \(R^+\) and the ancestry-cluster
indicators are Gram kernels, the Schur product theorem proves positive
semidefiniteness.  Finite tensor representations in the spin variables
and independent Gaussian marks at each ancestor give measurable fields.
Choose these marks independently of the entire Poisson tree, the
source-field marks, the constraint vectors, and the channel marks.
The source fields have the same construction using their trial lookup
covariances.  On \([0,1]^2\), \(0\le K_j\le1\) and
\(\operatorname{Lip}(K_j)\le1\), so
\begin{equation}\label{eq:RJ-mixture-summability}
 \sum_{j\ge1}4^{-j}
 \bigl(\|K_j\|_\infty+\operatorname{Lip}(K_j)\bigr)
 \le\frac23.
\end{equation}
For each fixed \(N\), \(V_N\) is therefore an \(L^2\) Gaussian
limit of finite mixtures, uniformly over bounded amplitudes.
The amplitudes are suppressed in \(V_N\).  Choose these processes
once for each \(N\); inserting or deleting constraints leaves \(V_N\)
unchanged.

The lower-bound model is \(t=1\); without \(V_N\) it is the original
zero-field model.  For that endpoint we fix \(d=2\).

\begin{proposition}[Perturbation, positivity, and synchronization]
\label{prop:0916-selection}
\label{prop:overview-lower-selection}
\label{prop:overview-Guerra-selection}
The series \eqref{eq:RJ-perturbation} is well defined.  Adding
\(V_N\) changes \(N^{-1}\mathbb E\log Z\) by \(o(1)\),
uniformly in the amplitudes and in \(t\).
Let \(M/N\to\alpha\), and fix \(t\in[0,1]\).
For every \(n\ge2\), bounded continuous \(F\) of
\((R_{\ell m},r_{\ell m})_{\ell,m\le n}\), and bounded continuous
\(f\) on \([-1,1]\times[0,1]\),
\begin{equation}\label{eq:overview-RJ-GG}
 \mathbb E_{\mathbf x}\left|
 \begin{aligned}
 &\mathbb E\langle Ff(R_{1,n+1},r_{1,n+1})\rangle-\frac1n\mathbb E\langle F\rangle
                  \mathbb E\langle f(R_{12},r_{12})\rangle-\frac1n\sum_{\ell=2}^n
       \mathbb E\langle Ff(R_{1\ell},r_{1\ell})\rangle
 \end{aligned}
 \right|\longrightarrow0.
\end{equation}
The inner expectations keep the amplitudes fixed.
The same assertions hold for the Ising coordinate-deleted measures
specified in Section~\ref{sec:overview-N-cavity}, whose constraints
retain the denominator of the larger dimension.
Every averaged replicated-array limit at fixed amplitudes and time
that satisfies the joint \((R,r)\) GG identities has \(R_{12}\ge0\)
and the RPC representation
\eqref{eq:overview-RJ-synchronization} for a deterministic \(q\in\Q\),
with diagonal \((1,1)\) and the ancestry law in
\eqref{eq:exact-genealogy-marginal}.
\end{proposition}

The scalar covariance terms give positivity of the limiting spin overlap;
the mixed terms give joint identities for spin overlap and ancestry.  The common
subsequence selection below applies simultaneously to the row deletions
and the coordinate-deleted Ising families required by the overlap equation.

\subsection{Finite ancestry and pressure concentration}
\label{sec:simple-rpc-row-tools}

Integrating spins and terminal Gaussians gives the marked-RPC integral
\begin{equation*}
 L_N(\eta)=\int e^{\mathcal H_{N,M,t}(\sigma,\eta,\boldsymbol z^0)}
       \mu_N^\kappa(\dd\sigma)\gamma^{\otimes M}(\dd\boldsymbol z^0),
 \qquad
 Z_N=\int L_N(\eta)\,\mathfrak R_d(\dd\eta).
\end{equation*}
Conditional on the channel marks, constraint vectors, and common
Gaussian increments, the ancestor marks are independent of the Poisson
tree.  The bound \(|u_k(x)|\le L|x|\) and Gaussian exponential moments
give every positive and inverse moment of the unnormalized weight,
averaged against the spin, cascade, and terminal-Gaussian product
measure.  For fixed \(N,M\), these bounds are uniform in finite
truncations of \(V_N\), and verify the conditional moment hypotheses
of \cref{rpc:prop-quasistationarity}.  Its genealogy conclusion gives
\begin{equation}\label{eq:exact-genealogy-marginal}
 \mathbb E\left\langle F((r_{\ell m})_{\ell,m\le n})\right\rangle_{N,M,t}
 =\mathbb E_{\mathfrak R_d}F((r_{\ell m})_{\ell,m\le n}),
\end{equation}
for every bounded Borel genealogy test \(F\), with fixed amplitudes.
The covariance tail of the perturbation tends uniformly to zero at
fixed \(N\), so \cref{lem:tilted-comparison}(ii) passes this identity
from finite perturbation sums to the full series.  The same argument
applies after fixed dimension or constraint shifts, constraint deletion,
and coordinate restriction.

\paragraph{Cost and concentration of the perturbation.}
Write \(a_N=N^{-1/8}\), and let \(Z_N^{\rm pert}\) and \(Z_N^0\)
be the normalizers with and without \(V_N\).  At fixed \(d\), uniformly over the
amplitudes, \(t\in[0,1]\), and constraint densities in a fixed compact
subset of \((0,\infty)\), we claim
\begin{align}
 N^{-1}\left|\mathbb E\log Z_N^{\rm pert}
                  -\mathbb E\log Z_N^0\right|
 &\le Ca_N^2=o(1),\notag\\
 \operatorname{Var}(N^{-1}\log Z_N^{\rm pert})
 &\le C_d/N.\label{eq:rj-pressure-variance}
\end{align}
For the first bound, apply \cref{lem:tilted-comparison}(ii) to
\(0\) and \(V_N\), using the normalized unperturbed measure as base.
By \eqref{eq:RJ-mixture-summability}, \(\delta\le CNa_N^2\);
dividing by \(N\) gives the claim.  The second bound follows from a
concentration estimate for marked cascades.

\begin{lemma}[Pressure concentration on a finite RPC]
\label{lem:finite-rpc-pressure-concentration}
Let a finite RPC have first mass \(m>0\), and let
\(\mathcal Z(g)=\sum_\eta v_\eta L(g,Y_\eta)\) satisfy the
conditional moment hypotheses of \cref{rpc:prop-quasistationarity}.
Here \(g\) is a standard Gaussian vector independent of the Poisson
and branch-mark trees, whose laws do not depend on \(g\).
If \(\log L(g,y)\) is \(D\)-Lipschitz in \(g\), uniformly in
the branch marks \(y\), then
\begin{equation}\label{eq:finite-rpc-pressure-concentration}
 \operatorname{Var}(\log\mathcal Z)\le D^2+Cm^{-2},
\end{equation}
where \(C\) is universal, independent of the later masses and the weight.
\end{lemma}

\begin{proof}
We separate the Gaussian fluctuations of the conditional mean from
the cascade fluctuations at fixed \(g\).  Put \(b(g)=\log S_0(g)\),
with \(S_0\) from the marked recursion.
For unnormalized cascade weights \(w_\eta\), set
\(T=\sum_\eta w_\eta\) and
\(T'=\sum_\eta w_\eta L(g,Y_\eta)/S_0(g)\).
Conditional only on \(g\), \cref{rpc:prop-quasistationarity} gives
\begin{equation}\label{eq:rpc-pressure-centered-totals}
 T'\stackrel{\rm d}=T,\qquad
 \log\mathcal Z-b(g)=\log T'-\log T.
\end{equation}
The total-mass calculation in \cite[Lemma~2.4]{PanchenkoBook}
identifies \(T\) as a deterministic scale times a positive
\(m\)-stable variable.  Later levels enter only through finite
fractional moments, since their masses are strictly increasing.

For any \(Q>0\) with \(\E e^{-sQ}=e^{-cs^m}\), take
\(E\sim\operatorname{Exp}(1)\) independent of \(Q\).  Then
\[
 \mathbb P(E/Q>s)=e^{-cs^m},\qquad
 \log(E/Q)\stackrel{\rm d}=m^{-1}(\log E'-\log c),
 \quad E'\sim\operatorname{Exp}(1).
\]
Since \(\log E\) and \(\log(E/Q)\) have finite second moments,
so does \(\log Q\).  Independence of \(E\) and \(Q\) now yields
\[
 \operatorname{Var}(\log Q)
 =(m^{-2}-1)\operatorname{Var}(\log E)\le Cm^{-2}.
\]
Centering the two identically distributed logarithms in
\eqref{eq:rpc-pressure-centered-totals} therefore gives
\[
 \E[\log\mathcal Z\mid g]=b(g),\qquad
 \operatorname{Var}(\log\mathcal Z\mid g)\le Cm^{-2}.
\]
No independence of \(T,T'\) is needed.
Finally, the bound
\(f(g,y)\le f(\widetilde g,y)+D\|g-\widetilde g\|\)
passes through every logarithmic moment transform in the backward
recursion by order preservation and translation equivariance.
Thus \(b\) is \(D\)-Lipschitz.  Gaussian Poincar\'e and total
variance prove \eqref{eq:finite-rpc-pressure-concentration}.
\end{proof}

Condition on all channel marks; the following constants are uniform
in those marks.  The common Gaussian vector consists of the constraint
vectors, common trial-row and spin-source increments, and the root
noises of \(V_N\).  All higher ancestral increments are independent
branch marks; terminal Gaussians are integrated inside \(L_N\).
Using \(\|\sigma\|^2=N\), \(|u_k'|\le L\), and
\eqref{eq:RJ-mixture-summability}, the squared Lipschitz constant of
the Hamiltonian in the common Gaussian vector is at most
\[
 D_N^2\le tML^2+(1-t)q^0(0)ML^2
       +(1-t)Na^0(0)+CNa_N^2\le C'N.
\]
Logarithmic spin and terminal integration preserves this bound.
The first cascade mass is \(m=1/d\), so
\cref{lem:finite-rpc-pressure-concentration} gives
\[
 \operatorname{Var}(\log Z_N^{\rm pert}\mid\text{channel marks})
 \le CN+Cd^2.
\]
The bound is uniform over finite perturbation sums and bounded
amplitudes.  To pass to the full sum, let \(V_N^{(J)}\) retain its
first \(J\) fields and let \(\langle\cdot\rangle_J\) be the associated
Gibbs average.  Independence and the vanishing diagonal variance of
the Gaussian tail give, at fixed \(N,M\),
\[
 \mathbb E\left|\log Z_N^{\rm pert}-\log Z_N^{(J)}\right|^2
 \le\mathbb E\left\langle
       e^{2|V_N-V_N^{(J)}|}-1\right\rangle_J\longrightarrow0.
\]
Here \(|\log\langle e^V\rangle|\le\log\langle e^{|V|}\rangle\)
and Jensen's inequality give the displayed bound.  Thus it also holds
for the full perturbation, after fixed constraint or dimension shifts.

To include mark fluctuations, resample one mark and delete its complete
row.  Under the deleted law \(\Gamma\), the change in \(\log Z\)
is at most \(2A\), where \(A=\log\Gamma(e^{L|X|})\) and
\(X\) is the fresh completed row input.  By
\eqref{eq:fresh-marked-row-bound}, \(\E A^2\le\E e^{2L|Z|}\)
for a standard Gaussian \(Z\).  Conditional Jensen and Efron--Stein
therefore bound the variance of the conditional mean given the marks
by \(CM\).  Returning to the full disorder law, total variance gives
\begin{equation}\label{eq:iid-variance}
 \Var(\log Z_{N,M,t})\le CN+Cd^2+CM=O_d(N),
\end{equation}
which proves \eqref{eq:rj-pressure-variance} and supplies concentration
for the joint GG identities below.  For the original centered model,
omit the cascade, source fields, and perturbation.  The same Gaussian
gradient bound and mark-resampling argument give
\(\Var(\log Z_{N,M}^{\kappa,\nu})\le ML^2+CM\).
Dividing by \(N^2\) proves \eqref{eq:unperturbed-iid-variance}.

\subsection{Positivity, Ghirlanda--Guerra identities, and synchronization}
\label{sec:simple-rj-perturbation}

The following representation identifies the array limits once the joint
GG identities hold.

\begin{lemma}[Uniform refinement of a finite ancestry array]
\label{lem:finite-rpc-representation}
Let \((R,r)\) be jointly weakly exchangeable positive semidefinite
arrays with diagonal \((1,1)\), satisfying the joint GG identities,
and suppose that \(r\) has the ancestry law of \(\mathfrak R_d\).
Then \(R_{12}\ge0\) almost surely and the joint array has the
representation \eqref{eq:overview-RJ-synchronization} for a deterministic
\(q\in\Q\).  This path is uniquely determined almost everywhere by
the law of \(R_{12}\).

Every bounded Borel function \(h\) that is constant almost everywhere
on each common fiber of \(q\) and \(\rho_d\) admits a Borel factorization
\(h(u)=\bar h(q(u),\rho_d(u))\) almost everywhere, with
\(\bar h:[0,1]^2\to\mathbb R\).  If a nondecreasing covariance
path \(c\) factors through \((q,\rho_d)\), then so does \(p_c\).
In particular, for the trial paths of
Section~\ref{sec:overview-N-cavity}, \(t\in[0,1]\), and \(\alpha>0\),
\[
 c=(1-t)q^0+tq,\qquad p_c,\qquad
 a=(1-t)a^0+t\alpha p_c,\qquad \widehat q_a
\]
all have such factorizations.  Their evaluations at \(U_{12}\) are
therefore measurable functions of the retained pair \((R_{12},r_{12})\), although
the uniform refinement \(U\) itself need not be observable.
\end{lemma}

\begin{proof}
The scalar GG identities for \(R\) and the positivity principle
\cite[Theorem~2.16]{PanchenkoBook} give
\begin{equation*}
 \mathbb P(R_{12}<0)=0.
\end{equation*}
Put \(S=(R+r)/2\).  Panchenko's synchronization theorem
\cite[Theorem~4]{PanchenkoMultiSpecies}, applied with weights
\(1/2,1/2\), gives deterministic nondecreasing \(2\)-Lipschitz maps
\(L_1,L_2\) such that
\[
 R_{\ell m}=L_1(S_{\ell m}),\qquad
 r_{\ell m}=L_2(S_{\ell m}).
\]
The scalar array \(S\) also satisfies GG.  The characterization and
existence results \cite[Theorems~2.13, 2.17 and Remark~2.1]{PanchenkoBook}
represent its off-diagonal law as \(s_0(U_{\ell m})\), where \(s_0\)
is the nondecreasing quantile of \(S_{12}\) and \(U\) is the
uniform limiting genealogy array from finite RPCs.  These results apply to the off-diagonal Gram--de Finetti
array via the Dovbysh--Sudakov representation, as in the proof of
Theorem~2.17; the specified true diagonal is kept separately.
Thus \(q=L_1\circ s_0\) and \(\rho=L_2\circ s_0\) give the joint
representation.  Since \(U_{12}\) is uniform, the nondecreasing
\(\rho\) is the quantile of \(r_{12}\), hence equals \(\rho_d\)
almost everywhere.  Likewise \(q\) is the quantile of \(R_{12}\).
Choose right-continuous versions and set \(q(1)=1\); then
\begin{equation*}
 ((R_{\ell m},r_{\ell m}))_{\ell<m}
 \stackrel{\rm d}=
 ((q(U_{\ell m}),\rho_d(U_{\ell m})))_{\ell<m},\qquad q\in\Q.
\end{equation*}
The diagonals on both sides are \((1,1)\), independently of the
off-diagonal endpoint values.  The relation \(s_0=(q+\rho_d)/2\)
holds almost everywhere.

