\chapter{Fixed low temperature}\label{sec:cold-fixed}

In this section, we determine the total-variation profile of the dynamics at a fixed rescaled inverse temperature $c>1$, which is case (5) of \cref{thm:intro-main}. The result is \cref{cq-full-profile}, with its consequence for cutoff \cref{cq-cutoff-clock}. Both are deduced in \cref{sec:coldf-goe-proof} from a deterministic statement about sequences of spectra, \cref{thm:coldf-deterministic}, whose proof occupies the rest of the section. Along the way, we prove \cref{mesocold-prop:stationary-graph}, which compares the Gibbs measure, written in graph coordinates over the top eigencoordinate, with a product of Gaussian laws. That estimate is uniform in the top fraction $\theta_N$, and \cref{sec:cold-meso} uses it when $\theta_N\to0$.

We use the conventions of \cref{sec:model}: the reflection symmetry \cref{lem:model-symmetry}, the facts on total variation in \cref{lem:model-tv}, the profiles and their properties in \cref{cot-profile}, the passage from pointwise convergence to profiles and mixing times in \cref{hp-monotone-profile-transfer}, and the removal of localizations in \cref{lem:model-localization}. The random-matrix inputs to the proof of \cref{cq-full-profile} are the eigenvalue gaps at fixed confidence (\cref{sp-fixed-confidence-geometry}) and rigidity and convergence at the edge (\cref{sp-goe-rigidity}), from which we deduce a law of large numbers for the inverse gaps (\cref{lem:coldf-inverse-gap-lln}). We also use the representation of the dynamics through the unnormalized process (\cref{cq-ce-source-valid-unshifted-matrix-lift}) and the following results of \cref{sec:scores}: the Bismut-type bound for scores over a final time interval (\cref{csr-thm:reset}), the generator and the scores in graph coordinates and the score of a marginal (\cref{lem:score-chart-generator,lem:score-chart,lem:score-marginal-score}), the Gaussian logarithmic Sobolev inequality \eqref{eq:score-gaussian-lsi}, the time derivative of the conditional relative entropy (\cref{prop:score-entropy-identity}), the bounds on weighted logarithms (\cref{cce-lem:weighted-logs}), the passage from weak convergence to total variation by Gaussian smoothing (\cref{csh-thm:weak-to-tv,csh-lem:convolution-fisher}), and the final comparison of product measures (\cref{ccf-lem:finite-product-closure}).

\section{Results and outline}\label{sec:coldf-results}\label{cq-statements}

\subsection{The main result}

Fix $c>1$ and let $W_N$ be a GOE matrix with eigenvalues $\lambda_1\ge\cdots\ge\lambda_N$. As in \eqref{eq:model-raw} and \eqref{eq:model-cold-parameters}, we set
\[
a_i=\frac c2(\lambda_1-\lambda_i),\qquad
F(x)=\sum_{i=2}^Na_ix_i^2,\qquad
\theta_N=1-\frac1N\sum_{i=2}^N\frac1{2a_i},\qquad
R_N=Na_2,
\]
so that $L=\Delta_{S_N}-\nabla_{S_N}F\cdot\nabla_{S_N}$ and $\pi_{N,c}\propto e^{-F}\nu_N$. Almost surely the eigenvalues are distinct, so $0=a_1<a_2<\cdots<a_N$. Let $\theta=1-1/c$, and define the centers
\begin{equation}\label{eq:coldf-centers}
T_N''=\begin{cases}
\dfrac{\log R_N}{4a_2},&R_N>1,\\
1,&R_N\le1,
\end{cases}
\qquad
T_N=\begin{cases}
\dfrac{\log(2\theta_NR_N)}{4a_2},&\theta_N>0\text{ and }2\theta_NR_N>1,\\
1,&\text{otherwise}.
\end{cases}
\end{equation}
Both centers are positive. The events on which the second formulas are used have probabilities tending to zero (\cref{lem:coldf-goe} below), so that $T_N''=\log R_N/(4a_2)$ and $T_N=\log(2\theta_NR_N)/(4a_2)$ with probability tending to one. For $t<0$ we set $D_N(t)=D_N(0)$, so that expressions such as $D_N(T_N''+u/a_2)$ are defined for all $u\in\R$; this convention does not change the mixing times. Recall the profiles $\cV$ and $\cV_\theta$ from \eqref{eq:model-cauchy-profiles}, and recall from \cref{cot-profile} that $\cV_\theta(u)=\cV(u-\log(2\theta)/4)$.

\begin{theorem}[Profile at fixed low temperature]\label{cq-full-profile}
Fix $c>1$ and let $\theta=1-1/c$. Then $D_N$ has cutoff profile $\cV$ with center $T_N$ and window $a_2^{-1}$, in the sense of \cref{def:model-cutoff}. Equivalently, for every compact set $K\subset\R$,
\begin{equation}\label{eq:coldf-main}
\sup_{u\in K}\big|D_N(T_N''+u/a_2)-\cV_\theta(u)\big|\longrightarrow0
\end{equation}
in probability. For every compact interval $I\subset(0,1)$,
\[
\sup_{\eps\in I}\big|a_2\big(t_N(\eps)-T_N''\big)-\cV_\theta^{-1}(\eps)\big|\to0,
\qquad
\sup_{\eps\in I}\big|a_2\big(t_N(\eps)-T_N\big)-\cV^{-1}(\eps)\big|\to0,
\]
in probability, and $a_2t_N(\eps)/\log N\to1/12$ in probability for every $\eps\in(0,1)$. For finitely many fixed values $c_1,\dots,c_m>1$, these conclusions hold simultaneously for the same matrices $W_N$.
\end{theorem}

The centers are random through the first spectral gap $\lambda_1-\lambda_2$, which is of order $N^{-2/3}$, and through $\theta_N$. By \cref{sp-goe-rigidity}, $N^{2/3}a_2$ converges in distribution to $c\Delta_2/2$, where $\Delta_2$ is almost surely positive and finite (see \cref{lem:coldf-goe}). We make no statement with a deterministic center.

\begin{corollary}[Cutoff at fixed low temperature]\label{cq-cutoff-clock}
Fix $c>1$. For every fixed $\eta\in(0,1)$,
\[
D_N\big((1-\eta)T_N''\big)\to1,\qquad D_N\big((1+\eta)T_N''\big)\to0
\]
in probability. For every $\eps\in(0,1)$, we have $t_N(\eps)=T_N''+O_{\PP}(a_2^{-1})$, and $\log(a_2^{-1})/\log N\to2/3$ in probability. Further, $D_N(T_N''+\omega_N/a_2)\to0$ in probability for every deterministic sequence $\omega_N\to\infty$.
\end{corollary}

The proofs of \cref{cq-full-profile,cq-cutoff-clock} are given in \cref{sec:coldf-goe-proof}.

\subsection{Deterministic hypotheses}

The dynamical estimates of this section are statements about a fixed sequence of spectra. Throughout the section, $N\ge3$, and $0=a_1<a_2\le a_3\le\cdots\le a_N$ are real numbers, which may depend on $N$. We define $F$, $L$, $\pi$, $\theta_N$, and $R_N$ by the formulas above, write $P_t=e^{tL}$, and set
\[
D_N(t)=\|\nu_NP_t-\pi\|_{\TV},\qquad t_N(\eps)=\inf\{t\ge0:D_N(t)\le\eps\},
\]
with $D_N(t)=D_N(0)$ for $t<0$. For the model, these are the quantities of \cref{sec:model}, by \eqref{eq:model-raw} and the reduction to eigencoordinates.

\begin{definition}[Hypotheses at low temperature]\label{def:coldf-hypotheses}
Let $\theta_*\in(0,1)$ and $B,c_*,C_*>0$. We say that $(a_i)$ satisfies $\mathrm H(\theta_*,B,c_*,C_*)$ if
\begin{equation}\label{eq:coldf-H}
\theta_N\ge\theta_*,\qquad a_N\le B,\qquad
a_j\ge c_*(j-1)^{2/3}N^{-2/3}\quad(2\le j\le N),\qquad
a_2\le C_*N^{-2/3}.
\end{equation}
For $k\ge3$, $\varpi>0$, and $D_k\ge1$, we say that $(a_i)$ satisfies $\mathrm H_k(\varpi,D_k)$ if
\begin{equation}\label{eq:coldf-Hk}
\frac{a_3}{a_2}\ge1+\varpi,\qquad \frac{a_k}{a_2}\le D_k.
\end{equation}
A sequence of spectra satisfies these hypotheses if it does so for all sufficiently large $N$, with constants independent of $N$.
\end{definition}

Recall that the first condition in \eqref{eq:coldf-H} says that the Gaussian vector $(Z_i)_{i\ge2}$ with independent coordinates of variances $(2a_i)^{-1}$ has mean squared norm at most $(1-\theta_*)N$. The other conditions give a lower bound of order $((j-1)/N)^{2/3}$ on each $a_j$ and the two-sided bound $c_*N^{-2/3}\le a_2\le C_*N^{-2/3}$; an upper bound on $a_k/a_2$ is imposed separately in $\mathrm H_k(\varpi,D_k)$. The condition $\mathrm H_3(\varpi,D_3)$ is used in two places. Its lower bound separates the third gap from the second, and in \cref{thm:coldf-finite-coordinates} it ensures that the standardized coordinates $x_j$, $j\ge3$, have standard Gaussian limits. Its upper bound controls the singular factor of the Gibbs measure in graph coordinates in \cref{mesocold-prop:stationary-graph}. We record some elementary consequences.

\begin{lemma}[Consequences of the hypotheses]\label{lem:coldf-consequences}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$. Then the following hold.
\begin{enumerate}
\item We have $c_*N^{1/3}\le R_N\le C_*N^{1/3}$, and $F\le BN$ on $S_N$.
\item We have $a_2^2\sum_{j=2}^Na_j^{-2}\le C_0$, where $C_0=(C_*/c_*)^2\sum_{n\ge1}n^{-4/3}$.
\item For every $t>0$, we have $\sum_{j=2}^Ne^{-4a_jt}\le C_1Nt^{-3/2}$, where $C_1$ depends only on $c_*$.
\item We have $\theta_N\le1-(N-1)/(2BN)$.
\end{enumerate}
\end{lemma}

\begin{proof}
The first claim follows from $a_2\ge c_*N^{-2/3}$, $a_2\le C_*N^{-2/3}$, and $F(x)\le a_N|x|^2$. For the second, we have $a_2^2a_j^{-2}\le C_*^2c_*^{-2}(j-1)^{-4/3}$. For the third, we note that the function $u\mapsto e^{-4c_*tu^{2/3}N^{-2/3}}$ is decreasing on $[0,\infty)$, which implies that
\[
\sum_{j=2}^Ne^{-4a_jt}\le\sum_{n\ge1}e^{-4c_*tn^{2/3}N^{-2/3}}\le\int_0^\infty e^{-4c_*tu^{2/3}N^{-2/3}}\,du=\frac{3\sqrt\pi}{4}\,(4c_*t)^{-3/2}N,
\]
where we substituted $w=4c_*tu^{2/3}N^{-2/3}$ and used $\int_0^\infty w^{1/2}e^{-w}\,dw=\sqrt\pi/2$. The last claim follows from $a_j\le B$.
\end{proof}

\subsection{The deterministic theorem}

For $u\in\R$, we write
\[
t_u=T_N''+\frac u{a_2}
\]
for the time at offset $u$. Under $\mathrm H(\theta_*,B,c_*,C_*)$, we have $R_N\ge c_*N^{1/3}>1$ for large $N$, and then $t_u=\log R_N/(4a_2)+u/a_2$ and $a_2t_u=\log R_N/4+u\to\infty$. In particular, $t_u>0$ for large $N$.

\begin{theorem}[Deterministic profile]\label{thm:coldf-deterministic}
Let $(a_i)$ be a sequence of spectra satisfying $\mathrm H(\theta_*,B,c_*,C_*)$ and $\mathrm H_3(\varpi,D_3)$, and suppose that $\theta_N\to\theta$ for some $\theta\in(0,1)$. Then the following hold.
\begin{enumerate}
\item For every compact set $K\subset\R$, we have $\sup_{u\in K}|D_N(t_u)-\cV_\theta(u)|\to0$.
\item For every compact interval $I\subset(0,1)$, we have $\sup_{\eps\in I}|a_2(t_N(\eps)-T_N'')-\cV_\theta^{-1}(\eps)|\to0$.
\item Let $T_N$ be as in \eqref{eq:coldf-centers}, so that $T_N=\log(2\theta_NR_N)/(4a_2)$ for large $N$. For every compact set $K\subset\R$ and every compact interval $I\subset(0,1)$, we have $\sup_{u\in K}|D_N(T_N+u/a_2)-\cV(u)|\to0$ and $\sup_{\eps\in I}|a_2(t_N(\eps)-T_N)-\cV^{-1}(\eps)|\to0$.
\end{enumerate}
\end{theorem}

The second part gives exact limits for the differences of mixing times on the scale $a_2^{-1}$: for $0<\eps<\eps'<1$, we have $a_2(t_N(\eps)-t_N(\eps'))\to\cV_\theta^{-1}(\eps)-\cV_\theta^{-1}(\eps')>0$. The proof of \cref{thm:coldf-deterministic} is completed in \cref{sec:coldf-assembly}.

