\section{From binary patterns to retained patterns at fixed overlap}
\label{sec:reduction}

\newcommand{\thrGood}{\mathrm{Good}_N}
\newcommand{\thrk}{\boldsymbol{k}}
\newcommand{\thrB}{\mathbb{B}}
\newcommand{\thrD}{\mathbb{D}}
\newcommand{\throp}{\mathrm{op}}
\newcommand{\thrF}{\mathsf{F}}

The reduction of binary patterns to finitely many retained patterns in a Gaussian bulk was first carried out in \cite{BD}, for the unconstrained free energy, and in \cite{ABCHIM} with cutoffs on the auxiliary Gaussian variables. As the capacity depends on the free energy at fixed overlap, we prove in this section the reduction on every slice $m_1=m_k$, in both directions, with errors independent of the slice, and with concentration strong enough for a union over all patterns and all slices; this is Theorem~\ref{thm:reduction}. Combined with Theorem~\ref{thm:gaussian-slice}, it gives the limit of the free energy at fixed overlap, $\LamR(m)=\sup_r\Theta_r(m)$.

The reduction extends to slices because a gauge transformation turns the pattern that defines the slice into the constant vector, so that the slice becomes a deterministic measure on the cube, and the comparison lemmas of \cite[Section~6]{ABCHIM} treat an arbitrary weight on the cube, independent of the disorder.

\subsection{Notation}\label{subsec:red-notation}

We fix $\beta>0$, $\alpha>0$ and $M=M(N)$ with $M/N\to\alpha$, and put $K:=M-1$. We also fix $0<\alpha_-<\alpha<\alpha_+$ and $N_0$ such that $(M-r)/N\in[\alpha_-,\alpha_+]$ for the finitely many $r$ in use and all $N\ge N_0$; all statements are for $N\ge N_0$.

Here a matrix $X$ is $N\times K'$, with columns $X^\mu$, and acts through $X^T\sigma\in\R^{K'}$, so that the rows of Theorem~\ref{thm:gaussian-slice} are the columns here. We write $\varepsilon$ for the $N\times K$ matrix with i.i.d.\ symmetric $\pm1$ entries and $G$ for a matrix with i.i.d.\ standard Gaussian entries, of a size given by the context. We denote by $\gamma_{K'}$ the standard Gaussian measure on $\R^{K'}$ and by $P_N$ the uniform law on $\Cube$.

Throughout, $\rho$ denotes a nonzero finite positive deterministic measure on $\Cube$, and we set
\[
f^\rho(X):=\frac1N\log\sum_{\sigma\in\Cube}\rho(\sigma)\exp\Bigl(\frac{\beta}{2N}|X^T\sigma|^2\Bigr).
\]
The exponent lies in $[0,\frac\beta2\|X\|_\throp^2]$, so
\begin{equation}\label{eq:f-bound}
f^\rho(0)\le f^\rho(X)\le f^\rho(0)+\frac{\beta}{2N}\|X\|_\throp^2,\qquad f^\rho(0)=\frac1N\log\rho(\Cube).
\end{equation}
As before, $\Slice(k)$ denotes the set of configurations with exactly $k$ plus signs and $m_k:=2k/N-1$. We put $\rho_k:=2^{-N}\one_{\Slice(k)}$, so that $f^{\rho_k}(0)\in[-\log2,0]$.

Recall that the overlap is $m_\mu(\sigma)=\xi^\mu\cdot\sigma/N$. The gauge $\sigma\mapsto\mathrm{diag}(\xi^1)\sigma$ sends $\{m_1(\sigma)=m_k\}$ to $\Slice(k)$, and the vectors $\mathrm{diag}(\xi^1)\xi^\mu$, $\mu\ge2$, are i.i.d.\ Rademacher vectors independent of $\xi^1$. On $\Slice(k)$ we have $(\one\cdot\sigma)^2=N^2m_k^2$. With $\varepsilon$ the matrix of the gauged patterns,
\begin{equation}\label{eq:gauge}
p_N(k):=\frac1N\E\log2^{-N}\sum_{m_1(\sigma)=m_k}\exp\Bigl(\frac{\beta}{2N}\sum_{\mu\le M}(\xi^\mu\cdot\sigma)^2\Bigr)=\frac{\beta m_k^2}{2}+\E f^{\rho_k}(\varepsilon).
\end{equation}

