\chapter{Approach to the critical temperature from above}\label{sec:hot-meso}

In this section, we prove \cref{thm:intro-main}(3). Let $c_N=1-\delta_N$, where $\delta_N\to0$ and $M_N=N\delta_N^3\to\infty$. \Cref{mgf-all-mesoscopic-profile} below states that $D_N$ has the cutoff profile $\Psi$ with window $\delta_N^{-2}$ and center
\[
T_N=\frac{\log M_N-\frac32\log\log M_N+\log B_{c_N}}{4\delta_N^2},\qquad B_c=\frac{\sqrt c}{4\sqrt\pi\,(1+c)^2}.
\]
No condition is imposed on the rate at which $M_N$ tends to infinity.

The proof compares the time-$t$ law $\nu_NP_t$ with the Bingham comparison law $\mu_t$ of \cref{sec:heat}, and it has two independent parts. The first part concerns only Gaussian laws conditioned on $S_N$. Both $\mu_t$ and $\pi_{N,c_N}$ are laws of this kind, and by \cref{gp-pinned-lan-quantitative-third} the distance between them is close to $2\Phi(\sqrt{\cI_N(t)}/2)-1$, where $\cI_N(t)$ is the information of the covariance evolution (\cref{def:model-information}). Recall from \cref{mc-exact-info} that $\cI_N(t)$ is determined by the mean $\bar h$ of the heat vector and by its second derivative. In \cref{sec:hotm-information}, we compute $\bar h$ and $\bar h''$ at times of order $\delta_N^{-2}\log M_N$. At such times only the precisions within a distance of order $\delta_N^2/\log M_N$ of the smallest precision matter. We describe them through a rescaled spectral measure (\cref{mc-measure-theorem}). This gives the asymptotics of $\cI_N$ (\cref{mc-information}) and the profile of $\mu_t$ (\cref{mc-proxy-profile}). The second part, in \cref{sec:hotm-comparison}, is deterministic. For a matrix whose precisions near the smallest precision $d$ are spaced like the eigenvalues of the GOE near the edge, \cref{mgf-global-bridge} bounds $\|\nu_NP_t-\mu_t\|_{\TV}$ by $Ce^{-(d-2/N)t}$ for $t\le C_Td^{-1}\log M$, where $M=Nd^{3/2}$. The constant $C$ does not depend on $N$ or $d$, and the bound holds as soon as $M$ exceeds a fixed threshold. We combine the two parts in \cref{sec:hotm-proof}.

We use the following earlier results. From \cref{sec:model}, we use the properties of $\Psi$ (\cref{cot-profile}), \cref{lem:model-localization}, and \cref{cor:model-profile-cutoff}; from \cref{sec:spectral}, \cref{sp-canonical-saddle,sp-raw-gap-resolvent,sp-fixed-confidence-geometry,mc-raw-edge-measure}; and from \cref{sec:conditioning,sec:likelihood}, \cref{prop:cond-bingham,gp-pinned-lan-quantitative-third}. From \cref{sec:heat}, we use the covariance evolution together with \cref{hi3-finite-heat,hi3-information-envelopes,iw-all-time-heat,mc-exact-info,pr-real-measure-transfer}; from \cref{sec:comparison}, \cref{ht-forward-defect,pc-centered-defect-transfer,rp-general-flux}; and from \cref{sec:process}, \cref{hfk-pointwise-fk,iw-chemical-occupation,iw-early-pin-simultaneous-centers,hmr-main}. \Cref{mgf-global-bridge} is used again in \cref{sec:outer-hot}, where $M$ stays fixed while $N\to\infty$, and its statement is therefore nonasymptotic.

\section{Setting and main result}\label{sec:hotm-setting}

\paragraph{The covariance evolution.}
Let $A$ be a real symmetric $N\times N$ matrix with precisions $p_1\le\cdots\le p_N$ and Lagrange multiplier $z$ (\cref{def:model-multiplier}). Recall that $q_i=1/p_i$, $\sum_iq_i=N$, and $d=p_1$. We work in the eigencoordinates of $A$. As in \cref{sec:heat}, we let
\[
H=-2\diag(p_1,\dots,p_N)+\frac2N\ind\ind^\T,\qquad h(t)=e^{tH}\ind,\qquad \bar h(t)=\frac1N\sum_{i=1}^Nh_i(t),
\]
so that $h_i'(t)=-2p_ih_i(t)+2\bar h(t)$. The variances of the covariance evolution of $A$ are $v_i(t)=h_i(t)/\bar h(t)$, and the Bingham comparison law at time $t$ is $\mu_t=\mu_{v(t)}$, the Gaussian law with variances $v_i(t)$ conditioned on $S_N$ (\cref{prop:cond-bingham}). We have $v_i(0)=1$ and $\mu_0=\nu_N$. The numbers $v_i(t)$ are the variances of the unconditioned Gaussian vector in the definition of $\mu_t$; they need not equal the second moments of $\mu_t$ or those of $\nu_NP_t$. Recall the relative errors and, from \cref{def:model-information}, the information $\cI(t)=\cI(q,v(t))$:
\[
e_i(t)=\frac{v_i(t)}{q_i}-1=p_iv_i(t)-1,\qquad \cI(t)=\frac12\sum_{i=1}^Ne_i(t)^2 .
\]

The matrix $H$ is negative semidefinite, its kernel is spanned by $q=(q_1,\dots,q_N)$, and $-H\ge2d$ on the orthogonal complement of $q$ (the second part of \cref{lem:heat-algebra}). Let $\vartheta=N/\sum_iq_i^2$. We let $\rho$ be the positive measure on $[2d,\infty)$ such that
\begin{equation}\label{eq:hotm-spectral}
\frac1N\ind^\T g(-H)\ind=\vartheta\,g(0)+\int g(x)\,\rho(\dd x)
\end{equation}
for every function $g$ on the spectrum of $-H$; this is the measure $\rho$ of \cref{lem:heat-algebra}. In other words, $\vartheta\delta_0+\rho$ is the spectral measure of $-H$ at the unit vector $N^{-1/2}\ind$, and $\vartheta=(\ind^\T q)^2/(N|q|^2)$ is the mass of its atom at zero. The choice $g(x)=e^{-tx}$ gives
\begin{equation}\label{eq:hotm-U}
\bar h(t)=\vartheta+\int_{[2d,\infty)}e^{-tx}\,\rho(\dd x),\qquad \rho([2d,\infty))=1-\vartheta,
\end{equation}
which is \eqref{mc-positive-measure}, and by \cref{mc-exact-info} we have
\begin{equation}\label{eq:hotm-info}
\cI(t)=\frac{N\,\bar h''(2t)}{8\,\bar h(t)^2}\qquad(t\ge0).
\end{equation}
When $A=c_NW_N$, we write $d_N$, $\vartheta_N$, $\rho_N$, and $\cI_N$ for these quantities.

\paragraph{The regime.}
In \cref{sec:hotm-setting,sec:hotm-information,sec:hotm-proof}, we assume that
\begin{equation}\label{eq:hotm-regime}
c_N=1-\delta_N,\qquad 0<\delta_N<1,\qquad \delta_N\to0,\qquad M_N=N\delta_N^3\to\infty,
\end{equation}
where the sequence $\delta_N$ is deterministic and need not be monotone. We write $\ell_N=\log(2+M_N)$. The matrix is $A=c_NW_N$, its precisions are $p_i=z_N-c_N\lambda_i$, and we set
\[
\gamma_i=p_i-d_N=c_N(\lambda_1-\lambda_i)\qquad(1\le i\le N).
\]
We have $\gamma_1=0$, and almost surely $\gamma_i>0$ for $i\ge2$. Probabilities refer to $W_N$, and $o_\PP$ and $O_\PP$ are as in \cref{sec:model-quantifiers}.

\paragraph{The main result.}
The following theorem is \cref{thm:intro-main}(3). Recall the profile $\Psi(u)=2\Phi(e^{-2u}/2)-1$ from \eqref{hp-universal-profile}.

\begin{theorem}[Profile for the approach from above]\label{mgf-all-mesoscopic-profile}
Assume \eqref{eq:hotm-regime}. For $N$ so large that $M_N>e$, define
\begin{equation}\label{mc-profile-center}
B_c=\frac{\sqrt c}{4\sqrt\pi\,(1+c)^2},\qquad
T_N=\frac{\log M_N-\frac32\log\log M_N+\log B_{c_N}}{4\delta_N^2},\qquad
T_N(u)=T_N+\frac u{\delta_N^2}.
\end{equation}
Then for every $u_0>0$,
\begin{equation}\label{mgf-actual-profile-convergence}
\sup_{|u|\le u_0}\big|D_N\big(T_N(u)\big)-\Psi(u)\big|\longrightarrow0
\end{equation}
in probability. For every $\eps\in(0,1)$,
\begin{equation}\label{mgf-accuracy-offsets}
\delta_N^2\big(t_N(\eps)-T_N\big)\longrightarrow-\frac12\log\Big[2\,\Phi^{-1}\Big(\frac{1+\eps}2\Big)\Big]=\Psi^{-1}(\eps)
\end{equation}
in probability. Further, $4\delta_N^2T_N/\log M_N\to1$, and $D_N$ has cutoff profile $\Psi$ with center $T_N$ and window $\delta_N^{-2}$ in the sense of \cref{def:model-cutoff}; in particular, there is cutoff.
\end{theorem}

The numbers $T_N$ and $\delta_N^{-2}$ are deterministic. As $\delta_N^2T_N\to\infty$, all the times $T_N(u)$ with $|u|\le u_0$ are positive for large $N$. The comparison of $T_N$ with the center at fixed temperature is discussed in \cref{sec:intro-discussion}.

\paragraph{Spectral inputs.}
We use the following consequences of \cref{sec:spectral}. Recall from \eqref{sp-log-precise-saddle} in \cref{sp-canonical-saddle} that
\begin{equation}\label{mc-canonical-input}
\ell_N\,\Big|\frac{d_N}{\delta_N^2}-1\Big|\longrightarrow0,\qquad
\ell_N\,\Big|\frac{1-c_N^2}N\sum_{i=1}^Nq_i^2-1\Big|\longrightarrow0
\end{equation}
in probability. As $1-c_N^2=\delta_N(2-\delta_N)$ and $\vartheta_N=N/\sum_iq_i^2$, we obtain
\begin{equation}\label{eq:hotm-canonical-consequences}
\frac{d_N}{\delta_N^2}\to1,\qquad \frac{Nd_N^{3/2}}{M_N}\to1,\qquad \frac{\vartheta_N}{\sqrt{d_N}}\to2
\end{equation}
in probability. By the third and fourth parts of \cref{sp-raw-gap-resolvent}, we have
\begin{equation}\label{mc-raw-trace}
\frac1N\sum_{i=2}^N\frac1{\gamma_i}=\frac1{c_N}+\frac{\Xi_N}{c_NN^{1/3}},\qquad
\frac1{\delta_N}\Big(\frac1N\sum_{i=2}^N\frac1{\gamma_i}-1\Big)=\frac1{c_N}+\frac{\Xi_N}{c_NM_N^{1/3}}\longrightarrow1,
\end{equation}
where the limit holds in probability. Here $\Xi_N$ is the regularized sum of inverse gaps of \cref{sp-raw-gap-resolvent}, which is tight; its convergence was proved by Landon and Sosoe \cite[Theorem~6.1]{LandonSosoe2022}.
Finally, by \eqref{sp-indexed-raw-gaps} in \cref{sp-fixed-confidence-geometry}, and since $1/2\le c_N\le1$ for large $N$, for every $\eta\in(0,1)$ there exist a constant $K_\eta\ge1$ and events $\mathcal \mathsf G^\eta$ with $\liminf_N\PP(\mathcal \mathsf G^\eta)\ge1-\eta$ on which
\begin{equation}\label{mc-indexed-input}
K_\eta^{-1}\Big(\frac{i-1}N\Big)^{2/3}\le\gamma_i\le K_\eta\Big(\frac{i-1}N\Big)^{2/3}\qquad(2\le i\le N).
\end{equation}
The constant $K_\eta$ depends on $\eta$. As in \cref{sec:model-quantifiers}, we take $N\to\infty$ at fixed $\eta$ and remove the localization at the end with \cref{lem:model-localization}.