For the factorization, put \(s=(q+\rho_d)/2\).  Monotonicity makes
its fibers exactly the common fibers of \(q\) and \(\rho_d\).
Only countably many have positive length.  A generalized inverse of
\(s\), with the corresponding almost-everywhere constants assigned
at those fiber values, gives a Borel \(H\) with \(h=H\circ s\)
almost everywhere.  Set \(\bar h(x,y)=H((x+y)/2)\).
If \(c\) factors through \((q,\rho_d)\), it is constant almost
everywhere on each common fiber; the plateau property of \(p_c\)
gives the second assertion.  On such a fiber the lookup identities
\(q^{0,\sharp}\circ\rho_d=q^0\) and
\(a^{0,\sharp}\circ\rho_d=a^0\) show that the trial paths are
constant there as well.  Thus \(c\) is constant there, and the plateau
property of \(p_c\) in \cref{lem:rpc-constraint-insertion} implies the
same for \(p_c\) and \(a\).
\Cref{prop:scalar-coordinate-laws} gives the same for \(\widehat q_a\).
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:0916-selection}]
Apply the argument proving \cite[Theorems~3.2--3.3]{PanchenkoBook}
to one perturbation component at a time.  For fixed \(j\), the field
\(g_{N,j}=G_{j,N}/\sqrt N\) has covariance \(K_j\), bounded by one
with deterministic diagonal \(1/k_j\), and coefficient
\(s_{N,j}x_j\), where \(s_{N,j}=2^{-j}N^{3/8}\to\infty\).
Fix all other amplitudes and absorb the remaining Hamiltonian into
the random reference measure.  On the fixed finite-depth label tree, the selected
Gaussian noise is independent of this reference environment.
These are precisely the covariance and independence properties used
in the cited proof: Gaussian integration by parts is conditional on
the reference environment, while its convexity estimate uses pressure
concentration over all disorder.

By \eqref{eq:iid-variance}, uniformly over amplitudes in \([0,3]\),
\[
 \mathfrak v_N:=\sup_{\mathbf x}
 \mathbb E|\log Z_{N,M,t}-\mathbb E\log Z_{N,M,t}|=O_d(\sqrt N),
 \qquad
 \frac{\mathfrak v_N}{s_{N,j}^2}=O_{d,j}(N^{-1/4})\longrightarrow0.
\]
The cited argument therefore makes the GG defect with pair test
\(K_j(R^+,r)\) tend to zero after averaging \(x_j\), for every
bounded test of the full overlap--ancestry array.  Averaging the
remaining amplitudes preserves this conclusion.  The rational mixed
powers defining \(K_j\), together with constants, span all bivariate
polynomials, so uniform approximation gives \eqref{eq:overview-RJ-GG}.

For the Ising coordinate-deleted measures, the constraint coefficient is
\(\sqrt{t/D}\) in dimension \(D-1\), and the completed variance is
\(1-t/D\).  The Gaussian gradient bound, the cascade moment bounds,
and the independence of the perturbing noises used above are unchanged.
Thus their pressure concentration and GG defect estimates hold with
the same orders, uniformly in \(t\).  Their fresh-constraint covariance
rules converge uniformly to \(tR+(1-t)q^{0,\sharp}(r)\).

\paragraph{Common subsequence selection.}
\label{par:common-cavity-selection}
Fix a starting dimension sequence and finitely many changes in the
constraint count, including the coordinate-deleted Ising families when
they are needed.  Enumerate countable dense families of GG tests for these
systems, with nonnegative defects \(e_{N,j}\).  Each defect has
vanishing parameter average: the parameter law \(\mu\) is the amplitude
law for the lower specialization, or its product with Lebesgue measure
on \(t\in(0,1)\) for the interpolation.  The latter assertion follows
by dominated convergence; the lower endpoint is selected separately.
Choose \(N_k\) so that
\[
 \sum_{j=1}^k\int e_{N_k,j}\,\dd\mu\le2^{-k}.
\]
Tonelli's theorem gives \(e_{N_k,j}\to0\) for every \(j\) outside one
\(\mu\)-null set.  Uniform approximation gives all continuous GG
tests.  The same selection accommodates countably many fixed constraint-count
changes or coordinate-deleted families, without a rate uniform over them.  At each retained
parameter, compactness gives a further, possibly parameter-dependent,
subsequence on which any fixed finite collection of arrays converges
jointly.  Their spin paths are not yet asserted to agree.

\paragraph{Positivity and synchronization of the limits.}
The joint GG identities of any limiting array extend to bounded Borel
tests: for each replica count they express equality of finite Borel
measures on a compact space.  Positive semidefiniteness, joint weak
exchangeability, and the diagonals pass to the limit, and
\eqref{eq:exact-genealogy-marginal} preserves the finite ancestry law.
Lemma~\ref{lem:finite-rpc-representation} therefore gives positivity
and the representation \eqref{eq:overview-RJ-synchronization}.

No constraint derivative enters the perturbation or the selection.
\end{proof}


\paragraph{The perturbation under coordinate insertion.}
The processes \(V_N\) are independent of the constraints and spatial
source fields, and their covariance depends on the spins only through
their overlap.  The averaged Ising measures are therefore invariant
under coordinate permutations.  For insertion of \(K\) coordinates,
write \(D=N+K\).  The embeddings are
\[
 \iota_\tau^{\rm I}\sigma=(\sigma,\tau),\qquad
 \iota_\tau^{\rm S}\sigma=
 \left(\sqrt{\frac{D-\|\tau\|^2}{N}}\,\sigma,\tau\right).
\]
For fixed \(K\) and \(\tau,\tau'\in[-R,R]^K\), they change spin
overlaps by \(O_{K,R}(N^{-1})\) and leave ancestry unchanged.
The covariance scale \(N^{3/4}\) and
\eqref{eq:RJ-mixture-summability} give
\begin{equation*}
 \sup_{s,s',\tau,\tau'}\left|
 \operatorname{Cov}(V_D(\iota_\tau s),V_D(\iota_{\tau'}s'))
 -\operatorname{Cov}(V_N(s),V_N(s'))\right|
 \le C_{K,R}N^{-1/4}.
\end{equation*}
This is the comparison used in Section~5.  Also,
\(\sum_j4^{-j}x_j^2\|\partial_RK_j((1+R)/2,r)\|_\infty\le C\).
After spherical and Gaussian integration by parts, this bound makes
the perturbation contribution to the tangent identity \(O(N^{-1/4})\).
One may first truncate the series: covariance comparison passes its
left side to the limit, and the derivative tail bounds its integrated
right side.  The common selection above supplies all array hypotheses
needed in Sections~2 and~5.


\section{Quenched large deviations}
\label{sec:quenched-ldp-proof}

We prove \cref{thm:quenched-empirical-ldp}.  The new step is a uniform
curvature bound for the pressure as a function of finitely many channel
coefficients.  It gives the differentiability needed for the
G\"artner--Ellis theorem.  The spectral bound
\eqref{eq:simple-spectral-empirical} then permits passage from finitely
many empirical averages to the full empirical measure.

\subsection{Differentiability of the channel pressure}

For a bounded function \(f\), put
\(\operatorname{osc}(f)=\sup f-\inf f\).

\begin{proposition}[Differentiability in bounded channel directions]
\label{prop:channel-pressure-differentiability}
Fix \(\alpha>0\), \(\kappa\in\{\mathrm S,\mathrm I\}\), and
\(f_1,\ldots,f_k\in\mathrm{BL}(\R)\).  Define
\begin{equation*}
 \Lambda_f(\lambda)
 =\frac1\alpha\mathcal P_{\kappa,u_\lambda}(\alpha),
 \qquad u_\lambda=\sum_{i=1}^k\lambda_i f_i,
 \qquad C_f=\frac14\sum_{i=1}^k\operatorname{osc}(f_i)^2.
\end{equation*}
Then \(\Lambda_f\) is convex and belongs to
\(C^{1,1}(\R^k)\), with
\begin{equation*}
 \|\nabla\Lambda_f(\lambda)-\nabla\Lambda_f(\lambda')\|_2
 \le C_f\|\lambda-\lambda'\|_2.
\end{equation*}
\end{proposition}

\begin{proof}
First let \(q\in\Q\) be a step path, and choose a finite grid
\(\pi\) containing its jump points.  On the finite cascade
\(\mathfrak R_\pi\), use the completed Gaussian field \(X^q\) from
\eqref{eq:overview-completed-row-field}, whose construction does not
depend on \(\lambda\), and define
\[
 G_{q,\lambda}^{\pi}(\dd\eta,\dd z)
 =\frac{e^{u_\lambda(X^q(\eta,z))}}
        {Z_{q,\lambda}^{\pi}}\,\mathfrak R_\pi(\dd\eta)\gamma(\dd z),
 \qquad
 Z_{q,\lambda}^{\pi}
 =\int e^{u_\lambda(X^q(\eta,z))}
               \mathfrak R_\pi(\dd\eta)\gamma(\dd z).
\]
The finite-step one-constraint identity \eqref{eq:rpc-insertion-log}
gives \(\Phi_{u_\lambda}(q)=\E\log Z_{q,\lambda}^{\pi}\); for
bounded Lipschitz channels the identity follows also by uniform
mollification.
Differentiation in \(\lambda\) is justified by boundedness of the
\(f_i\).  For \(h\in\R^k\), writing
\(h\cdot f=\sum_i h_i f_i\), it gives
\begin{equation}\label{eq:rpc-channel-curvature}
 \begin{aligned}
 D_\lambda^2\Phi_{u_\lambda}(q)[h,h]=\E\operatorname{Var}_{G_{q,\lambda}^{\pi}}
                         \bigl(h\cdot f(X^q)\bigr)\le\tfrac14\operatorname{osc}(h\cdot f)^2
 \le C_f\|h\|_2^2.
 \end{aligned}
\end{equation}
The first inequality is the variance bound for a random variable taking
values in an interval of length \(\operatorname{osc}(h\cdot f)\).
Thus \(\lambda\mapsto\Phi_{u_\lambda}(q)-C_f\|\lambda\|_2^2/2\)
is concave for every step path \(q\).

For an arbitrary \(q\in\Q\), choose step paths \(q_n\to q\) in
\(L^1\).  On each compact set \(K\subset\R^k\),
\(L_K:=\sup_{\lambda\in K}\Lip(u_\lambda)<\infty\), and scalar
path continuity \eqref{rpc:Phi-L1} gives
\[
 \sup_{\lambda\in K}
 |\Phi_{u_\lambda}(q_n)-\Phi_{u_\lambda}(q)|
 \le\tfrac12 L_K^2\|q_n-q\|_{L^1}\longrightarrow0.
\]
Concavity passes to this limit.  In particular, for every \(q\in\Q\),
\begin{equation*}
 \Phi_{u_{\lambda+h}}(q)+\Phi_{u_{\lambda-h}}(q)
       -2\Phi_{u_\lambda}(q)\le C_f\|h\|_2^2.
\end{equation*}
Only this semiconcavity bound is needed for general paths.

The main variational formula now shows that
\begin{equation}\label{eq:channel-pressure-semiconcavity}
 \Lambda_f(\lambda)-\tfrac{C_f}{2}\|\lambda\|_2^2
 =\inf_{\substack{q\in\Q\\\mathcal A_\kappa(q)<\infty}}
 \left\{\frac{\mathcal A_\kappa(q)}\alpha
       +\Phi_{u_\lambda}(q)-\tfrac{C_f}{2}\|\lambda\|_2^2\right\}
\end{equation}
is concave: each function inside the infimum is concave by the
finite-step bound \eqref{eq:rpc-channel-curvature} and the preceding
limit argument.  The set of finite-cost paths is
nonempty and independent of \(\lambda\).  On the other hand,
\(\Lambda_f\) is finite and convex, since it is the pointwise limit
of the convex functions
\(M_N^{-1}\E\log Z_{N,M_N}^{\kappa,u_\lambda}\).

For completeness, if a finite convex function \(g\) satisfies that
\(g-C\|\cdot\|_2^2/2\) is concave, then its central differences
are at most \(C\|h\|_2^2\).  For any two subgradients \(p,p'\)
at \(x\), the subgradient inequalities therefore give
\(t(p-p')\cdot h\le Ct^2\|h\|_2^2\).  Letting \(t\downarrow0\)
shows that the subgradient is unique, so \(g\) is differentiable.
Convolution with a compactly supported smooth probability density preserves convexity and
the same upper curvature bound; hence
\(0\le D^2g_\varepsilon\le C I_k\).
The gradients converge locally to \(\nabla g\), proving that
\(\nabla g\) is \(C\)-Lipschitz.  Apply this fact to
\eqref{eq:channel-pressure-semiconcavity}.
No selection or uniqueness of a minimizing path is used.
\end{proof}

\subsection{finite dimensional large deviations}

Fix \(\kappa,\alpha\), and abbreviate
\[
 \Lambda(u)=\frac{\mathcal P_{\kappa,u}(\alpha)}\alpha,
 \qquad
 \Lambda_N(u)=\frac1{M_N}\log Z_{N,M_N}^{\kappa,u}.
\]
All finite dimensional large-deviation principles must hold on one
probability-one event.  We construct this event for a countable class of
channels, then extend the pressure limits to every bounded Lipschitz
channel using the spectral bound.

Let \(\mathbf G_N\) be the \(M_N\times N\) matrix with rows
\(g_k^{\mathsf T}\), and fix
\(C>(1+\alpha^{-1/2})^2\).  The Gaussian matrix tail estimate in
\cref{lem:spectral-moments} and Borel--Cantelli imply, almost surely,
\begin{equation}\label{eq:ldp-eventual-compact-support}
 \sup_{\sigma\in\Sigma_N^\kappa}\int x^2\,L_{N,\sigma}(\dd x)
 \le\frac{\|\mathbf G_N\|_{\rm op}^2}{M_N}\le C
 \quad\text{for all sufficiently large }N.
\end{equation}
Thus \(\rho_N^\kappa\) is eventually supported on
\[
 K_C=\left\{\mu\in\mathcal P(\R):\int x^2\,\mu(\dd x)\le C\right\}.
\]
This set is weakly compact, by tightness and lower semicontinuity of
the second moment.

For \(u,v\in\mathrm{BL}(\R)\), set
\(d_C(u,v)=\sup_{\mu\in K_C}\int|u-v|\,\dd\mu\).
When \(\rho_N^\kappa(K_C)=1\), direct comparison of the integrands gives
\begin{equation}\label{eq:ldp-channel-modulus}
 |\Lambda_N(u)-\Lambda_N(v)|\le d_C(u,v),
 \qquad |\Lambda(u)-\Lambda(v)|\le d_C(u,v).
\end{equation}
To justify the deterministic second inequality, first fix \(u,v\).
The first inequality holds with probability tending to one, while
\(|\Lambda_N(u)-\Lambda_N(v)|\le\|u-v\|_\infty\) always.
Take expectations and use \cref{thm:iid-channels}.