For $1\le r\le M$, with $\zeta^2,\dots,\zeta^r$ i.i.d.\ Rademacher vectors, $G'$ a Gaussian $N\times(M-r)$ matrix and $X_r:=(\zeta^2,\dots,\zeta^r,G')$, we set
\[
p^{(r)}(\rho):=\E f^\rho(X_r),\qquad p^{(r)}_N(k):=\frac{\beta m_k^2}{2}+p^{(r)}(\rho_k).
\]
Thus $r$ counts pattern $1$ and $r-1$ further Rademacher patterns, and $p^{(M)}_N(k)=p_N(k)$. For $r=1$, with $Y_{N,k}:=2^{-N}\sum_{\Slice(k)}\exp(\frac{\beta}{2N}|G^T\sigma|^2)$ and $G$ Gaussian $N\times K$, we have $p^{(1)}_N(k)=\beta m_k^2/2+\frac1N\E\log Y_{N,k}$.

Recall that $d_r:=2^{r-1}$ for $r\ge1$, that the index $\eta$ ranges over $\{-1,1\}^{r-1}$, with coordinates $\eta_j$, $j=2,\dots,r$, and that $\Eeta$ is the uniform average. For $\tv=(t_\eta)\in[-1,1]^{d_r}$, we set, as before,
\[
\ov_j(\tv):=\Eeta[\eta_jt_\eta],\qquad V(\tv)^2:=(\Eeta t_\eta)^2+|\ov(\tv)|^2,\qquad P_r(m):=\{\tv\in[-1,1]^{d_r}:\Eeta t_\eta=m\}.
\]
With $\Phi^{(r)}$ from Theorem~\ref{thm:gaussian-slice}, so that $\Phi^{(1)}=\Phi$, let
\[
\Fr(\tv):=\frac\beta2V(\tv)^2+\Phi^{(r)}(\tv),\qquad\Theta_r(m):=\max_{\tv\in P_r(m)}\Fr(\tv),\qquad\LamR(m):=\sup_{r\ge1}\Theta_r(m),
\]
and $\Lam(m):=\Theta_1(m)=\beta m^2/2+\Phi(m)$.

At zero temperature, we put $L_A(X):=\frac1N\max_{\sigma\in A}|X^T\sigma|$ for nonempty $A\subset\Cube$, and
\[
e_N(k):=\frac{1}{2N^2}\E\max_{m_1(\sigma)=m_k}\sum_{\mu\le M}(\xi^\mu\cdot\sigma)^2=\frac{m_k^2}{2}+\frac12\E L_{\Slice(k)}(\varepsilon)^2.
\]
Keeping the largest term gives, pointwise on matrices,
\begin{equation}\label{eq:sandwich}
\frac\beta2L_A(X)^2-\log2\le f^{2^{-N}\one_A}(X)\le\frac\beta2L_A(X)^2.
\end{equation}

\subsection{The reduction}

The main result of this section is the following.