\subsection{Outline of the proof}\label{sec:coldf-outline}

We first describe the mechanism, which is visible in the unnormalized process of \cref{sec:process}. Let $Y$ solve
\begin{equation}\label{eq:coldf-Y}
dY_j=-2a_jY_j\,dt+\sqrt{2/N}\,|Y|\,dB_j,\qquad Y(0)=X_0\sim\nu_N,
\end{equation}
so that $X=\sqrt N\,Y/|Y|$ is the diffusion generated by $L$ and started from $\nu_N$. The first coordinate $Y_1$ is a martingale, because $a_1=0$, while the coordinates $Y_j$, $j\ge2$, are pulled toward zero at rate $2a_j$.
\begin{itemize}
\item \emph{Gaussian approximation at time $t_{\mathrm p}$.} At the time $t_{\mathrm p}=(\log N)/100$, the coordinates $(Y_1(t_{\mathrm p}),Y_2(t_{\mathrm p}))$ are close in total variation to a pair $\theta^{-1/2}(g_1,g_2)$ of independent centered Gaussian variables with variance $\theta^{-1}$. They are built from the initial point and the noise up to time $t_{\mathrm p}$.
\item \emph{Growth of the top coordinate.} The martingale $Y_1$ barely moves after $t_{\mathrm p}$, while $|Y|^2$ relaxes to $Y_1(t_{\mathrm p})^2/\theta_N$. After normalization, the top eigencoordinate carries the fraction $\theta_N$ of the squared norm, and $X_1$ grows from order one to $\sqrt{\theta N}$.
\item \emph{The ratio of the first two coordinates.} The second coordinate decays like $e^{-2a_2t}$. At time $t_u$, its standardized value $\sqrt{2a_2}\,X_2(t_u)$ is approximately
\[
\sqrt{2a_2N\theta_N}\,e^{-2a_2t_u}\,\frac{Y_2(t_{\mathrm p})}{|Y_1(t_{\mathrm p})|}+G
=\sqrt{2\theta_N}\,e^{-2u}\,\frac{Y_2(t_{\mathrm p})}{|Y_1(t_{\mathrm p})|}+G,
\]
where $G$ is a standard Gaussian variable produced by the noise in coordinate $2$ during the last time interval of length of order $a_2^{-1}$, independent of $(g_1,g_2)$. The ratio $g_2/|g_1|$ has the standard Cauchy law, which gives the profile $\cV_\theta$.
\item \emph{The remaining coordinates.} We retain all coordinates $x_j$ with $a_j\le Aa_2$, where $A=16/r$ and $r>0$ is a fixed constant chosen in \cref{sec:coldf-selected}; by \cref{lem:coldf-bounded-head}, their number is bounded. On the support of a cutoff function that vanishes on $\{x_1^2<rN/2\}$, the Gaussian logarithmic Sobolev inequality bounds the conditional relative entropy of the other coordinates given the retained ones by $(32a_2)^{-1}$ times its dissipation. The resulting contraction rate $16a_2$ is four times the rate $4a_2$ that sets the profile, and it absorbs an initial value of order $N$. We show that the conditional law of the coordinates $x_j$ with $a_j>Aa_2$ given the retained ones is close to its equilibrium at time $t_u$.
\end{itemize}

The ratio of two independent centered Gaussian variables is a classical construction of the Cauchy law; see \cite{Marsaglia1965}. The exponential relaxation of the dynamics on the time scale $N^{2/3}$, governed by the random first spectral gap, was identified at zero temperature by Fyodorov, Perret, and Schehr \cite{FyodorovPerretSchehr2015}; see also the study of finite-size effects by Barbier, de Freitas Pimenta, Cugliandolo, and Stariolo \cite{BarbierDeFreitasPimentaCugliandoloStariolo2021}. The hitting-time bounds of orders $N^{2/3}$ and $N^{2/3}\log N$ in \cite[Section~3, Theorem~3.3]{Yu2024} concern gradient descent on the sphere started from the uniform distribution, and the time it needs to reach a fixed overlap with an extreme eigenvector. In a spherical quadratic model with a large spike, Gheissari, Grant, and Yu proved fast mixing from symmetric initial laws at low temperature \cite[Theorem~1.2]{GheissariGrantYu2026}.
The renewal structure of the mean squared norm in \cref{sec:coldf-process} is the finite-dimensional counterpart of the self-consistent equation of Cugliandolo and Dean \cite[(II.4)--(II.7)]{CugliandoloDean1995}.

The proof proceeds in the following steps.
\begin{enumerate}
\item In \cref{sec:coldf-stationary}, we compare the Gibbs measure in graph coordinates with the Gaussian product law $G=\bigotimes_{j\ge2}\cN(0,(2a_j)^{-1})$, including the singular factor $(1-|y|^2/N)^{-1/2}$ of the graph parametrization (\cref{mesocold-prop:stationary-graph}).
\item In \cref{sec:coldf-process}, we compute the mean of $|Y|^2$ through a renewal equation (\cref{lem:coldf-renewal}). In \cref{sec:coldf-entrance}, we deduce that at the time $T_{\mathrm e}=\chi_N/a_2$, with $\chi_N\to\infty$ slowly, the diffusion lies in a fixed spherical cap $\{x_1^2\ge\sigma N\}$ outside an event of small fixed probability (\cref{prop:coldf-entrance}). We also show that the uniformly initialized path, reweighted by a bounded measurable function of its past at entrance, stays in a slightly larger cap for times of order $\log R_N/a_2$ (\cref{prop:coldf-residence}).
\item In \cref{sec:coldf-bounded-head}, we show that for every fixed $A>1$ the number of coordinates $x_j$ with $a_j\le Aa_2$ is bounded (\cref{lem:coldf-bounded-head}). For densities that vanish outside a cap, we prove a logarithmic Sobolev inequality for the conditional relative entropy of the other coordinates given these, with constant of order $(Aa_2)^{-1}$ (\cref{lem:coldf-coarse-lsi}).
\item In \cref{sec:coldf-gaussian}, we prove the Gaussian limit of the first coordinates of $Y(t_{\mathrm p})$ (\cref{prop:coldf-initial-coordinates}). In \cref{sec:coldf-finite}, we prove that $X_1(t_u)^2/N\to\theta$ (\cref{prop:coldf-normalization}) and identify the weak limit of finitely many standardized coordinates (\cref{thm:coldf-finite-coordinates}).
\item In \cref{sec:coldf-selected}, we weight the path law by a smooth function of the position at the time $T_{\mathrm e}$, with values in $[0,1]$, which vanishes outside a cap and has mean at least $1-\eps$. We multiply the endpoint law by a cutoff function that depends only on $x_1^2$ and vanishes outside a larger cap. We describe the density of this law and prove bounds on the cutoff, on the density, and on the scores (\cref{lem:coldf-selected-density,lem:coldf-collar,lem:coldf-cutoff,lem:coldf-scores,lem:coldf-density,lem:coldf-marginal-score}).
\item In \cref{sec:coldf-entropy}, we show that the relative entropy of the conditional law of the coordinates $x_j$ with $a_j>Aa_2$ given the retained ones, under the weighted law multiplied by the cutoff and normalized, with respect to its Gaussian equilibrium, tends to zero at the times $t_u$ (\cref{lem:coldf-initial-entropy,prop:coldf-conditional-entropy}). The initial value is of order $N$, and the contraction rate is $16a_2$.
\item In \cref{sec:coldf-head}, we upgrade the weak limit of finitely many standardized coordinates to convergence in total variation (\cref{prop:coldf-finite-head-tv}), and \cref{cq-hc-head-smoothing} is its case of the second coordinate. In \cref{sec:coldf-assembly}, we combine these estimates and prove \cref{thm:coldf-deterministic}. The number of retained coordinates may depend on $N$, and we pass to subsequences on which it is constant.
\end{enumerate}

\Cref{tab:coldf-parameters} lists the parameters of the proof in the order in which they are chosen. All constants in the estimates below depend only on the parameters chosen before them in the table.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.48\textwidth}>{\raggedright\arraybackslash}p{0.2\textwidth}@{}}
\toprule
Parameter & Role & Depends on\\
\midrule
$\theta_*,B,c_*,C_*$; $\varpi,D_3$ & spectral constants, \eqref{eq:coldf-H}--\eqref{eq:coldf-Hk} & fixed\\
$\eps\in(0,1/8)$ & loss allowed in the weight $\Psi$ of \eqref{eq:coldf-weight} & fixed\\
$h,M$; $\sigma_{\mathrm e},\sigma_0$ & bounds of \cref{prop:coldf-entrance} with loss $\eps/2$; caps $\{x_1^2\ge\sigma_{\mathrm e}N\}$ and $\{x_1^2>\sigma_0N\}$ for the weight & $\eps$, spectral constants\\
$r$ & cutoff cap, \eqref{eq:coldf-r}; residence in $\{x_1^2\ge2rN\}$ & $\sigma_0,\theta_*$\\
$A=16/r$, $M_A$; $J_N$ & threshold for the retained coordinates and bound on their number; last retained index, \eqref{eq:coldf-bounded-head} & $r,c_*,C_*$; $J_N$ also on $N$ and the spectrum\\
$\psi,\chi,\eta$ & cutoff functions in \eqref{eq:coldf-eta} & $r$\\
$1/8$ & Young's inequality for the two gradient terms in the entropy estimate & numerical\\
$u$, or a compact set $K$ & offset of the time $t_u$ & fixed\\
$N\to\infty$ & determines $\chi_N=\sqrt{\log\log N}$, $T_{\mathrm e}=\chi_N/a_2$, $t_0$, $t_1$, $t_{\mathrm p}=(\log N)/100$, and $p_N$ in \eqref{eq:coldf-pN} & all of the above\\
$M_*\to\infty$, $r_0\downarrow0$ & last noise interval and restriction to $\{|Y_1(t_{\mathrm p})|\ge r_0\}$ in \cref{thm:coldf-finite-coordinates} & after $N\to\infty$\\
$\delta\downarrow0$ & loss in the weight used in \cref{prop:coldf-finite-head-tv}; $\delta$ is fixed before $N\to\infty$, and the smoothing scale of \cref{csh-thm:weak-to-tv} tends to zero before $\delta\downarrow0$ & after $N\to\infty$\\
$\eps\downarrow0$ & removal of the weight & last\\
\bottomrule
\end{tabular}
\caption{Parameters in the proof of \cref{thm:coldf-deterministic}, in the order in which they are chosen.}
\label{tab:coldf-parameters}
\end{table}