Let \(\mathscr D\) be the countable rational vector space generated
by the constant function one and compactly supported piecewise linear
functions with rational breakpoints and values.  It is dense for
\(d_C\) in \(\mathrm{BL}(\R)\): cutting off \(u\) outside
\([-R-1,R+1]\), while leaving it unchanged on \([-R,R]\), costs at
most \(C\|u\|_\infty/R^2\); the resulting compactly supported
function can be approximated uniformly by rational piecewise linear
functions.  Bounded-channel almost-sure convergence in
\cref{thm:iid-channels}, applied to a deterministic channel, gives one
event of probability one on which \(\Lambda_N(v)\to\Lambda(v)\)
for all \(v\in\mathscr D\), and
\eqref{eq:ldp-eventual-compact-support} holds.  On this event,
\eqref{eq:ldp-channel-modulus} and approximation give
\begin{equation}\label{eq:ldp-simultaneous-pressure-limits}
 \Lambda_N(u)\longrightarrow\Lambda(u)
 \qquad\text{for every }u\in\mathrm{BL}(\R).
\end{equation}
Indeed, the limit superior of the absolute difference is at most
\(2d_C(u,v)\) for each \(v\in\mathscr D\).

Fix an environment in this event.  For any finite family
\(f=(f_1,\ldots,f_k)\) of bounded Lipschitz functions, define
\[
 T_f(\mu)=\left(\int f_1\,\dd\mu,\ldots,\int f_k\,\dd\mu\right),
 \qquad Y_N(\sigma)=T_f(L_{N,\sigma}).
\]
Its logarithmic moment generating function satisfies the exact identity
\begin{equation*}
 \frac1{M_N}\log\int e^{M_N\lambda\cdot Y_N(\sigma)}
                       \mu_N^\kappa(\dd\sigma)
 =\Lambda_N(\lambda\cdot f)
 \longrightarrow\Lambda_f(\lambda).
\end{equation*}
The limit is finite and differentiable on all of \(\R^k\), by
\cref{prop:channel-pressure-differentiability}; there is no boundary
of its effective domain at which a steepness condition is needed.
The vectors \(Y_N\) lie in a fixed compact rectangle.  The
G\"artner--Ellis theorem \cite[Theorem~2.3.6]{DemboZeitouni}
therefore gives a full large-deviation principle at speed \(M_N\),
with good rate
\begin{equation}\label{eq:ldp-finite-moment-rate}
 J_f(z)=\sup_{\lambda\in\R^k}
              \{\lambda\cdot z-\Lambda_f(\lambda)\}.
\end{equation}
This conclusion holds for every such finite family on the same event.

\subsection{The empirical measure principle and nonlinear free energies}

We continue on this event and write
\(\mathcal J=\mathcal J_{\kappa,\alpha}\).
Its definition \eqref{eq:quenched-ldp-rate} is a supremum of continuous
affine functions of \(\mu\), so \(\mathcal J\) is convex and lower
semicontinuous.  It is nonnegative because \(\Lambda(0)=0\).
It is infinite outside \(K_C\): if \(\mu(x^2)>C\), choose
\(R<\infty\) with \(\mu(f_R)>C\), where
\(f_R(x)=\min\{x^2,R\}\) is bounded and Lipschitz.  Since
\(\nu(f_R)\le C\) for \(\nu\in K_C\),
\eqref{eq:ldp-simultaneous-pressure-limits} gives
\(\Lambda(t f_R)\le tC\) for every \(t>0\).  Consequently
\[
 \mathcal J(\mu)\ge t\{\mu(f_R)-C\}\longrightarrow\infty.
\]
Thus all sublevel sets of \(\mathcal J\) are compact.

\emph{Passage to empirical measures.}
Enumerate \(\mathscr D=\{f_j:j\ge1\}\).  The continuous coordinate map
\[
 T:\mathcal P(\R)\longrightarrow\R^{\mathbb N},\qquad
 T(\mu)=(\mu(f_j))_{j\ge1},
\]
restricts to a homeomorphism of \(K_C\) onto its compact image in the
product topology, since the functions in \(\mathscr D\) separate measures.
Every finite projection of \(T_\#\rho_N^\kappa\) satisfies the good
LDP \eqref{eq:ldp-finite-moment-rate} on the same quenched event.
The Dawson--G\"artner theorem
\cite[Theorem~4.6.1]{DemboZeitouni} gives a good LDP in the product
space, with rate
\(z\mapsto\sup_kJ_{(f_1,\ldots,f_k)}(z_1,\ldots,z_k)\).
The laws are eventually supported on the closed set \(T(K_C)\).
Restricting to that set and transporting by \(T^{-1}\) gives the
LDP on \(K_C\), with rate
\[
 \sup_{u\in\operatorname{span}_{\R}\mathscr D}
             \{\mu(u)-\Lambda(u)\}=\mathcal J(\mu).
\]
The equality follows from \(d_C\)-density and
\eqref{eq:ldp-channel-modulus}.  Eventual support in \(K_C\) and
\(\mathcal J=\infty\) outside it extend this to the weak topology
on \(\mathcal P(\R)\), proving the asserted large-deviation principle.

\emph{Nonlinear free energies.}
For bounded weakly continuous \(\Psi\), Varadhan's lemma
\cite[Theorem~4.3.1]{DemboZeitouni} now gives
\[
 \frac1{M_N}\log\int e^{M_N\Psi(\mu)}\,\rho_N^\kappa(\dd\mu)
 \longrightarrow\sup_\mu\{\Psi(\mu)-\mathcal J(\mu)\}.
\]
The integral on the left equals the spin integral in
\eqref{eq:nonlinear-perceptron-free energy}.  Multiplication by
\(M_N/N\to\alpha\) proves that formula; replacing normalized
Ising counting measure by the sum adds \(\log2\).
This completes the proof of \cref{thm:quenched-empirical-ldp}.

\section{Applications and singular limits}
\label{sec:applications-proofs}

We first prove the Gaussian-field extension, then pass to zero
temperature for scalar projection pursuit and justify the radial
passage used in \cref{thm:st-free energy}.  Removing quadratic
truncations and hardening distance penalties give the ground-state
and capacity results.  Throughout this section, \(\mathbf G_N\)
denotes the \(M_N\times N\) matrix with rows \(g_k^{\mathsf T}\).

\subsection{Gaussian external fields}
\label{sec:gaussian-field-proof}

We now prove the \hyperref[par:gaussian-field-extension]{Gaussian-field extension},
using the notation of \eqref{eq:intro-gaussian-field-partition}--\eqref{eq:intro-field-value}.

\begin{proof}[Proof of the {\hyperref[par:gaussian-field-extension]{Gaussian-field extension}}]
Put \(\ell_b(x)=bx\).  On a step path, a Gaussian recursion step
of mass \(m\) and variance \(\Delta\) contributes \(b^2m\Delta/2\).
Summing and then using path continuity gives
\[
 \Phi_{\ell_b}(q)=\frac{b^2}{2}\int_0^1\gamma_q(s)\,\dd s
                =\frac{b^2}{2}\mathsf L(q).
\]
Apply \cref{thm:iid-channels} at zero field to
\(T_N=\lfloor(\alpha+1)N\rfloor\) rows, each having channel
\(u_\Theta\) with probability \(\alpha/(\alpha+1)\) and
\(\ell_b\) otherwise.  This is a Borel family indexed by \(\mathbb R\):
assign label \(e^\theta>0\) to \(u_\theta\), label \(0\) to \(\ell_b\),
and the zero channel to negative labels.
This mixture has a common Lipschitz bound and
the required origin moment.  Its variational value is
\[
 \inf_{q\in\Q}\left\{\mathcal A_\kappa(q)
       +\alpha\E_\nu\Phi_{u_\Theta}(q)+\frac{b^2}{2}\mathsf L(q)\right\}.
\]
Conditional on the counts \(M'_N,K'_N=T_N-M'_N\), the linear rows sum
to \(b\sqrt{K'_N/N}\,g^0\cdot\sigma\).  Thus the mixture partition
\(Z_N^{\rm mix}\) can be coupled to the desired partition \(Z_N(b)\)
using one original-row stream and one \(g^0\), both independent of the
counts and of each other; \(g^0\) is standard Gaussian even when
\(K'_N=0\), when its coefficient vanishes.

For a fresh original row, centering the channel in
\eqref{eq:fresh-marked-row-bound} bounds its logarithmic increment
in \(L^p\) by \(C_p+\|u_\Theta(0)\|_p\), \(p\in\{1,2\}\),
uniformly over the independent deleted law.  Conditional-count Minkowski and
\(N^{-1}|(b'-b)g^0\cdot\sigma|
\le|b'-b|\|g^0\|_2/\sqrt N\) give
\[
 \left\|\frac{\log Z_N^{\rm mix}-\log Z_N(b)}N\right\|_p
 \le\frac{C_p}{N}\|M'_N-M_N\|_p
       +|b|\left\|\sqrt{K'_N/N}-1\right\|_p=o(1).
\]
The last step uses the binomial count moments and \(M_N/N\to\alpha\).
This proves the asserted \(L^1\) and \(L^2\) limits; the spherical
entropy representation is inherited from the mixture theorem.
For a deterministic channel, the squared gradient norms of the
normalized pressure in the rows and in \(g^0\) are bounded by
\(M_NL^2/N^2\) and \(b^2/N\), respectively.  Gaussian Poincar\'e gives
\begin{equation}\label{eq:gaussian-field-variance}
 \Var(F_{N,\kappa}^{u,b})
 \le\frac{M_N\Lip(u)^2}{N^2}+\frac{b^2}{N}.
\end{equation}
\end{proof}

\subsection{Zero temperature and the Montanari--Zhou formula}
\label{sec:mz-proof}

The zero-temperature passage uses control values for integrable
coefficients and measurable marked terminals.  We prove their stability
and integrability by extending the standard step-coefficient Parisi
control argument; see
\cite[Lemma~18]{JagannathTobasco}.

\begin{lemma}[Scalar control with integrable coefficients]
\label{lem:mz-control}
Let \(\lambda\in L^1_+([0,1])\), \(a\ge0\), and let
\(K:\mathbb R\to\mathbb R\) be \(L\)-Lipschitz.  Define
\begin{equation*}
 \mathcal T_\lambda^a K(t,x)
 =\sup_v\E\left[
 K\left(x+\sqrt a(B_1-B_t)+\int_t^1\lambda(s)v_s\,\dd s\right)
 -\frac12\int_t^1\lambda(s)v_s^2\,\dd s\right].
\end{equation*}
Here \(B\) is standard Brownian motion.  The controls are progressively
measurable for its increments after \(t\), also when \(a=0\), and satisfy
\(\E\int_t^1\lambda(s)v_s^2\,\dd s<\infty\).
The value is finite and \(L\)-Lipschitz in \(x\), with
\begin{equation}\label{eq:mz-control-growth}
 |\mathcal T_\lambda^aK(t,x)-K(x)|
 \le L\sqrt{\frac{2a(1-t)}\pi}
       +\frac{L^2}{2}\int_t^1\lambda(s)\,\dd s.
\end{equation}
For bounded \(K\), it is also bounded in absolute value by
\(\|K\|_\infty\).  The supremum is unchanged by restricting to
\(|v|\le L\).
The terminal map preserves order and is a contraction in the sup norm.
For another such triple \((\widetilde\lambda,\widetilde a,\widetilde K)\)
with the same Lipschitz bound \(L\),
\begin{equation}\label{eq:mz-control-stability}
 \begin{aligned}
 \|\mathcal T_\lambda^aK(t,\cdot)
      -\mathcal T_{\widetilde\lambda}^{\widetilde a}
                         \widetilde K(t,\cdot)\|_\infty
 \le{}&\|K-\widetilde K\|_\infty
       +\frac32L^2\|\lambda-\widetilde\lambda\|_{L^1(t,1)}\\
 &+L\sqrt{\frac{2(1-t)}{\pi}}\,
                       |\sqrt a-\sqrt{\widetilde a}|.
 \end{aligned}
\end{equation}
For step \(\lambda\) and \(a>0\), the value is the backward
Cole--Hopf/heat recursion solving, between the step endpoints,
\[
 \partial_t f+\frac a2f_{xx}+\frac{\lambda(t)}2f_x^2=0,
 \qquad f(1,\cdot)=K.
\]
For general \(\lambda\), it is the uniform limit on
\([0,1]\times\mathbb R\) of these values for any nonnegative step
approximation in \(L^1\).  This approximation-independent extension is
the \emph{stochastic-control solution}.  It satisfies dynamic programming
at every deterministic intermediate time.  At zero diffusion,
\begin{equation}\label{eq:mz-zero-diffusion}
 \mathcal T_\lambda^0K(t,x)
 =\mathsf H_{\int_t^1\lambda(s)\,\dd s}K(x),
 \qquad \mathsf H_0K=K.
\end{equation}
If \((\theta,x)\mapsto K_\theta(x)\) is jointly Borel measurable on
\(\mathbb R^2\), with common Lipschitz bound \(L\), then
\((\theta,t,x)\mapsto\mathcal T_\lambda^aK_\theta(t,x)\) is jointly
Borel measurable.  For any Borel probability measure \(\nu\) on
\(\mathbb R\) satisfying \(\int_{\mathbb R}|K_\theta(0)|\,\nu(\dd\theta)<\infty\),
these values are integrable at every \((t,x)\), and the preceding
comparison bounds may be integrated over the mark.
\end{lemma}

\begin{proof}
\emph{Clipping and comparison.}
Weighted Cauchy--Schwarz makes the drift of every admissible control
integrable.  Since \(|K(x)|\le|K(0)|+L|x|\), its terminal payoff and
cost are integrable.  For \(w=(-L)\vee(v\wedge L)\), the terminal payoff lost
by clipping is at most \(L\int_t^1\lambda|v-w|\), whereas
\[
 \frac12\int_t^1\lambda(v^2-w^2)
 \ge L\int_t^1\lambda|v-w|.
\]
Thus clipping improves the payoff pathwise.  The zero control gives
the lower growth bound.  The upper bound follows from
\(L|v|-v^2/2\le L^2/2\), proving \eqref{eq:mz-control-growth}.
For bounded terminals the zero control and nonnegative cost give the
sup-norm bound.  Comparing translations and terminals gives the
Lipschitz bound, order preservation, and contraction.  For the same
clipped control, changing \(\lambda\)
changes the terminal expectation by at most
\(L^2\|\lambda-\widetilde\lambda\|_1\), and the cost by at most half
this amount.  Changing \(a\) changes only the Brownian displacement,
with expected magnitude
\(\sqrt{2(1-t)/\pi}\,|\sqrt a-\sqrt{\widetilde a}|\).
Taking suprema proves \eqref{eq:mz-control-stability}.
All these arguments also hold on any subinterval.