\begin{theorem}\label{thm:reduction}
There are constants $C_1,c_1,L,\Lambda>0$, depending only on $\beta,\alpha_\pm$, such that the following hold.
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item (Lower bound.) For fixed $r$, $N$ large and every $\rho$,
\[
\E f^\rho(\varepsilon)\ge p^{(r)}(\rho)-C_1N^{-1/2}.
\]
So $p_N(k)\ge p^{(r)}_N(k)-C_1N^{-1/2}$ for all $k$.
\item (Upper bound.) For $\delta\in(0,1]$ put $K_\delta:=\lfloor L^2/\delta^2\rfloor$. For $N$ large and every $\rho$,
\[
\E f^\rho(\varepsilon)\le\max_{1\le r\le1+K_\delta}p^{(r)}(\rho)+C_1\delta^2+\mathrm{err}_N(\delta),
\]
where $\mathrm{err}_N(\delta)\to0$ as $N\to\infty$ for fixed $\delta$, and it does not depend on $\rho$. In particular $p_N(k)\le\max_{r\le1+K_\delta}p^{(r)}_N(k)+C_1\delta^2+\mathrm{err}_N(\delta)$ for all $k$.
\item (Limit form.)
\begin{enumerate}[label=(\alph*),leftmargin=*]
\item For fixed $r$, $\sup_k|p^{(r)}_N(k)-\Theta_r(m_k)|\to0$.
\item $\Theta_r$ is continuous, and $\LamR+I$ is H\"older of exponent $1/2$ on $[-1,1]$.
\item $\sup_k|p_N(k)-\LamR(m_k)|\to0$. So if $m_{k_N}\to m$,
\[
\lim_Np_N(k_N)=\LamR(m)=\sup_{r\ge1}\ \sup_{\tv\in P_r(m)}\Bigl\{\frac\beta2\bigl(m^2+|\ov(\tv)|^2\bigr)+\Phi^{(r)}(\tv)\Bigr\}\ge\frac{\beta m^2}{2}+\Phi(m).
\]
\end{enumerate}
\item (Concentration.) For every $\rho$ and $t>0$,
\begin{equation}\label{eq:conc}
\P\Bigl(|f^\rho(\varepsilon)-\E f^\rho(\varepsilon)|>t+C_1e^{-c_1N}\Bigr)\le C_1e^{-c_1N}+C_1e^{-c_1Nt^2}.
\end{equation}
With $f^{\mu,k}:=\frac1N\log2^{-N}\sum_{m_\mu(\sigma)=m_k}\exp\bigl(\frac{\beta}{2N}\sum_{\mu'\ne\mu}(\xi^{\mu'}\cdot\sigma)^2\bigr)$, a union over all patterns and all slices costs a factor $M(N+1)$ in \eqref{eq:conc}. At zero temperature, for every nonempty $A\subset\Cube$,
\begin{equation}\label{eq:conc-L}
\P\bigl(|L_A(\varepsilon)-\E L_A(\varepsilon)|>t\bigr)\le C_1e^{-c_1Nt^2},\qquad0\le\frac12\E L_A(\varepsilon)^2-\frac12\bigl(\E L_A(\varepsilon)\bigr)^2\le\frac{C_1}N,
\end{equation}
and $\E L_A(\varepsilon)\le C_1$.
\end{enumerate}
\end{theorem}

\subsection{Inputs}

For an $N\times K'$ matrix $A$, a weight $w:\Cube\to[0,\infty)$ with $\int w\dd P_N>0$, and a Borel set $D\subset\R^{K'}$ of positive Gaussian measure, we put \cite[(6.1)]{ABCHIM}
\[
\thrF(A,w,D):=\frac1N\log\iint_{\Cube\times D}w(\sigma)\exp\Bigl(\sqrt{\frac\beta N}\,\sigma^TAz\Bigr)\dd P_N(\sigma)\dd\gamma_{K'}(z).
\]
By the Gaussian integral, $\thrF(A,2^N\rho,\R^{K'})=f^\rho(A)$, and for random weights we assume that $\log\int w\dd P_N$ is integrable. With $\thrB(L):=\{|z|\le L\sqrt N\}$, let $\thrD(\delta,L)$ be the intersection of $\thrB(L)$ with the cube $[-\delta\sqrt N,\delta\sqrt N]^{K'}$. Let $A$ have independent entries, uniform on $\{-1,1\}$ or standard Gaussian, and $K'\le\alpha_+N$. We use the following from \cite[Section~6]{ABCHIM}.
\begin{itemize}[leftmargin=*]
\item (6.3). $\P(\|A\|_\throp>C(\sqrt N+\sqrt{K'})+t)\le2e^{-ct^2}$, and $\E\|A\|_\throp^q\le C_q(N+K')^{q/2}$.
\item (6.4), (6.5). If $U$ is a convex, globally $\ell$-Lipschitz function of independent signs, then $\P(|U-\E U|>t)\le Ce^{-ct^2/\ell^2}$. For $J$ such functions, with no independence between them, $\E\max_jU_j\le\max_j\E U_j+C\ell\sqrt{\log(2J)}$.
\item Lemma 6.1. If $w$ is independent of $\varepsilon$ and $G$, $\delta\in(0,1]$, $L>0$ and $D\subset\thrD(\delta,L)$, then $|\E\thrF(\varepsilon,w,D)-\E\thrF(G,w,D)|\le C_\beta L^2\delta^2$.
\item Lemma 6.2. There are $L_0\ge1$, $\Lambda<\infty$, depending only on $\beta,\alpha_+$, such that $\P(\|A\|_\throp>\Lambda\sqrt N)\le Ce^{-cN}$, and such that for $L\ge L_0$ and every $w$, on $\{\|A\|_\throp\le\Lambda\sqrt N\}$,
\begin{equation}\label{eq:trunc}
0\le\thrF(A,w,\R^{K'})-\thrF(A,w,\thrB(L))\le Ce^{-cN}.
\end{equation}
The difference is at most $\frac{\beta}{2N}\|A\|_\throp^2+\frac{\log2}N$ for all $A$, and its expectation is at most $Ce^{-cN}$, also for random $w$.
\item (6.20). If $\alpha_-N\le K'$, $\delta\sqrt N\ge2$, $L\ge L_0$ and $w$ is independent of $G$, then $0\le\E\thrF(G,w,\R^{K'})-\E\thrF(G,w,\thrD(\delta,L))\le C/(\delta^2N)+Ce^{-cN}$.
\end{itemize}
From now on, we fix $L:=L_0$.