\paragraph{Deterministic inputs.}
Two results of \cref{sec:heat,sec:likelihood} about a fixed matrix are used in the following form. Let $K\ge1$, and suppose that $0<d\le1$, $Nd^{3/2}\ge1$, and $K^{-1}((i-1)/N)^{2/3}\le p_i-d\le K((i-1)/N)^{2/3}$ for $2\le i\le N$. By \eqref{hi3-late-coordinate} in \cref{hi3-information-envelopes}, there exists a constant $C_K$, depending only on $K$, such that
\begin{equation}\label{mc-late-input}
|e_i(d^{-1}+s)|\le C_K\,\frac d{p_i}\,(1+ds)\,e^{-2ds}\qquad(s\ge0,\ 1\le i\le N).
\end{equation}
The second input is \cref{gp-pinned-lan-quantitative-third}: there exists a universal constant $\eta_{\mathrm{lik}}>0$, and for every $B<\infty$ a constant $C_B$, with the following property. Let $v,\tilde v$ be variance sequences on $\{1,\dots,N\}$ with $\sum_iv_i=\sum_i\tilde v_i=N$, write $\tilde v_i=v_i(1+\tilde e_i)$, and suppose that $\max_i|\tilde e_i|\le1/4$, $\eta_v\le\eta_{\mathrm{lik}}$, and $I=\frac12\sum_i\tilde e_i^2\le B$. Then
\begin{equation}\label{mc-pin-input}
\Big|\,\|\mu_{\tilde v}-\mu_v\|_{\TV}-\Big(2\Phi\Big(\frac{\sqrt I}2\Big)-1\Big)\Big|\le C_B\Big(\eta_v+\max_i|\tilde e_i|\Big).
\end{equation}
Here $\eta_v=\max_iv_i/(2\sum_jv_j^2)^{1/2}$ is the largest standardized variance of \eqref{gp-standardization}.

\section{The information at a smaller scale}\label{sec:hotm-information}

We now compute the information at times $t$ with $\delta_N^2t\to\infty$. At such a time, the relative error $e_i(t)$ is small unless $p_i-d_N$ is at most of order $1/t$, which is much smaller than $\delta_N^2$. These coordinates correspond to the eigenvalues within a distance of order $1/t$ of $\lambda_1$, and there are of order $Nt^{-3/2}$ of them. Throughout this section, we fix deterministic times $\bar t_N>0$ that satisfy \eqref{mc-time-hypotheses} with $t_N=\bar t_N$, that is,
\[
\delta_N^2\bar t_N\longrightarrow\infty,\qquad \delta_N^2\bar t_N=o\big(M_N^{2/3}\big),
\]
and we set $\bar w_N=d_N\bar t_N$, which is random. Recall from \eqref{mc-scales} that $\bar w_N/(\delta_N^2\bar t_N)\to1$ in probability, that $\bar t_N=o(N^{2/3})$ and $N\bar t_N^{-3/2}\to\infty$, and that $\bar t_N/(N\delta_N)=\delta_N^2\bar t_N/M_N\to0$. In \cref{mc-proxy-profile} below, we take $\bar t_N=T_N$, for which $\delta_N^2\bar t_N\sim\frac14\log M_N$.

\paragraph{Counts at the scale $1/\bar t_N$.}
The eigenvalue counts at this scale are given by \cref{mc-raw-edge-measure}, applied with $t_N=\bar t_N$. Recalling that $c_N(\lambda_1-\lambda_i)=\gamma_i$, we may state it as follows: the positive measures
\begin{equation}\label{eq:hotm-chi}
\omega_N^{\mathrm{gap}}=\frac{\bar t_N^{3/2}}N\sum_{i=1}^N\delta_{\bar t_N\gamma_i}
\end{equation}
on $[0,\infty)$ satisfy, for every continuous function $g$ on $[0,\infty)$ with $\sup_{y\ge0}(1+y)^2|g(y)|<\infty$,
\begin{equation}\label{eq:hotm-chi-limit}
\int g\,\dd\omega_N^{\mathrm{gap}}\longrightarrow\frac1\pi\int_0^\infty g(y)\sqrt y\,\dd y\quad\text{in probability}.
\end{equation}
The atom of $\omega_N^{\mathrm{gap}}$ at zero comes from $i=1$ and has mass $\bar t_N^{3/2}/N$, which tends to zero.

\paragraph{The rescaled transient measure.}
We rescale $\rho_N$ at the same scale. Let $\mathfrak m_N$ be the finite positive measure on $[0,\infty)$ defined by
\begin{equation}\label{mc-nu-definition}
\int g\,\dd\mathfrak m_N=d_N\,\bar t_N^{3/2}\int g\Big(\bar t_N\Big(\frac x2-d_N\Big)\Big)\,\rho_N(\dd x)
\end{equation}
for bounded measurable $g$. It is supported on $[0,\infty)$ because $\rho_N$ is supported on $[2d_N,\infty)$. We substitute $x=2d_N+2y/\bar t_N$ in \eqref{eq:hotm-U}, so that $\bar t_Nx=2\bar w_N+2y$ and $x^2=4d_N^2(1+y/\bar w_N)^2$. We also use $d_N\bar t_N^{3/2}=d_N^{-1/2}\bar w_N^{3/2}$ and $\bar h''(t)=\int x^2e^{-tx}\rho_N(\dd x)$. For every $a>0$, this gives
\begin{equation}\label{eq:hotm-U-measure}
\begin{aligned}
\bar h(a\bar t_N)-\vartheta_N&=\sqrt{d_N}\,\bar w_N^{-3/2}e^{-2a\bar w_N}\int e^{-2ay}\,\mathfrak m_N(\dd y),\\
\bar h''(2a\bar t_N)&=4d_N^{5/2}\,\bar w_N^{-3/2}e^{-4a\bar w_N}\int\Big(1+\frac y{\bar w_N}\Big)^2e^{-4ay}\,\mathfrak m_N(\dd y).
\end{aligned}
\end{equation}
These are exact identities for each $N$.

We say that positive measures $\mathfrak m_N$ on $[0,\infty)$ converge vaguely in probability to $\mathfrak m$ if $\int g\,\dd\mathfrak m_N\to\int g\,\dd\mathfrak m$ in probability for every continuous, compactly supported $g$ on $[0,\infty)$.

\begin{theorem}[The rescaled transient measure]\label{mc-measure-theorem}
Assume \eqref{eq:hotm-regime}, and let $\bar t_N>0$ be deterministic times that satisfy \eqref{mc-time-hypotheses} with $t_N=\bar t_N$. Then, in probability,
\begin{equation}\label{mc-measure-limit}
\mathfrak m_N\longrightarrow\frac{\sqrt y}\pi\,\dd y\quad\text{vaguely},\qquad
\int\frac{\mathfrak m_N(\dd y)}{1+y^2}\longrightarrow\frac1{\sqrt2}.
\end{equation}
Moreover,
\begin{align}
\bar h(\bar t_N)-\vartheta_N&=\frac{\sqrt{d_N}\,\bar w_N^{-3/2}e^{-2\bar w_N}}{4\sqrt{2\pi}}\big(1+o_\PP(1)\big),\label{mc-j-asymptotic}\\
\bar h''(2\bar t_N)&=\frac{d_N^{5/2}\,\bar w_N^{-3/2}e^{-4\bar w_N}}{4\sqrt\pi}\big(1+o_\PP(1)\big).\label{mc-j-derivative}
\end{align}
\end{theorem}

The limit $\pi^{-1}\sqrt y\,\dd y$ is the image of the semicircle density near the edge, $\rho_{\mathrm{sc}}(2-r)\approx\pi^{-1}\sqrt r$, under the rescaling of \eqref{eq:hotm-chi}. We derive \eqref{mc-j-derivative} from the exact formula for $\bar h''$ in \eqref{eq:hotm-U-measure}, and not by differentiating \eqref{mc-j-asymptotic}.

\begin{proof}
The proof has four steps. We first express the transforms $\int(s+y)^{-2}\mathfrak m_N(\dd y)$ through the gaps $\gamma_i$, using an exact identity for the resolvent of $H$ at a negative real argument. We then compute their limits with \eqref{eq:hotm-chi-limit} and \eqref{mc-raw-trace}, identify the limit of $\mathfrak m_N$ with \cref{pr-real-measure-transfer}, and insert the result into \eqref{eq:hotm-U-measure}.

\paragraph{Step 1: a resolvent identity.}
For $\xi\in(-2d_N,0)$, let
\[
\mathsf G(\xi)=\frac1N\sum_{i=1}^N\frac1{p_i+\xi/2},
\]
as in \eqref{eq:heat-GS}; it has no poles on this interval. Such $\xi$ do not lie in $\{0\}\cup[-2p_N,-2d_N]$, and the sixth part of \cref{lem:heat-algebra} gives $\mathsf G(\xi)\ne1$ and
\begin{equation}\label{eq:hotm-resolvent}
\frac{\vartheta_N}\xi+\int\frac{\rho_N(\dd x)}{\xi+x}=\frac{\mathsf G(\xi)}{2\big(1-\mathsf G(\xi)\big)}\qquad(-2d_N<\xi<0).
\end{equation}
For these $\xi$, the left side is $N^{-1}\ind^\T(\xi-H)^{-1}\ind$, a resolvent of the finite matrix $H$ at a negative argument, and not a Laplace transform of $\bar h$.

Both sides of \eqref{eq:hotm-resolvent} are smooth functions of $\xi\in(-2d_N,0)$. Differentiating, and using $\mathsf G'(\xi)=-\frac1{2N}\sum_i(p_i+\xi/2)^{-2}$, we obtain
\begin{equation}\label{eq:hotm-squared-resolvent}
\int\frac{\rho_N(\dd x)}{(\xi+x)^2}=\frac1{4\big(1-\mathsf G(\xi)\big)^2}\,\frac1N\sum_{i=1}^N\frac1{(p_i+\xi/2)^2}-\frac{\vartheta_N}{\xi^2}\qquad(-2d_N<\xi<0).
\end{equation}

\paragraph{Step 2: the squared transforms of $\mathfrak m_N$.}
For $s>0$, write $\mathcal T_N(s)=\int(s+y)^{-2}\mathfrak m_N(\dd y)$, which is finite. Fix $s>0$. On the event $\{\bar w_N>s\}$, whose probability tends to one, we set $\xi_s=-2d_N+2s/\bar t_N$, which lies in $(-2d_N,0)$. Then $p_i+\xi_s/2=\gamma_i+s/\bar t_N$ and $s+\bar t_N(x/2-d_N)=\frac{\bar t_N}2(x+\xi_s)$. Using \eqref{mc-nu-definition} and \eqref{eq:hotm-squared-resolvent}, we find
\begin{equation}\label{mc-real-squared-transform}
\mathcal T_N(s)=\frac{4d_N}{\sqrt{\bar t_N}}\int\frac{\rho_N(\dd x)}{(x+\xi_s)^2}
=\frac{d_N}{\big(1-\mathsf G(\xi_s)\big)^2}\int\frac{\omega_N^{\mathrm{gap}}(\dd y)}{(s+y)^2}-\frac{4d_N\vartheta_N}{\sqrt{\bar t_N}\,\xi_s^2}.
\end{equation}
In the second equality, we used that
\[
\frac1{\sqrt{\bar t_N}}\,\frac1N\sum_{i=1}^N\frac1{(\gamma_i+s/\bar t_N)^2}=\frac{\bar t_N^{3/2}}N\sum_{i=1}^N\frac1{(s+\bar t_N\gamma_i)^2}=\int\frac{\omega_N^{\mathrm{gap}}(\dd y)}{(s+y)^2}.
\]

\paragraph{Step 3: limits of the squared transforms.}
We treat the three factors in \eqref{mc-real-squared-transform} in turn. First, the function $g(y)=(s+y)^{-2}$ satisfies $(1+y)^2g(y)\le\max(1,s^{-2})$, and \eqref{eq:hotm-chi-limit} implies that
\begin{equation}\label{mc-real-raw-square}
\int\frac{\omega_N^{\mathrm{gap}}(\dd y)}{(s+y)^2}\longrightarrow\frac1\pi\int_0^\infty\frac{\sqrt y}{(s+y)^2}\,\dd y=\frac1{2\sqrt s}
\end{equation}
in probability. The value of the integral follows from the substitution $y=sr^2$ and $\int_0^\infty r^2(1+r^2)^{-2}\dd r=\pi/4$.