\subsection{Proof of the main result}\label{sec:coldf-goe-proof}

We now deduce \cref{cq-full-profile,cq-cutoff-clock} from \cref{thm:coldf-deterministic}. The spectral input consists of the eigenvalue gaps at fixed confidence of \cref{sp-fixed-confidence-geometry}, rigidity and convergence at the edge (\cref{sp-goe-rigidity}), and the following law of large numbers for the inverse gaps, which we deduce from them.

\begin{lemma}[A law of large numbers for the inverse gaps]\label{lem:coldf-inverse-gap-lln}
Let $\lambda_1\ge\cdots\ge\lambda_N$ be the eigenvalues of $W_N$. Then
\begin{equation}\label{eq:coldf-inverse-gap-lln}
\frac1N\sum_{i=2}^N\frac1{\lambda_1-\lambda_i}\longrightarrow1\quad\text{in probability}.
\end{equation}
\end{lemma}

The left side of \eqref{eq:coldf-inverse-gap-lln} is defined on the event $\{\lambda_1>\lambda_2\}$, and we set it equal to zero on its complement. Since the ordered eigenvalues are continuous functions of the entries of $W_N$, this defines a random variable. The complement has probability zero, because the eigenvalues of $W_N$ are almost surely distinct: the discriminant of the characteristic polynomial is a nonzero polynomial in the independent entries, whose zero set has Lebesgue measure zero, and the entries have a joint density. The limit $1$ equals $\lim_{E\downarrow2}\mathfrak m_{\mathrm{sc}}(E)$, with $\mathfrak m_{\mathrm{sc}}$ as in \eqref{eq:spec-stieltjes}.

\begin{proof}
We write $I_N$ for the left side of \eqref{eq:coldf-inverse-gap-lln}. Fix $\eta\in(0,1)$, and let $\mathcal G_N^{\eta/2}$ and $c_{\eta/2}$ be as in \cref{sp-fixed-confidence-geometry}. Then $\liminf_N\PP(\mathcal G_N^{\eta/2})\ge1-\eta/2$, and by \eqref{sp-indexed-raw-gaps}, on $\mathcal G_N^{\eta/2}$ and for large $N$,
\begin{equation}\label{eq:coldf-lln-lower}
\lambda_1-\lambda_i\ge c_{\eta/2}(i-1)^{2/3}N^{-2/3}\qquad(2\le i\le N).
\end{equation}
In particular $\lambda_1>\lambda_2$ there. Fix $\alpha\in(0,1/2)$ and let $m=\lfloor\alpha N\rfloor$, so that $m\ge2$ for large $N$. We split the sum defining $I_N$ at the index $m$.

\paragraph{The first indices.} By \eqref{eq:coldf-lln-lower}, on $\mathcal G_N^{\eta/2}$,
\begin{equation}\label{eq:coldf-lln-edge}
0\le\frac1N\sum_{i=2}^m\frac1{\lambda_1-\lambda_i}\le\frac{N^{-1/3}}{c_{\eta/2}}\sum_{k=1}^{m-1}k^{-2/3}\le\frac3{c_{\eta/2}}\Big(\frac mN\Big)^{1/3}\le\frac3{c_{\eta/2}}\,\alpha^{1/3}.
\end{equation}
In the second inequality, we compared the sum with $\int_0^{m-1}s^{-2/3}\,ds=3(m-1)^{1/3}$.

\paragraph{Comparison with the quantiles.} By \cref{sp-goe-rigidity}, choose $C=C_\eta$ and $N_0$ such that the event $F=F_C$ in \eqref{sgl-rigidity-input} has probability at least $1-\eta/2$ for $N\ge N_0$. On $F$, we have $|\lambda_i-\gamma_i|\le C N^{-5/12}$ for all $i$, and \cref{lem:spec-quantiles} with $i=1$ gives $|\lambda_1-2|\le C N^{-5/12}+7N^{-2/3}$. Then
\[
\max_{2\le i\le N}\big|(\lambda_1-\lambda_i)-(2-\gamma_i)\big|\le e_N,\qquad e_N=2C N^{-5/12}+7N^{-2/3}.
\]
For $i>m$, we have $i/N>\alpha$, and \cref{lem:spec-quantiles} gives $2-\gamma_i\ge d_\alpha$ with $d_\alpha=(3\pi\alpha/2)^{2/3}$. For large $N$, we have $e_N\le d_\alpha/2$, so that $\lambda_1-\lambda_i\ge d_\alpha/2$ for $i>m$. Writing the difference of the inverses over a common denominator, we obtain, on $F$ and for large $N$,
\begin{equation}\label{eq:coldf-lln-bulk}
\frac1N\sum_{i=m+1}^N\Big|\frac1{\lambda_1-\lambda_i}-\frac1{2-\gamma_i}\Big|\le\frac{2e_N}{d_\alpha^2}.
\end{equation}

\paragraph{The sum over the quantiles.} The map $x\mapsto\int_x^2\rho_{\mathrm{sc}}(y)\,dy$ is a continuous, strictly decreasing bijection from $[-2,2]$ onto $[0,1]$. Let $Q_{\mathrm{sc}}\colon[0,1]\to[-2,2]$ be its inverse, which is continuous, and let $\phi(s)=(2-Q_{\mathrm{sc}}(s))^{-1}$ for $0<s\le1$. Then $\gamma_i=Q_{\mathrm{sc}}(i/N)$. Since $\rho_{\mathrm{sc}}(x)\le\pi^{-1}\sqrt{2-x}$, we have $s\le2(2-Q_{\mathrm{sc}}(s))^{3/2}/(3\pi)$, that is,
\begin{equation}\label{eq:coldf-lln-quantile}
\phi(s)\le\Big(\frac2{3\pi s}\Big)^{2/3}\qquad(0<s\le1),
\end{equation}
which is the lower bound of \cref{lem:spec-quantiles} for a continuous index. The function $\phi$ is continuous on $[\alpha/2,1]$, and we let $\omega_\alpha$ be its modulus of continuity there. For large $N$, we have $m/N\ge\alpha/2$, and comparing each term $N^{-1}\phi(i/N)$ with $\int_{(i-1)/N}^{i/N}\phi$, we find
\[
\Big|\frac1N\sum_{i=m+1}^N\phi\Big(\frac iN\Big)-\int_{m/N}^1\phi(s)\,ds\Big|\le\omega_\alpha(1/N).
\]
Since $|m/N-\alpha|\le1/N$ and $\phi\le C_\alpha=(4/(3\pi\alpha))^{2/3}$ on $[\alpha/2,1]$ by \eqref{eq:coldf-lln-quantile}, we obtain
\begin{equation}\label{eq:coldf-lln-riemann}
\Big|\frac1N\sum_{i=m+1}^N\frac1{2-\gamma_i}-\int_\alpha^1\phi(s)\,ds\Big|\le\omega_\alpha(1/N)+\frac{C_\alpha}N .
\end{equation}
Substituting $x=Q_{\mathrm{sc}}(s)$, we find
\begin{equation}\label{eq:coldf-lln-integral}
\int_0^1\phi(s)\,ds=\int_{-2}^2\frac{\rho_{\mathrm{sc}}(x)}{2-x}\,dx=1 .
\end{equation}
For the last equality, we substitute $x=2\cos t$ with $0\le t\le\pi$, under which $\rho_{\mathrm{sc}}(x)\,dx/(2-x)$ becomes $\pi^{-1}(1+\cos t)\,dt$, up to orientation. By \eqref{eq:coldf-lln-quantile}, we have
\[
0\le1-\int_\alpha^1\phi(s)\,ds=\int_0^\alpha\phi(s)\,ds\le C_{\mathrm{sc}}\,\alpha^{1/3},\qquad C_{\mathrm{sc}}=3\Big(\frac2{3\pi}\Big)^{2/3}.
\]

\paragraph{Conclusion.} Combining \eqref{eq:coldf-lln-edge}, \eqref{eq:coldf-lln-bulk}, \eqref{eq:coldf-lln-riemann}, and the last display, we find that on $\mathcal G_N^{\eta/2}\cap F$, for large $N$,
\[
|I_N-1|\le\Big(\frac3{c_{\eta/2}}+C_{\mathrm{sc}}\Big)\alpha^{1/3}+\frac{2e_N}{d_\alpha^2}+\omega_\alpha(1/N)+\frac{C_\alpha}N .
\]
Let $\gamma>0$. We choose $\alpha$ such that the first term is less than $\gamma/2$. The other terms are deterministic and tend to zero as $N\to\infty$, and we obtain $\limsup_N\PP(|I_N-1|>\gamma)\le\limsup_N\PP((\mathcal G_N^{\eta/2}\cap F)^c)\le\eta$. As $\eta\in(0,1)$ is arbitrary, this proves \eqref{eq:coldf-inverse-gap-lln}.
\end{proof}

In the proof, the confidence $\eta$ is fixed before the index $m=\lfloor\alpha N\rfloor$, because the first indices are controlled by the lower bounds \eqref{eq:coldf-lln-lower} at fixed confidence; the convergence of the empirical eigenvalue distribution alone would not control them. The constants $c_\eta$ need not be bounded below as $\eta\to0$, and the proof gives no rate of convergence.

\begin{lemma}[Hypotheses for the GOE]\label{lem:coldf-goe}
Fix $c>1$, let $a_i=c(\lambda_1-\lambda_i)/2$, and let $\theta=1-1/c$. Then $\theta_N\to\theta$ in probability, and $N^{2/3}a_2$ and $(N^{2/3}a_2)^{-1}$ are tight. Further, for every $\eta\in(0,1)$ there exist constants $\theta_*\in(0,\theta)$, $B,c_*,C_*,\varpi>0$, and $D_3\ge1$, depending only on $c$ and $\eta$, and events $\mathcal G_N^\eta$ with $\liminf_N\PP(\mathcal G_N^\eta)\ge1-\eta$, such that for all sufficiently large $N$, on $\mathcal G_N^\eta$, the spectrum has distinct eigenvalues and satisfies $\mathrm H(\theta_*,B,c_*,C_*)$ and $\mathrm H_3(\varpi,D_3)$.
\end{lemma}

\begin{proof}
Since $2a_i=c(\lambda_1-\lambda_i)$, we have $\theta_N=1-c^{-1}N^{-1}\sum_{i\ge2}(\lambda_1-\lambda_i)^{-1}$, and \cref{lem:coldf-inverse-gap-lln} implies that $\theta_N\to1-1/c=\theta$ in probability. By the second part of \cref{sp-goe-rigidity}, $N^{2/3}a_2=c\Delta_{2,N}/2\Rightarrow c\Delta_2/2$, where $\Delta_2$ is almost surely positive and finite. This implies that $N^{2/3}a_2$ and its reciprocal are tight; for the reciprocal, we use the portmanteau theorem \cite[Section~2]{Billingsley1999} for the closed sets $[0,1/M]$, $M>0$. In particular, $R_N=N^{1/3}\cdot N^{2/3}a_2\to\infty$ and $\theta_NR_N\to\infty$ in probability, so that the events on which \eqref{eq:coldf-centers} uses the value $1$ have probabilities tending to zero.