We will also need a Gaussian form of the maximum bound. Let $\thrk$ run over a finite set of size $J$, let $w_{\thrk}$ be deterministic weights and $a_{\thrk}$ real numbers. Each $\thrF(G,w_{\thrk},\thrB(L))$ is globally Lipschitz in $G$, with constant $\ell=L\sqrt{\beta/N}$. Indeed, its gradient has entries $N^{-1}\sqrt{\beta/N}\langle\sigma_iz_\mu\rangle$, so its squared norm is at most $\beta N^{-2}\langle|z|^2\rangle\le\beta L^2/N$. Using Gaussian concentration, \eqref{eq:trunc} and the deterministic bound on its complement, we obtain
\begin{equation}\label{eq:max-exchange}
\E\max_{\thrk}\bigl\{a_{\thrk}+\thrF(G,w_{\thrk},\R^{K'})\bigr\}\le\max_{\thrk}\bigl\{a_{\thrk}+\E\thrF(G,w_{\thrk},\R^{K'})\bigr\}+C\sqrt{\frac{\log(2J)}N}.
\end{equation}

\subsection{The lower and upper bounds}

\begin{proof}[Proof of Theorem~\ref{thm:reduction}\textup{(i)}]
We keep $r-1$ columns inside the weight and compare the others under a cutoff, which only lowers the free energy on the Rademacher side.

We split $\varepsilon=(\zeta,\varepsilon')$, with $\zeta$ the first $r-1$ columns and $\varepsilon'$ the other $M-r$, and put $w(\sigma):=2^N\rho(\sigma)\exp(\frac\beta{2N}\sum_j(\zeta^j\cdot\sigma)^2)$, which is a function of $\zeta$ only, with $\log\int w\dd P_N$ bounded. Then $f^\rho(\varepsilon)=\thrF(\varepsilon',w,\R^{M-r})$ and $f^\rho(\zeta,G')=\thrF(G',w,\R^{M-r})$, so, with $\delta:=N^{-1/4}$,
\begin{align*}
\E f^\rho(\varepsilon)&\ge\E\thrF(\varepsilon',w,\thrD(\delta,L))\ge\E\thrF(G',w,\thrD(\delta,L))-C_\beta L^2\delta^2\\
&\ge\E\thrF(G',w,\R^{M-r})-C_\beta L^2\delta^2-\frac C{\delta^2N}-Ce^{-cN}=p^{(r)}(\rho)-O(N^{-1/2}).
\end{align*}
The first step restricts a positive integrand, the second is Lemma~6.1 of \cite{ABCHIM}, and the third is (6.20) of \cite{ABCHIM}. The constants do not depend on $\rho$, and the statement on $p_N$ follows from \eqref{eq:gauge}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:reduction}\textup{(ii)}]
Large hidden coordinates cannot be replaced by a Lindeberg step, but there are few of them. We thus cut $z$ to a ball and split the ball according to the set $S$ of large coordinates; the columns in $S$ go into the weight, and the others meet only small coordinates.