\emph{Step coefficients.}
For \(a>0\), on an interval of length \(\Delta\) where \(\lambda=m>0\),
the backward operator is
\[
 F(x)\longmapsto
 \frac a m\log\E\exp\left\{\frac m a
                       F(x+\sqrt{a\Delta}\,G)\right\},
 \qquad G\sim N(0,1).
\]
For \(m=0\), use \(\E F(x+\sqrt{a\Delta}\,G)\).
Mollify \(K\) uniformly, preserving its Lipschitz bound.
The resulting recursion solves the PDE, has \(|f_x|\le L\),
and has bounded \(f_{xx}\) for each fixed mollification.
For \(X_s=x+\sqrt a(B_s-B_t)+\int_t^s\lambda(u)v_u\,\dd u\),
apply It\^o's formula on the successive intervals, stopping first
when \(|X_s|\) exceeds a finite level.  For clipped controls, linear
growth of \(f\), bounded \(f_x\), and the integrable Brownian maximum
justify removing the stopping.  This gives
\[
 \E\left[K(X_1)-\frac12\int_t^1\lambda(s)v_s^2\,\dd s\right]
 =f(t,x)-\frac12\E\int_t^1
               \lambda(s)\bigl(v_s-f_x(s,X_s)\bigr)^2\,\dd s.
\]
Here \(K\) denotes the mollified terminal.  The feedback
\(v_s=f_x(s,X_s)\) has a bounded Lipschitz drift and gives equality.
Thus the recursion equals the control value.  Applying the same
verification on subintervals gives dynamic programming.
Terminal contraction removes the mollification.

At \(a=0\), put \(A=\int_t^1\lambda\).
If \(A>0\), the displacement \(z=\int_t^1\lambda v\) satisfies
\(\int_t^1\lambda v^2\ge z^2/A\) pathwise; constant controls \(v=z/A\)
give the reverse inequality after taking the supremum over deterministic
\(z\).  This proves \eqref{eq:mz-zero-diffusion}.  If \(A=0\), the
value is \(K\).  Splitting the quadratic cost between two intervals
gives the Hopf--Lax semigroup identity and hence dynamic programming.

\emph{Integrable coefficients and marks.}
Take nonnegative step functions \(\lambda_n\to\lambda\) in \(L^1\);
interval averages preserve monotonicity when \(\lambda\) is nondecreasing.
Estimate \eqref{eq:mz-control-stability} gives uniform convergence to
the control value, independently of the approximation.
The same estimate on subintervals and terminal contraction pass dynamic
programming to the limit.  For marked terminals, the finite-step
Gaussian recursions are jointly Borel measurable in \((\theta,t,x)\).
At zero diffusion the Hopf--Lax supremum can be taken over rational
displacements.  The uniform limits therefore establish measurability
for integrable coefficients.  Clipped payoffs are bounded in absolute
value by
\[
 |K_\theta(0)|+L|x|+L\sqrt a\,|B_1-B_t|
                  +\tfrac32L^2\|\lambda\|_1.
\]
The assumed first moment makes this bound integrable over marks and
Brownian paths.  Fubini and dominated convergence therefore apply;
no measurable choice of an optimizing control is needed.
\end{proof}

Write \(\mathcal T_\lambda=\mathcal T_\lambda^1\).
For \(H\in C_b(\mathbb R)\) with \(\|H\|_\infty\le B\), maximizing displacements in
\(\mathsf H_cH\) have magnitude at most \(2\sqrt{Bc}\).
Comparison using an optimizer gives a spatial increment bound
\(2\sqrt{B/c}|h|+h^2/(2c)\); subdivision removes the quadratic term.
Thus \(\mathsf H_cH\) is bounded by \(B\) and
\(2\sqrt{B/c}\)-Lipschitz.  More generally, if \(H\) is
\(L\)-Lipschitz, its Hopf--Lax transform is finite, \(L\)-Lipschitz, and
\(H\le\mathsf H_cH\le H+cL^2/2\).
If \((\theta,x)\mapsto H_\theta(x)\) is jointly Borel measurable on
\(\mathbb R^2\), with each \(H_\theta\) continuous, the transform
is jointly Borel measurable by its rational-displacement supremum.
Consequently \cref{lem:mz-control} defines
\(f_{\mu,c}^{H_\theta}=\mathcal T_\mu(\mathsf H_cH_\theta)\),
with the mark expectation taken after this scalar evolution.

\begin{lemma}[Zero-temperature variational comparison]
\label{lem:mz-variational-comparison}
Fix \(\alpha>0\), a nonempty parameter set \(\Lambda\),
\(b:\Lambda\to\mathbb R\), and a Borel probability measure \(\nu\)
on \(\mathbb R\).  For each \(\lambda\in\Lambda\), let
\((\theta,z)\mapsto H_{\theta,\lambda}(z)\) be jointly Borel measurable
on \(\mathbb R^2\), and assume
\[
 \sup_{\theta\in\mathbb R,\,\lambda\in\Lambda}\Lip(H_{\theta,\lambda})\le L,
 \qquad \int_{\mathbb R}|H_{\theta,\lambda}(0)|\,\nu(\dd\theta)<\infty
 \quad\text{for every }\lambda.
\]
Let \(\mathscr C\) be the nondecreasing right-continuous functions
\(\gamma:[0,1]\to[0,1]\) equal to one on some terminal interval,
and write \(D_\gamma(t)=\int_t^1\gamma(s)\,\dd s\).  Define
\begin{equation}\label{eq:mz-variational-objectives}
 \begin{aligned}
 \mathcal J_\beta(\lambda,\gamma)
 &=\E\mathcal T_{\beta\gamma}H_{\Theta,\lambda}(0,0)
  +\frac1{2\alpha\beta}\int_0^1
       \left\{\frac1{D_\gamma(t)}-\frac1{1-t}\right\}\dd t+b(\lambda),\\
 \mathcal F(\lambda,\mu,c)
 &=\E\mathcal T_\mu(\mathsf H_cH_{\Theta,\lambda})(0,0)
  +\frac1{2\alpha}\int_0^1\frac{\dd t}{c+\int_t^1\mu(s)\,\dd s}
  +b(\lambda).
 \end{aligned}
\end{equation}
For every fixed \((\lambda,\mu,c)\in\Lambda\times\mathscr U\times(0,\infty)\),
there are \(\gamma_\beta\in\mathscr C\) such that
\(\mathcal J_\beta(\lambda,\gamma_\beta)\to\mathcal F(\lambda,\mu,c)\).
Conversely, for every \(\beta\ge2\) and \(\gamma\in\mathscr C\),
\(\mathcal F(\lambda,\beta\gamma,\beta^{-1})
\le\mathcal J_\beta(\lambda,\gamma)
+C_{\alpha,L}(1+\log\beta)/\beta\), uniformly in
\(\lambda\) and the mark law.  Consequently, in
\(\mathbb R\cup\{-\infty\}\),
\begin{equation*}
 \lim_{\beta\to\infty}\inf_{\lambda\in\Lambda,\,\gamma\in\mathscr C}
                \mathcal J_\beta(\lambda,\gamma)
 =\inf_{\lambda\in\Lambda,\,\mu\in\mathscr U,\,c>0}
                \mathcal F(\lambda,\mu,c).
\end{equation*}
\end{lemma}

\begin{proof}
\emph{Uniform reverse comparison.}
For \(c>0\), use the Hopf--Lax bound
\(H\le\mathsf H_cH\le H+cL^2/2\) and the elementary inequality
\[
 \frac1{c+x}-\frac1x+\frac1y\le\frac1{c+y}
 \qquad(0<x\le y).
\]
Apply the latter with \(x=\beta D_\gamma(t)\le y=\beta(1-t)\)
and integrate.  Order preservation and translation equivariance of
\(\mathcal T_{\beta\gamma}\) then give
\begin{equation*}
 \mathcal F(\lambda,\beta\gamma,c)
 \le\mathcal J_\beta(\lambda,\gamma)
      +\frac{cL^2}{2}
      +\frac1{2\alpha\beta}\log\left(1+\frac\beta c\right).
\end{equation*}
Take \(c=\beta^{-1}\).  The error is independent of \(\lambda\),
terminal offsets, and the mark law, so taking infima proves the lower limit.

\emph{Recovery.}
Fix \((\lambda,\mu,c)\), taking \(\mu\) right-continuous.
For \(\beta>c\), set \(T=1-c/\beta\), \(\mu_\beta=\mu\wedge\beta\),
and \(\gamma_\beta=\mu_\beta/\beta\) below \(T\), and one above.
This is admissible, and its entropy is
\begin{equation*}
 \frac1{2\alpha\beta}\int_0^1
   \left\{\frac1{D_{\gamma_\beta}(t)}-\frac1{1-t}\right\}\dd t
 =\frac1{2\alpha}\int_0^T
       \frac{\dd t}{c+\int_t^T\mu_\beta(s)\dd s}
       +\frac{\log(c/\beta)}{2\alpha\beta}.
\end{equation*}
It converges to the entropy in \(\mathcal F\), since
\(\mu_\beta\to\mu\) in \(L^1\) and the denominators are at least \(c\).
For the channel, split at \(T\) and apply \cref{lem:mz-control}:
remove diffusion from the terminal interval, whose drift mass is \(c\);
replace \(\mu_\beta\) by \(\mu\) below \(T\); and restore the omitted
\(\mu\)-evolution on \([T,1]\).  Terminal contraction bounds the total error by
\[
 2L\E|G|\sqrt{c/\beta}
 +\frac32L^2\|\mu_\beta-\mu\|_1
 +\frac{L^2}{2}\int_T^1\mu(s)\dd s\longrightarrow0.
\]
This proves recovery uniformly in the mark.  Taking the infimum over
fixed candidates proves the upper limit, also when it is \(-\infty\).
\end{proof}

\begin{proposition}[Marked spherical ground states]
\label{prop:marked-ground-state}
Let \(\nu\) be a Borel probability measure on \(\mathbb R\), and let
\((\theta,x)\mapsto H_\theta(x)\) be jointly Borel measurable on
\(\mathbb R^2\), with \(\sup_{\theta\in\mathbb R}\|H_\theta\|_\infty\le B\)
and \(\sup_{\theta\in\mathbb R}\Lip(H_\theta)\le L\).
For i.i.d.\ marks \(\Theta_k\sim\nu\), independent
of the Gaussian rows and \(M_N/N\to\alpha\in(0,\infty)\), almost surely,
\begin{equation*}
 \begin{aligned}
 E_N&:=\max_{\|v\|_2=1}\frac1{M_N}\sum_{k\le M_N}H_{\Theta_k}(g_k\cdot v)
 \longrightarrow \inf_{\mu\in\mathscr U,\ c>0}\mathcal F_\nu(\mu,c),\\
 \mathcal F_\nu(\mu,c)&:=\E_\nu f_{\mu,c}^{H_\Theta}(0,0)
       +\frac1{2\alpha}\int_0^1\frac{\dd t}{c+\int_t^1\mu(s)\,\dd s}.
 \end{aligned}
\end{equation*}
Convergence also holds in every finite \(L^p\).
\end{proposition}

\begin{proof}
\emph{Finite temperature.}
The class \(\mathscr C\) in \cref{lem:mz-variational-comparison}
consists precisely of the distribution functions \(\gamma_q\) with
\(q(1-)<1\), up to endpoint conventions.  This restriction does not
change the finite-temperature infimum: replacing \(q\) by
\(q\wedge(1-\varepsilon)\) on \([0,1)\), retaining \(q(1)=1\),
decreases the spherical entropy, while
\eqref{rpc:Phi-L1}, applied to \(\beta H_\Theta\), changes its
\(\beta^{-1}\)-scaled channel term by at most
\(\beta L^2\varepsilon/2\).  Let \(\varepsilon\downarrow0\).
The recursion \eqref{rpc:channel-semigroup} and
\cref{lem:mz-control} give
\(\beta^{-1}\Phi_{\beta H}(q)=\mathcal T_{\beta\gamma_q}H(0,0)\),
first on finite grids and then by approximation.
Consequently \cref{thm:iid-channels} and
\eqref{eq:spherical-entropy-classical} give, for the partition function with
channels \(\beta H_{\Theta_k}\),
\begin{equation}\label{eq:mz-finite-temperature}
 \lim_N\frac{\log Z_N(\beta)}{\beta M_N}
 =\inf_{\gamma\in\mathscr C}\mathcal J_\beta(\gamma),
\end{equation}
where \(\mathcal J_\beta\) is \eqref{eq:mz-variational-objectives}
with a singleton parameter set and zero penalty.  The limit holds
almost surely, simultaneously for integer \(\beta\).
By \cref{lem:mz-variational-comparison},
\(\lim_{\beta\to\infty}\inf_\gamma\mathcal J_\beta(\gamma)
=\inf_{\mu,c}\mathcal F_\nu(\mu,c)\).

\emph{From pressure to the maximum.}
The objective defining \(E_N\) has Lipschitz constant at most
\(L\|\mathbf G_N\|_{\rm op}/\sqrt{M_N}\).
The cap estimate used in \eqref{eq:spherical-zero-temperature-comparison}
therefore gives, for \(0<\eta\le1\),
\[
 0\le E_N-\frac{\log Z_N(\beta)}{\beta M_N}
 \le\frac{L\|\mathbf G_N\|_{\rm op}}{\sqrt{M_N}}\eta
           +\frac{N}{\beta M_N}\log\frac8\eta.
\]
Gaussian operator-norm tails imply that the first coefficient is
eventually bounded almost surely.  Take \(\eta=\beta^{-1}\), let
\(N\to\infty\) on the common event in
\eqref{eq:mz-finite-temperature}, then let integer \(\beta\to\infty\).
This proves the almost-sure limit.  Boundedness gives every finite
\(L^p\) convergence.
\end{proof}

\begin{proof}[Proof of \cref{thm:montanari-zhou}]
First suppose \(h\) is bounded, jointly Borel measurable, and uniformly
\(L\)-Lipschitz in its prediction coordinate.  Rotate the
teacher coordinates to write \(X_i=(G_i,Z_i)\), where
\(G_i\sim N(0,I_k)\) and \(Z_i\sim N(0,I_{d-k})\) are independent,
and \(Y_i=\varphi(G_i,\varepsilon_i)\).  For
\(r\in B_k(1)\), set \(s_r=\sqrt{1-|r|^2}\) and
\[
 H_{r,(G,Y)}(z)=h(Y,r\cdot G+s_rz),\qquad
 M_{n,d}(r)=\max_{\|v\|_2=1}\frac1n\sum_i
                         H_{r,(G_i,Y_i)}(Z_i\cdot v).
\]
These channels are uniformly bounded and \(L\)-Lipschitz, and their
marks are independent of the residual rows.  Identify \((G,Y)\in\mathbb R^{k+1}\)
with a real label through a fixed Borel isomorphism, the same for every
\(r\); the resulting channels are jointly Borel measurable with the same bounds.
Since
\(n/(d-k)\to\alpha\), \cref{prop:marked-ground-state} gives
\(M_{n,d}(r)\to V(r)\) almost surely at each fixed \(r\), with
\(V(r)\) given by its marked functional.  To maximize over the teacher
overlap, we need uniform convergence.  Comparing the objectives on the
same disorder gives
\begin{equation*}
 |M_{n,d}(r)-M_{n,d}(r')|
 \le L|r-r'|\left(\frac1n\sum_i|G_i|^2\right)^{1/2}
    +L|s_r-s_{r'}|\frac{\|Z\|_{\rm op}}{\sqrt n}.
\end{equation*}
The two random coefficients are eventually bounded almost surely,
and \(|s_r-s_{r'}|\le\sqrt{2|r-r'|}\).
The deterministic limit inherits this modulus.  Intersect this
probability-one event with the convergence events for a countable dense
set of overlaps.  Finite nets then give uniform convergence on
\(B_k(1)\).  The original maximum is exactly
\(\max_r M_{n,d}(r)\), so its limit is \(\sup_r V(r)\).