Second, we show that $(1-\mathsf G(\xi_s))/\delta_N\to-1$ in probability. Separating the term $i=1$, we have
\[
\mathsf G(\xi_s)=\frac{\bar t_N}{Ns}+\frac1N\sum_{i=2}^N\frac1{\gamma_i+s/\bar t_N}.
\]
Fix $\eta\in(0,1)$ and work on $\mathcal \mathsf G^\eta$. The function $\gamma\mapsto b/(\gamma(\gamma+b))$, with $b=s/\bar t_N$, is decreasing on $(0,\infty)$. Using \eqref{mc-indexed-input}, we bound the summand with index $i\ge2$ in the following sum by the value of this function at $K_\eta^{-1}((i-1)/N)^{2/3}$. Comparing this right Riemann sum of a decreasing function with the corresponding integral, we find
\[
\begin{aligned}
0\le\frac1N\sum_{i=2}^N\Big(\frac1{\gamma_i}-\frac1{\gamma_i+b}\Big)
&\le\int_0^1\frac{b\,\dd y}{K_\eta^{-1}y^{2/3}\big(K_\eta^{-1}y^{2/3}+b\big)}\\
&\le\frac32K_\eta^{3/2}\,b\int_0^\infty\frac{r^{-1/2}\,\dd r}{r+b}=\frac{3\pi}2K_\eta^{3/2}\sqrt b .
\end{aligned}
\]
In the second inequality, we substituted $r=K_\eta^{-1}y^{2/3}$ and extended the range of integration. Recall that $\bar t_N/(N\delta_N)=\delta_N^2\bar t_N/M_N$, and note that $\sqrt{s/\bar t_N}/\delta_N=(s/(\delta_N^2\bar t_N))^{1/2}$. We conclude that on $\mathcal \mathsf G^\eta$,
\[
\Big|\frac{\mathsf G(\xi_s)-1}{\delta_N}-\frac1{\delta_N}\Big(\frac1N\sum_{i=2}^N\frac1{\gamma_i}-1\Big)\Big|\le\frac{\delta_N^2\bar t_N}{sM_N}+\frac{3\pi}2K_\eta^{3/2}\Big(\frac s{\delta_N^2\bar t_N}\Big)^{1/2} .
\]
The right side is deterministic and tends to zero by \eqref{mc-time-hypotheses}. Together with \eqref{mc-raw-trace} and \cref{lem:model-localization}, this gives $(\mathsf G(\xi_s)-1)/\delta_N\to1$ in probability. As $d_N/\delta_N^2\to1$ as well, we have
\[
\frac{d_N}{\big(1-\mathsf G(\xi_s)\big)^2}=\frac{d_N/\delta_N^2}{\big((1-\mathsf G(\xi_s))/\delta_N\big)^2}\longrightarrow1
\]
in probability.

Third, on the event $\{\bar w_N\ge2s\}$ we have $|\xi_s|=2d_N(1-s/\bar w_N)\ge d_N$, so that
\[
0\le\frac{4d_N\vartheta_N}{\sqrt{\bar t_N}\,\xi_s^2}\le\frac{4\vartheta_N}{d_N\sqrt{\bar t_N}}=\frac{4\vartheta_N}{\sqrt{d_N}}\,\bar w_N^{-1/2},
\]
which tends to zero in probability by \eqref{eq:hotm-canonical-consequences}, since $\bar w_N\to\infty$. Inserting these three limits into \eqref{mc-real-squared-transform}, we find that for every fixed $s>0$,
\begin{equation}\label{eq:hotm-transform-limit}
\mathcal T_N(s)\longrightarrow\frac1{2\sqrt s}=\int_0^\infty\frac{\mathfrak m(\dd y)}{(s+y)^2}\quad\text{in probability},\qquad \mathfrak m(\dd y)=\frac{\sqrt y}\pi\,\dd y .
\end{equation}

\paragraph{Step 4: identification of the limit.}
The measures $\mathfrak m_N$ and $\mathfrak m$ have finite transforms at $s=1$, and \eqref{eq:hotm-transform-limit} holds for every rational $s\ge1$. \Cref{pr-real-measure-transfer}, in its version for convergence in probability, then implies that $\mathfrak m_N\to\mathfrak m$ vaguely in probability, and that for every continuous $f$ on $[0,1]$,
\begin{equation}\label{eq:hotm-weighted-convergence}
\int\frac{f\big((1+y)^{-1}\big)}{(1+y)^2}\,\mathfrak m_N(\dd y)\longrightarrow\int\frac{f\big((1+y)^{-1}\big)}{(1+y)^2}\,\mathfrak m(\dd y)\quad\text{in probability}.
\end{equation}
The test function $(1+y^2)^{-1}$ has the form in \eqref{eq:hotm-weighted-convergence} with $f(r)=(r^2+(1-r)^2)^{-1}$, which is continuous on $[0,1]$, since $(1+y)^2/(1+y^2)=f((1+y)^{-1})$. This gives the second claim in \eqref{mc-measure-limit}, with the value $\pi^{-1}\int_0^\infty\sqrt y(1+y^2)^{-1}\dd y=1/\sqrt2$. More generally, if $g$ is continuous on $[0,\infty)$ and $(1+y)^2g(y)\to0$ as $y\to\infty$, then $f(r)=r^{-2}g(r^{-1}-1)$ for $r\in(0,1]$ and $f(0)=0$ define a continuous function on $[0,1]$, and by \eqref{eq:hotm-weighted-convergence} we have
\begin{equation}\label{eq:hotm-test-convergence}
\int g\,\dd\mathfrak m_N\longrightarrow\frac1\pi\int_0^\infty g(y)\sqrt y\,\dd y\quad\text{in probability}.
\end{equation}
This applies to $e^{-2y}$ and $e^{-4y}$, and we obtain
\begin{equation}\label{eq:hotm-exponential-tests}
\begin{gathered}
\int e^{-2y}\,\mathfrak m_N(\dd y)\to\frac1\pi\int_0^\infty e^{-2y}\sqrt y\,\dd y=\frac1{4\sqrt{2\pi}},\\
\int e^{-4y}\,\mathfrak m_N(\dd y)\to\frac1\pi\int_0^\infty e^{-4y}\sqrt y\,\dd y=\frac1{16\sqrt\pi}.
\end{gathered}
\end{equation}

The test function in the formula for $\bar h''$ in \eqref{eq:hotm-U-measure} depends on $\bar w_N$. On $\{\bar w_N\ge1\}$, we have $0\le(1+y/\bar w_N)^2-1\le3(y+y^2)/\bar w_N$, which gives
\[
\Big|\int\Big(1+\frac y{\bar w_N}\Big)^2e^{-4y}\,\mathfrak m_N(\dd y)-\int e^{-4y}\,\mathfrak m_N(\dd y)\Big|
\le\frac3{\bar w_N}\sup_{y\ge0}\big((1+y)^2(y+y^2)e^{-4y}\big)\,\mathcal T_N(1).
\]
The right side tends to zero in probability, since $\mathcal T_N(1)\to1/2$ and $\bar w_N\to\infty$. Inserting this and \eqref{eq:hotm-exponential-tests} into \eqref{eq:hotm-U-measure} with $a=1$, we obtain \eqref{mc-j-asymptotic} and \eqref{mc-j-derivative}, since $4\cdot(16\sqrt\pi)^{-1}=(4\sqrt\pi)^{-1}$. This completes the proof.
\end{proof}

\begin{corollary}[Information at a smaller scale]\label{mc-information}
Assume \eqref{eq:hotm-regime}, and let $\bar t_N>0$ be deterministic times that satisfy \eqref{mc-time-hypotheses} with $t_N=\bar t_N$. Then
\begin{equation}\label{mc-information-asymptotic}
\cI_N(\bar t_N)=\frac{Nd_N^{3/2}}{128\sqrt\pi}\,\bar w_N^{-3/2}e^{-4\bar w_N}\big(1+o_\PP(1)\big).
\end{equation}
Moreover, $\max_i|e_i(\bar t_N)|\to0$ in probability, and $\eta_q=\max_iq_i/(2\sum_iq_i^2)^{1/2}=O_\PP(M_N^{-1/2})$.
\end{corollary}

\begin{proof}
By \eqref{mc-j-asymptotic} and $\bar w_N\to\infty$, we have $(\bar h(\bar t_N)-\vartheta_N)/\sqrt{d_N}\to0$ in probability. As $\vartheta_N/\sqrt{d_N}\to2$, we have $\bar h(\bar t_N)=\vartheta_N(1+o_\PP(1))$. Inserting this and \eqref{mc-j-derivative} into \eqref{eq:hotm-info}, we find
\[
\cI_N(\bar t_N)=\frac{N\,d_N^{5/2}\,\bar w_N^{-3/2}e^{-4\bar w_N}}{32\sqrt\pi\,\vartheta_N^2}\big(1+o_\PP(1)\big),
\]
and since $d_N/\vartheta_N^2\to1/4$, this is \eqref{mc-information-asymptotic}.

For the relative errors, fix $\eta\in(0,1)$ and work on the event $\mathcal \mathsf G^\eta\cap\{d_N\le1,\ Nd_N^{3/2}\ge1,\ \bar w_N\ge1\}$, whose probability is at least $1-\eta-o(1)$. On this event, the hypotheses of \eqref{mc-late-input} hold with $K=K_\eta$. Taking $s=\bar t_N-d_N^{-1}\ge0$ in \eqref{mc-late-input}, for which $1+d_Ns=\bar w_N$, and using $d_N\le p_i$, we obtain
\[
\max_i|e_i(\bar t_N)|\le C_{K_\eta}\,e^2\,\bar w_Ne^{-2\bar w_N},
\]
which tends to zero in probability. Removing the localization with \cref{lem:model-localization}, we conclude that $\max_i|e_i(\bar t_N)|\to0$ in probability.

Finally, $\max_iq_i=1/d_N$ and $\sum_iq_i^2=N/\vartheta_N$, so that
\[
\eta_q=\frac1{d_N}\Big(\frac{\vartheta_N}{2N}\Big)^{1/2}=\big(Nd_N^{3/2}\big)^{-1/2}\Big(\frac{\vartheta_N}{2\sqrt{d_N}}\Big)^{1/2}.
\]
The last factor tends to one, and $Nd_N^{3/2}/M_N\to1$, by \eqref{eq:hotm-canonical-consequences}.
\end{proof}

In \eqref{mc-information-asymptotic}, the factor $Nd_N^{3/2}\bar w_N^{-3/2}$ equals $N\bar t_N^{-3/2}$, which is of the order of the number of indices with $\gamma_i\le1/\bar t_N$, and the factor $e^{-4\bar w_N}$ comes from the decay of the relative errors $e_i(t)$ at the rate $2d_N$. The center $T_N$ of \cref{mgf-all-mesoscopic-profile} is the time at which the right side of \eqref{mc-information-asymptotic} is close to one.

\begin{theorem}[The comparison law at the times $T_N(u)$]\label{mc-proxy-profile}
Assume \eqref{eq:hotm-regime}, let $T_N(u)$ be as in \eqref{mc-profile-center}, and let $\mu_t$ be the Bingham comparison law of $c_NW_N$. Then for every $u_0>0$,
\[
\sup_{|u|\le u_0}\Big|\,\big\|\mu_{T_N(u)}-\pi_{N,c_N}\big\|_{\TV}-\Psi(u)\Big|\longrightarrow0\quad\text{in probability}.
\]
\end{theorem}

\begin{proof}
The proof has three steps. We first prove that the information at the times $T_N(u)$ converges to $e^{-4u}$, uniformly in $u$, using one measure $\mathfrak m_N$ for all $u$. We then check the remaining hypotheses of \eqref{mc-pin-input}, and finally we apply it. Fix $u_0>0$, and let $N$ be so large that $M_N>e$ and $T_N(-u_0)>0$.

\paragraph{Step 1: the information.}
Recall that $B_{c_N}\to B_1=1/(16\sqrt\pi)$, so that
\[
\delta_N^2T_N=\frac14\log M_N-\frac38\log\log M_N+\frac14\log B_{c_N}=\frac14\log M_N\,\big(1+o(1)\big).
\]
In particular, $\bar t_N=T_N$ satisfies \eqref{mc-time-hypotheses} with $t_N=T_N$. Let $\mathfrak m_N$ and $\bar w_N=d_NT_N$ be defined with this choice. For $|u|\le u_0$, let $a_N(u)=T_N(u)/T_N=1+u/(\delta_N^2T_N)$. These numbers are deterministic, $\sup_{|u|\le u_0}|a_N(u)-1|\to0$, and $a_N(u)\in[1/2,2]$ for large $N$. Note that $a_N(u)\bar w_N=d_NT_N(u)$. Then the identities \eqref{eq:hotm-U-measure} with $a=a_N(u)$ read
\begin{equation}\label{eq:hotm-uniform-U}
\begin{aligned}
\bar h\big(T_N(u)\big)&=\vartheta_N+\sqrt{d_N}\,\bar w_N^{-3/2}e^{-2d_NT_N(u)}\,\Lambda^{(1)}_N(u),\\
\bar h''\big(2T_N(u)\big)&=4d_N^{5/2}\,\bar w_N^{-3/2}e^{-4d_NT_N(u)}\,\Lambda^{(2)}_N(u),
\end{aligned}
\end{equation}
where
\[
\Lambda^{(1)}_N(u)=\int e^{-2a_N(u)y}\,\mathfrak m_N(\dd y),\qquad
\Lambda^{(2)}_N(u)=\int\Big(1+\frac y{\bar w_N}\Big)^2e^{-4a_N(u)y}\,\mathfrak m_N(\dd y).
\]
The mean value theorem gives $|e^{-2ay}-e^{-2y}|\le2|a-1|ye^{-y}$ and $|e^{-4ay}-e^{-4y}|\le4|a-1|ye^{-2y}$ for $a\ge1/2$ and $y\ge0$, and for $\bar w_N\ge1$ we have $0\le(1+y/\bar w_N)^2-1\le3(y+y^2)/\bar w_N$. Multiplying these bounds by $(1+y)^2$ and integrating against $(1+y)^{-2}\mathfrak m_N(\dd y)$, we find that
\[
\begin{aligned}
&\sup_{|u|\le u_0}\Big|\Lambda^{(1)}_N(u)-\int e^{-2y}\,\dd\mathfrak m_N\Big|+\sup_{|u|\le u_0}\Big|\Lambda^{(2)}_N(u)-\int e^{-4y}\,\dd\mathfrak m_N\Big|\\
&\qquad\le C\Big(\sup_{|u|\le u_0}|a_N(u)-1|+\bar w_N^{-1}\Big)\,\mathcal T_N(1),
\end{aligned}
\]
where $C$ is a numerical constant and $\mathcal T_N(1)=\int(1+y)^{-2}\mathfrak m_N(\dd y)$ as in the proof of \cref{mc-measure-theorem}. The right side tends to zero in probability, because $\mathcal T_N(1)\to1/2$, and \eqref{eq:hotm-exponential-tests} implies that
\begin{equation}\label{eq:hotm-uniform-tests}
\sup_{|u|\le u_0}\Big|\Lambda^{(1)}_N(u)-\frac1{4\sqrt{2\pi}}\Big|\to0,\qquad
\sup_{|u|\le u_0}\Big|\Lambda^{(2)}_N(u)-\frac1{16\sqrt\pi}\Big|\to0
\end{equation}
in probability.