Let $w:=2^N\rho$. By Lemma~6.2 of \cite{ABCHIM}, $\E f^\rho(\varepsilon)\le\E\thrF(\varepsilon,w,\thrB(L))+Ce^{-cN}$. For $z\in\thrB(L)$, the set $S(z):=\{\mu:|z_\mu|>\delta\sqrt N\}$ has at most $K_\delta$ elements, so $\thrB(L)$ is covered by the sets $D_S:=\{|z_S|\le L\sqrt N\}\times\thrD_{S^c}(\delta,L)$, $S\subset\{2,\dots,M\}$, $|S|\le K_\delta$, where $\thrD_{S^c}$ is the set $\thrD$ in the coordinates outside $S$. Let $\mathcal S$ be the family of these $S$, of size $J\le(K_\delta+1)M^{K_\delta}$, and put $F_S(X):=\thrF(X,w,D_S)$. Then
\[
\thrF(\varepsilon,w,\thrB(L))\le\max_{S\in\mathcal S}F_S(\varepsilon)+\frac{\log J}N.
\]
Each $F_S$ is convex in $X$, as a log-Laplace transform of functions linear in $X$. As before, its gradient has entries $N^{-1}\sqrt{\beta/N}\langle\sigma_iz_\mu\rangle$, with squared norm at most $\beta N^{-2}\langle|z|^2\rangle\le2\beta L^2/N$, because $|z|^2\le2L^2N$ on $D_S$. Hence (6.5) of \cite{ABCHIM} gives $\E\max_SF_S(\varepsilon)\le\max_S\E F_S(\varepsilon)+CL\sqrt{\beta\log(2J)/N}$.

We next fix $S$ with $|S|=s$ and put
\[
w_S(\sigma):=w(\sigma)\int_{|z_S|\le L\sqrt N}\exp\Bigl(\sqrt{\frac\beta N}\,\sigma^T\varepsilon_Sz_S\Bigr)\dd\gamma_S(z_S),
\]
which is a function of $\varepsilon_S$ only, and so is independent of $\varepsilon_{S^c}$ and of a Gaussian matrix $G_{S^c}$ in the other columns. By Jensen and the symmetry of the ball, $\int w_S\dd P_N\ge\gamma_S(|z_S|\le L\sqrt N)\int w\dd P_N$, and $\varepsilon_S$ takes finitely many values, so $\log\int w_S\dd P_N$ is bounded. By Fubini, $F_S(\varepsilon)=\thrF(\varepsilon_{S^c},w_S,\thrD_{S^c}(\delta,L))$. Lemma~6.1 of \cite{ABCHIM}, and then enlarging the domain to the whole space twice, give
\[
\E F_S(\varepsilon)\le\E\thrF(G_{S^c},w_S,\thrD_{S^c}(\delta,L))+C_\beta L^2\delta^2\le\E f^\rho(\varepsilon_S,G_{S^c})+C_\beta L^2\delta^2.
\]
Since the columns are i.i.d.\ and $\rho$ does not see their labels, we have $\E f^\rho(\varepsilon_S,G_{S^c})=p^{(s+1)}(\rho)$. Collecting the terms gives (ii), with $\mathrm{err}_N(\delta)=Ce^{-cN}+\frac{\log J}N+CL\sqrt{\beta\log(2J)/N}$.
\end{proof}

\subsection{Concentration}\label{subsec:red-conc}

\begin{proof}[Proof of Theorem~\ref{thm:reduction}\textup{(iv)}]
Let $w=2^N\rho$ and $F^L(X):=\thrF(X,w,\thrB(L))$. As in the last proof, $F^L$ is convex and globally $L\sqrt{\beta/N}$-Lipschitz, so (6.4) of \cite{ABCHIM} applies. By Lemma~6.2 of \cite{ABCHIM}, $0\le f^\rho-F^L\le Ce^{-cN}$ on $\{\|\varepsilon\|_\throp\le\Lambda\sqrt N\}$ and $0\le\E f^\rho-\E F^L\le Ce^{-cN}$, which gives \eqref{eq:conc}.