For \(a=|r|^2<1\), change variables in the marked functional by
\[
 t_{\rm new}=a+(1-a)t,\qquad x_{\rm new}=r\cdot G+\sqrt{1-a}\,x,
 \qquad c_{\rm new}=(1-a)c,
 \qquad \mu_{\rm new}(t)=\mu\left(\frac{t-a}{1-a}\right).
\]
The last formula applies on \([a,1)\), with zero extension below
\(a\).  For step coefficients it transforms both the scalar evolution
and its reciprocal-integral term into
\eqref{eq:mz-supervised-functional}; \eqref{eq:mz-control-stability}
extends the identity to integrable coefficients.  Conversely every admissible
coefficient restricted to \([a,1)\) has an admissible inverse under
this change, so the infima agree.  At \(|r|=1\) the channel is constant
in its residual argument and its value is \(\E h(Y,r\cdot G)\).
In \eqref{eq:mz-supervised-functional} the entropy interval is then
empty; letting \(c\downarrow0\) gives the same value.  Applying
\cref{prop:marked-ground-state} directly to a deterministic channel
gives the unlabelled assertion.

Finally let \(h\in C_b(\R^2)\), \(\|h\|_\infty\le B\), and
regularize only the prediction coordinate:
\[
 h_L^-(y,z)=\inf_w\{h(y,w)+L|z-w|\},\qquad
 h_L^+(y,z)=\sup_w\{h(y,w)-L|z-w|\}.
\]
These functions satisfy \(-B\le h_L^-\le h\le h_L^+\le B\)
and are uniformly \(L\)-Lipschitz in \(z\).  Taking the extrema
over rational \(w\) proves joint measurability.  For each fixed
\(R,y\), continuity of \(h(y,\cdot)\) gives
\[
 \eta_L(R,y):=\sup_{|z|\le R}(h_L^+-h_L^-)(y,z)\longrightarrow0,
 \qquad 0\le\eta_L(R,y)\le2B.
\]
The supremum can again be taken over rational points, so \(\eta_L\)
is measurable.  Writing \(X\) for the full covariate matrix,
\begin{equation*}
 \sup_{\|w\|_2=1}\frac1n\sum_i
       (h_L^+-h_L^-)(Y_i,w\cdot X_i)
 \le\frac1n\sum_i\eta_L(R,Y_i)
           +\frac{2B\|X\|_{\rm op}^2}{nR^2}.
\end{equation*}
For each integer \(L,R\), bounded-variable concentration and
Borel--Cantelli give convergence of the empirical first term to
\(\E\eta_L(R,Y)\), also for triangular arrays.  Dominated convergence
makes this expectation tend to zero as \(L\to\infty\); Gaussian
operator-norm control bounds the other coefficient almost surely.
Thus the difference of the two envelope optimization limits tends to
zero, first as \(L\to\infty\), then \(R\to\infty\).
The functional for \(h\) is well defined by \cref{lem:mz-control}
and the Hopf--Lax terminal bounds.  Its terminal transform, scalar
evolution, infimum, and outer supremum preserve order, so its value
lies between the two envelope values.  The same squeeze proves
convergence of the empirical maximum to this value.  The one-variable
envelopes give the unlabelled case.  Boundedness gives convergence in
every finite \(L^p\).
\end{proof}

\subsection{Radial passage for Gaussian spins}
\label{sec:gaussian-spins}

The following lemma justifies the radial optimization used in
\cref{thm:st-free energy}.

\begin{lemma}[Radial Laplace passage]
\label{lem:st-radial-laplace}
Let \(u,h,c\) and \(M_N\) satisfy the assumptions of
\cref{thm:st-free energy}.
On the same disorder, define
\begin{align*}
 Y_N(s)&:=\frac1N\log Z_{N,M_N}^{\mathrm S,u_{s^2},hs},
 \\
 Y(s)&:=\inf_{q\in\Q}\left\{
   \mathcal A_{\rm S}(q)+\alpha\Phi_{u_{s^2}}(q)
       +\frac{h^2s^2}{2}\mathsf L(q)\right\}.
\end{align*}
Then
\begin{equation}\label{eq:st-quenched-laplace-principle}
 \frac1N\log\int_0^\infty s^{N-1}e^{N[Y_N(s)-cs^2]}\,\dd s
 \xrightarrow{L^1}\sup_{s>0}\{Y(s)+\log s-cs^2\}.
\end{equation}
The supremum is finite.
\end{lemma}

\begin{proof}
\emph{Shell convergence.}
For each \(s>0\), the
\hyperref[par:gaussian-field-extension]{Gaussian-field extension} and
\eqref{eq:gaussian-field-variance} give \(Y_N(s)\to Y(s)\) in \(L^2\).
Writing \(L=\Lip(u)\), \(u_0=u(0)\), and \(\alpha_N=M_N/N\), comparison of
the Gibbs weights gives
\begin{align}
 |Y_N(s)-Y_N(t)|&\le D_N|s-t|,
 &D_N&=L\sqrt{\alpha_N}\frac{\|\mathbf G_N\|_{\rm op}}{\sqrt N}
              +|h|\frac{\|g^0\|_2}{\sqrt N},
 \notag\\
 |Y_N(s)-\alpha_Nu_0|&\le sD_N,
 &&s,t\ge0.
 \label{eq:st-shell-linear-envelope}
\end{align}
Here \(Y_N(0)=\alpha_Nu_0\), and \cref{lem:spectral-moments} and Gaussian
norm moments give \(\sup_N\E D_N^p<\infty\) for every finite \(p\).
The expected first bound makes \(Y\) locally Lipschitz; finite nets then give
\begin{equation}\label{eq:st-local-uniform-shell-limit}
 \sup_{s\in[a,A]}|Y_N(s)-Y(s)|\longrightarrow0
 \quad\text{in probability and in }L^1,
 \qquad 0<a<A<\infty.
\end{equation}

\emph{Radial localization.}
Putting \(\rho=s^2\) in \eqref{eq:st-polar-disintegration} gives
\begin{equation}\label{eq:st-polar-s-variable}
 \mathcal Z_{N,M_N}^{\rm ST}(u,h,c)
 =2C_N\int_0^\infty
       s^{N-1}e^{N[Y_N(s)-cs^2]}\,\dd s.
\end{equation}
Choose \(\bar\alpha\ge\sup_N\alpha_N\).  On \(\{D_N\le B\}\),
\eqref{eq:st-shell-linear-envelope} implies, for \(0<a<1\),
\begin{equation*}
 \int_0^a s^{N-1}e^{N[Y_N(s)-cs^2]}\,\dd s
 \le\frac{a^N}{N}e^{N(\bar\alpha|u_0|+Ba)}.
\end{equation*}
For \(A\) large enough that
\(\log s+\bar\alpha|u_0|+Bs\le cs^2/2\) on \([A,\infty)\),
\begin{equation*}
 \int_A^\infty s^{N-1}e^{N[Y_N(s)-cs^2]}\,\dd s
 \le\int_A^\infty s^{-1}e^{-Ncs^2/2}\,\dd s
 \le\frac{e^{-NcA^2/2}}{NcA^2}.
\end{equation*}
These tails have arbitrarily negative exponential rates as
\(a\downarrow0\) and \(A\uparrow\infty\).  Apply the Laplace principle on
\([a,A]\) using \eqref{eq:st-local-uniform-shell-limit}, then remove the
cutoffs and let \(B\uparrow\infty\), using tightness of \(D_N\).
This proves \eqref{eq:st-quenched-laplace-principle} in probability.
The limiting envelope \(|Y(s)-\alpha u_0|\le Cs\) makes the
supremum finite and sends its objective to \(-\infty\) at both radial
ends; a fixed positive \(s\) supplies the lower bound.

\emph{Uniform integrability.}
The nonquadratic part differs from \(M_Nu_0\) by at most
\(\sqrt N D_N\|x\|_2\).  Completing the square and applying Jensen
under \(N(0,(2c)^{-1}I_N)\) give, respectively,
\begin{align*}
 \frac1N\log\mathcal Z_{N,M_N}^{\rm ST}(u,h,c)
 &\le\frac12\log\frac{2\pi}{c}+\alpha_Nu_0+\frac{D_N^2}{2c},
 \\
 \frac1N\log\mathcal Z_{N,M_N}^{\rm ST}(u,h,c)
 &\ge\frac12\log\frac{\pi}{c}+\alpha_Nu_0-\frac{D_N}{\sqrt{2c}}.
\end{align*}
The moment bounds on \(D_N\) give uniform integrability of
\(N^{-1}\log\mathcal Z_{N,M_N}^{\rm ST}\).  By
\eqref{eq:st-polar-s-variable} and \eqref{eq:st-polar-constant}, the
normalized log radial integral differs from it by the bounded
deterministic sequence \(N^{-1}\log(2C_N)\).  It is therefore uniformly
integrable as well, which upgrades the limit to \(L^1\).
\end{proof}

\paragraph{Constant-path restriction.}
For \(q=\overline q_{x/\rho}\), \(0\le x<\rho\), the objective in
\eqref{eq:st-variational-value}, apart from
\(\tfrac12\log(2\pi e)\), becomes
\begin{align*}
 &\alpha\E_Z\log\E_\xi
   \exp\{u(\sqrt x\,Z+\sqrt{\rho-x}\,\xi)\}
 +\frac12\left\{\frac{x}{\rho-x}+\log(\rho-x)\right\}
 \\
 &\hspace{13em}-c\rho+\frac{h^2}{2}(\rho-x).
\end{align*}
where \(Z,\xi\) are independent standard Gaussians.  For each fixed \(\rho\), its
infimum over \(0\le x<\rho\) is the constant-path inner problem; the
replica-symmetric restriction of \(\mathcal P_{\rm ST}\) is obtained after
taking the outer supremum over \(\rho\).
\Cref{thm:st-free energy} does not assert constant-path optimality or
uniqueness of its stationary equations.
\subsection{Quadratic penalties and hard constraints}
\label{sec:singular-limits}

This subsection proves two singular limits of the Lipschitz-channel formula.
The first gives quadratic free energies and their zero-temperature ground
states.  The second hardens distance penalties into exact closed constraints.

\subsubsection{Quadratic penalties and ground states}
\label{sec:harmonic-ground-states}

Fix \(\theta\in\mathbb R\) and \(\beta>0\), and put
\begin{equation*}
 \ell_\theta(x)=(\theta-x)_+,
 \qquad
 u_{\beta,\theta}^{\mathrm{har}}(x)
 =-\frac{\beta}{2}\ell_\theta(x)^2.
\end{equation*}
For \(R>0\), its capped Lipschitz approximation is
\begin{equation*}
 u_{\beta,\theta,R}(x)
 =-\frac{\beta}{2}\bigl(\ell_\theta(x)\wedge R\bigr)^2.
\end{equation*}
Define
\begin{align*}
 \Phi_{\beta,\theta}^{\mathrm{har}}(q)
 &=\lim_{R\to\infty}\Phi_{u_{\beta,\theta,R}}(q),
 \\
 \mathcal P_{\kappa,\beta,\theta}^{\mathrm{har}}(\alpha)
 &=\inf_{q\in\Q}\left\{
   \mathcal A_\kappa(q)+\alpha\Phi_{\beta,\theta}^{\mathrm{har}}(q)
  \right\},
 \\
 F_{N,\kappa}^{\mathrm{har}}(\beta,\theta)
 &=\frac1N\log\int_{\Sigma_N^\kappa}
   \exp\left\{-\frac\beta2\sum_{k=1}^{M_N}
        (\theta-h_k(\sigma))_+^2\right\}\mu_N^\kappa(\dd\sigma),
 \\
 e_{N,\kappa}(\theta)
 &=\frac1{2N}\min_{\sigma\in\Sigma_N^\kappa}
   \sum_{k=1}^{M_N}(\theta-h_k(\sigma))_+^2.
\end{align*}

\begin{theorem}[Quadratic free energy and ground state]
\label{thm:harmonic-free energy-ground-state}
Let \(M_N/N\to\alpha\in(0,\infty)\), \(\theta\in\mathbb R\), and
\(\kappa\in\{\mathrm S,\mathrm I\}\).  For every \(\beta>0\),
\begin{equation}
 F_{N,\kappa}^{\mathrm{har}}(\beta,\theta)
 \longrightarrow
 \mathcal P_{\kappa,\beta,\theta}^{\mathrm{har}}(\alpha)
 \qquad\text{in }L^1,
 \label{eq:harmonic-free energy-limit}
\end{equation}
and
\begin{equation}
 \mathcal P_{\kappa,\beta,\theta}^{\mathrm{har}}(\alpha)
 =\lim_{R\to\infty}\mathcal P_{\kappa,u_{\beta,\theta,R}}(\alpha).
 \label{eq:harmonic-variational-truncation}
\end{equation}
There is a deterministic \(e_{\mathrm{gs},\kappa}(\alpha,\theta)\ge0\)
such that
\begin{equation}
 e_{N,\kappa}(\theta)\longrightarrow
 e_{\mathrm{gs},\kappa}(\alpha,\theta)
 \qquad\text{in }L^1,
 \label{eq:quadratic-ground-state-limit}
\end{equation}
and
\begin{equation}
 e_{\mathrm{gs},\kappa}(\alpha,\theta)
 =\lim_{\beta\to\infty}
  -\frac1\beta\mathcal P_{\kappa,\beta,\theta}^{\mathrm{har}}(\alpha).
 \label{eq:quadratic-zero-temperature-identity}
\end{equation}
The proof first sends \(N\to\infty\) at fixed \((\beta,R)\), then
\(R\to\infty\) at fixed \(\beta\), and finally \(\beta\to\infty\).
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:harmonic-free energy-ground-state}]
\emph{Finite temperature.}
Put
\[
 d_R(x)=u_{\beta,\theta,R}(x)-u_{\beta,\theta}^{\mathrm{har}}(x)
       =\frac\beta2\bigl(\ell_\theta(x)^2-R^2\bigr)_+,
 \qquad \varepsilon_R=\E d_R(Z)\longrightarrow0,
 \quad Z\sim N(0,1).
\]
We first compare one row.  Let \(\Gamma\) be a possibly random probability
measure under which \(X\) has annealed standard-Gaussian marginal.  Write
\(\mathcal N_R=\Gamma(e^{u_{\beta,\theta,R}(X)})\),
\(\mathcal N_\infty=\Gamma(e^{u_{\beta,\theta}^{\rm har}(X)})\), and
\(\Gamma_R=\mathcal N_R^{-1}e^{u_{\beta,\theta,R}(X)}\Gamma\).  Jensen gives
\[
 0\le\log\mathcal N_R-\log\mathcal N_\infty
 =-\log\Gamma_R(e^{-d_R(X)})
 \le\Gamma_R(d_R(X))\le\Gamma(d_R(X)).
\]
The last inequality follows because \(d_R\) increases and
\(e^{u_{\beta,\theta,R}}\) decreases as functions of \(\ell_\theta^2\).
Consequently
\begin{equation}\label{eq:harmonic-one-row-capped-comparison}
 0\le\E(\log\mathcal N_R-\log\mathcal N_\infty)
       \le\varepsilon_R.
\end{equation}
For the finite-volume capped pressure \(F_{N,\kappa}^R\), replace
the rows successively.  The normalized measure with the current row deleted
is independent of that row, whose marginal at each spin is standard normal.
Thus \eqref{eq:harmonic-one-row-capped-comparison} gives
\begin{equation}\label{eq:harmonic-finite-volume-telescope}
 0\le\E\bigl[F_{N,\kappa}^R(\beta,\theta)
                  -F_{N,\kappa}^{\rm har}(\beta,\theta)\bigr]
       \le\frac{M_N}{N}\varepsilon_R,
\end{equation}
with a pointwise nonnegative difference.