Now fix $\eta\in(0,1)$. Let $\mathcal G_N'$ be the events of \cref{sp-fixed-confidence-geometry} with $\eta/3$ in place of $\eta$, and let $c_{\eta/3},C_{\eta/3}$ be the corresponding constants in \eqref{sp-indexed-raw-gaps}. We set
\[
c_*=\frac c2\,c_{\eta/3},\qquad B=C_*=\frac c2\,C_{\eta/3},\qquad\theta_*=\frac\theta2,\qquad D_3=2^{2/3}\frac{C_*}{c_*}.
\]
On $\mathcal G_N'$, for large $N$, we have $c_*(i-1)^{2/3}N^{-2/3}\le a_i\le C_*(i-1)^{2/3}N^{-2/3}$ for $2\le i\le N$. In particular, $a_N\le C_*=B$, $a_2\le C_*N^{-2/3}$, and $a_3/a_2\le D_3$, where $D_3\ge1$ because $C_*\ge c_*$. Further, $a_3/a_2=\Delta_{3,N}/\Delta_{2,N}$ converges in distribution to $\Delta_3/\Delta_2$, by the second part of \cref{sp-goe-rigidity} and the continuous mapping theorem \cite[Section~2]{Billingsley1999}, and $\Delta_3/\Delta_2>1$ almost surely, because the reflected GOE Airy point process is simple. We choose $\varpi>0$ such that $\PP(\Delta_3/\Delta_2\le1+\varpi)<\eta/3$, which is possible because these probabilities decrease to $\PP(\Delta_3/\Delta_2\le1)=0$ as $\varpi\downarrow0$. The portmanteau theorem for the closed set $(-\infty,1+\varpi]$ gives $\limsup_N\PP(a_3/a_2\le1+\varpi)<\eta/3$. Let $\mathcal G_N^\eta$ be the intersection of $\mathcal G_N'$, the events $\{a_3/a_2>1+\varpi\}$ and $\{\theta_N\ge\theta/2\}$, and the event that the eigenvalues are distinct. Since $\PP(\theta_N<\theta/2)\to0$ and the eigenvalues are almost surely distinct, we have $\liminf_N\PP(\mathcal G_N^\eta)\ge1-2\eta/3\ge1-\eta$. These events are measurable, since they are defined by inequalities between the ordered eigenvalues. For large $N$, on $\mathcal G_N^\eta$, the spectrum has distinct eigenvalues and satisfies $\mathrm H(\theta_*,B,c_*,C_*)$ and both inequalities of $\mathrm H_3(\varpi,D_3)$, with constants depending only on $c$ and $\eta$.
\end{proof}

\begin{proof}[Proof of \cref{cq-full-profile}]
Fix $u\in\R$ and $\gamma>0$. We first show that $D_N(t_u)\to\cV_\theta(u)$ in probability. Because $\theta_N\to\theta$ in probability, there exists a deterministic sequence $\zeta_N\downarrow0$ such that $\PP(|\theta_N-\theta|>\zeta_N)\to0$. Fix $\eta\in(0,1)$ and let $E_N=\mathcal G^\eta_N\cap\{|\theta_N-\theta|\le\zeta_N\}$. We claim that $\PP(E_N\cap\{|D_N(t_u)-\cV_\theta(u)|>\gamma\})\to0$. Suppose that this fails. Then there exist $\alpha>0$ and a subsequence $(N_k)$ along which this probability is at least $\alpha$; in particular, the event is nonempty. Recall that $D_N(t)$ depends on $W_N$ only through its eigenvalues (\cref{sec:model-model}). We may therefore choose, for each $k$, a spectrum in this event. The resulting deterministic sequence of spectra, indexed by $N_k$, satisfies $\mathrm H(\theta_*,B,c_*,C_*)$ and $\mathrm H_3(\varpi,D_3)$ with $\theta_{N_k}\to\theta$, and $|D_{N_k}(t_u)-\cV_\theta(u)|>\gamma$ for all $k$. This contradicts the first part of \cref{thm:coldf-deterministic}, whose proof applies verbatim along any subsequence of dimensions. We conclude from \cref{lem:model-localization} that $D_N(t_u)\to\cV_\theta(u)$ in probability.

Next, $D_N$ is nonincreasing (\cref{lem:model-tv}), $D_N(t)\to0$ as $t\to\infty$ (see the proof of \cref{cor:model-profile-cutoff}), and $a_2T_N''=\log R_N/4$ on $\{R_N>1\}$, which tends to infinity in probability by \cref{lem:coldf-goe}. Recall from \cref{cot-profile} that $\cV_\theta$ is continuous and strictly decreasing from $1$ to $0$. Using \cref{hp-monotone-profile-transfer} with $b_N=T_N''$, $w_N=a_2^{-1}$, and $V=\cV_\theta$, we obtain \eqref{eq:coldf-main} and the convergence of $a_2(t_N(\eps)-T_N'')$, uniformly for $\eps\in I$.

For the center $T_N$, we work on the event $\{R_N>1,\ \theta_N>0,\ 2\theta_NR_N>1\}$, whose probability tends to one by \cref{lem:coldf-goe}. On this event, $T_N+u/a_2=t_{u+\ell_N}$ with $\ell_N=\log(2\theta_N)/4$, and $\ell_N\to\log(2\theta)/4$ in probability. By \eqref{eq:coldf-main} applied on a compact neighborhood of $u+\log(2\theta)/4$ and the continuity of $\cV_\theta$, we have $D_N(T_N+u/a_2)\to\cV_\theta(u+\log(2\theta)/4)=\cV(u)$ in probability, where we used the third part of \cref{cot-profile}. On the same event, $a_2T_N=\log(2\theta_NR_N)/4$, which tends to infinity in probability, so that $a_2^{-1}/T_N\to0$. Then \cref{hp-monotone-profile-transfer} with $b_N=T_N$ and $V=\cV$ implies that $D_N$ has cutoff profile $\cV$ with center $T_N$ and window $a_2^{-1}$, together with the convergence of $a_2(t_N(\eps)-T_N)$.

For the last claim, $a_2t_N(\eps)=\log R_N/4+\cV_\theta^{-1}(\eps)+o_{\PP}(1)$ and $\log R_N=(\log N)/3+\log(N^{2/3}a_2)=(\log N)/3+O_{\PP}(1)$ by \cref{lem:coldf-goe}, so $a_2t_N(\eps)/\log N\to1/12$ in probability. Finally, each of the conclusions above is a convergence in probability of a function of $W_N$. For finitely many temperatures, each of them holds separately for the same matrices $W_N$, and they therefore hold simultaneously.
\end{proof}

\begin{proof}[Proof of \cref{cq-cutoff-clock}]
Fix $\eta\in(0,1)$ and $A>0$. As in the proof of \cref{cq-full-profile}, $a_2T_N''\to\infty$ in probability. Then $(1-\eta)T_N''\le t_{-A}$ with probability tending to one, and on this event $D_N((1-\eta)T_N'')\ge D_N(t_{-A})$, because $D_N$ is nonincreasing. Then \eqref{eq:coldf-main} implies that $D_N(t_{-A})\to\cV_\theta(-A)$ in probability, and $\cV_\theta(-A)\to1$ as $A\to\infty$. This proves the first limit, and the second is proved in the same way with $t_A$ (this is the argument of the third part of \cref{hp-monotone-profile-transfer}). The same bracketing with $t_A$ proves the claim for $\omega_N\to\infty$. The statement $t_N(\eps)=T_N''+O_{\PP}(a_2^{-1})$ is part of \cref{cq-full-profile}, and $\log(a_2^{-1})/\log N\to2/3$ because $N^{2/3}a_2$ and its reciprocal are tight (\cref{lem:coldf-goe}).
\end{proof}
\section{The Gibbs measure in graph coordinates}\label{sec:coldf-stationary}

In this section, we compare the Gibbs measure with a Gaussian product law. The comparison is uniform in the top fraction $\theta_N$, which may tend to zero, so that it also applies in \cref{sec:cold-meso}. We use only the spectral quantities, not the dynamics.

\paragraph{Graph coordinates.}
Let $Q_{\{1\}}(x)=(|x_1|,x_2,\dots,x_N)$ be the map that folds the first eigencoordinate (see \cref{sec:model-model}). We identify the closed upper hemisphere $\{x\in S_N:x_1\ge0\}$ with the closed ball $\{y\in\R^{N-1}:|y|^2\le N\}$ through
\[
y=(x_2,\dots,x_N),\qquad x_1=\sqrt{N-|y|^2}.
\]
We call $y$ the graph coordinates and write $y=(y_2,\dots,y_N)$, so that $y_j=x_j$. For a probability measure $\mu$ on $S_N$, we write $\mu^+$ for the image of $\mu$ under $x\mapsto(x_2,\dots,x_N)$; when $\mu$ is invariant under the reflection $R_1$ of the first eigencoordinate, this is the image of $\mu$ under $Q_{\{1\}}$ in graph coordinates, and by the second part of \cref{lem:model-symmetry} we have
\begin{equation}\label{eq:coldf-fold}
\|\mu-\mu'\|_{\TV}=\|\mu^+-\mu'^+\|_{\TV}
\end{equation}
for any two $R_1$-invariant probability measures $\mu,\mu'$ on $S_N$; see also \eqref{eq:score-fold-tv}. On the open upper hemisphere, the surface measure of $S_N$ is $\sqrt N\,x_1^{-1}\,dy$. In particular, $\nu_N^+$ has a Lebesgue density proportional to $(1-|y|^2/N)^{-1/2}$ on $\{|y|^2<N\}$.

Let $G$ be the centered Gaussian measure on $\R^{N-1}$ with independent coordinates of variances $(2a_j)^{-1}$, $2\le j\le N$, that is, $G(dy)=\prod_{j=2}^N\sqrt{a_j/\pi}\,e^{-a_jy_j^2}\,dy_j$. Write
\[
J(y)=\begin{cases}
\big(1-|y|^2/N\big)^{-1/2},&|y|^2<N,\\
0,&|y|^2\ge N,
\end{cases}
\qquad Z_G=\E_G[J].
\]
Recall that $\pi\propto e^{-F}\nu_N$ and $F(x)=\sum_{j\ge2}a_jy_j^2$. It follows that the folded Gibbs measure $\pi^+$ satisfies
\begin{equation}\label{mesocold-eq:stationary-graph}
\frac{d\pi^+}{dG}(y)=\frac{J(y)}{Z_G}.
\end{equation}
This identity is exact; the factor $J$ is the Jacobian of the graph parametrization. Under $G$, the squared radius $S=|y|^2$ has mean $\E_GS=\sum_{j\ge2}(2a_j)^{-1}=(1-\theta_N)N$. When $\theta_N>0$, a typical sample of $G$ lies inside the ball, and the remaining squared radius, about $\theta_NN$, corresponds to the top coordinate. We set
\[
m_N=\theta_NNa_2=\theta_NR_N.
\]

\begin{lemma}[Sums of Gaussian squares]\label{lem:coldf-gaussian-squares}
Assume $a_2^2\sum_{j\ge2}a_j^{-2}\le C_0$, and let $S=|y|^2$ under $G$. Then the following hold.
\begin{enumerate}
\item We have $\E_GS=(1-\theta_N)N$ and $\Var_G(S)=\sum_{j\ge2}a_j^{-2}/2\le C_0/(2a_2^2)$.
\item For $|\lambda|\le a_2/4$, we have
\begin{equation}\label{mesocold-eq:centered-mgf}
\log\E_G\,e^{\lambda(S-\E_GS)}\le\frac{\lambda^2}2\sum_{j\ge2}a_j^{-2}\le\frac{C_0}{32}.
\end{equation}
In particular, $G(|S-\E_GS|>x)\le2e^{C_0/32}e^{-a_2x/4}$ for every $x\ge0$.
\item Let $0\le\lambda\le a_2/4$, and let $G_\lambda(dy)=e^{\lambda|y|^2}G(dy)/\E_Ge^{\lambda S}$. Under $G_\lambda$, the coordinates are independent centered Gaussian variables with variances $(2(a_j-\lambda))^{-1}$, and $S$ has a density bounded by $\sqrt{(a_2-\lambda)(a_3-\lambda)}\le\sqrt{a_2a_3}$.
\end{enumerate}
\end{lemma}

\begin{proof}
We write $S=\sum_{j\ge2}(2a_j)^{-1}g_j^2$ with independent standard Gaussian variables $g_j$. The first part follows from $\E g^2=1$ and $\Var(g^2)=2$. For the second, the identity $\E e^{sg^2}=(1-2s)^{-1/2}$ for $s<1/2$ gives
\[
\log\E_Ge^{\lambda(S-\E_GS)}=\frac12\sum_{j\ge2}\big[-\log(1-x_j)-x_j\big],\qquad x_j=\frac{\lambda}{a_j}\in[-\tfrac14,\tfrac14].
\]
For $|x|\le1/4$, we have $-\log(1-x)-x=\sum_{n\ge2}x^n/n\le(x^2/2)\sum_{n\ge0}|x|^n\le x^2$. We obtain \eqref{mesocold-eq:centered-mgf}, where in the last step we used $\lambda^2\le a_2^2/16$. The tail bound follows from Markov's inequality applied to $e^{\pm a_2(S-\E_GS)/4}$.