Under the gauge $\mathrm{diag}(\xi^\mu)$ for pattern $\mu$, the other patterns become i.i.d.\ Rademacher, so $f^{\mu,k}$ has the law of $f^{\rho_k}(\varepsilon)$. A union bound over $M(N+1)$ events gives the second claim, and no independence between patterns is used. The term of pattern $\mu$ itself is the constant $\beta m_k^2/2$ and does not fluctuate.

The function $L_A$ is convex, as a maximum of norms of linear maps, and $|L_A(X)-L_A(X')|\le\|X-X'\|_{\mathrm{HS}}/\sqrt N$ globally. Hence (6.4) of \cite{ABCHIM} with $\ell=1/\sqrt N$ gives the tail in \eqref{eq:conc-L}, and integrating the tail gives the variance. Finally, $L_A(\varepsilon)\le\|\varepsilon\|_\throp/\sqrt N$, and (6.3) of \cite{ABCHIM} bounds its mean.
\end{proof}

\subsection{The limit}

We now prove Theorem~\ref{thm:reduction}(iii). Given the retained patterns, the sites split into $d_r$ types, and the retained term depends on $\sigma$ only through the magnetizations of the types, so a slice splits into polynomially many type slices. On each type slice, only the Gaussian bulk remains and Theorem~\ref{thm:gaussian-slice} applies, giving $\Theta_r$ for fixed $r$. The two sides of (i) and (ii) then squeeze $p_N(k)$ between $\Theta_r$ and $\sup_r\Theta_r$.

\begin{lemma}\label{lem:polytope}
$m\mapsto P_r(m)$ is Lipschitz in Hausdorff distance for the sup norm, with constant $d_r$. So $\Theta_r$ is uniformly continuous on $[-1,1]$.
\end{lemma}

\begin{proof}
For $\tv\in P_r(m)$ and $m'>m$, the vector $\tv':=\tv+\frac{m'-m}{1-m}(\one-\tv)$ lies in $P_r(m')$. Since $0\le1-t_\eta\le d_r(1-m)$, we get $|\tv'-\tv|_\infty\le d_r(m'-m)$, and the case $m'<m$ is symmetric. Moreover, $\Fr$ is uniformly continuous, by Theorem~\ref{thm:gaussian-slice}(d) and continuity of $I$.
\end{proof}

\begin{lemma}\label{lem:fixed-r}
For fixed $r$, $\sup_k|p^{(r)}_N(k)-\Theta_r(m_k)|\to0$.
\end{lemma}

\begin{proof}
We condition on $\zeta=(\zeta^2,\dots,\zeta^r)$, and we denote by $\eta(i):=(\zeta^j_i)_j$ the type of site $i$. Put $I_\eta:=\{i:\eta(i)=\eta\}$, $n_\eta:=|I_\eta|$ and $w^N_\eta:=n_\eta/N$, and let $\thrGood:=\{|w^N_\eta-1/d_r|\le N^{-1/3}\text{ for all }\eta\}$. By Hoeffding, $\P(\thrGood^c)\le2d_r\exp(-2N^{1/3})$. We take $N$ so large that $n_\eta\ge N/(2d_r)$ on $\thrGood$.