For a step path \(q\), the completed field on its finite cascade has
the same standard-Gaussian marginal under the averaged unweighted
measure.  Jensen therefore gives
\[
 -\frac\beta2\E(\theta-Z)_+^2
 \le\Phi_{u_{\beta,\theta,R}}(q)\le0.
\]
At fixed \(\beta,R\), the capped channel is Lipschitz, so
\eqref{rpc:Phi-L1} extends these bounds to every \(q\in\Q\).
They then persist in the decreasing limit
\(\Phi_{\beta,\theta}^{\rm har}(q)\).  Commuting \(\inf_R\) and \(\inf_q\)
on \(\{q:\mathcal A_\kappa(q)<\infty\}\) therefore gives
\[
 \mathcal P_{\kappa,u_{\beta,\theta,R}}(\alpha)
 \downarrow\inf_{q:\mathcal A_\kappa(q)<\infty}
   \{\mathcal A_\kappa(q)+\alpha\Phi_{\beta,\theta}^{\rm har}(q)\}
 =\mathcal P_{\kappa,\beta,\theta}^{\rm har}(\alpha).
\]
The values are finite by the Jensen bound, \(\mathcal A_\kappa\ge0\),
and the trial path \(\overline q_0\).  This proves
\eqref{eq:harmonic-variational-truncation}.  At fixed \(R\),
\cref{thm:iid-channels} gives convergence of
\(F_{N,\kappa}^R\) to its variational value in \(L^2\).
The triangle inequality and \eqref{eq:harmonic-finite-volume-telescope},
followed by \(R\to\infty\), prove \eqref{eq:harmonic-free energy-limit}.

\emph{Zero temperature.}
Set \(X_{N,\kappa}(\beta)=-F_{N,\kappa}^{\rm har}(\beta,\theta)/\beta\).
Since the prior is normalized, H\"older's inequality makes this
nonincreasing in \(\beta\), with
\(X_{N,\kappa}(\beta)\ge e_{N,\kappa}(\theta)\ge0\).
A minimizing Ising atom has mass \(2^{-N}\), so
\begin{equation*}
 0\le X_{N,\mathrm I}(\beta)-e_{N,\mathrm I}(\theta)
       \le\frac{\log2}{\beta}.
\end{equation*}
For the spherical prior write \(x=\sigma/\sqrt N\), put
\(\theta_+=\max\{\theta,0\}\), and set
\[
 \mathscr L_{N,\theta}(x)=\frac1{2N}
             \|(\theta\mathbf1-\mathbf G_Nx)_+\|_2^2,
 \qquad
 L_{N,\theta}=\frac{\|\mathbf G_N\|_{\rm op}}{\sqrt N}
 \left(\theta_+\sqrt{\frac{M_N}{N}}
                +\frac{\|\mathbf G_N\|_{\rm op}}{\sqrt N}\right).
\]
Contraction of the positive-part map and
\(\|(\theta\mathbf1-\mathbf G_Nx)_+\|_2
\le\theta_+\sqrt{M_N}+\|\mathbf G_N\|_{\rm op}\)
show that \(\mathscr L_{N,\theta}\) is \(L_{N,\theta}\)-Lipschitz on the
unit sphere.  For \(N\ge2\) and \(0<r\le1\), its Euclidean radius-\(r\)
cap about a minimizer has normalized measure at least \((r/8)^{N-1}\),
by integration in the tangent-plane chart of the hemisphere.  Restricting
the partition function to this cap yields
\begin{equation}\label{eq:spherical-zero-temperature-comparison}
 0\le X_{N,\mathrm S}(\beta)-e_{N,\mathrm S}(\theta)
       \le L_{N,\theta}r+\frac1\beta\log\frac8r.
\end{equation}
By \cref{lem:spectral-moments}, \(\sup_N\E L_{N,\theta}<\infty\).
Taking \(r=\beta^{-1}\) gives, for both priors and \(\beta\ge1\),
\begin{equation}\label{eq:uniform-zero-temperature-error}
 \limsup_{N\to\infty}\E|X_{N,\kappa}(\beta)-e_{N,\kappa}(\theta)|
       \le\frac{C_{\alpha,\theta}+\log(8\beta)}{\beta}
       \longrightarrow0.
\end{equation}
The finite-temperature limit
\(e_\kappa^{\rm soft}(\beta)=-\mathcal P_{\kappa,\beta,\theta}^{\rm har}(\alpha)/\beta\)
inherits nonnegativity and monotonicity; denote its limit by
\(e_{\mathrm{gs},\kappa}(\alpha,\theta)\).
Using \eqref{eq:harmonic-free energy-limit} in
\eqref{eq:uniform-zero-temperature-error}, first let \(N\to\infty\), then
\(\beta\to\infty\).  The triangle inequality proves
\eqref{eq:quadratic-ground-state-limit} and
\eqref{eq:quadratic-zero-temperature-identity}.
\end{proof}

\subsubsection{Closed-set hardening}
\label{sec:hard-constraints-capacities}

For the proof, abbreviate the soft pressure and its limiting value by
\begin{align*}
 F_{N,\kappa}^{\beta,C}
 &=\frac1N\log\int_{\Sigma_N^\kappa}
   \exp\left\{-\beta\sum_{k=1}^{M_N}d_C(h_k(\sigma))\right\}
   \mu_N^\kappa(\dd\sigma),
 \\
 \mathcal P_{\kappa,\beta,C}(\alpha)
 &=\mathcal P_{\kappa,u_{\beta,C}}(\alpha).
\end{align*}

We now prove \cref{thm:closed-set-hardening}.  The single-constraint estimate in the
proof is formulated for weighted measures because, after some constraints are
hardened, the remaining measure is neither uniform nor a product measure.

\begin{proposition}[Sequential hardening preserves the limiting entropy]
\label{prop:hard-entropy-preserving-hybrid}
Fix a tail-accessible closed set $C$ and
$\kappa\in\{\mathrm S,\mathrm I\}$, and suppose that, for some finite
$\bar\alpha>0$, \(M_N\le\bar\alpha N\) for all large $N$.  Assume that, for some $c_0>0$
with $c_0<\log2$ when $\kappa=\mathrm I$ and for every fixed $\beta>0$,
\begin{equation}\label{eq:hard-soft-lower-input}
 \Pp\bigl(F_{N,\kappa}^{\beta,C}<-c_0\bigr)\longrightarrow0.
\end{equation}
Then, for every $\varepsilon>0$,
\begin{equation*}
 \lim_{\beta\to\infty}\limsup_{N\to\infty}
 \Pp\left(F_{N,\kappa}^{\beta,C}
              -\operatorname{Ent}_{N,\kappa}(C)>\varepsilon\right)=0.
\end{equation*}
\end{proposition}

\begin{proof}
\emph{Single-constraint claim.}
Fix $\rho,\zeta>0$ and a tail-accessible closed set $C$, and choose
$\theta_C\in\R$ such that $[\theta_C,\infty)\subset C$.  There are constants
$b_{\rm row},c,A,w_0>0$ and $N_0<\infty$, depending only on
$(\rho,\zeta,C,\theta_C)$, such that the following holds for every
$N\ge N_0$.  Let $\nu$ be any deterministic probability measure on a subset $\Omega_N\subset
S_N$ satisfying
\begin{equation}\label{eq:hard-small-ball-hypothesis}
 \sup_{\sigma_0\in\Omega_N}
 \nu\left\{\sigma:\|\sigma-\sigma_0\|_2\le\rho\sqrt N\right\}
 \le e^{-\zeta N}.
\end{equation}
For $g\sim\mathsf N_N$, set
\begin{align*}
 X_g(\sigma)&=N^{-1/2}g\cdot\sigma,\\
 p_{\nu,C}(g)&=\nu\{X_g\in C\},\\
 s_{\nu,\beta,C}(g)&=\int e^{-\beta d_C(X_g(\sigma))}\nu(\dd\sigma),\\
 D_{\nu,\beta,C}(g)&=\log\frac{s_{\nu,\beta,C}(g)}{p_{\nu,C}(g)},
\end{align*}
We suppress the common argument \(g\) in the remainder of this claim and
its proof.  Set $D_{\nu,\beta,C}=+\infty$ when $p_{\nu,C}=0$.  Uniformly for
$w_0\le w\le b_{\rm row}N$,
\begin{equation}\label{eq:hard-one-row-tail}
 \Pp_g\bigl(p_{\nu,C}(g)\le e^{-w}\bigr)
 \le Ae^{-cw}+Ae^{-cN}.
\end{equation}
Moreover, there is a deterministic function
$\omega_{\rho,\zeta,C}(\beta)\downarrow0$ as $\beta\to\infty$
such that, for each fixed $\beta$ and all sufficiently large $N$,
\begin{equation}\label{eq:hard-one-row-mean}
 \E_g(D_{\nu,\beta,C}\wedge b_{\rm row}N)
 \le\omega_{\rho,\zeta,C}(\beta)+ANe^{-cN}.
\end{equation}
All constants are uniform over $\nu$ satisfying
\eqref{eq:hard-small-ball-hypothesis}.
The same conclusions hold conditionally if $\nu$ is random, independent of
$g$, and \eqref{eq:hard-small-ball-hypothesis} holds almost surely.

\emph{Proof of the claim.}
Decrease $b_{\rm row}$ so that $2b_{\rm row}<\zeta$.  Given
$w\in[w_0,b_{\rm row}N]$, let $K=\lfloor e^w/4\rfloor$ and sample
$\sigma^1,\ldots,\sigma^K$ independently from $\nu$.  A union bound and
\eqref{eq:hard-small-ball-hypothesis} show that the $\nu^{\otimes K}$-probability of a pair
within distance $\rho\sqrt N$ is at most
$K^2e^{-\zeta N}\le e^{-cN}$.  Conditional on pairwise separation, the
centered Gaussian variables $X_g(\sigma^i)$ have variance one and
canonical distances at least $\rho$.  Sudakov minoration gives
$\E_g\max_{i\le K}X_g(\sigma^i)\ge c_\rho\sqrt{\log K}$, and the Gaussian
concentration inequality for the maximum then gives, after increasing
$w_0$,
\begin{equation}\label{eq:hard-separated-sample}
 \Pp_g\left(\left.\max_{i\le K}X_g(\sigma^i)<\theta_C
       \,\right|\,\sigma^1,\ldots,\sigma^K\right)\le e^{-cw}
\end{equation}
for every pairwise separated realization
$(\sigma^1,\ldots,\sigma^K)$.
On the other hand, on $\{p_{\nu,C}(g)\le e^{-w}\}$, conditional independence
of the samples gives
\[
 \Pp\left(X_g(\sigma^i)\notin C\text{ for every }i\le K\mid g\right)
 =(1-p_{\nu,C}(g))^K\ge\frac34
\]
once $w_0$ is large.  Averaging first in the samples and then in $g$, and
noting that $x\notin C$ implies $x<\theta_C$, combining this inequality
with \eqref{eq:hard-separated-sample} proves
\eqref{eq:hard-one-row-tail}.

Write $\Delta_{\nu,\beta,C}(g)=s_{\nu,\beta,C}(g)-p_{\nu,C}(g)$ and
$Y_{\nu,C}(g)=-\log p_{\nu,C}(g)$, with
$Y_{\nu,C}=+\infty$ when $p_{\nu,C}=0$.  Since $X_g(\sigma)$ is standard Gaussian for every
$\sigma\in S_N$, Fubini gives the uniform estimate
\begin{equation}\label{eq:hard-soft-excess-mean}
 \E_g \Delta_{\nu,\beta,C}
 =\varepsilon_C(\beta)
 :=\E\left[e^{-\beta d_C(Z)}\1_{\{Z\notin C\}}\right]
 \longrightarrow0.
\end{equation}
Here dominated convergence applies because closedness gives
$d_C(Z)>0$ on $\{Z\notin C\}$.
For $w\in[w_0,b_{\rm row}N)$,
\begin{equation}\label{eq:hard-one-row-split}
 D_{\nu,\beta,C}\wedge b_{\rm row}N
 \le e^w \Delta_{\nu,\beta,C}
 +(Y_{\nu,C}\wedge b_{\rm row}N)\1_{\{Y_{\nu,C}>w\}}.
\end{equation}
Indeed, on $\{Y_{\nu,C}\le w\}$ use
$\log(1+\Delta_{\nu,\beta,C}/p_{\nu,C})
\le \Delta_{\nu,\beta,C}/p_{\nu,C}
\le e^w\Delta_{\nu,\beta,C}$, while on $\{Y_{\nu,C}>w\}$ use
$s_{\nu,\beta,C}\le1$, hence $D_{\nu,\beta,C}\le Y_{\nu,C}$.
Integrating \eqref{eq:hard-one-row-tail} yields
\[
 \E_g\bigl[(Y_{\nu,C}\wedge b_{\rm row}N)
             \1_{\{Y_{\nu,C}>w\}}\bigr]
 \le A(1+w)e^{-cw}+ANe^{-cN}.
\]
For large $\beta$, take
\[
 w_\beta=w_0\vee\frac12\log\frac1{\varepsilon_C(\beta)};
\]
for fixed $\beta$ this lies below $b_{\rm row}N$ when $N$ is large.  Then
$e^{w_\beta}\varepsilon_C(\beta)\to0$ and
$(1+w_\beta)e^{-cw_\beta}\to0$.  Equations
\eqref{eq:hard-soft-excess-mean} and \eqref{eq:hard-one-row-split} prove
\eqref{eq:hard-one-row-mean} with some
$\omega_{\rho,\zeta,C}(\beta)\downarrow0$.