Combining \eqref{eq:hotm-uniform-U} and \eqref{eq:hotm-uniform-tests} with $\sqrt{d_N}/\vartheta_N\to1/2$ and $\bar w_N\to\infty$, we find that $\sup_{|u|\le u_0}|\bar h(T_N(u))/\vartheta_N-1|\to0$ in probability. Inserting \eqref{eq:hotm-uniform-U} into \eqref{eq:hotm-info} and using $d_N/\vartheta_N^2\to1/4$, we find, uniformly for $|u|\le u_0$,
\[
\cI_N\big(T_N(u)\big)=\frac{Nd_N^{3/2}}8\,\bar w_N^{-3/2}e^{-4d_NT_N(u)}\,\Lambda^{(2)}_N(u)\,\big(1+o_\PP(1)\big).
\]
We now replace $d_N$ by $\delta_N^2$ in the exponent. We have $4d_NT_N(u)-4\delta_N^2T_N(u)=(d_N/\delta_N^2-1)(4\delta_N^2T_N+4u)$, and $|4\delta_N^2T_N+4u|\le C\ell_N$ for $|u|\le u_0$. Using \eqref{mc-canonical-input}, we see that the difference is $o_\PP(1)$, uniformly in $|u|\le u_0$. From the definition of $T_N$, we have
\[
e^{-4\delta_N^2T_N(u)}=\frac{(\log M_N)^{3/2}}{M_N\,B_{c_N}}\,e^{-4u}.
\]
Further, $\bar w_N=d_NT_N=\frac14\log M_N\,(1+o_\PP(1))$ and $Nd_N^{3/2}=M_N(1+o_\PP(1))$. Combining these estimates with \eqref{eq:hotm-uniform-tests}, we conclude that uniformly for $|u|\le u_0$,
\[
\begin{aligned}
\cI_N\big(T_N(u)\big)&=\frac{M_N}8\cdot\frac8{(\log M_N)^{3/2}}\cdot\frac{(\log M_N)^{3/2}}{M_N\,B_{c_N}}\,e^{-4u}\cdot\frac1{16\sqrt\pi}\,\big(1+o_\PP(1)\big)\\
&=\frac{e^{-4u}}{16\sqrt\pi\,B_{c_N}}\big(1+o_\PP(1)\big).
\end{aligned}
\]
As $16\sqrt\pi\,B_{c_N}\to1$, this proves
\begin{equation}\label{mc-uniform-information}
\sup_{|u|\le u_0}\big|\cI_N\big(T_N(u)\big)-e^{-4u}\big|\longrightarrow0\quad\text{in probability}.
\end{equation}

\paragraph{Step 2: the relative errors and the largest standardized variance.}
\Cref{mc-information} implies that $\eta_q\to0$ in probability, and $\eta_q$ does not depend on $u$. For the relative errors, fix $\eta\in(0,1)$ and work on $\mathcal \mathsf G^\eta\cap\{d_N\le1,\ Nd_N^{3/2}\ge1,\ d_NT_N(-u_0)\ge1\}$. As in the proof of \cref{mc-information}, \eqref{mc-late-input} yields
\[
\begin{aligned}
\sup_{|u|\le u_0}\max_i\big|e_i\big(T_N(u)\big)\big|&\le C_{K_\eta}e^2\sup_{|u|\le u_0}d_NT_N(u)\,e^{-2d_NT_N(u)}\\
&=C_{K_\eta}e^2\,d_NT_N(-u_0)\,e^{-2d_NT_N(-u_0)},
\end{aligned}
\]
where we used that $r\mapsto re^{-2r}$ is decreasing for $r\ge1/2$. The right side is $o_\PP(1)$, and \cref{lem:model-localization} implies that the supremum over $|u|\le u_0$ of $\max_i|e_i(T_N(u))|$ tends to zero in probability.

\paragraph{Step 3: conclusion.}
Let $B=e^{4u_0}+1$. By Steps 1 and 2, the event that $\eta_q\le\eta_{\mathrm{lik}}$, $\max_i|e_i(T_N(u))|\le1/4$, and $\cI_N(T_N(u))\le B$ for all $|u|\le u_0$ has probability tending to one. On this event we apply \eqref{mc-pin-input} with $v=q$ and $\tilde v=v(T_N(u))$. Both sequences have sum $N$, the relative errors are $\tilde e_i=e_i(T_N(u))$, and $I=\cI_N(T_N(u))$. Recall that $\mu_q=\pi_{N,c_N}$ by \cref{prop:cond-bingham}. Then \eqref{mc-pin-input} gives
\[
\begin{aligned}
&\sup_{|u|\le u_0}\Big|\,\big\|\mu_{T_N(u)}-\pi_{N,c_N}\big\|_{\TV}-\Big(2\Phi\Big(\tfrac12\sqrt{\cI_N(T_N(u))}\Big)-1\Big)\Big|\\
&\qquad\le C_B\Big(\eta_q+\sup_{|u|\le u_0}\max_i\big|e_i\big(T_N(u)\big)\big|\Big),
\end{aligned}
\]
and the right side tends to zero in probability. Finally, $\Phi$ is Lipschitz with constant $(2\pi)^{-1/2}$ and $|\sqrt a-\sqrt b|\le|a-b|^{1/2}$ for $a,b\ge0$, so that
\[
\Big|2\Phi\Big(\tfrac12\sqrt{\cI_N(T_N(u))}\Big)-2\Phi\Big(\tfrac12e^{-2u}\Big)\Big|\le(2\pi)^{-1/2}\big|\cI_N\big(T_N(u)\big)-e^{-4u}\big|^{1/2}.
\]
The supremum over $|u|\le u_0$ of the right side tends to zero in probability, by \eqref{mc-uniform-information}. As $2\Phi(e^{-2u}/2)-1=\Psi(u)$, this completes the proof.
\end{proof}

\section{The time-\texorpdfstring{$t$}{t} law and the Bingham comparison law}\label{sec:hotm-comparison}

In this section, the matrix $A$ is deterministic, and all statements are deterministic. We compare $\nu_NP_t^A$ with the Bingham comparison law $\mu_t$ of $A$ on the whole time interval relevant for the profile. The main result is \cref{mgf-global-bridge}.

\paragraph{Hypotheses and constants.}
Let $A$ be a real symmetric $N\times N$ matrix with precisions $p_1\le\cdots\le p_N$ and Lagrange multiplier $z$, and let $d=p_1$. Fix $K\ge1$ and $d_0\in(0,1)$. We assume that
\begin{equation}\label{eq:hotm-geometry}
0<d\le d_0,\qquad K^{-1}\Big(\frac{i-1}N\Big)^{2/3}\le p_i-d\le K\Big(\frac{i-1}N\Big)^{2/3}\qquad(2\le i\le N),
\end{equation}
and we write
\[
M=Nd^{3/2}.
\]
These are the hypotheses \eqref{hi3-indexed-geometry} of \cref{sec:heat}, together with $d\le d_0$, and they are the hypotheses of the estimates for the unnormalized process in \cref{sec:process}. The largest precision satisfies $p_N\le d_0+K$. Adding a multiple of the identity to $A$ changes neither $L_A$ nor the covariance evolution, and the results below depend on $A$ only through its precisions and its eigenbasis.

We also fix $C_T>0$ and set
\[
T_*=C_T\,d^{-1}\log M,\qquad \bar r=d-\frac2N .
\]
The rate $\bar r$ is the rate in the weighted gradient estimate of \cref{hfk-pointwise-fk}; it is not the rate $r_0$ of \cref{sec:comparison}.
In this section, $C$ denotes a constant that depends only on $K$, $d_0$, and $C_T$, and $M_0$ a threshold with the same dependence; when a constant also depends on a moment order, we say so. Their values may change from line to line. Every statement holds for every $N\ge2$, every matrix $A$ that satisfies \eqref{eq:hotm-geometry} with $M\ge M_0$, and every time in the stated range, with the same constants.

\paragraph{The coefficient differences and the forward equation.}
We use the notation of \cref{sec:hotm-setting} for the covariance evolution of $A$, and let $\zeta(t)=\bar h'(t)/\bar h(t)$. Summing the equations $h_i'=-2p_ih_i+2\bar h$ over $i$, we have $N^{-1}\sum_ip_ih_i=\bar h-\bar h'/2$, and hence $N^{-1}\sum_ip_iv_i(t)=1-\zeta(t)/2$. Writing $\alpha_i=z-p_i$ for the eigenvalues of $A$, the moving multiplier of \cref{sec:heat} is therefore $z(t)=1+N^{-1}\sum_i\alpha_iv_i(t)=z+\zeta(t)/2$. The matrix $C_t=A-B_t$ of \cref{ht-forward-defect}, where $B_t=\diag(b_i(t))$ with $b_i(t)=z(t)-v_i(t)^{-1}$, is diagonal, and its entries are the coefficient differences $C_i(t)=\alpha_i-b_i(t)$. Since $\alpha_i=z-p_i$ and $z(t)=z+\zeta(t)/2$, they satisfy
\begin{equation}\label{eq:hotm-field}
C_i(t)=\frac1{v_i(t)}-p_i-\frac{\zeta(t)}2=\frac{v_i'(t)}{2v_i(t)},\qquad\text{so that}\qquad\sum_{i=1}^NC_i(t)v_i(t)=\frac12\sum_{i=1}^Nv_i'(t)=0.
\end{equation}
For the second expression of $C_i(t)$, note that $v_i'/v_i=h_i'/h_i-\zeta=-2p_i+2/v_i-\zeta$. By \cref{ht-forward-defect}, the forward-equation defect (\cref{def:comp-defect}) of the family $(\mu_t)_{t\ge0}$ is
\[
\begin{gathered}
R_t=-\frac{Q_B(t)Q_C(t)-\mu_t\big(Q_B(t)Q_C(t)\big)}N,\\
Q_B(t)=N-\sum_{i=1}^N\frac{x_i^2}{v_i(t)},\qquad Q_C(t)=\sum_{i=1}^NC_i(t)x_i^2,
\end{gathered}
\]
where we used that $Q_B(t)=x^\T B_tx-N(z(t)-1)$ equals $N-\sum_ix_i^2/v_i(t)$ on $S_N$. In other words,
\begin{equation}\label{eq:hotm-forward}
\partial_t\mu_t=L_A^*\mu_t+R_t\,\mu_t,\qquad \mu_t(R_t)=0,\qquad \mu_0=\nu_N .
\end{equation}

\paragraph{The Poisson equation.}
The law $\mu_t$ is the Gibbs measure $\pi_{B_t}$ of the generator $L_{B_t}$ (\cref{prop:cond-bingham}). By \cref{lem:comp-poisson}, there exists a unique smooth solution $\psi_t$ of
\[
-L_{B_t}\psi_t=R_t,\qquad \mu_t(\psi_t)=0,
\]
and we write $\cE(t)=\mu_t\Gamma(\psi_t)$ for its energy. Recall from \cref{cor:comp-poisson-flux} that for every smooth $g$,
\begin{equation}\label{eq:hotm-flux}
\mu_t(R_tg)=\mu_t\Gamma(\psi_t,g),
\end{equation}
Thus $R_t=-\operatorname{div}_{\mu_t}(\nabla_{S_N}\psi_t)$, where $\operatorname{div}_{\mu_t}j=g_t^{-1}\operatorname{div}_{S_N}(g_tj)$ and $g_t=\dd\mu_t/\dd\nu_N$; its size is measured by $\cE(t)$.