We call an integer vector $\thrk=(k_\eta)$ admissible if $0\le k_\eta\le n_\eta$ and $\sum_\eta k_\eta=k$. The admissible vectors form a nonempty set of at most $(N+1)^{d_r}$ elements. With $t_\eta:=2k_\eta/n_\eta-1$, let $S(\thrk)$ be the type slice. On $S(\thrk)$, we have $\zeta^j\cdot\sigma=N\ov^N_j(\tv)$ with $\ov^N_j(\tv):=\sum_\eta w^N_\eta\eta_jt_\eta$. We set
\[
\Gamma_N(\thrk):=\frac1N\log2^{-N}\sum_{S(\thrk)}\exp\Bigl(\frac{\beta}{2N}|G'^T\sigma|^2\Bigr),\qquad\Upsilon_N:=\frac{\beta m_k^2}{2}+f^{\rho_k}(\zeta,G'),
\]
\[
V_N(\thrk)^2:=m_k^2+|\ov^N(\tv)|^2,\qquad B_N(\thrk):=\frac\beta2V_N(\thrk)^2+\Gamma_N(\thrk).
\]
Pointwise,
\begin{equation}\label{eq:type-sum}
\max_{\thrk}B_N(\thrk)\le\Upsilon_N\le\max_{\thrk}B_N(\thrk)+\frac{d_r\log(N+1)}{N}.
\end{equation}

For uniformity in the type proportions, let $\mathrm{err}'_N$ be the supremum over counts $(n_\eta)$ in $\thrGood$ and over $\thrk$ of $|\E_{G'}\Gamma_N(\thrk)-\Phi^{(r)}(\tv)|$. Since the rows of $G'$ are exchangeable, the law of $\Gamma_N$ depends on the partition only through $(n_\eta)$. Choosing for each $N$ counts that nearly attain the supremum gives a sequence of partitions of $[N]$ with $|I_\eta|/N\to1/d_r$, and $(M-r)/N\to\alpha$. Hence Theorem~\ref{thm:gaussian-slice}(a) applies, and $\mathrm{err}'_N\to0$.

For the upper bound on $\thrGood$, note that $|\ov^N_j(\tv)-\ov_j(\tv)|\le d_rN^{-1/3}$ and $|\Eeta t_\eta-m_k|\le d_rN^{-1/3}$, so $\E_{G'}B_N(\thrk)\le\Fr(\tv)+C_rN^{-1/3}+\mathrm{err}'_N$, with $\Fr(\tv)\le\Theta_r(\Eeta t_\eta)\le\Theta_r(m_k)+\varpi_r(d_rN^{-1/3})$, where $\varpi_r$ is the modulus of continuity of $\Theta_r$ from Lemma~\ref{lem:polytope}. Given $\zeta$, the type slices are deterministic and only $G'$ is random, so using \eqref{eq:max-exchange} with $J\le(N+1)^{d_r}$ and \eqref{eq:type-sum}, we obtain
\[
\E_{G'}\Upsilon_N\le\max_{\thrk}\E_{G'}B_N(\thrk)+O\bigl(\sqrt{\log N/N}\bigr)\le\Theta_r(m_k)+o(1),
\]
uniformly in $k$ and in good $\zeta$.

For the lower bound on $\thrGood$, let $\tv^o$ attain $\Theta_r(m_k)$ and put $\bar k_\eta:=n_\eta(1+t^o_\eta)/2$, so $|\sum_\eta\bar k_\eta-k|\le(d_r/2)N^{2/3}$. Rounding with $\sum_\eta k_\eta=k$ inside the box is possible since $0\le k\le\sum_\eta n_\eta$, with $\sum_\eta|k_\eta-\bar k_\eta|\le(d_r/2)N^{2/3}+d_r$, and hence $|\tv(\thrk)-\tv^o|_\infty\le C_rN^{-1/3}$. By uniform continuity of $\Fr$, we obtain $\E_{G'}\Upsilon_N\ge\max_{\thrk}\E_{G'}B_N(\thrk)\ge\Theta_r(m_k)-o(1)$.

It remains to treat bad partitions, for which \eqref{eq:f-bound} gives $-\log2\le\Upsilon_N\le\beta r/2+\frac{\beta}{2N}\|G'\|_\throp^2$. Hence $\E[(|\Upsilon_N|+|\Theta_r(m_k)|)\one_{\thrGood^c}]\le C\,\P(\thrGood^c)^{1/2}\to0$, uniformly in $k$, which proves the lemma.
\end{proof}