For the third part, the density of $G_\lambda$ is proportional to $\prod_j e^{-(a_j-\lambda)y_j^2}$, and $a_j-\lambda\ge3a_j/4>0$. If $y_j$ has density $\sqrt{p/\pi}\,e^{-py^2}$, then $y_j^2$ has density $\sqrt{p/\pi}\,s^{-1/2}e^{-ps}$ on $(0,\infty)$. With $p_j=a_j-\lambda$, we find that the density of $y_2^2+y_3^2$ at $v>0$ is
\[
\frac{\sqrt{p_2p_3}}{\pi}\int_0^v\frac{e^{-p_2w-p_3(v-w)}}{\sqrt{w(v-w)}}\,dw\le\frac{\sqrt{p_2p_3}}{\pi}\int_0^v\frac{dw}{\sqrt{w(v-w)}}=\sqrt{p_2p_3}.
\]
The density of $S$ is the convolution of this density with the law of the independent variable $\sum_{j\ge4}y_j^2$, and convolution with a probability measure does not increase the supremum of a density.
\end{proof}

\begin{proposition}[The Gibbs measure in graph coordinates]\label{mesocold-prop:stationary-graph}
Let $B,D,C_0>0$, and assume that $\theta_N>0$ and
\begin{equation}\label{eq:coldf-stationary-hyp}
a_N\le B,\qquad a_3\le Da_2,\qquad a_2^2\sum_{j=2}^Na_j^{-2}\le C_0.
\end{equation}
There exist constants $c_0,C'>0$, depending only on $B$, $D$, and $C_0$, such that
\begin{equation}\label{eq:coldf-stationary-bound}
\big|\sqrt{\theta_N}\,Z_G-1\big|+\|\pi^+-G\|_{\TV}\le\frac{C'}{m_N}+C'e^{-c_0m_N}.
\end{equation}
Further, consider a sequence of spectra satisfying \eqref{eq:coldf-stationary-hyp} with fixed constants and $m_N\to\infty$. Then $x_1^2/(\theta_NN)\to1$ in probability under $\pi$, and for every fixed $k\ge2$ the law of $(\sqrt{2a_j}\,x_j)_{j=2}^k$ under $\pi$ converges in total variation to the standard Gaussian law on $\R^{k-1}$.
\end{proposition}

The estimate \eqref{eq:coldf-stationary-bound} is uniform in $\theta_N$: the constants do not depend on a lower bound for the top fraction. The concentration of $S$ alone does not control the singular factor $J$ near the sphere $|y|^2=N$. For this part, we use the bound on the density of $S$ under an exponential tilt from \cref{lem:coldf-gaussian-squares}.

\begin{proof}
Let $W=\sqrt{\theta_N}\,J$, so that $\sqrt{\theta_N}\,Z_G=\E_GW$. On $\{S<N\}$, we have $1-S/N=\theta_N-(S-\E_GS)/N$, so that
\[
W=\Big(1-\frac{S-\E_GS}{\theta_NN}\Big)^{-1/2}.
\]
We bound $\E_G|W-1|$ separately on three events. Here and below, $C$ and $c_0$ denote positive constants depending only on $B$, $D$, and $C_0$.

\paragraph{The central event.} On $\{|S-\E_GS|\le\theta_NN/2\}$, the derivative of $s\mapsto(1-s/(\theta_NN))^{-1/2}$ at $s=S-\E_GS$ is at most $(\theta_NN)^{-1}2^{-1}2^{3/2}\le2/(\theta_NN)$. As $W=1$ at $S=\E_GS$, the first part of \cref{lem:coldf-gaussian-squares} implies that
\[
\E_G\big[|W-1|;\,|S-\E_GS|\le\theta_NN/2\big]\le\frac{2}{\theta_NN}\,\E_G|S-\E_GS|\le\frac{2}{\theta_NN}\Big(\frac{C_0}{2a_2^2}\Big)^{1/2}\le\frac{C}{m_N}.
\]

\paragraph{The lower complement.} On $\{S<\E_GS-\theta_NN/2\}$, we have $1-S/N>3\theta_N/2$, which gives $W\le1$ and $|W-1|\le1$. The probability of this event is at most $Ce^{-m_N/8}$, by the second part of \cref{lem:coldf-gaussian-squares} with $x=\theta_NN/2$.

\paragraph{The upper complement.} Let $\ell=\theta_NN/2$, so that $\E_GS+\ell=N-\ell$. On $\{S\ge N\}$ we have $W=0$, and on $\{N-\ell<S<N\}$ we have $|W-1|\le W+1$. The probability of $\{S>N-\ell\}$ is at most $Ce^{-m_N/8}$, as before. It remains to bound $\E_G[W;\,N-\ell<S<N]$. Let $\lambda=a_2/4$ and let $G_\lambda$ be the tilted measure of \cref{lem:coldf-gaussian-squares}. For a nonnegative function $\phi$,
\[
\E_G[\phi(S)]=\E_G\big[e^{\lambda(S-\E_GS)}\big]\,\E_{G_\lambda}\big[\phi(S)e^{-\lambda(S-\E_GS)}\big]\le e^{C_0/32}\,\E_{G_\lambda}\big[\phi(S)e^{-\lambda(S-\E_GS)}\big].
\]
We use this identity with $\phi(S)=W\ind_{\{N-\ell<S<N\}}$, and we write $S=N-r$ with $0<r<\ell$. Then $S-\E_GS=2\ell-r$ and $W=\sqrt{\theta_NN}\,r^{-1/2}$. The density of $S$ under $G_\lambda$ is at most $\sqrt{a_2a_3}\le\sqrt Da_2$, and we find
\[
\E_G[W;\,N-\ell<S<N]\le e^{C_0/32}\sqrt D\,a_2\sqrt{\theta_NN}\,e^{-\lambda\ell}\int_0^\ell e^{-\lambda(\ell-r)}r^{-1/2}\,dr.
\]
On $[0,\ell/2]$ the integrand is at most $e^{-\lambda\ell/2}r^{-1/2}$, and on $[\ell/2,\ell]$ it is at most $(\ell/2)^{-1/2}e^{-\lambda(\ell-r)}$. The integral is then at most $2\sqrt{\ell}\,e^{-\lambda\ell/2}+\sqrt2\,\ell^{-1/2}\lambda^{-1}$. Using $\lambda\ell=m_N/8$, $a_2\sqrt{\theta_NN}\sqrt\ell=m_N/\sqrt2$, and $a_2\sqrt{\theta_NN}\,\ell^{-1/2}\lambda^{-1}=4\sqrt2$, we obtain
\[
\E_G[W;\,N-\ell<S<N]\le C\big(m_Ne^{-3m_N/16}+e^{-m_N/8}\big)\le Ce^{-c_0m_N}.
\]

\paragraph{Conclusion.} Combining the three bounds, we find
\begin{equation}\label{eq:coldf-W}
\E_G|W-1|\le\frac{C}{m_N}+Ce^{-c_0m_N}.
\end{equation}
This bounds $|\sqrt{\theta_N}Z_G-1|=|\E_G(W-1)|$. For the total-variation distance, by \eqref{mesocold-eq:stationary-graph} and the first part of \cref{lem:model-tv}, we have
\[
\|\pi^+-G\|_{\TV}=\frac12\E_G\Big|\frac{W}{\E_GW}-1\Big|\le\frac{\E_G|W-1|+|\E_GW-1|}{2\,\E_GW}\le\frac{\E_G|W-1|}{\E_GW}.
\]
If $\E_G|W-1|\le1/2$, then $\E_GW\ge1/2$, and the right side is at most $2\E_G|W-1|$. Otherwise, the total-variation distance is at most $1\le2\E_G|W-1|$. In both cases, \eqref{eq:coldf-W} gives \eqref{eq:coldf-stationary-bound} after enlarging $C'$.

\paragraph{The limits as $m_N\to\infty$.} Under $\pi$, we have $x_1^2=N-|y|^2$. Under $G$, the first part of \cref{lem:coldf-gaussian-squares} gives
\[
\E_G\Big|\frac{N-S}{\theta_NN}-1\Big|^2=\frac{\Var_G(S)}{\theta_N^2N^2}\le\frac{C_0}{2m_N^2}\to0.
\]
Using $\|\pi^+-G\|_{\TV}\to0$, we find that the event $\{|x_1^2/(\theta_NN)-1|>\gamma\}$ has $\pi$-probability tending to zero for every $\gamma>0$. The image of $G$ under $y\mapsto(\sqrt{2a_j}\,y_j)_{j=2}^k$ is the standard Gaussian law, and the image of $\pi$ is the image of $\pi^+$, since the map does not involve $x_1$. The second part of \cref{lem:model-tv} implies that the total-variation distance between the two images is at most $\|\pi^+-G\|_{\TV}\to0$.