\emph{Sequential hardening.}
Choose $c_1>c_0$, with $c_1<\log2$ when $\kappa=\mathrm I$.
We first verify the spread hypothesis in
the single-constraint claim.  If a probability measure $\nu$ on
$S_N$ satisfies $\dd\nu/\dd\mu_N^{\rm S}\le e^{c_1N}$, then a spherical
cap estimate gives, for fixed $0<\rho<1$,
\[
 \sup_{\sigma_0\in S_N}\nu\{\|\sigma-\sigma_0\|_2\le\rho\sqrt N\}
 \le e^{c_1N}C_\rho\sqrt N\,\rho^{N-3}.
\]
In the spherical case, choose $\rho$ sufficiently small and then
$\zeta>0$ so that this bound is at most $e^{-\zeta N}$ for all large
$N$.  On the cube, if
$\max_{\sigma\in\Sigma_N^{\rm I}}\nu(\sigma)\le e^{-sN}$ for some
$s>0$, a ball of radius $\rho\sqrt N$
contains at most
\[
 \exp\{N\mathsf h(\rho^2/4)+o(N)\}
\]
vertices, where
\(\mathsf h(x)=-x\log x-(1-x)\log(1-x)\) is binary entropy.  Choosing $\rho$ so that
$\mathsf h(\rho^2/4)<s/2$ and taking any $0<\zeta<s/2$ gives
\eqref{eq:hard-small-ball-hypothesis} for all large $N$.  In the
application below, $s=\log2-c_1>0$ in the Ising case.  Fix the resulting
pair $(\rho,\zeta)$ for the chosen prior.

Write
\[
 U_k(\sigma)=e^{-\beta d_C(h_k(\sigma))},
 \qquad I_k(\sigma)=\1_{\{h_k(\sigma)\in C\}}.
\]
For $0\le m\le M_N$, introduce the soft--hard hybrids
\begin{equation*}
 Z_m=\int
 \left(\prod_{k\le m}I_k(\sigma)\right)
 \left(\prod_{m<k\le M_N}U_k(\sigma)\right)
 \mu_N^\kappa(\dd\sigma).
\end{equation*}
Thus $N^{-1}\log Z_0=F_{N,\kappa}^{\beta,C}$ and
$N^{-1}\log Z_{M_N}=\operatorname{Ent}_{N,\kappa}(C)$.  Delete row $m$ and let
\begin{align*}
 \widehat Z_m
 &=\int\left(\prod_{k<m}I_k\right)
        \left(\prod_{m<k\le M_N}U_k\right)\dd\mu_N^\kappa,\\
 \nu_m(\dd\sigma)
 &=\widehat Z_m^{-1}
   \left(\prod_{k<m}I_k(\sigma)\right)
   \left(\prod_{m<k\le M_N}U_k(\sigma)\right)
   \mu_N^\kappa(\dd\sigma)
\end{align*}
when $\widehat Z_m>0$; otherwise set $\nu_m=\mu_N^\kappa$.
Conditional on all rows except $g_m$, $\nu_m$ is independent of $g_m$.
When $\widehat Z_m>0$,
\begin{equation*}
 Z_{m-1}=\widehat Z_m s_{\nu_m,\beta,C}(g_m),
 \qquad Z_m=\widehat Z_m p_{\nu_m,C}(g_m).
\end{equation*}
Let $D_m=D_{\nu_m,\beta,C}(g_m)$.  Whenever $Z_{m-1}>0$,
$D_m=\log Z_{m-1}-\log Z_m\in[0,\infty]$.

On $\{\widehat Z_m\ge e^{-c_1N}\}$ the unnormalized density defining
$\nu_m$ is at most one.  Hence
\begin{align*}
 \frac{\dd\nu_m}{\dd\mu_N^{\rm S}}&\le e^{c_1N}
 &&(\kappa=\mathrm S),\\
 \max_\sigma\nu_m(\sigma)&\le2^{-N}e^{c_1N}
 =e^{-(\log2-c_1)N}&&(\kappa=\mathrm I).
\end{align*}
The preceding spread estimates and
the single-constraint claim, conditionally on all constraints except $g_m$,
give constants $b_{\rm row},c,A>0$ and
$\omega_{\rho,\zeta,C}(\beta)\downarrow0$, uniform in $m$,
such that
\begin{equation}\label{eq:hard-active-cost}
 \E\left[(D_m\wedge b_{\rm row}N)
       \1_{\{\widehat Z_m\ge e^{-c_1N}\}}\right]
 \le\omega_{\rho,\zeta,C}(\beta)+ANe^{-cN}.
\end{equation}

To telescope even when a hard constraint makes a partition function
zero, truncate logarithms at a common floor.  For \(z\ge0\), put
\(\ell_N(z)=\log(z\vee e^{-c_1N})\) and
\(L_m=\ell_N(Z_{m-1})-\ell_N(Z_m)\ge0\).
Both hybrids lie below the floor when \(\widehat Z_m<e^{-c_1N}\);
otherwise \(L_m\le D_m\wedge c_1N\), since \(Z_{m-1}\le1\).  Thus
\[
 L_m\le C_1(D_m\wedge b_{\rm row}N)
             \1_{\{\widehat Z_m\ge e^{-c_1N}\}},
 \qquad C_1=\max\{1,c_1/b_{\rm row}\}.
\]
The event on the right is measurable without row \(m\).
Sum \eqref{eq:hard-active-cost} and telescope to obtain
\begin{equation*}
 \frac1N\E[\ell_N(Z_0)-\ell_N(Z_{M_N})]
 \le C_1\bar\alpha\,\omega_{\rho,\zeta,C}(\beta)+o_N(1).
\end{equation*}
Fix \(0<\eta<\min\{\varepsilon,c_1-c_0\}\).
If \(Z_0\ge e^{-c_0N}\) and the floored loss is at most \(\eta N\),
then \(\ell_N(Z_{M_N})\ge-(c_0+\eta)N>-c_1N\); hence both endpoint
logs are unclipped and their difference is at most \(\eta N\).
Markov's inequality gives
\[
 \Pp\left(F_{N,\kappa}^{\beta,C}
               -\operatorname{Ent}_{N,\kappa}(C)>\varepsilon\right)
 \le\Pp(Z_0<e^{-c_0N})
       +\frac{C_1\bar\alpha\,\omega_{\rho,\zeta,C}(\beta)+o_N(1)}\eta.
\]
Apply \eqref{eq:hard-soft-lower-input}, first letting \(N\to\infty\)
at fixed \(\beta\), then \(\beta\to\infty\).
\end{proof}

\begin{proof}[Proof of \cref{thm:closed-set-hardening}]
At fixed $q$, monotonicity of the scalar recursion gives
$\Phi_{u_{\beta,C}}(q)\downarrow\Phi_{\infty,C}(q)$.
Hence the soft variational values decrease to a limit no smaller than the
hard value.  Conversely, for every $q$ with
$\mathcal A_\kappa(q)<\infty$,
\[
 \mathcal P_{\kappa,\beta,C}(\alpha)
 \le \mathcal A_\kappa(q)+\alpha\Phi_{u_{\beta,C}}(q).
\]
Letting $\beta\to\infty$ and then taking the infimum proves
\eqref{eq:hard-variational-convergence}, including the value $-\infty$.

Since a hard indicator is bounded above by its soft distance weight,
\begin{equation}\label{eq:hard-below-soft}
 \operatorname{Ent}_{N,\kappa}(C)\le
 F_{N,\kappa}^{\beta,C}.
\end{equation}
Write $P_\infty=\mathcal P_{\kappa,\infty,C}(\alpha)$ and
$P_\beta=\mathcal P_{\kappa,\beta,C}(\alpha)$.  Suppose first that
$P_\infty> -\infty$ for spherical spins, or that
$P_\infty> -\log2$ for Ising spins.  Choose $c_0>-P_\infty$, with
$c_0<\log2$ in the Ising case.
Because $P_\beta\ge P_\infty$, the main free energy theorem at the fixed
finite Lipschitz channel $u_{\beta,C}$ implies
\eqref{eq:hard-soft-lower-input}.  Apply
\cref{prop:hard-entropy-preserving-hybrid}.
Choose first $\beta$ with $P_\beta-P_\infty$ small and then send
$N\to\infty$.  The hybrid controls the lower tail of
$\operatorname{Ent}_{N,\kappa}(C)$,
while \eqref{eq:hard-below-soft} controls the upper tail.  Thus
$\operatorname{Ent}_{N,\kappa}(C)\to P_\infty$ in probability, proving the finite
spherical case and the Ising entropy assertions.

If $P_\infty=-\infty$ for spherical spins, fix $L>0$ and choose a finite
$\beta$ with $P_\beta<-L-1$.  The fixed-$\beta$ soft formula and
\eqref{eq:hard-below-soft} give
$\Pp(\operatorname{Ent}_{N,\mathrm S}(C)>-L)\to0$; then let $L\to\infty$.

Finally suppose $P_\infty<-\log2$ for Ising spins.  Choose $\eta>0$ and
then a finite $\beta$ so that
 $P_\beta<-\log2-3\eta$.  For all large $N$,
\[
 \E F_{N,\mathrm I}^{\beta,C}
 <-\log2-2\eta.
\]
On the satisfiable event, one configuration has zero penalty and hence
$F_{N,\mathrm I}^{\beta,C}\ge-\log2$.  Since
$\Lip(u_{\beta,C})=\beta$, Gaussian concentration with row-matrix
Lipschitz constant $\beta\sqrt{M_N}/N$ gives
\[
 \Pp\bigl(\mathcal F_{M_N,N}^{\mathrm I}(C)\ne\varnothing\bigr)
 \le2\exp\left\{-\frac{\eta^2N^2}{2M_N\beta^2}\right\}.
\]
As $M_N/N\to\alpha$, this proves
\eqref{eq:hard-ising-exponential-unsat}.
\end{proof}

\subsection{Capacity consequences}
\label{sec:ising-capacities}

This subsection proves the general Ising capacity theorem, the spherical
half-space threshold, and the explicit spherical trial bound summarized
in \cref{sec:introduction}.

\subsubsection{Ising capacities for tail-accessible sets}

\begin{proof}[Proof of \cref{thm:closed-set-ising-capacity}]
Fix a tail-accessible closed set \(C\) and \(\beta>0\).  For a step
path \(q\), choose a finite grid \(\pi\) containing its jump points.
Jensen's inequality on the finite cascade gives
\[
 \Phi_{u_{\beta,C}}(q)
 \le\log\E\int e^{-\beta d_C(X^q(\eta,z))}
             \mathfrak R_\pi(\dd\eta)\mathsf N_1(\dd z)
 =\log\E e^{-\beta d_C(Z)}.
\]
The equality uses the standard-Gaussian marginal of the completed field
under the averaged unweighted measure.  At this fixed \(\beta\),
\(\Lip(u_{\beta,C})=\beta\), so \eqref{rpc:Phi-L1} extends the
inequality to every \(q\in\Q\) by step-path approximation.
Only then let \(\beta\to\infty\); dominated convergence yields
\begin{equation}\label{eq:ising-hard-annealed-bound}
 \Phi_{\infty,C}(q)\le\log\pi_C<0.
\end{equation}

Put \(B_C(q)=-\Phi_{\infty,C}(q)\),
\(r_C=\alpha_c^{\rm I}(C)\), and
\[
 s_C(\alpha)=\log2+\mathcal P_{\mathrm I,\infty,C}(\alpha)
 =\inf_{\substack{q\in\Q:\ \mathcal A_{\rm I}(q)<\infty}}
   \{\log2+\mathcal A_{\rm I}(q)-\alpha B_C(q)\},
 \qquad \alpha>0.
\]
If \(0<\alpha<r_C\), the ratio definition implies that \(B_C(q)\)
is finite for every finite-cost path, and
\[
 \log2+\mathcal A_{\rm I}(q)-\alpha B_C(q)
 \ge(r_C-\alpha)B_C(q)
 \ge(-\log\pi_C)(r_C-\alpha)>0.
\]
The last bound is uniform in \(q\), so \(s_C(\alpha)>0\).
If \(\alpha>r_C\), some trial ratio is smaller than \(\alpha\),
giving \(s_C(\alpha)<0\); if its denominator is infinite, the trial
value is \(-\infty\).  Thus
\[
 s_C(\alpha)>0\quad(0<\alpha<r_C),\qquad
 s_C(\alpha)<0\quad(\alpha>r_C).
\]
Together with \(\mathcal P_{\mathrm I,\infty,C}(0)=0\), these
inequalities prove \eqref{eq:closed-set-ising-capacity-boundary},
including the possibility \(r_C=0\) at this stage.
The path $\overline q_0$ has
$\mathcal A_{\rm I}(\overline q_0)=0$.  Its scalar recursion has no common
Gaussian increment, and hence
\[
 \Phi_{u_{\beta,C}}(\overline q_0)
 =\log\E e^{-\beta d_C(Z)}\longrightarrow\log\pi_C.
\]
Therefore $B_C(\overline q_0)=-\log\pi_C$, giving the upper bound in
\eqref{eq:closed-set-capacity-upper-bound}.

Apply \cref{thm:closed-set-hardening} to any sequence
\(M_N/N\to\alpha>0\).  For \(\alpha<r_C\), it gives
\(N^{-1}\log\#\mathcal F_{M_N,N}^{\rm I}(C)\to s_C(\alpha)>0\)
in probability, hence satisfiability with high probability.  For
\(\alpha>r_C\), it gives exponential decay of the satisfiability
probability.  This proves \eqref{eq:closed-set-ising-sat-unsat}.

For strict positivity, choose \(\theta_C\) with
\([\theta_C,\infty)\subseteq C\).  The small-density theorem of
Bolthausen--Nakajima--Sun--Xu applies to the half-space indicator
\(U(x)=\1_{\{x\ge\theta_C\}}\) and gives satisfiability with high
probability at some density \(\alpha_0>0\)
\cite[Theorem~1.1 and Proposition~1.3]{BolthausenNakajimaSunXu}.
Every such solution also satisfies the constraints in \(C\).
The supercritical conclusion just proved therefore forces
\(r_C\ge\alpha_0>0\), completing
\eqref{eq:closed-set-capacity-upper-bound}.

For fixed $N$, each of the $2^N$ configurations survives the first $m$
independent rows with probability $\pi_C^m$, and hence
\[
 \Pp\bigl(\mathfrak M_N^{\rm I}(C)\ge m\bigr)
 \le 2^N\pi_C^m.
\]
Letting $m\to\infty$ shows
$\mathfrak M_N^{\rm I}(C)<\infty$ almost surely.  For fixed
\(0<a_-<r_C<a_+\), row-count monotonicity and the fixed-load limits give
\[
 \Pp\bigl(\mathfrak M_N^{\rm I}(C)\ge\lfloor a_-N\rfloor\bigr)
 \longrightarrow1,\qquad
 \Pp\bigl(\mathfrak M_N^{\rm I}(C)\ge\lceil a_+N\rceil\bigr)
 \longrightarrow0.
\]
Let \(a_-\uparrow r_C\) and \(a_+\downarrow r_C\).
The integer roundings contribute at most \(N^{-1}\), proving
\eqref{eq:closed-set-random-capacity-limit}.
\end{proof}

The same results apply to \(C=[-K,0]\cup[K,\infty)\), \(K>0\).
Here \(\pi_C=1/2\), so \(\alpha_c^{\rm I}(C)\le1\); the full-path
formula does not by itself establish the optimized value one predicted in
\cite{BexSerneelsVandenBroeck}.  For the square-wave set
\[
 C=\bigcup_{j\in\mathbb Z}[(2j+1)\delta,(2j+2)\delta],\qquad\delta>0,
\]
\cref{thm:iid-channels} gives the soft formula for
\(-\beta\operatorname{dist}(\,\cdot\,,C)\), but the hardening theorem
does not apply because \(C\) contains no half-line.  The present results
also do not cover the correlated-label teacher--student model of
\cite{BenedettiEtAl}.