\begin{lemma}\label{lem:theta-regular}
Put $U_N(k):=p_N(k)-\beta m_k^2/2+I(m_k)$ and $C_3:=\sup_N\E\|\varepsilon\|_\throp^2/N$, which is finite by (6.3) of \cite{ABCHIM}. Then for all $k,k'$,
\[
|U_N(k)-U_N(k')|\le\sqrt2\,\beta C_3|m_k-m_{k'}|^{1/2}+\frac{2\log(N+1)}N.
\]
\end{lemma}

\begin{proof}
For a matrix $X$, let $\mathrm{Av}_k(X)$ be the uniform average over $\Slice(k)$ of $\exp(\frac{\beta}{2N}|X^T\sigma|^2)$. If $k<k'$ and $\sigma\le\sigma'$ coordinatewise, with $\sigma\in\Slice(k)$ and $\sigma'\in\Slice(k')$, then
\[
\bigl||X^T\sigma|^2-|X^T\sigma'|^2\bigr|\le4\|X\|_\throp^2\sqrt{(k'-k)N}.
\]
Since the bipartite graph $\{\sigma\le\sigma'\}$ between $\Slice(k)$ and $\Slice(k')$ is biregular, averaging over neighbors gives
\[
|\log\mathrm{Av}_k(X)-\log\mathrm{Av}_{k'}(X)|\le2\beta\|X\|_\throp^2\sqrt{(k'-k)/N},\qquad(k'-k)/N=|m_k-m_{k'}|/2.
\]
By \eqref{eq:gauge}, $U_N(k)=\frac1N\log(2^{-N}|\Slice(k)|)+I(m_k)+\E\frac1N\log\mathrm{Av}_k(\varepsilon)$, and $|\frac1N\log(2^{-N}|\Slice(k)|)+I(m_k)|\le\log(N+1)/N$.
\end{proof}

We now complete the proof of \textup{(iii)}. Part (a) is Lemma~\ref{lem:fixed-r}, and $\Theta_r$ is continuous by Lemma~\ref{lem:polytope}. Let $m_{k_N}\to m$. It follows from (i) and Lemma~\ref{lem:fixed-r} that $\liminf_Np_N(k_N)\ge\Theta_r(m)$ for every $r$, so $\liminf_Np_N(k_N)\ge\LamR(m)$. Conversely, for fixed $\delta$, applying (ii) and Lemma~\ref{lem:fixed-r} to the finitely many $r\le1+K_\delta$, we obtain
\[
\limsup_Np_N(k_N)\le\max_{r\le1+K_\delta}\Theta_r(m)+C_1\delta^2\le\LamR(m)+C_1\delta^2.
\]
Letting $\delta\to0$, we obtain $p_N(k_N)\to\LamR(m)$, which is finite by \eqref{eq:f-bound}. Taking two such sequences in Lemma~\ref{lem:theta-regular} shows that $\LamR-\beta m^2/2+I$ is H\"older of exponent $1/2$, which gives (b). If (c) failed, there would be $\epsilon>0$ and $k_N$, along a subsequence, with $|p_N(k_N)-\LamR(m_{k_N})|\ge\epsilon$ and $m_{k_N}\to m$, which contradicts the limit we just proved and the continuity of $\LamR$. This completes the proof of \textup{(iii)}. \qed
\begin{corollary}\label{cor:energy-limit}
$\sup_k|e_N(k)-\eR(m_k)|\to0$, where $\eR:=\lim_{\beta\to\infty}\LamR_\beta/\beta$, the limit being uniform on $[-1,1]$.
\end{corollary}

\begin{proof}
By \eqref{eq:sandwich}, $\beta e_N(k)-\log2\le p_N(k)\le\beta e_N(k)$, and Theorem~\ref{thm:reduction}(iii)(c) gives
\[
\limsup_N\sup_k\bigl|e_N(k)-\LamR_\beta(m_k)/\beta\bigr|\le\frac{\log2}{\beta}.
\]
Comparing two values of $\beta$ and using continuity of $\LamR_\beta$ on $[-1,1]$ gives
\[
\sup_m\bigl|\LamR_\beta(m)/\beta-\LamR_{\beta'}(m)/\beta'\bigr|\le\log2\,\Bigl(\frac1\beta+\frac1{\beta'}\Bigr).
\]
Hence the limit exists uniformly.
\end{proof}