The next lemma restates \cref{rp-general-flux} in terms of a scale $d_*$, which is the square of the parameter $\delta$ there. With $\delta=d_*^{1/2}$, the variance conditions at scale $\delta$ with constants $(c_0,C_0)$ (\cref{def:comp-variance-conditions}) read
\begin{equation}\label{eq:hotm-variance-conditions}
\begin{gathered}
c_0\le v_i\le C_0\,d_*^{-1}\quad(1\le i\le N),\qquad c_0Nd_*^{-1/2}\le\sum_iv_i^2\le C_0Nd_*^{-1/2},\\
\sum_iv_i^3\le C_0Nd_*^{-3/2},\qquad \sum_iv_i^4\le C_0Nd_*^{-5/2},
\end{gathered}
\end{equation}
together with the existence of a set $\mathcal S\subset\{1,\dots,N\}$ with $|\mathcal S|\ge c_0Nd_*^{3/2}$, such that $v_i\ge c_0d_*^{-1}$ for $i\in\mathcal S$ and $\sum_{i\notin\mathcal S}v_i^2\ge c_0\sum_iv_i^2$.

\begin{lemma}[The Poisson equation at a given scale]\label{mgf-general-flux}
Let $0<c_0<1\le C_0$. There exist constants $C_*$ and $M_*^0$, depending only on $c_0$ and $C_0$, with the following property. Let $v$ be a variance sequence on $\{1,\dots,N\}$ with $\sum_iv_i=N$, let $d_*\in(0,1]$ with $M_*=Nd_*^{3/2}\ge M_*^0$, and suppose that $v$ satisfies the variance conditions at scale $d_*^{1/2}$ with constants $(c_0,C_0)$. Let $C_1,\dots,C_N$ be real numbers with $\sum_iC_iv_i=0$, let $Q_B=N-\sum_ix_i^2/v_i$, $Q_C=\sum_iC_ix_i^2$, and $R=-N^{-1}(Q_BQ_C-\mu_v(Q_BQ_C))$, let $\psi$ be the solution of $-L_B\psi=R$ with $\mu_v(\psi)=0$, where $B=-\diag(v_i^{-1})$, and let
\[
I_C=\sum_iC_i^2v_i^2,\qquad J_C=\sum_iC_i^2,\qquad K_C=\sum_iC_i^2v_i .
\]
Then
\[
\mu_v\Gamma(\psi)\le\frac{C_*}{M_*}\Big(K_C+\frac{J_C}{d_*}+d_*I_C\Big).
\]
\end{lemma}

\begin{proof}
This is \cref{rp-general-flux} with $\delta=d_*^{1/2}\in(0,1]$. Indeed, $N\delta^3=M_*$, the conditions \eqref{rp-covariance-hypotheses} and \eqref{eq:comp-soft-set} at scale $\delta$ are \eqref{eq:hotm-variance-conditions} and the conditions on $\mathcal S$, and the bound \eqref{rp-unaligned-energy} is $\frac{C_*}{M_*}(K_C+J_Cd_*^{-1}+d_*I_C)$. The only assumption on the numbers $C_i$ is $\sum_iC_iv_i=0$.
\end{proof}

The proof of \cref{rp-general-flux} uses the lower bound of order $d_*$ on the spectral gap of $L_{B}$ from \cref{ht-many-soft-sharp-gaps}; this is where the set $\mathcal S$ enters. The next lemma turns bounds on the coefficient differences into bounds on the energy $\cE$.

\begin{lemma}[Energy bounds from bounds on the coefficient differences]\label{mgf-field-envelopes}
Assume \eqref{eq:hotm-geometry}. Let $v(t)$, $t\ge0$, be variance sequences with $\sum_iv_i(t)=N$, and let $C_i(t)$ be real numbers with $\sum_iC_i(t)v_i(t)=0$. Suppose that there exist $\Lambda_0\ge1$, $0<c_0<1\le C_0$, and an integer $j\ge0$ such that the following hold.
\begin{enumerate}
\item For $0\le t\le d^{-1}$ and all $i$, $|C_i(t)|\le \Lambda_0(1+t)^{-1}(1+p_it)^{-1}$.
\item For $s\ge0$ and all $i$, $|C_i(d^{-1}+s)|\le \Lambda_0\,d^2p_i^{-1}(1+ds)^je^{-2ds}$.
\item For $t\ge0$ and all $i$, $\Lambda_0^{-1}(1+t)/(1+p_it)\le v_i(t)\le \Lambda_0(1+t)/(1+p_it)$.
\item For $0\le t\le d^{-1}$, the sequence $v(t)$ satisfies the variance conditions at scale $(1+t)^{-1/2}$ with constants $(c_0,C_0)$. For $t\ge d^{-1}$, it satisfies the variance conditions at scale $d^{1/2}$ with the same constants.
\end{enumerate}
For $t\ge0$, let $\psi_t$ be the solution of the Poisson equation of \cref{mgf-general-flux} defined from $v=v(t)$ and $C_i=C_i(t)$, and let $\cE(t)=\mu_{v(t)}\Gamma(\psi_t)$. There exist $C$ and $M_0$, depending only on $K$, $d_0$, $\Lambda_0$, $c_0$, $C_0$, and $j$, such that if $M\ge M_0$, then
\[
\cE(t)\le\frac C{1+t}\quad(0\le t\le d^{-1}),\qquad \cE(d^{-1}+s)\le Cd\,(1+ds)^{2j}e^{-4ds}\quad(s\ge0).
\]
\end{lemma}

\begin{proof}
We first record two sums over the precisions. For every integer $k\ge2$, we claim that
\begin{equation}\label{eq:hotm-indexed-sums}
\sum_{i=1}^N\frac1{(1+p_it)^k}\le C_kN(1+t)^{-3/2}\quad(0\le t\le d^{-1}),\qquad
\sum_{i=1}^N\frac1{p_i^k}\le C_kNd^{3/2-k},
\end{equation}
provided $M\ge2^{3/2}$, where $C_k$ depends only on $K$ and $k$. For the first bound, we note that the summand is decreasing in $p_i$. Using \eqref{eq:hotm-geometry} and comparing a right Riemann sum with an integral, we find for $t>0$ that
\[
\sum_{i=1}^N\frac1{(1+p_it)^k}\le1+N\int_0^\infty\frac{\dd y}{(1+K^{-1}y^{2/3}t)^k}=1+\frac32K^{3/2}\Big(\int_0^\infty\frac{r^{1/2}\,\dd r}{(1+r)^k}\Big)Nt^{-3/2},
\]
where we substituted $r=K^{-1}y^{2/3}t$. The last integral is finite since $k>3/2$. For $t\le1$ the sum is at most $N\le2^{3/2}N(1+t)^{-3/2}$. For $1\le t\le d^{-1}$, we have $t^{-3/2}\le2^{3/2}(1+t)^{-3/2}$, and since $1+t\le2d^{-1}$, also $N(1+t)^{-3/2}\ge2^{-3/2}M\ge1$, which absorbs the term $1$. The second bound follows in the same way from
\[
\sum_{i=1}^N\frac1{p_i^k}\le d^{-k}+N\int_0^\infty\frac{\dd y}{(d+K^{-1}y^{2/3})^k}=d^{-k}+\frac32K^{3/2}\Big(\int_0^\infty\frac{r^{1/2}\,\dd r}{(1+r)^k}\Big)Nd^{3/2-k},
\]
together with $d^{-k}=Nd^{3/2-k}/M$.

Let $0\le t\le d^{-1}$, $d_*=(1+t)^{-1}$, and $M_t=Nd_*^{3/2}=N(1+t)^{-3/2}$. By (i), (iii), and \eqref{eq:hotm-indexed-sums},
\begin{align*}
I_C&\le \Lambda_0^4\sum_i\frac1{(1+p_it)^4}\le CM_t,\\
J_C&\le\frac{\Lambda_0^2}{(1+t)^2}\sum_i\frac1{(1+p_it)^2}\le C(1+t)^{-2}M_t,\\
K_C&\le\frac{\Lambda_0^3}{1+t}\sum_i\frac1{(1+p_it)^3}\le C(1+t)^{-1}M_t,
\end{align*}
where $I_C,J_C,K_C$ are defined from $v(t)$ and $(C_i(t))$ as in \cref{mgf-general-flux}. As $1+t\le2d^{-1}$, we have $M_t\ge2^{-3/2}M$, and $M_t\ge M_*^0$ when $M_0$ is large. Then (iv) and \cref{mgf-general-flux} imply that
\[
\cE(t)\le\frac{C}{M_t}\Big(\frac{M_t}{1+t}+(1+t)\frac{M_t}{(1+t)^2}+\frac{M_t}{1+t}\Big)=\frac{3C}{1+t}.
\]

Let $t=d^{-1}+s$ with $s\ge0$, let $d_*=d$, and write $L_j(s)=(1+ds)^je^{-2ds}$. Using (iii) and $p_it\ge1$, we have $v_i(t)\le \Lambda_0(1+t)/(p_it)\le2\Lambda_0/p_i$. By (ii) and \eqref{eq:hotm-indexed-sums},
\[
\begin{gathered}
I_C\le Cd^4L_j(s)^2\sum_i\frac1{p_i^4}\le CML_j(s)^2,\qquad
J_C\le Cd^4L_j(s)^2\sum_i\frac1{p_i^2}\le Cd^2ML_j(s)^2,\\
K_C\le Cd^4L_j(s)^2\sum_i\frac1{p_i^3}\le CdML_j(s)^2 .
\end{gathered}
\]
With (iv) and \cref{mgf-general-flux} for $M_*=M$, we obtain
\[
\cE(d^{-1}+s)\le\frac CM\big(dM+d^{-1}d^2M+d\,M\big)L_j(s)^2=3Cd\,(1+ds)^{2j}e^{-4ds}.
\]
This completes the proof.
\end{proof}

We now verify the hypotheses of \cref{mgf-field-envelopes} for the covariance evolution.

\begin{lemma}[Coefficient differences along the covariance evolution]\label{mgf-exact-flow-envelopes}
Assume \eqref{eq:hotm-geometry}. There exist $\Lambda_0\ge1$, $0<c_0<1\le C_0$, and $M_0$, depending only on $K$ and $d_0$, such that if $M\ge M_0$, then the variances $v_i(t)$ of the covariance evolution of $A$ and the entries $C_i(t)$ in \eqref{eq:hotm-field} satisfy (i)--(iv) of \cref{mgf-field-envelopes} with $j=1$. Moreover, after enlarging $M_0$, there exists $C$, depending only on $K$ and $d_0$, such that
\begin{equation}\label{eq:hotm-energy}
\cE(t)\le\frac C{1+t}\quad(0\le t\le d^{-1}),\qquad \cE(d^{-1}+s)\le Cd\,(1+ds)^2e^{-4ds}\quad(s\ge0).
\end{equation}
\end{lemma}

\begin{proof}
The proof has five steps: bounds on $\bar h$ and its derivatives, the comparison (iii) of the variances, the bound (i) on the coefficient differences at times $t\le d^{-1}$, the bound (ii) at later times, and the variance conditions (iv). Throughout, $M\ge M_0$, and we write $a(t)=\sqrt d+(1+t)^{-1/2}$. Positive lower-bound constants denoted by $c_*$ depend only on $K$ and $d_0$ and may change from line to line.