\end{proof}

The following consequence makes precise the statement that the Gibbs measure places the fraction $\theta_N$ of $|x|^2$ on the top eigencoordinate.

\begin{corollary}[The top fraction under the Gibbs measure]\label{cor:coldf-top-fraction}
Under the hypotheses of \cref{mesocold-prop:stationary-graph}, for every $\gamma>0$,
\begin{equation}\label{eq:coldf-top-fraction}
\pi\Big(\Big|\frac{x_1^2}{\theta_NN}-1\Big|>\gamma\Big)\le\frac{C'}{m_N}+C'e^{-c_0m_N}+\frac{C_0}{2\gamma^2m_N^2}.
\end{equation}
For the GOE model, in either case (F) or case (M) of \cref{sp-condensed-geometry}, the left side of \eqref{eq:coldf-top-fraction} with $\pi=\pi_{N,c_N}$ tends to zero in probability for every $\gamma>0$. In particular, at a fixed temperature $c>1$, we have $\pi_{N,c}(|x_1^2/N-(1-1/c)|>\gamma)\to0$ in probability for every $\gamma>0$.
\end{corollary}

\begin{proof}
Recall that $x_1^2=N-|y|^2$ and $\E_G|y|^2=(1-\theta_N)N$. By Chebyshev's inequality and \cref{lem:coldf-gaussian-squares}, we have $G(|N-|y|^2-\theta_NN|>\gamma\theta_NN)\le\Var_G(|y|^2)/(\gamma\theta_NN)^2\le C_0/(2\gamma^2m_N^2)$. The event in \eqref{eq:coldf-top-fraction} is invariant under $R_1$, and its $\pi$-probability equals its $\pi^+$-probability. Then \eqref{eq:coldf-stationary-bound} implies \eqref{eq:coldf-top-fraction}. For the GOE model, $m_N\to\infty$ in probability by the first part of \cref{sp-condensed-geometry}, and on the events of its second part (intersected with $\{\theta_N>0\}$ in case (M)) the hypotheses \eqref{eq:coldf-stationary-hyp} hold with $D=2^{2/3}C_*/c_*$ and $C_0=4(C_*/c_*)^2$, by \eqref{eq:spec-cold-consequences}. The right side of \eqref{eq:coldf-top-fraction} then tends to zero in probability, by \cref{lem:model-localization}. In case (F), we also use $\theta_N\to1-1/c$ in probability.
\end{proof}

The statement and proof of \cref{mesocold-prop:stationary-graph} use only \eqref{eq:coldf-stationary-hyp}. Under $\mathrm H(\theta_*,B,c_*,C_*)$ and $\mathrm H_3(\varpi,D_3)$, these hold with $D=D_3$ and $C_0$ from \cref{lem:coldf-consequences}, and $m_N\ge\theta_*c_*N^{1/3}$. The following static bounds on a cap give an upper bound on the density of the Gibbs measure in graph coordinates away from the sphere $\{|y|^2=N\}$, which we use in \cref{sec:coldf-selected}.

\begin{lemma}[The Gibbs measure on a cap]\label{lem:coldf-cap-gibbs}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$, and let $0<r\le\theta_*/2$. Let $\mathcal C_r=\{x\in S_N:x_1\ge\sqrt{rN}\}$, let $D_r=\{y:|y|^2\le(1-r)N\}$ be its image in graph coordinates, and let $\pi_{\mathcal C_r}$ be the probability measure $\pi$ conditioned on $\mathcal C_r$. Then, for $N$ larger than a constant depending only on the spectral constants and $r$,
\begin{equation}\label{eq:coldf-cap-gibbs}
G(D_r^c)\le e^{C_0/32-m_N/8},\qquad
\frac{d\pi_{\mathcal C_r}}{dG}=\frac{J\ind_{D_r}}{\E_G[J\ind_{D_r}]}\le2r^{-1/2}.
\end{equation}
Further, $Z_G\ge\theta_N\ge\theta_*$, and $d\pi^+/dG\le r^{-1/2}\theta_*^{-1}$ on $D_r$.
\end{lemma}

\begin{proof}
Since $r\le\theta_N/2$, we have $D_r^c\subset\{S-\E_GS>\theta_NN/2\}$, and the first bound follows from \eqref{mesocold-eq:centered-mgf} and Markov's inequality applied to $e^{a_2(S-\E_GS)/4}$. The formula for the density follows from \eqref{mesocold-eq:stationary-graph}. On $D_r$, we have $1\le J\le r^{-1/2}$. This gives $\E_G[J\ind_{D_r}]\ge G(D_r)\ge1/2$ for large $N$, and the bound on the density follows. We also have $G(S\ge N)\le\E_GS/N=1-\theta_N$ by Markov's inequality, which implies $Z_G\ge G(S<N)\ge\theta_N$. The last claim follows from \eqref{mesocold-eq:stationary-graph}.
\end{proof}
\section{The unnormalized process and its mean}\label{sec:coldf-process}

From now on, we assume $\mathrm H(\theta_*,B,c_*,C_*)$ and $N\ge3$. Unless said otherwise, constants denoted $C$ and $c_0$ depend only on $\theta_*$, $B$, $c_*$, and $C_*$, and they may change from line to line.

\paragraph{The unnormalized process.}
We use \cref{cq-ce-source-valid-unshifted-matrix-lift} with the quadratic potential coefficients $\rho_i=a_i$. Let $\mathcal B=(\mathcal B_{ij})_{i,j=1}^N$ be a matrix of independent standard Brownian motions, independent of $X_0\sim\nu_N$, and let $\Phi$ solve the linear matrix equation
\begin{equation}\label{eq:coldf-matrix-flow}
d\Phi=-2D_a\Phi\,dt+\sqrt{2/N}\,d\mathcal B\,\Phi,\qquad \Phi(0)=I,\qquad D_a=\diag(a_1,\dots,a_N).
\end{equation}
By the fifth part of \cref{cq-ce-source-valid-unshifted-matrix-lift}, $\Phi(t)$ is invertible for all $t\ge0$, and $Y=\Phi X_0$ is the unnormalized process with initial value $X_0$, with respect to the filtration $(\mathcal F_t)$ generated by $X_0$ and $\mathcal B$. That is, there exists a standard Brownian motion $B=(B_1,\dots,B_N)$ for this filtration such that $Y$ solves \eqref{eq:coldf-Y}. The first two parts of that lemma imply that $Y$ never vanishes, and $X=\sqrt N\,Y/|Y|$ is the diffusion generated by $L$, started from $\nu_N$; it solves
\begin{equation}\label{eq:coldf-X}
dX_j=\sqrt2\,\big(dB_j-X_jN^{-1}X\cdot dB\big)+\Big[-2a_j+\frac{2F(X)}N-\frac{N-1}N\Big]X_j\,dt.
\end{equation}
In particular, the law of $X_t$ is $\nu_NP_t$. We also know from the first part of the same lemma that $Y$ is a strong solution, and $Y$ and $X$ are adapted to the filtration generated by $X_0$ and $B$. Note that $Y_1$ is a martingale, since $a_1=0$. We write
\[
\mathsf v(t)=\E|Y(t)|^2,\qquad h_i(t)=\E Y_i(t)^2 .
\]
The process $Y$ has moments of all orders at every fixed time, since $|Y(t)|\le\|\Phi(t)\|\sqrt N$ and $\Phi$ solves a linear equation.

\paragraph{The renewal kernel.}
Let
\[
k(t)=\frac2N\sum_{j=2}^Ne^{-4a_jt},\qquad
\Upsilon=\sum_{n\ge1}k^{*n},\qquad
H(t)=1+\int_0^t\Upsilon(s)\,ds,
\]
where $k^{*n}$ is the $n$-fold convolution of $k$ on $[0,\infty)$. Note that $\int_0^\infty k=(2/N)\sum_{j\ge2}(4a_j)^{-1}=1-\theta_N\le1-\theta_*$. In the terminology of renewal theory, $k$ is a defective renewal kernel.

\begin{lemma}[Renewal estimates]\label{lem:coldf-renewal}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$. There exists $C>0$, depending only on $\theta_*$ and $c_*$, such that the following hold.
\begin{enumerate}
\item For $t\ge0$, we have $k(t)\le C(1+t)^{-3/2}$, $\Upsilon(t)\le C(1+t)^{-3/2}$, $\int_t^\infty\Upsilon\le C(1+t)^{-1/2}$, $\int_0^\infty\Upsilon=(1-\theta_N)/\theta_N$, and $H(t)\le1/\theta_N\le1/\theta_*$.
\item We have the identity
\begin{equation}\label{eq:coldf-renewal}
\mathsf v=H+\frac N2\,\Upsilon+\frac2N\,H*\mathsf v,
\end{equation}
and, for $0\le t\le N$,
\begin{equation}\label{eq:coldf-v}
\mathsf v(t)\le Ce^{2t/(\theta_*N)}\big[1+N(1+t)^{-3/2}\big],\qquad \int_0^t\mathsf v(s)\,ds\le C(t+N).
\end{equation}
\item For $1\le t\le N$, we have $|h_1(t)-\theta_N^{-1}|\le C[(1+t)^{-1/2}+t/N]$. For every $i$ and $t\ge0$, we have $0\le h_1(t)-h_i(t)\le4a_ith_1(t)$.
\item The first column $\Phi e_1$ satisfies $\E|\Phi(t)e_1|^2\le\theta_*^{-1}e^{2t/(\theta_*N)}$, and $\E|\Phi_{11}(t)-1|^2\le Ct/N$ for $0\le t\le N$.
\end{enumerate}
\end{lemma}

\begin{proof}
\emph{First part.} By the third part of \cref{lem:coldf-consequences}, $k(t)\le2C_1t^{-3/2}$, and $k(t)\le2$ for all $t$. Together, these bounds imply $k(t)\le C(1+t)^{-3/2}$. For $n\ge1$, write
\[
k^{*n}(t)=\int_{\{s\in[0,\infty)^n:\,s_1+\cdots+s_n=t\}}\prod_{i=1}^nk(s_i),
\]
where the integral is with respect to the Lebesgue measure in the variables $(s_1,\dots,s_{n-1})$, with $s_n=t-s_1-\cdots-s_{n-1}$. On this simplex at least one coordinate is at least $t/n$. It follows that $k^{*n}(t)$ is at most the sum over $i$ of the integrals over $\{s_i\ge t/n\}$. The integrand is symmetric under permutations of the coordinates, and each of these integrals equals the one with $i=n$. In that integral we bound $k(s_n)\le C(1+t/n)^{-3/2}$ and integrate $s_1,\dots,s_{n-1}$ over $[0,\infty)$. We obtain
\[
k^{*n}(t)\le nC\Big(1+\frac tn\Big)^{-3/2}(1-\theta_N)^{n-1}\le Cn^{5/2}(1+t)^{-3/2}(1-\theta_*)^{n-1},
\]
where we used $1+t/n\ge(1+t)/n$. Summing over $n$, we obtain the bound on $\Upsilon$, and integrating it, we find the bound on $\int_t^\infty\Upsilon$. As $\int k^{*n}=(1-\theta_N)^n$, we have $\int\Upsilon=(1-\theta_N)/\theta_N$, which implies $H(t)\le1+(1-\theta_N)/\theta_N=1/\theta_N$.

\emph{Second part.} Using Itô's formula and \eqref{eq:coldf-Y}, we have $d(Y_j^2)=-4a_jY_j^2\,dt+(2/N)|Y|^2\,dt+dM_j$ with a martingale $M_j$. Taking expectations, and using $\E X_0X_0^\T=I$, we find
\begin{equation}\label{eq:coldf-hi}
h_j(t)=e^{-4a_jt}+\frac2N\int_0^te^{-4a_j(t-s)}\mathsf v(s)\,ds .
\end{equation}
Summing over $j$ and separating $j=1$, for which $a_1=0$, we obtain the renewal equation
\[
\mathsf v=1+\frac N2k+k*\mathsf v+\frac2N\,1*\mathsf v,
\]
where $1*\mathsf v(t)=\int_0^t\mathsf v$. The measure $\delta_0+\Upsilon$ is the convolution inverse of $\delta_0-k$, since $\Upsilon=k+k*\Upsilon$. Convolving the renewal equation with $\delta_0+\Upsilon$, and using $(\delta_0+\Upsilon)*1=H$ and $(\delta_0+\Upsilon)*k=\Upsilon$, we obtain \eqref{eq:coldf-renewal}. The convolutions are well defined, because all terms are nonnegative.