\subsubsection{Spherical half-spaces}
\label{sec:halfspace-spherical-capacity}

In this subsection we abbreviate the half-space quantities by
\begin{gather*}
 u_{\beta,\theta}=u_{\beta,C_\theta},\qquad
 \Phi_{\infty,\theta}=\Phi_{\infty,C_\theta},\qquad
 \mathcal P_{\kappa,\infty,\theta}=\mathcal P_{\kappa,\infty,C_\theta},\\
 \alpha_c^{\rm I}(\theta)=\alpha_c^{\rm I}(C_\theta),\qquad
 \mathcal F_{M,N}^{\kappa}(\theta)=\mathcal F_{M,N}^{\kappa}(C_\theta),\qquad
 \mathfrak M_N^\kappa(\theta)=\mathfrak M_N^\kappa(C_\theta).
\end{gather*}
Write \(\varphi_{\rm G}(x)=(2\pi)^{-1/2}e^{-x^2/2}\) and
\(\overline\Phi_{\rm G}(x)=\Pp(Z\ge x)\), with \(Z\sim N(0,1)\).
\begin{equation*}
 \mathcal B_\theta(q)=-\Phi_{\infty,\theta}(q),
\end{equation*}
\begin{equation*}
 D_\theta=\E(\theta-Z)_+^2.
\end{equation*}

The hard-volume formula does not itself imply emptiness on the sphere.
The next three results supply the threshold geometry, margin transfer, and
robust thickening needed to prove \cref{thm:spherical-halfspace-capacity}.

\begin{lemma}[Constant-path bound and the spherical finiteness threshold]
\label{lem:hard-capacity-geometry}
Fix \(\theta\in\mathbb R\).
For $0\le x<1$, the constant path satisfies
\begin{align}
 \mathcal A_{\rm S}(\overline q_x)
 &=\frac12\left(\frac{x}{1-x}+\log(1-x)\right),
 \label{eq:spherical-rs-spin-cost}\\
 \Phi_{\infty,\theta}(\overline q_x)
 &=\E\log\overline\Phi_{\rm G}\left(
       \frac{\theta-\sqrt{x}Z}{\sqrt{1-x}}\right).
 \label{eq:constant-hard-channel}
\end{align}
The set of $\alpha$ for which
$\mathcal P_{\mathrm S,\infty,\theta}(\alpha)>-\infty$ is downward
closed.  Hence the spherical hard value is finite for
$\alpha<\alpha_c^{\rm S}(\theta)$ and equals $-\infty$ for
$\alpha>\alpha_c^{\rm S}(\theta)$.  Moreover,
\begin{equation*}
 \alpha_c^{\rm S}(\theta)\le D_\theta^{-1}<\infty.
\end{equation*}
\end{lemma}

\begin{proof}
The spherical identity in \eqref{eq:spherical-rs-spin-cost} follows from
\eqref{eq:spherical-step-entropy}.  For the channel, conditional on the
common field $\sqrt xZ$, the soft normalizer decreases to
\[
 \overline\Phi_{\rm G}\left(
       \frac{\theta-\sqrt xZ}{\sqrt{1-x}}\right).
\]
Its logarithm has integrable negative part by Mills' inequalities, which
proves \eqref{eq:constant-hard-channel}.  The same inequalities and
dominated convergence give, as $x\uparrow1$,
\[
 (1-x)\mathcal A_{\rm S}(\overline q_x)\longrightarrow\frac12,
 \qquad
 (1-x)\mathcal B_\theta(\overline q_x)
 \longrightarrow\frac12D_\theta.
\]
Thus constant paths make the spherical objective tend to $-\infty$ when
$\alpha>D_\theta^{-1}$.

Since \(\Phi_{\infty,\theta}(q)\le0\), the spherical hard value is
nonincreasing in \(\alpha\).  Finiteness at \(\alpha_2\) therefore gives a
finite lower bound at every \(0<\alpha_1<\alpha_2\), while the trial path
\(\overline q_0\) gives a finite upper bound.  This proves downward closure.
\end{proof}

\begin{proposition}[Supercriticality persists after lowering the margin]
\label{prop:spherical-margin-relaxation}
For every \(\theta\in\mathbb R\) and every
$\alpha>\alpha_c^{\rm S}(\theta)$, there exists $\delta>0$ such
that
\begin{equation}\label{eq:spherical-relaxed-supercritical}
 \mathcal P_{\mathrm S,\infty,\theta-\delta}(\alpha)=-\infty.
\end{equation}
\end{proposition}

\begin{proof}
We bound the cost of shifting the margin by the spherical spin cost.
First, every path of finite spherical spin cost satisfies
\begin{align}
 \mathcal A_{\rm S}(q)
 &\ge\frac12\left(\frac1{\mathsf L(q)}-1+\log\mathsf L(q)\right),
 \notag\\
 \frac1{\mathsf L(q)}&\le4\bigl(1+\mathcal A_{\rm S}(q)\bigr).
 \label{eq:spherical-susceptibility-bound}
\end{align}
In particular, \(\mathsf L(q)>0\).  To prove these, first suppose
\(q(1-)<1\), put \(\ell=\mathsf L(q)=\int_0^1\gamma_q\), and write
\(D(t)=\int_t^1\gamma_q(s)\,\dd s\).  Since
\(D(t)\le\min\{\ell,1-t\}\), \eqref{eq:spherical-entropy-classical} gives
\[
 2\mathcal A_{\rm S}(q)
 =\int_0^1\left\{\frac1{D(t)}-\frac1{1-t}\right\}\dd t
 \ge\int_0^{1-\ell}\left\{\frac1\ell-\frac1{1-t}\right\}\dd t
 =\ell^{-1}-1+\log\ell.
\]
For general \(q\), cap it at \(1-\eta\) on \([0,1)\), retaining
the diagonal value \(q_\eta(1)=1\).  The conjugate formula
\eqref{eq:spherical-conjugate-identification} gives
\(\mathcal A_{\rm S}(q_\eta)\le\mathcal A_{\rm S}(q)\), while
\(\mathsf L(q_\eta)\to\mathsf L(q)\).  The preceding bound rules
out \(\mathsf L(q)=0\) and passes to the limit.
Applying \(x\le4+2(x-1-\log x)\), \(x\ge1\), proves
\eqref{eq:spherical-susceptibility-bound}.

The second estimate, for every \(\delta,\varepsilon>0\), is
\begin{equation}\label{eq:spherical-margin-shift}
 \mathcal B_{\theta+\delta}(q)
 \le(1+\varepsilon)\mathcal B_\theta(q)
 +\left(1+\frac1\varepsilon\right)\frac{\delta^2}{2\mathsf L(q)}.
\end{equation}
The scalar recursion and \cref{lem:mz-control}, as in the
finite-temperature passage above, identify
\(\Phi_{u_{\beta,\vartheta}}(q)
 =\mathcal T_{\gamma_q}u_{\beta,\vartheta}(0,0)\) directly for the
Lipschitz terminal \(u_{\beta,\vartheta}(x)=-\beta(\vartheta-x)_+\).
Put \(\ell=\mathsf L(q)\).  For any admissible control \(v\), set
\(X^v=B_1+\int_0^1\gamma_qv\) and \(w=v+\delta/\ell\).
The shifted control has finite weighted quadratic cost and is therefore
admissible; the optional clipping bound need not be imposed.  For \(p>1\),
\[
 u_{\beta,\theta+\delta}(X^w)
 =u_{\beta,\theta}(X^v)\ge p\,u_{\beta,\theta}(X^v),\qquad
 \int_0^1\gamma_qw^2
 \le p\int_0^1\gamma_qv^2+\frac{p\delta^2}{(p-1)\ell}.
\]
The first inequality uses nonpositivity of the terminal and the second
is Young's inequality.  Taking control suprema, then letting
\(\beta\to\infty\) and setting \(p=1+\varepsilon\), proves
\eqref{eq:spherical-margin-shift}.

Choose \(\alpha_c^{\rm S}(\theta)<\alpha_0<\alpha\), and then
\(\delta>0\) so small that
\[
 \alpha_0\left\{\frac{1+\delta}{\alpha}
                  +2\delta(1+\delta)\right\}<1.
\]
If the hard value at \((\alpha,\theta-\delta)\) were finite, some
\(C_0\ge0\) would give, for every finite-cost path,
\[
 \mathcal B_{\theta-\delta}(q)
 \le\frac{\mathcal A_{\rm S}(q)+C_0}{\alpha}.
\]
Apply \eqref{eq:spherical-margin-shift} from \(\theta-\delta\) to
\(\theta\), with \(\varepsilon=\delta\), and use
\eqref{eq:spherical-susceptibility-bound}.  The result is
\[
 \mathcal B_\theta(q)
 \le\left\{\frac{1+\delta}{\alpha}+2\delta(1+\delta)\right\}
          \mathcal A_{\rm S}(q)+C_\delta.
\]
Thus \(\mathcal A_{\rm S}(q)-\alpha_0\mathcal B_\theta(q)\)
is uniformly bounded below, contradicting
\cref{lem:hard-capacity-geometry}.  This proves
\eqref{eq:spherical-relaxed-supercritical}.
\end{proof}

\begin{proposition}[Feasibility at margin \(\theta\) gives exponential volume at margin \(\theta-\delta\)]
\label{prop:spherical-robust-thickening}
Suppose $M_N/N\to\alpha\in(0,\infty)$.  For every
$\theta\in\R$ and $\delta>0$, there is a finite
$C=C(\alpha,\theta,\delta)$ such that
\begin{equation}\label{eq:spherical-thickening}
 \Pp\left(
 \mathcal F_{M_N,N}^{\rm S}(\theta)\ne\varnothing,\quad
 \frac1N\log\mu_N^{\rm S}
   \bigl(\mathcal F_{M_N,N}^{\rm S}(\theta-\delta)\bigr)<-C
 \right)\longrightarrow0.
\end{equation}
\end{proposition}

\begin{proof}
Fix $\bar\alpha>\alpha$.  Gaussian norm tails show that, with probability
tending to one, $M_N\le\bar\alpha N$ and
$\max_{k\le M_N}\|g_k\|\le2\sqrt N$.  The remaining argument is
deterministic in any rows satisfying these bounds.  Identify $S_N$ with
the unit sphere by $s=\sigma/\sqrt N$, so constraints become
$g_k\cdot s\ge\theta$ and $\mu_N^{\rm S}$ remains normalized surface measure.
For an independent $W\sim N(0,I_N/N)$, \v{S}id\'ak's inequality
\cite[Corollary~1, p.~628]{Sidak1967} gives
\[
 \Pp_W\left(\max_k|g_k\cdot W|\le L\right)
 \ge p_{L/2}^{M_N}\ge e^{-\kappa_LN},\qquad
 p_x=\Pp(|Z|\le x),\quad \kappa_L=-\bar\alpha\log p_{L/2}.
\]
Here each $g_k\cdot W$ has variance at most four; arbitrary correlations
are allowed.  Also $\Pp_W(\|W\|>2)\le e^{-c_0N}$ for a universal
$c_0>0$.  Fix $L$ large enough that $\kappa_L<c_0/2$.  For all large $N$,
\[
 E=\{\max_k|g_k\cdot W|\le L,\ \|W\|\le2\},\qquad
 \Pp_W(E)\ge\tfrac12e^{-\kappa_LN}.
\]
Take any feasible unit vector $s$ and set
\[
 r=\min\left\{\frac14,
       \frac\delta{2(L+2|\theta|+1)}\right\},\qquad Y=s+rW.
\]
On $E$, $1-2r\le\|Y\|\le1+2r$, and for every row,
\[
 g_k\cdot\frac{Y}{\|Y\|}-\theta
 \ge\frac{\theta(1-\|Y\|)-rL}{\|Y\|}
 \ge-\frac{r(L+2|\theta|)}{1-2r}\ge-\delta.
\]
Let $A$ be the cone over the feasible set at margin $\theta-\delta$ and
write $P=N(0,r^2I_N/N)$, $Q_s=N(s,r^2I_N/N)$.  Then
$Q_s(A)\ge\frac12e^{-\kappa_LN}$, while isotropy identifies $P(A)$ with
the normalized spherical volume.  The Gaussian likelihood ratio satisfies
$\E_P[(\dd Q_s/\dd P)^2]=e^{N/r^2}$, so Cauchy--Schwarz gives
\[
 \mu_N^{\rm S}(\mathcal F_{M_N,N}^{\rm S}(\theta-\delta))
 =P(A)\ge e^{-N/r^2}Q_s(A)^2
 \ge\tfrac14e^{-N(r^{-2}+2\kappa_L)}.
\]
This holds for every feasible $s$ with the same constants, proving
\eqref{eq:spherical-thickening}.
\end{proof}

\begin{proof}[Proof of \cref{thm:spherical-halfspace-capacity}]
Let
$M_N=\lfloor\alpha N\rfloor$.  By
\cref{lem:hard-capacity-geometry}, the spherical variational hard value is
finite for $\alpha<\alpha_c^{\rm S}(\theta)$.  The hardening theorem then
gives convergence of the log-volume to this finite value, so the feasible
set, on whose complement the log-volume is $-\infty$, is
nonempty with probability tending to one.  If
$\alpha>\alpha_c^{\rm S}(\theta)$, choose $\delta>0$ from
\cref{prop:spherical-margin-relaxation}.  The hard-wall theorem at margin
$\theta-\delta$ and \cref{prop:spherical-robust-thickening} imply
$\Pp(\mathcal F_{M_N,N}^{\rm S}(\theta)\ne\varnothing)\to0$.

For fixed $N$, cover the unit sphere by finitely many Euclidean caps
$\{s:\|s-v_j\|_2<r\}$ with unit centers $v_j$ and fixed $0<r<1$.
Choose $K$ so large that
$-K(1-r^2/2)+1<\theta$.  If
$\|g+Kv_j\|_2<1$, then throughout the $j$th cap
\[
 g\cdot s\le-Kv_j\cdot s+1
 \le-K(1-r^2/2)+1<\theta.
\]
Each such open Gaussian ball has positive probability, so an infinite
i.i.d.\ row sequence almost surely visits all finitely many of them.
Once it has done so, every point of the sphere violates at least one of
the rows already seen.  Thus
$\mathfrak M_N^{\rm S}(\theta)<\infty$ almost surely.
Finally, row-count monotonicity sandwiches the lower tail of
$\mathfrak M_N^{\rm S}(\theta)/N$ by failure of satisfiability at
$\alpha_c^{\rm S}(\theta)-\varepsilon/2$ and its upper tail by
satisfiability at $\alpha_c^{\rm S}(\theta)+\varepsilon/2$; the first event
is omitted when the lower load is nonpositive.  The fixed-load statements
make both probabilities vanish, proving \eqref{eq:random-capacity-limit}.
\end{proof}

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\end{document}