\paragraph{Step 1: the function $\bar h$.}
Recall from \cref{iw-all-time-heat} that, for all $t\ge0$ and all $i$,
\begin{equation}\label{eq:hotm-heat-profile}
\bar h(t)\asymp a(t),\qquad h_i(t)\asymp a(t)\,\frac{1+t}{1+p_it},
\end{equation}
where $X\asymp Y$ means $C^{-1}Y\le X\le CY$. Also $\vartheta\asymp\sqrt d$ by \cref{hi3-finite-heat}, and differentiating \eqref{eq:hotm-U}, we have $\bar h^{(n)}(t)=(-1)^n\int x^ne^{-tx}\rho(\dd x)$ for $n\ge1$. For $t\ge1$, we use $x^ne^{-tx}\le(2n/(et))^ne^{-tx/2}$ and $\bar h(t/2)\le C\bar h(t)$, which follows from \eqref{eq:hotm-heat-profile} and $a(t/2)\le\sqrt2\,a(t)$. For $t\le1$, we use that $\rho$ is supported in $[0,\|H\|]$, that $\|H\|\le2p_N+2\le2(d_0+K+1)$, and that $\bar h(t)\ge \bar h(1)\ge c_*$. This gives
\begin{equation}\label{eq:hotm-U-derivatives}
|\bar h^{(n)}(t)|\le C_n(1+t)^{-n}\,\bar h(t)\qquad(t\ge0,\ n=1,2).
\end{equation}
For $t=d^{-1}+s$, we use $e^{-x(d^{-1}+s)}\le e^{-xd^{-1}}e^{-2ds}$ on $[2d,\infty)$ and the previous bound at time $d^{-1}$ to obtain $|\bar h^{(n)}(d^{-1}+s)|\le C_nd^n\bar h(d^{-1}/2)e^{-2ds}\le C_nd^{n+1/2}e^{-2ds}$. Recall that $\zeta=\bar h'/\bar h$. As $\bar h$ is positive and decreasing, we have $\zeta\le0$. By the Cauchy--Schwarz inequality for the measure $\vartheta\delta_0+\rho$, we also have $\bar h''\bar h\ge(\bar h')^2$, that is, $\zeta'\ge0$. Using \eqref{eq:hotm-U-derivatives} and $\bar h\ge\vartheta\ge c_*\sqrt d$, we conclude that
\begin{equation}\label{eq:hotm-zeta}
\begin{gathered}
|\zeta(t)|\le\frac C{1+t},\qquad 0\le\zeta'(t)\le\frac C{(1+t)^2}\qquad(t\ge0),\\
0\le\zeta'(d^{-1}+s)\le Cd^2e^{-2ds}\qquad(s\ge0).
\end{gathered}
\end{equation}

\paragraph{Step 2: the variances.}
The comparison (iii) follows from \eqref{eq:hotm-heat-profile}, since $v_i=h_i/\bar h$. In particular, $c_*\le v_i(t)\le C(1+t)$ for all $t$. For $t\ge d^{-1}$, we have $p_it\ge1$, so that $(1+t)/(1+p_it)$ lies between $1/(2p_i)$ and $2/p_i$; hence $v_i(t)\asymp p_i^{-1}$ and $1+e_i(t)=p_iv_i(t)\ge c_*$.

\paragraph{Step 3: the coefficient differences at times $t\le d^{-1}$.}
Recall from \eqref{eq:hotm-field} that $2C_i(t)=h_i'(t)/h_i(t)-\zeta(t)$. Fix $t\ge1$ and an index $i$ with $p_it\ge1$. For $r\ge0$, let $g(r)=h_i'(r)-\zeta(t)h_i(r)$. Differentiating $h_i'=-2p_ih_i+2\bar h$, we find $g'(r)=-2p_ig(r)+2\bar h(r)(\zeta(r)-\zeta(t))$. Solving this linear equation on $[t/2,t]$ and substituting $r=t-s$, we obtain
\[
h_i'(t)-\zeta(t)h_i(t)=e^{-p_it}g(t/2)+2\int_0^{t/2}e^{-2p_is}\,\bar h(t-s)\big(\zeta(t-s)-\zeta(t)\big)\,\dd s.
\]
We divide by $h_i(t)$. From \eqref{eq:hotm-heat-profile}, we have $h_i(t/2)\le Ch_i(t)$ and, as $1+p_it\le2p_it$, also $\bar h(t/2)\le C\bar h(t)\le Cp_ih_i(t)$. With $|h_i'|\le2p_ih_i+2\bar h$ and \eqref{eq:hotm-zeta}, the first term divided by $h_i(t)$ is at most $C(p_i+t^{-1})e^{-p_it}\le C/(p_it^2)$, where we used $r^2e^{-r}\le C$ and $re^{-r}\le C$ with $r=p_it\ge1$. In the integral, $|\zeta(t-s)-\zeta(t)|\le Cs/t^2$ for $s\le t/2$ by \eqref{eq:hotm-zeta}, and $\bar h(t-s)\le \bar h(t/2)\le Cp_ih_i(t)$. As $\int_0^\infty se^{-2p_is}\dd s=(4p_i^2)^{-1}$, the integral divided by $h_i(t)$ is at most $C/(p_it^2)$. We conclude that
\[
|C_i(t)|\le\frac C{p_it^2}\le\frac{4C}{(1+t)(1+p_it)}\qquad(t\ge1,\ p_it\ge1).
\]
If $t\ge1$ and $p_it<1$, then $2C_i(t)=-2p_i+2\bar h(t)/h_i(t)-\zeta(t)$, where $p_i<t^{-1}$, $\bar h(t)/h_i(t)\le C(1+p_it)/(1+t)\le2C/t$ by \eqref{eq:hotm-heat-profile}, and $|\zeta(t)|\le C/t$. Together, these bounds give $|C_i(t)|\le C/t\le4C(1+t)^{-1}(1+p_it)^{-1}$. If $t<1$, then $|C_i(t)|\le v_i(t)^{-1}+p_i+|\zeta(t)|\le C$, while $(1+t)(1+p_it)\le2(1+d_0+K)$. This proves (i) for $0\le t\le d^{-1}$.

\paragraph{Step 4: the coefficient differences at times $t\ge d^{-1}$.}
Let $t=d^{-1}+s$ with $s\ge0$. The hypotheses of \eqref{mc-late-input} hold by \eqref{eq:hotm-geometry}. Using also $1+e_i(t)\ge c_*$ from Step 2, we find
\begin{equation}\label{eq:hotm-late-precision}
\Big|\frac1{v_i(t)}-p_i\Big|=\frac{p_i|e_i(t)|}{1+e_i(t)}\le Cd\,(1+ds)\,e^{-2ds}.
\end{equation}
Differentiating \eqref{eq:hotm-field} and using $v_i'=2C_iv_i$, we find the linear equation
\[
C_i'(t)=-\frac{2C_i(t)}{v_i(t)}-\frac{\zeta'(t)}2 .
\]
For $0\le s_1\le s_2$, \eqref{eq:hotm-late-precision} implies that $\int_{s_1}^{s_2}2v_i(d^{-1}+r)^{-1}\dd r\ge2p_i(s_2-s_1)-2C\int_0^\infty d(1+dr)e^{-2dr}\dd r=2p_i(s_2-s_1)-\frac32C$. Hence the integrating factor of this equation from $d^{-1}+s_1$ to $d^{-1}+s_2$ is at most $Ce^{-2p_i(s_2-s_1)}$. From (i) at $t=d^{-1}$, we have $|C_i(d^{-1})|\le C(1+d^{-1})^{-1}(1+p_id^{-1})^{-1}\le Cd^2/p_i$. With \eqref{eq:hotm-zeta}, we obtain
\[
\begin{aligned}
|C_i(d^{-1}+s)|&\le C\frac{d^2}{p_i}e^{-2p_is}+C\int_0^se^{-2p_i(s-r)}d^2e^{-2dr}\,\dd r\\
&\le C\frac{d^2}{p_i}e^{-2ds}+Cd^2e^{-2ds}\min\Big\{s,\frac1{2(p_i-d)}\Big\},
\end{aligned}
\]
where the minimum is $s$ if $p_i=d$. In the last step, we wrote $e^{-2p_i(s-r)-2dr}=e^{-2ds}e^{-2(p_i-d)(s-r)}$ and integrated. If $p_i\le2d$, then $d^2s\le2(d^2/p_i)\,ds$. If $p_i>2d$, then $p_i-d>p_i/2$ and $d^2/(2(p_i-d))\le d^2/p_i$. In both cases,
\[
|C_i(d^{-1}+s)|\le C\frac{d^2}{p_i}(1+ds)\,e^{-2ds},
\]
which is (ii) with $j=1$.

\paragraph{Step 5: the variance conditions.}
Let $0\le t\le d^{-1}$, $d_*=(1+t)^{-1}$, and $M_*=N(1+t)^{-3/2}\ge2^{-3/2}M$. We have $c_*\le v_i(t)\le Cd_*^{-1}$ by Step 2, and (iii) and \eqref{eq:hotm-indexed-sums} imply that $\sum_iv_i(t)^k\le C(1+t)^k\sum_i(1+p_it)^{-k}\le CN(1+t)^{k-3/2}$ for $k=2,3,4$; these are the required upper bounds. For the lower bound on $\sum_iv_i^2$ and for the set $\mathcal S$, consider the indices with $i-1\le\epsilon N(1+t)^{-3/2}$, where $\epsilon\in(0,1]$. For such $i$, we have $p_i\le d+K\epsilon^{2/3}(1+t)^{-1}\le(2+K)(1+t)^{-1}$ by \eqref{eq:hotm-geometry} and $d\le2(1+t)^{-1}$, and then $v_i(t)\ge c_*(1+t)$ by (iii). Taking $\epsilon=K^{-3/2}$, we obtain at least $K^{-3/2}M_*$ indices with $v_i(t)\ge c_*(1+t)$, which gives $\sum_iv_i(t)^2\ge c_*N(1+t)^{1/2}$. Let $\mathcal S$ consist of the first $\lceil\epsilon M_*\rceil$ indices, with $\epsilon\le K^{-3/2}$ to be chosen. Then $v_i(t)\ge c_*(1+t)=c_*\,d_*^{-1}$ for $i\in\mathcal S$, and if $\epsilon M_*\ge1$,
\[
\sum_{i\in\mathcal S}v_i(t)^2\le2\epsilon M_*\,C^2(1+t)^2=2\epsilon C^2N(1+t)^{1/2}.
\]
We choose $\epsilon$, depending only on $K$ and $d_0$, so small that the right side is at most half of the lower bound on $\sum_iv_i(t)^2$; then the complement of $\mathcal S$ carries at least half of $\sum_iv_i(t)^2$. The condition $\epsilon M_*\ge1$ holds when $M\ge M_0$.

Let $t\ge d^{-1}$, $d_*=d$, and $M_*=M$. Step 2 and \eqref{eq:hotm-indexed-sums} yield $c_*\le v_i(t)\le C/p_i\le Cd^{-1}$ and $\sum_iv_i(t)^k\le C\sum_ip_i^{-k}\le CNd^{3/2-k}$. The indices with $i-1\le\epsilon M$ satisfy $p_i\le(1+K\epsilon^{2/3})d$, so that $v_i(t)\ge c_*d^{-1}$. The rest of the argument is the same as above. This proves (iv), with constants depending only on $K$ and $d_0$.

Finally, Steps 2--5 and \cref{mgf-field-envelopes} with $j=1$ yield \eqref{eq:hotm-energy}.
\end{proof}

\paragraph{The initial law weighted by $1/\varphi$.}
As in \eqref{eq:comp-vartheta}, let
\[
\varphi(x)=\frac1N\sum_{i=1}^Nq_ix_i^2,\qquad\text{and recall}\qquad \frac1\vartheta=\frac1N\sum_{i=1}^Nq_i^2 ,
\]
which is the value of $\varphi$ at $x_i^2=q_i$. On $S_N$ we have $1/p_N\le\varphi\le1/d$, and computing its tangential gradient, we find
\[
\Gamma(\log\varphi)(x)=\frac4N\Big(\frac{N\sum_iq_i^2x_i^2}{(\sum_iq_ix_i^2)^2}-1\Big)\ge0 .
\]
We write $X=(X_s)_{s\ge0}$ for the diffusion with generator $L_A$, and $\E_x$ and $\E_\mu$ for expectations when $X_0=x$ and when $X_0$ has law $\mu$. The following three estimates are proved in \cref{sec:process} for the unnormalized process $Y$ with quadratic potential coefficients $\rho_i=p_i/2$, the half-precisions. Its normalization $\sqrt NY/|Y|$ is the diffusion $X$ started from $\sqrt NY_0/|Y_0|$, and the quantities below depend only on the path of $X$. For every $\lambda>0$ and $r>0$, there exist $C$ and $M_0$, depending only on $K$, $d_0$, $C_T$, $\lambda$, and $r$, such that if $M\ge M_0$, then for every $u\ge0$,
\begin{align}
\sup_{x\in S_N}\E_x\exp\Big(\lambda\int_0^{T_*}\Gamma(\log\varphi)(X_s)\,\dd s\Big)&\le C,\label{mgf-occupation-input}\\
\E_{\mu_u}\Big[\sup_{0\le s\le T_*}\big(\vartheta\varphi(X_s)\big)^r\Big]&\le C,\label{mgf-maximal-input}\\
\mu_u\Big[\big(\varphi \bar h(u)\big)^r+\big(\varphi \bar h(u)\big)^{-r}\Big]&\le C.\label{mgf-initial-source-input}
\end{align}
The bound \eqref{mgf-occupation-input} is \cref{iw-chemical-occupation}, and \eqref{mgf-maximal-input} is the last bound of \cref{hmr-main}. For \eqref{mgf-initial-source-input}, let $X_0$ have law $\mu_u$ and $Y_i(0)=X_i(0)/\varphi(X_0)^{1/2}$, as in \cref{iw-early-pin-simultaneous-centers}. As $h(0)=\ind$ and $|X_0|^2=N$, we have $\sum_ih_i(0)Y_i(0)^2=N/\varphi(X_0)$. Using \cref{iw-early-pin-simultaneous-centers} at time $0$, we obtain a random variable $C_{\mathrm{init}}\ge1$ with $\E C_{\mathrm{init}}^r\le C_r$ and $C_{\mathrm{init}}^{-1}\le(\varphi(X_0)\bar h(u))^{-1}\le C_{\mathrm{init}}$, which implies \eqref{mgf-initial-source-input}.