The first part shows that $g=H+(N/2)\Upsilon$ satisfies $g(t)\le C[1+N(1+t)^{-3/2}]$ and $\int_0^tg\le C(t+N)$. Using $H\le1/\theta_*$ in \eqref{eq:coldf-renewal}, we find $\mathsf v(t)\le g(t)+\alpha\int_0^t\mathsf v$ with $\alpha=2/(\theta_*N)$. By Gronwall's inequality,
\[
\mathsf v(t)\le g(t)+\alpha\int_0^te^{\alpha(t-s)}g(s)\,ds\le g(t)+\alpha e^{\alpha t}C(t+N),
\]
and for $t\le N$ we have $\alpha(t+N)\le4/\theta_*$. This proves the first bound in \eqref{eq:coldf-v}. Integrating the same inequality over $[0,t]$ and using $\alpha te^{\alpha t}\le Ce^{2/\theta_*}$ for $t\le N$ gives the second bound.

\emph{Third part.} By \eqref{eq:coldf-hi} with $j=1$ and by \eqref{eq:coldf-renewal},
\[
h_1(t)=1+\frac2N\int_0^t\mathsf v=H(t)+\frac2N\int_0^tH(s)\,ds+\frac4{N^2}\int_0^t(H*\mathsf v)(s)\,ds .
\]
We have $\theta_N^{-1}-H(t)=\int_t^\infty\Upsilon\le C(1+t)^{-1/2}$, the second term is at most $2t/(\theta_*N)$, and the third is at most $(4t/(\theta_*N^2))\int_0^t\mathsf v\le Ct(t+N)/N^2\le Ct/N$ for $t\le N$. This proves the bound on $h_1$. Further, \eqref{eq:coldf-hi} shows that $h_i(t)\le1+(2/N)\int_0^t\mathsf v=h_1(t)$ and $h_i(t)\ge e^{-4a_it}h_1(t)$, so that $0\le h_1-h_i\le(1-e^{-4a_it})h_1\le4a_ith_1$.

\emph{Fourth part.} The column $V=\Phi e_1$ solves $dV=-2D_aV\,dt+\sqrt{2/N}\,d\mathcal BV$ with $V(0)=e_1$. As in the second part, $f_j=\E V_j^2$ satisfies $f_j'=-4a_jf_j+(2/N)f$ with $f=\sum_jf_j$ and $f_j(0)=\ind_{\{j=1\}}$. Then $f=1+k*f+(2/N)1*f$, and convolving with $\delta_0+\Upsilon$ as before, we find $f=H+(2/N)H*f\le\theta_*^{-1}+(2/(\theta_*N))\int_0^tf$. Gronwall's inequality now gives the first claim. The entry $\Phi_{11}=V_1$ is a martingale with $\Phi_{11}(0)=1$ and quadratic variation $(2/N)\int_0^t|V(s)|^2\,ds$, which implies $\E|\Phi_{11}(t)-1|^2=(2/N)\int_0^tf\le Ct/N$ for $t\le N$.
\end{proof}

\section{Entrance into a cap and residence}\label{sec:coldf-entrance}\label{cq-ce}

For $\sigma\in(0,1)$, let $\mathcal C_\sigma=\{x\in S_N:x_1\ge\sqrt{\sigma N}\}$ be the spherical cap around the top eigenvector, and let $-\mathcal C_\sigma=R_1\mathcal C_\sigma$. In this section, we show that the diffusion enters $\mathcal C_\sigma\cup(-\mathcal C_\sigma)$ at a time much smaller than $\log R_N/a_2$, outside an event of small fixed probability (\cref{prop:coldf-entrance}), and that the original uniform-start path, after a bounded reweighting at a deterministic entrance time supported in these caps, stays in a slightly larger cap for times of order $\log R_N/a_2$ (\cref{prop:coldf-residence}).

\begin{proposition}[Entrance into a cap]\label{prop:coldf-entrance}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$, let $\chi_N\to\infty$ with $\chi_N=o(\log R_N)$, and let $T_{\mathrm e}=\chi_N/a_2$. For every $\eps>0$ there exist $h>0$ and $M<\infty$ with $h^2<M$, depending only on $\eps$, $\theta_*$, $c_*$, and $C_*$, such that
\begin{equation}\label{eq:coldf-entrance}
\PP\big(|Y_1(T_{\mathrm e})|\ge h,\ |Y(T_{\mathrm e})|^2\le M\big)\ge1-\eps-o(1).
\end{equation}
In particular, $\PP(X_1(T_{\mathrm e})^2\ge\sigma N)\ge1-\eps-o(1)$ with $\sigma=h^2/M$.
\end{proposition}

\begin{proof}
We have $T_{\mathrm e}\ge\chi_NN^{2/3}/C_*$ by \cref{lem:coldf-consequences}, and $T_{\mathrm e}/N=\chi_N/R_N\to0$. It follows that $N(1+T_{\mathrm e})^{-3/2}\le C_*^{3/2}\chi_N^{-3/2}\le C$, and by \eqref{eq:coldf-v}, we have $\E|Y(T_{\mathrm e})|^2\le C_2$ for large $N$, with $C_2$ depending only on $\theta_*$, $c_*$, and $C_*$. Then Markov's inequality gives $\PP(|Y(T_{\mathrm e})|^2>M)\le C_2/M$.

For the first coordinate, we condition on the matrix $\Phi$, which is independent of $X_0$. We have $Y_1(T_{\mathrm e})=\langle\varphi,X_0\rangle$, where $\varphi$ is the first row of $\Phi(T_{\mathrm e})$. For a fixed vector $\varphi\ne0$, rotation invariance of $\nu_N$ shows that $\langle\varphi,X_0\rangle$ has the law of $|\varphi|\,x_1$ under $\nu_N$. The density of $x_1$ under $\nu_N$, for $|x|<\sqrt N$, is
\[
\frac{\Gamma(N/2)}{\Gamma((N-1)/2)\sqrt{\pi N}}\Big(1-\frac{x^2}N\Big)^{(N-3)/2}\le\frac{\Gamma(N/2)}{\Gamma((N-1)/2)\sqrt{\pi N}}\le\frac1{\sqrt{2\pi}},
\]
and we set it equal to zero for $|x|\ge\sqrt N$. For the bounds, we used $N\ge3$ and $\Gamma(z+1/2)\le\sqrt z\,\Gamma(z)$ for $z=(N-1)/2$, which follows from the log-convexity of $\Gamma$. On the event $\{\Phi_{11}(T_{\mathrm e})\ge1/2\}$, we have $|\varphi|\ge1/2$, and then $\PP(|Y_1(T_{\mathrm e})|\le h\mid\Phi)\le4h/\sqrt{2\pi}\le2h$. The fourth part of \cref{lem:coldf-renewal} and Chebyshev's inequality give $\PP(\Phi_{11}(T_{\mathrm e})<1/2)\le4\E|\Phi_{11}(T_{\mathrm e})-1|^2\le CT_{\mathrm e}/N=o(1)$. Combining these bounds, we obtain
\[
\PP\big(|Y_1(T_{\mathrm e})|<h\big)\le2h+o(1).
\]
Let $\eps_0=\min\{\eps,1/2\}$ and choose $h=\eps_0/4$ and $M=\max\{2C_2/\eps_0,1\}$. Then $2h\le\eps_0/2$, $C_2/M\le\eps_0/2$, and $0<h^2/M\le1/64<1$. Thus the probability in \eqref{eq:coldf-entrance} is at least $1-\eps_0-o(1)$, which is positive for large $N$ and implies the stated bound for every $\eps>0$. On the event in \eqref{eq:coldf-entrance}, $X_1(T_{\mathrm e})^2=NY_1(T_{\mathrm e})^2/|Y(T_{\mathrm e})|^2\ge h^2N/M$.
\end{proof}

The estimate \eqref{eq:coldf-entrance} concerns the whole initial law $\nu_N$. It does not assert that a single fixed cap carries all but a vanishing fraction of the mass; the fraction left out is a fixed number $\eps$, which is sent to zero only after $N\to\infty$.

For $K\ge1$, let
\[
U_K=\frac{K\log R_N}{a_2}.
\]
Note that $U_K/N=K\log R_N/R_N\to0$, by \cref{lem:coldf-consequences}.

\begin{proposition}[Residence in a cap]\label{prop:coldf-residence}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$, and fix $\sigma\in(0,1)$. Let $\gamma=\theta_*/1024$, $\zeta=32/\theta_*$, and choose $\sigma'$ with
\begin{equation}\label{eq:coldf-sigma-prime}
0<\sigma'<\sigma,\qquad \sigma'\le\min\Big\{\frac{\theta_*}8,\ \frac{\theta_*\sigma}{16(1+\zeta)}\Big\}.
\end{equation}
Let $P$ be the law of the uniformly initialized path \eqref{eq:coldf-Y}, let $T\ge0$ be deterministic, and let $w$ be an $\mathcal F_T$-measurable random variable with $0\le w\le1$ and $\E_Pw>0$. Suppose that $w\ne0$ implies $X_1(T)^2\ge\sigma N$, $P$-almost surely, and set $Q=wP/\E_Pw$. For every $K\ge1$ there exists $c_0>0$, depending only on $K$, $\sigma'$, $\theta_*$, and $B$, such that for $N$ larger than a constant depending on the same quantities and $c_*,C_*$, uniformly in $T$ and $w$,
\begin{equation}\label{eq:coldf-residence}
\begin{gathered}
Q\big(\exists s\in[0,U_K]:X_1(T+s)^2<\sigma'N\big)\le\eps_N,\\
\eps_N=2e^{-c_0R_N/\log R_N}+(N^3+2)e^{-3\theta_*^2R_N/1024}+2(N^3+2)e^{-c_0N^3}.
\end{gathered}
\end{equation}
The reweighting changes the law up to time $T$; the Brownian increments after $T$ remain Brownian under $Q$.
\end{proposition}

Note that $\eps_N$ tends to zero faster than any power of $N$, by \cref{lem:coldf-consequences}. The constant $\sigma'$ depends only on $\sigma$ and $\theta_*$. The proof follows the ratios $X_j/X_1$, $j\ge2$, whose evolution does not contain the radial term $2F(X)/N$ of \eqref{eq:coldf-X}.

\begin{proof}
Under $Q$, restart at $T$ and reflect the first coordinate by $s_T=\operatorname{sgn}X_1(T)$. Since $w$ and $s_T$ are $\mathcal F_T$-measurable, the reflected increments $(s_T(B_1(T+t)-B_1(T)),B_2(T+t)-B_2(T),\dots)$ form a Brownian motion for the restarted filtration. In this proof only, write $X_t$ for this reflected restarted path; then $X_0\in\mathcal C_\sigma$, $Q$-almost surely. Let $\tau=\inf\{t\ge0:X_1(t)\le\sqrt{\sigma'N}\}$ and $U=U_K$. The proof has four steps: an exact formula for the ratios, a bound on an integrating factor, a bound on the noise at a fixed time, and a bound between the points of a fine grid.