The weighted gradient estimate \eqref{eq:hotm-fk} below controls $\varphi\,\Gamma(P_\tau^Af)$ rather than $\Gamma(P_\tau^Af)$. To bound $\mu_u\Gamma(P^A_\tau f)$, we divide by $\varphi$, which replaces the initial law $\mu_u$ by the following law.

\begin{lemma}[The initial law weighted by $1/\varphi$]\label{mgf-weighted-source-transfer}
Assume \eqref{eq:hotm-geometry}. For $u\ge0$, let
\[
Z_u=\mu_u\Big(\frac1{\vartheta\varphi}\Big),\qquad \tilde\mu_u(\dd x)=\frac1{Z_u}\,\frac1{\vartheta\varphi(x)}\,\mu_u(\dd x).
\]
There exist $C$ and $M_0$ such that if $M\ge M_0$, then for every $u\ge0$,
\[
\frac{\bar h(u)}{C\vartheta}\le Z_u\le\frac{C\,\bar h(u)}\vartheta,\qquad \Big\|\frac{\dd\tilde\mu_u}{\dd\mu_u}\Big\|_{L^2(\mu_u)}\le C .
\]
Further, for every $r>0$ and $\lambda>0$, there exist $C_{r,\lambda}$ and $M_0(r,\lambda)$ such that if $M\ge M_0(r,\lambda)$, then for every $u\ge0$,
\[
\E_{\tilde\mu_u}\Big[\sup_{0\le s\le T_*}\big(\vartheta\varphi(X_s)\big)^r\Big]\le C_{r,\lambda},\qquad
\E_{\tilde\mu_u}\exp\Big(\lambda\int_0^{T_*}\Gamma(\log\varphi)(X_s)\,\dd s\Big)\le C_{r,\lambda}.
\]
\end{lemma}

\begin{proof}
We have $Z_u=\vartheta^{-1}\bar h(u)\,\mu_u((\varphi \bar h(u))^{-1})$ and
\[
\frac{\dd\tilde\mu_u}{\dd\mu_u}=\frac{(\varphi \bar h(u))^{-1}}{\mu_u\big((\varphi \bar h(u))^{-1}\big)} .
\]
Using \eqref{mgf-initial-source-input} with $r=1$, we have $\mu_u((\varphi \bar h(u))^{-1})\le C$ and $\mu_u(\varphi \bar h(u))\le C$, and by Jensen's inequality, $\mu_u((\varphi \bar h(u))^{-1})\ge\mu_u(\varphi \bar h(u))^{-1}\ge C^{-1}$. This proves the bounds on $Z_u$. With $r=2$, \eqref{mgf-initial-source-input} implies that the $L^2(\mu_u)$ norm of $\dd\tilde\mu_u/\dd\mu_u$ is at most $C^{1/2}\cdot C$.

Let $\mathcal B\ge0$ be a measurable function of the path $(X_s)_{0\le s\le T_*}$. Conditionally on $X_0$, the law of the path is the same whether $X_0$ has law $\mu_u$ or $\tilde\mu_u$. Using the Cauchy--Schwarz inequality, we find
\[
\E_{\tilde\mu_u}\mathcal B=\E_{\mu_u}\Big[\frac{\dd\tilde\mu_u}{\dd\mu_u}(X_0)\,\mathcal B\Big]\le C\big(\E_{\mu_u}\mathcal B^2\big)^{1/2}.
\]
With $\mathcal B=\sup_{s\le T_*}(\vartheta\varphi(X_s))^r$, the first claim follows from \eqref{mgf-maximal-input} with order $2r$. The second claim holds for every initial law, since \eqref{mgf-occupation-input} is a supremum over initial points.
\end{proof}

We use the weighted gradient estimate of \cref{hfk-pointwise-fk}: for every smooth $g$ on $S_N$, every $x\in S_N$, and every $s\ge0$,
\begin{equation}\label{eq:hotm-fk}
\varphi(x)\,\Gamma(P_s^Ag)(x)\le e^{-2\bar rs}\,\E_x\Big[\exp\Big(2\int_0^s\Gamma(\log\varphi)(X_v)\,\dd v\Big)\varphi(X_s)\,\Gamma(g)(X_s)\Big].
\end{equation}
It follows from the Bochner identity for the Dirichlet form weighted by $\varphi$, in which the curvature term is at least $\bar r$ after a square is completed, together with the Feynman--Kac formula; the term $\Gamma(\log\varphi)$ is what remains of the square. It is controlled along the paths of the diffusion by \eqref{mgf-occupation-input}.

\begin{lemma}[Gradient bounds under the comparison laws]\label{lem:hotm-gradient}
Assume \eqref{eq:hotm-geometry}. There exist $C$ and $M_0$ such that if $M\ge M_0$, then $d/2\le\bar r\le d$, and for every $u\ge0$, every $0<\tau\le T_*$, and every smooth $f:S_N\to[0,1]$,
\begin{equation}\label{mgf-launched-reverse-variance}
\mu_u\Gamma(P_\tau^Af)\le\frac{C\,\bar r\,Z_u}{e^{2\bar r\tau}-1}\le\frac{C\,\bar r\,\bar h(u)}{\vartheta\,(e^{2\bar r\tau}-1)}.
\end{equation}
\end{lemma}

The factor $\bar h(u)$, which is of order $(1+u)^{-1/2}$ for $u\le d^{-1}$, comes from the normalization $Z_u$ of the weighted initial law. In the proof of \cref{mgf-global-bridge}, it makes the contribution of times $u\le d^{-1}$ bounded independently of $d$.

\begin{proof}
Note that $Nd=Md^{-1/2}\ge M$. Then $2/N\le d/2$ when $M\ge4$, and $d/2\le\bar r\le d$. The proof has three steps.

\paragraph{Step 1: integration of \eqref{eq:hotm-fk}.}
Fix $\tau\in(0,T_*]$ and a smooth $f:S_N\to[0,1]$. For $0\le s\le\tau$, we apply \eqref{eq:hotm-fk} to $g=P^A_{\tau-s}f$, multiply by $e^{2\bar rs}$, and integrate over $s\in[0,\tau]$. Since $P_s^AP_{\tau-s}^A=P_\tau^A$, the left side becomes $\varphi(x)\Gamma(P_\tau^Af)(x)(e^{2\bar r\tau}-1)/(2\bar r)$. On the right side, we use $\Gamma(\log\varphi)\ge0$ to bound the exponential by $e^{2\mathcal O_\tau}$ and $\varphi(X_s)$ by $\vartheta^{-1}\varphi^*_\tau$, where
\[
\mathcal O_\tau=\int_0^\tau\Gamma(\log\varphi)(X_v)\,\dd v,\qquad \varphi^*_\tau=\sup_{0\le s\le\tau}\vartheta\varphi(X_s),\qquad \mathcal V_\tau=\int_0^\tau\Gamma\big(P_{\tau-s}^Af\big)(X_s)\,\dd s .
\]
By Tonelli's theorem, we obtain for every $x\in S_N$
\begin{equation}\label{eq:hotm-fk-integrated}
\varphi(x)\,\Gamma(P_\tau^Af)(x)\,\frac{e^{2\bar r\tau}-1}{2\bar r}\le\frac1\vartheta\,\E_x\big[e^{2\mathcal O_\tau}\varphi^*_\tau\,\mathcal V_\tau\big].
\end{equation}

\paragraph{Step 2: the quadratic variation.}
Let $m_s=P_{\tau-s}^Af(X_s)$ for $0\le s\le\tau$. Using Itô's formula and the backward equation $\partial_sP_{\tau-s}^Af=-L_AP_{\tau-s}^Af$, we see that $m$ is a continuous martingale. Its quadratic variation is $[m]_s=2\int_0^s\Gamma(P_{\tau-v}^Af)(X_v)\,\dd v$, because the noise coefficient of the diffusion is $\sqrt2$; see \cite[Chapter~IV, Section~3]{RevuzYor1999}. In particular $[m]_\tau=2\mathcal V_\tau$. As $0\le f\le1$, we have $0\le m_s\le1$ and $\sup_s|m_s-m_0|\le1$, and the Burkholder--Davis--Gundy inequality with exponent $6$ \cite[Chapter~IV, Section~4]{RevuzYor1999} implies that
\begin{equation}\label{eq:hotm-bdg}
\E_x\mathcal V_\tau^3\le C\qquad(x\in S_N),
\end{equation}
where $C$ is a numerical constant. The same bound holds for every initial law.

\paragraph{Step 3: averaging over the initial law.}
We divide \eqref{eq:hotm-fk-integrated} by $\varphi(x)$ and integrate over $x$ with respect to $\mu_u$. By the definition of $\tilde\mu_u$, we have $\int\mu_u(\dd x)\,(\vartheta\varphi(x))^{-1}\E_x[\,\cdot\,]=Z_u\E_{\tilde\mu_u}[\,\cdot\,]$, and hence
\[
\mu_u\Gamma(P_\tau^Af)\,\frac{e^{2\bar r\tau}-1}{2\bar r}\le Z_u\,\E_{\tilde\mu_u}\big[e^{2\mathcal O_\tau}\varphi^*_\tau\mathcal V_\tau\big]
\le Z_u\big(\E_{\tilde\mu_u}e^{6\mathcal O_\tau}\big)^{1/3}\big(\E_{\tilde\mu_u}(\varphi^*_\tau)^3\big)^{1/3}\big(\E_{\tilde\mu_u}\mathcal V_\tau^3\big)^{1/3},
\]
where we used Hölder's inequality with three exponents equal to $3$. The quantities $\mathcal O_\tau$ and $\varphi^*_\tau$ are at most their values at $T_*$, because $\tau\le T_*$. Using \cref{mgf-weighted-source-transfer} with $\lambda=6$ and $r=3$, together with \eqref{eq:hotm-bdg}, we find that the three expectations are bounded by a constant. This proves the first inequality in \eqref{mgf-launched-reverse-variance}, and the second follows from the bound $Z_u\le C\bar h(u)/\vartheta$ of \cref{mgf-weighted-source-transfer}.
\end{proof}

We now prove the main estimate of this section. The constants are chosen in the following order.

\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Quantity & Role & Dependence \\
\midrule
$K$, $d_0$, $C_T$ & hypotheses \eqref{eq:hotm-geometry} and time $T_*$ & given \\
$\lambda=6$, $r=3$ & orders in \cref{mgf-weighted-source-transfer} & fixed \\
$M_0$ & threshold for \cref{mgf-exact-flow-envelopes,mgf-weighted-source-transfer,lem:hotm-gradient} & $K$, $d_0$, $C_T$ \\
$C$ & constant in \eqref{mgf-global-bridge-bound} & $K$, $d_0$, $C_T$ \\
$N$, $A$, $t$ & any, with $M\ge M_0$ and $0\le t\le T_*$ & last \\
\bottomrule
\end{tabular}
\end{center}

\begin{theorem}[{Distance to the Bingham comparison law on $[0,C_Td^{-1}\log M]$}]\label{mgf-global-bridge}
For every $K\ge1$, $d_0\in(0,1)$, and $C_T>0$, there exist constants $C\ge1$ and $M_0\ge4$, depending only on $K$, $d_0$, and $C_T$, with the following property. Let $N\ge2$, and let $A$ be a real symmetric $N\times N$ matrix whose precisions $p_1\le\cdots\le p_N$ satisfy \eqref{eq:hotm-geometry} and $M=Nd^{3/2}\ge M_0$, where $d=p_1$. Let $\mu_t$ be the Bingham comparison law of $A$. Then for $0\le t\le C_Td^{-1}\log M$,
\begin{equation}\label{mgf-global-bridge-bound}
\big\|\nu_NP_t^A-\mu_t\big\|_{\TV}\le\min\big\{1,\,C\,e^{-(d-2/N)t}\big\}.
\end{equation}
\end{theorem}

The bound contains no factor depending on $d$ or on $Nd^2$. At times with $dt\ge\frac18\log M$ and $t\le C_Td^{-1}\log M$, we have $2t/N\le2C_T\log M/(Nd)\le2C_T\log M/M$, and the right side of \eqref{mgf-global-bridge-bound} is at most $Ce^{2C_T}M^{-1/8}$; at $dt=\frac14\log M+O(\log\log M)$ it is $M^{-1/4}(\log M)^{O(1)}$. In both cases it tends to zero as $M\to\infty$, however slowly.

\begin{proof}
We first bound the distance by a Duhamel integral, using part (c) of \cref{pc-centered-defect-transfer}, and then estimate the integral. Let $M\ge M_0$, and let $T_*=C_Td^{-1}\log M$ and $\bar r=d-2/N\in[d/2,d]$ be as above.

\paragraph{Step 1: the Duhamel bound.}
Fix $t\in(0,T_*]$. We apply part (c) of \cref{pc-centered-defect-transfer} to the family $(\mu_u)_{u\ge0}$, whose forward-equation defect is $R_u$ by \eqref{eq:hotm-forward}, and to the semigroup $P^A_t$. By \cref{cor:comp-poisson-flux}, the flux hypothesis \eqref{pc-centered-defect-flux-input} holds with $j_u=\nabla_{S_N}\psi_u$ and with $\mathcal J(u)$ equal to the right side of \eqref{eq:hotm-energy}, that is, $\mathcal J(u)=C(1+u)^{-1}$ for $u\le d^{-1}$ and $\mathcal J(d^{-1}+s)=Cd(1+ds)^2e^{-4ds}$. For the gradient hypothesis, let
\[
\mathcal G(u,\tau)=\frac{C\,\bar r\,\bar h(u)}{\vartheta\,(e^{2\bar r\tau}-1)}\qquad(u,\tau\in(0,t)),
\]
where $C$ is the constant of \cref{lem:hotm-gradient}. For $\tau<t\le T_*$, that lemma states that $\mu_u\Gamma(P^A_\tau f)\le\mathcal G(u,\tau)$ for every smooth $f:S_N\to[0,1]$. Both $\mathcal J$ and $\mathcal G$ are Borel measurable. As $\mu_0=\nu_N$, \eqref{pc-centered-defect-flux-only-bound} gives
\[
\big\|\nu_NP_t^A-\mu_t\big\|_{\TV}\le\int_0^t\sqrt{\mathcal J(u)\,\mathcal G(u,t-u)}\,\dd u .
\]
We have $\mathcal G(u,t-u)=C\bar r\vartheta^{-1}\bar h(u)e^{-2\bar r(t-u)}(1-e^{-2\bar r(t-u)})^{-1}$, which yields
\begin{equation}\label{eq:hotm-duhamel}
\big\|\nu_NP_t^A-\mu_t\big\|_{\TV}\le C\,e^{-\bar rt}\int_0^t\frac{e^{\bar ru}\sqrt{\bar r\vartheta^{-1}\bar h(u)\mathcal J(u)}}{\sqrt{1-e^{-2\bar r(t-u)}}}\,\dd u\qquad(0<t\le T_*).
\end{equation}

\paragraph{Step 2: the numerator.}
We have $\bar r\le d$, $\vartheta^{-1}\le Cd^{-1/2}$ by \cref{hi3-finite-heat}, and $\bar h(u)\le C(\sqrt d+(1+u)^{-1/2})$ by \cref{iw-all-time-heat}. For $0\le u\le d^{-1}$, we have $\sqrt d\le\sqrt2(1+u)^{-1/2}$, $e^{\bar ru}\le e$, and $\mathcal J(u)=C(1+u)^{-1}$. For $u=d^{-1}+s$ with $s\ge0$, we have $\bar h(u)\le C\sqrt d$, $e^{\bar ru}\le e^{1+ds}$, and $\mathcal J(u)=Cd(1+ds)^2e^{-4ds}$. We conclude that
\begin{equation}\label{eq:hotm-numerator}
e^{\bar ru}\sqrt{\bar r\vartheta^{-1}\bar h(u)\mathcal J(u)}\le
\begin{cases}
Cd^{1/4}(1+u)^{-3/4}, & 0\le u\le d^{-1},\\
Cd\,(1+ds)\,e^{-ds}, & u=d^{-1}+s,\ s\ge0.
\end{cases}
\end{equation}
The first line integrates over $[0,d^{-1}]$ to at most $4Cd^{1/4}(1+d^{-1})^{1/4}\le4\cdot2^{1/4}C$, and the second over $s\in[0,\infty)$ to $C\int_0^\infty(1+r)e^{-r}\dd r=2C$. Thus
\begin{equation}\label{eq:hotm-numerator-integral}
\int_0^\infty e^{\bar ru}\sqrt{\bar r\vartheta^{-1}\bar h(u)\mathcal J(u)}\,\dd u\le C .
\end{equation}

\paragraph{Step 3: the denominator.}
Let $2/\bar r\le t\le T_*$, and split the integral in \eqref{eq:hotm-duhamel} at $u=t-1/\bar r$. For $u\le t-1/\bar r$, the denominator is at least $(1-e^{-2})^{1/2}$, and by \eqref{eq:hotm-numerator-integral} this part of the integral is at most $C(1-e^{-2})^{-1/2}$. For $t-1/\bar r\le u\le t$, we have $u\ge1/\bar r\ge d^{-1}$, so the second line of \eqref{eq:hotm-numerator} applies, and since $(1+r)e^{-r}\le1$, the numerator is at most $Cd$. By concavity, $1-e^{-2y}\ge(1-e^{-2})y$ for $0\le y\le1$, and this part of the integral is at most
\[
Cd\int_0^{1/\bar r}\frac{\dd v}{\sqrt{(1-e^{-2})\,\bar rv}}=\frac{2Cd}{\sqrt{1-e^{-2}}\;\bar r}\le\frac{4C}{\sqrt{1-e^{-2}}},
\]
where we used $\bar r\ge d/2$. By \eqref{eq:hotm-duhamel}, we conclude that $\|\nu_NP_t^A-\mu_t\|_{\TV}\le C'e^{-\bar rt}$ for $2/\bar r\le t\le T_*$, with $C'$ depending only on $K$, $d_0$, and $C_T$. If $0\le t<2/\bar r$, then $e^{-\bar rt}>e^{-2}$, and the total variation distance is at most $1\le e^2e^{-\bar rt}$. Taking $C=\max(C',e^2)$ proves \eqref{mgf-global-bridge-bound}.
\end{proof}

\begin{remark}[The parameters $M$ and $M_N$]\label{rem:hotm-M}
In \cref{mgf-global-bridge}, $M=Nd^{3/2}$ is defined from the smallest precision of a fixed matrix, and the estimate holds for every $N$ and every such matrix with $M\ge M_0$. For $A=c_NW_N$ in the regime \eqref{eq:hotm-regime}, $M=Nd_N^{3/2}$ is random, and it is related to the deterministic $M_N=N\delta_N^3$ by $Nd_N^{3/2}/M_N\to1$ in probability, by \eqref{eq:hotm-canonical-consequences}; more precisely, $\ell_N|Nd_N^{3/2}/M_N-1|\to0$. In \cref{sec:outer-hot}, \cref{mgf-global-bridge} is applied to deterministic matrices with $M$ fixed while $N\to\infty$. The estimate requires no relation between $N$ and $d$ beyond $M\ge M_0$, and it therefore applies however slowly $M_N$ tends to infinity.
\end{remark}

\section{Proof of the profile}\label{sec:hotm-proof}

\begin{proof}[Proof of \cref{mgf-all-mesoscopic-profile}]
We first prove \eqref{mgf-actual-profile-convergence}. Fix $u_0>0$ and $\eps_0>0$. The parameters are chosen in the following order.

\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Parameter & Role & When it is fixed \\
\midrule
$u_0$, $\eps_0$ & range of offsets and tolerance & first \\
$\eta\in(0,1)$ & confidence of the events $\mathcal \mathsf G^\eta$ in \eqref{mc-indexed-input} & second \\
$K=K_\eta$ & constant in \eqref{mc-indexed-input} & depends on $\eta$ \\
$d_0=1/2$, $C_T=1$ & constants in \eqref{eq:hotm-geometry} and in $T_*$ & absolute \\
$C$, $M_0$ & constants of \cref{mgf-global-bridge} & depend on $K_\eta$ \\
$N\to\infty$ & limit in probability & third \\
$\eta\to0$ & removal of the localization (\cref{lem:model-localization}) & last \\
\bottomrule
\end{tabular}
\end{center}

Let $X_N=\sup_{|u|\le u_0}|D_N(T_N(u))-\Psi(u)|$. Fix $\eta\in(0,1)$, and let $C$ and $M_0$ be the constants of \cref{mgf-global-bridge} for $K=K_\eta$, $d_0=1/2$, and $C_T=1$. Let $\mathcal E_N$ be the intersection of $\mathcal \mathsf G^\eta$ with the event that
\[
d_N\le\frac12,\qquad Nd_N^{3/2}\ge M_0,\qquad \frac18\log M_N\le d_NT_N(u)\le\log\big(Nd_N^{3/2}\big)\quad\text{for all }|u|\le u_0 .
\]
We claim that $\PP(\mathcal \mathsf G^\eta\setminus\mathcal E_N)\to0$. Indeed, by \eqref{eq:hotm-canonical-consequences}, $d_N\to0$ and $Nd_N^{3/2}/M_N\to1$ in probability. Further, $\delta_N^2T_N(u)=\frac14\log M_N(1+o(1))$ uniformly for $|u|\le u_0$, so $d_NT_N(u)/\log M_N=(d_N/\delta_N^2)\,\delta_N^2T_N(u)/\log M_N\to1/4$ in probability, uniformly for $|u|\le u_0$, while $\log(Nd_N^{3/2})/\log M_N\to1$.

On $\mathcal E_N$, the matrix $A=c_NW_N$ satisfies the hypotheses of \cref{mgf-global-bridge}. Indeed, its precisions in increasing order are $p_i=d_N+\gamma_i$, and \eqref{mc-indexed-input} and $d_N\le1/2$ give \eqref{eq:hotm-geometry}. Further $M=Nd_N^{3/2}\ge M_0$, and $0<T_N(u)\le d_N^{-1}\log M$ for $|u|\le u_0$. By \eqref{mgf-global-bridge-bound}, on $\mathcal E_N$,
\[
\sup_{|u|\le u_0}\big\|\nu_NP_{T_N(u)}-\mu_{T_N(u)}\big\|_{\TV}\le C\sup_{|u|\le u_0}e^{-d_NT_N(u)+2T_N(u)/N}\le C\,e^{2T_N(u_0)/N}M_N^{-1/8}.
\]
The number $T_N(u_0)/N$ is deterministic and is at most $C\delta_N^{-2}\log M_N/N=C\delta_N\log M_N/M_N$, which tends to zero. In particular, the supremum is at most $2CM_N^{-1/8}$ on $\mathcal E_N$ for large $N$. Using the triangle inequality, we have
\[
X_N\le\sup_{|u|\le u_0}\big\|\nu_NP_{T_N(u)}-\mu_{T_N(u)}\big\|_{\TV}+\sup_{|u|\le u_0}\Big|\,\big\|\mu_{T_N(u)}-\pi_{N,c_N}\big\|_{\TV}-\Psi(u)\Big| .
\]
For $N$ so large that $2CM_N^{-1/8}<\eps_0/2$, we therefore have
\[
\PP\big(\mathcal \mathsf G^\eta\cap\{X_N>\eps_0\}\big)\le\PP\big(\mathcal \mathsf G^\eta\setminus\mathcal E_N\big)+\PP\Big(\sup_{|u|\le u_0}\Big|\,\big\|\mu_{T_N(u)}-\pi_{N,c_N}\big\|_{\TV}-\Psi(u)\Big|>\frac{\eps_0}2\Big),
\]
and both terms tend to zero, the second by \cref{mc-proxy-profile}. Now \cref{lem:model-localization} implies that $X_N\to0$ in probability, which is \eqref{mgf-actual-profile-convergence}.

The center and the window are deterministic, and
\[
\begin{gathered}
\frac{4\delta_N^2T_N}{\log M_N}=1-\frac{3\log\log M_N}{2\log M_N}+\frac{\log B_{c_N}}{\log M_N}\to1,\\
\frac{\delta_N^{-2}}{T_N}=\frac4{\log M_N-\frac32\log\log M_N+\log B_{c_N}}\to0 .
\end{gathered}
\]
Together with \eqref{mgf-actual-profile-convergence}, this shows that $D_N$ has cutoff profile $\Psi$ with center $T_N$ and window $\delta_N^{-2}$ in the sense of \cref{def:model-cutoff}. Recall from \cref{cot-profile} that $\Psi$ is continuous and strictly decreasing from $1$ to $0$, with $\Psi^{-1}(\eps)=-\frac12\log[2\Phi^{-1}((1+\eps)/2)]$. Then \cref{cor:model-profile-cutoff} implies that $(t_N(\eps)-T_N)/\delta_N^{-2}\to\Psi^{-1}(\eps)$ in probability for every $\eps\in(0,1)$, which is \eqref{mgf-accuracy-offsets}, and there is cutoff.
\end{proof}