\paragraph{Step 1: the ratios.} For $t<\tau$, we have $X_1(t)^2\ge\sigma'N>0$, and we set $r_j=X_j/X_1$ for $j\ge2$ and $\mathcal S=\sum_{j\ge2}r_j^2=(N-X_1^2)/X_1^2$. The covariation of \eqref{eq:coldf-X} is $d\langle X_i,X_k\rangle=2(\delta_{ik}-X_iX_k/N)\,dt$. Itô's formula for the quotient gives
\begin{equation}\label{eq:coldf-ratio}
dr_j=\Big(-2a_j+\frac2{X_1^2}\Big)r_j\,dt+\frac{\sqrt2}{X_1}\,dB_j-\frac{\sqrt2\,r_j}{X_1}\,dB_1 .
\end{equation}
Indeed, the common factor $2F/N-(N-1)/N$ in the drifts of $X_j$ and $X_1$ cancels in the quotient, and so do the terms containing $X\cdot dB$ in the noise. The second-order terms are $-d\langle X_j,X_1\rangle/X_1^2=2X_j/(NX_1)\,dt$ and $X_j\,d\langle X_1\rangle/X_1^3=(2X_j/X_1^3-2X_j/(NX_1))\,dt$, whose sum is $2r_jX_1^{-2}\,dt$. Let $\cE$ solve
\[
d\cE_t=\frac{2}{X_1^2}\cE_t\,dt-\frac{\sqrt2}{X_1}\cE_t\,dB_1,\qquad \cE_0=1,
\]
so that $\log\cE_t=\int_0^tX_1^{-2}\,ds-\int_0^t\sqrt2X_1^{-1}\,dB_1$. Itô's formula shows that $d(1/\cE)=\sqrt2X_1^{-1}\cE^{-1}\,dB_1$, and then that $d(r_j/\cE)=-2a_j(r_j/\cE)\,dt+\sqrt2(X_1\cE)^{-1}\,dB_j$; the drift $2r_j/X_1^2$ cancels against the covariation of $r_j$ and $1/\cE$. Integrating, we obtain, for $t<\tau$,
\begin{equation}\label{eq:coldf-ratio-formula}
r_j(t)=\cE_t\big[e^{-2a_jt}r_j(0)+M_j(t)\big],\qquad
M_j(t)=\int_0^te^{-2a_j(t-s)}\frac{\sqrt2}{X_1(s)\cE_s}\,dB_j(s).
\end{equation}

\paragraph{Step 2: the integrating factor.} Let $\tau_{\cE}=\inf\{t:|\log\cE_t|\ge\gamma\}$ and $\widehat\tau=\tau\wedge\tau_{\cE}$. Before $\tau$, the drift of $\log\cE$ is at most $(\sigma'N)^{-1}$ per unit time and the quadratic variation of its martingale part grows at rate at most $2(\sigma'N)^{-1}$. For large $N$, we have $U/(\sigma'N)\le\gamma/2$. Then, on $\{\tau_{\cE}\le U\wedge\tau\}$, the martingale part reaches $\gamma/2$ in absolute value before time $U\wedge\tau$. We use the exponential inequality for a continuous martingale $m$ with $m_0=0$ and $\langle m\rangle_t\le y$, for $x,y>0$:
\begin{equation}\label{eq:coldf-exp-martingale}
\PP\Big(\sup_{s\le t}|m_s|\ge x\Big)\le2e^{-x^2/(2y)}.
\end{equation}
It follows from Doob's maximal inequality \cite[Chapter~II.1]{RevuzYor1999} applied to the supermartingales $\exp(\pm\lambda m_s-\lambda^2\langle m\rangle_s/2)$, with $\lambda=x/y$.
By \eqref{eq:coldf-exp-martingale},
\[
Q(\tau_{\cE}\le U\wedge\tau)\le2\exp\Big(-\frac{(\gamma/2)^2}{2\cdot2U/(\sigma'N)}\Big)=2\exp\Big(-\frac{\gamma^2\sigma'R_N}{16K\log R_N}\Big).
\]
In \eqref{eq:coldf-ratio-formula} and below, we replace the integrand of $M_j$ by zero after $\widehat\tau$. This defines $M_j(t)$ for all $t$, and \eqref{eq:coldf-ratio-formula} holds for $t<\widehat\tau$. The integrand is predictable and bounded by $\sqrt2e^\gamma(\sigma'N)^{-1/2}$.

\paragraph{Step 3: the noise at a fixed time.} Fix $t\le U$ and a deterministic vector $v=(v_j)_{j\ge2}$. The process $s\mapsto\sum_jv_j\int_0^{s}e^{-2a_j(t-s')}\sqrt2(X_1\cE)^{-1}\ind_{\{s'<\widehat\tau\}}\,dB_j(s')$, $s\le t$, is a continuous martingale, and since the $B_j$ are independent, its quadratic variation at time $t$ is at most $\sum_jd_jv_j^2$, where
\[
d_j=\frac{e^{2\gamma}}{2\sigma'Na_j},\qquad \sum_{j\ge2}d_j=\frac{e^{2\gamma}(1-\theta_N)}{\sigma'},\qquad \max_jd_j=\frac{e^{2\gamma}}{2\sigma'R_N}.
\]
Since the exponential of a continuous martingale minus half its quadratic variation is a supermartingale, we have $\E_Q\exp(\langle v,M(t)\rangle)\le\exp(\sum_jd_jv_j^2/2)$. We use the Gaussian linearization technique of \cite[Theorem~1 and Remark~2]{HsuKakadeZhang2012}. Let $\mathbf g$ be a standard Gaussian vector independent of everything else. For $u_0>0$ and $z\in\R^{N-1}$, we have $e^{u_0|z|^2}=\E_{\mathbf g}e^{\sqrt{2u_0}\langle\mathbf g,z\rangle}$. By Fubini's theorem,
\[
\E_Q e^{u_0|M(t)|^2}\le\E_{\mathbf g}\exp\Big(u_0\sum_jd_j\mathbf g_j^2\Big)=\prod_j(1-2u_0d_j)^{-1/2}
\]
if $2u_0\max_jd_j<1$. The auxiliary vector $\mathbf g$ is used only for this identity; we make no claim about the conditional law of $M(t)$. If $2u_0\max_jd_j\le1/2$, the inequality $-\log(1-x)\le x+x^2$ for $0\le x\le1/2$ gives $\log\E_Q e^{u_0|M(t)|^2}\le u_0\sum_jd_j+2u_0^2\max_jd_j\sum_jd_j$. We choose
\[
u_0=\frac{\theta_*\sigma'R_Ne^{-2\gamma}}{32},\qquad T_0=\frac{e^{2\gamma}(1-\theta_N+\theta_*/8)}{\sigma'}.
\]
Then $2u_0\max_jd_j=\theta_*/32$, $u_0(T_0-\sum_jd_j)=\theta_*^2R_N/256$, and $2u_0^2\max_jd_j\sum_jd_j\le\theta_*^2R_N/1024$. By Markov's inequality,
\begin{equation}\label{eq:coldf-M-tail}
Q\big(|M(t)|^2>T_0\big)\le\exp\Big(-\frac{\theta_*^2R_N}{256}+\frac{\theta_*^2R_N}{1024}\Big)=e^{-3\theta_*^2R_N/1024}.
\end{equation}
Now suppose that $t<\widehat\tau$ and $|M(t)|^2\le T_0$. By \eqref{eq:coldf-ratio-formula} and $(a+b)^2\le(1+\zeta)a^2+(1+\zeta^{-1})b^2$,
\[
\mathcal S(t)\le(1+\zeta)e^{2\gamma}\mathcal S(0)+(1+\zeta^{-1})e^{2\gamma}T_0 .
\]
Since $X_0\in\mathcal C_\sigma$ almost surely under $Q$, we have $\mathcal S(0)\le\sigma^{-1}$, and by \eqref{eq:coldf-sigma-prime} and $e^{2\gamma}\le2$, the first term is at most $\theta_*/(8\sigma')$. For the second term, we use $e^{4\gamma}\le1+\theta_*/128$, which gives $(1+\zeta^{-1})e^{4\gamma}\le(1+\theta_*/32)(1+\theta_*/128)\le1+\theta_*/16$, and $1-\theta_N+\theta_*/8\le1-7\theta_*/8$. The product of the last two bounds is at most $1-\theta_*/2$. We conclude that
\begin{equation}\label{eq:coldf-S-margin}
\mathcal S(t)\le\frac{1-3\theta_*/8}{\sigma'}\le\Big(\frac1{\sigma'}-1\Big)-\frac{\theta_*}{4\sigma'},
\end{equation}
where the last inequality uses $\sigma'\le\theta_*/8$. Recall that the boundary of $\mathcal C_{\sigma'}$ is $\{\mathcal S=1/\sigma'-1\}$. The bound \eqref{eq:coldf-S-margin} is a margin of $\theta_*/(4\sigma')$ below the boundary.

\paragraph{Step 4: between the points of a grid.} By Itô's formula and \eqref{eq:coldf-ratio},
\[
d\mathcal S=\Big[-4\sum_{j\ge2}a_jr_j^2+\frac{6\mathcal S}{X_1^2}+\frac{2(N-1)}{X_1^2}\Big]dt+dM_{\mathcal S},\qquad
d\langle M_{\mathcal S}\rangle=\frac{8(\mathcal S+\mathcal S^2)}{X_1^2}\,dt .
\]
Before $\tau$, we have $\mathcal S\le1/\sigma'$ and $X_1^2\ge\sigma'N$. It follows that the drift is at most a constant $C_{\sigma',B}$ in absolute value and the quadratic variation grows at rate at most $C_{\sigma'}/N$. We consider the stopped process $\mathcal S(t\wedge\widehat\tau)$ and the grid $t_i=iN^{-2}$, $0\le i\le\lfloor UN^2\rfloor+1$, which has at most $N^3+2$ points for large $N$, since $U\le N$. On each interval $[t_i,t_{i+1}]$, the drift changes $\mathcal S(\cdot\wedge\widehat\tau)$ by at most $C_{\sigma',B}N^{-2}\le\theta_*/(16\sigma')$ for large $N$, and \eqref{eq:coldf-exp-martingale} shows that the martingale part deviates from its value at $t_i$ by more than $\theta_*/(16\sigma')$ on $[t_i,t_{i+1}]$ with probability at most $2\exp(-c_0N^3)$. Let $\mathcal A$ be the event on which $\tau_{\cE}>U\wedge\tau$, the bound $|M(t_i)|^2\le T_0$ holds at every grid point $t_i\le U$, and
\[
\sup_{s\in[t_i,t_{i+1}]}\big|\mathcal S(s\wedge\widehat\tau)-\mathcal S(t_i\wedge\widehat\tau)\big|<\frac{\theta_*}{8\sigma'}
\]
for every grid interval. Combining Step 2, \eqref{eq:coldf-M-tail}, and a union bound, we find $Q(\mathcal A^c)\le\eps_N$.

We claim that $\tau>U$ on $\mathcal A$. Suppose that $\tau\le U$. We have $\tau>0$, because $X_0\in\mathcal C_\sigma$ almost surely and $\sigma'<\sigma$. Let $t_i$ be the last grid point with $t_i<\tau$. On $\mathcal A$, we have $\widehat\tau=\tau$, and in particular $t_i<\widehat\tau$, so that \eqref{eq:coldf-S-margin} holds at $t_i$. As $\tau\le t_{i+1}$, we have $\mathcal S(\tau)<\mathcal S(t_i)+\theta_*/(8\sigma')$, and by \eqref{eq:coldf-S-margin}, this implies $\mathcal S(\tau)<1/\sigma'-1$, which contradicts $X_1(\tau)^2=\sigma'N$. We conclude that $Q(\tau\le U)\le Q(\mathcal A^c)\le\eps_N$. If the original path has $X_1(T+s)^2<\sigma'N$ for some $s\le U$, continuity forces $\tau\le U$, which proves \eqref{eq:coldf-residence}.
\end{proof}

\begin{corollary}[Entrance and residence]\label{cor:coldf-entrance-residence}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$, let $T_{\mathrm e}$ and $\eps$ be as in \cref{prop:coldf-entrance}, and choose $h,M$ as in its proof. Let $\sigma=h^2/M$, choose $\sigma'$ as in \eqref{eq:coldf-sigma-prime}, and fix $K\ge1$. Then
\[
\PP\big(X_1(T_{\mathrm e})^2\ge\sigma N,\ X_1(t)^2\ge\sigma'N\text{ for all }t\in[T_{\mathrm e},T_{\mathrm e}+U_K]\big)\ge1-\eps-o(1).
\]
On this event, the sign of $X_1$ is constant on $[T_{\mathrm e},T_{\mathrm e}+U_K]$.
\end{corollary}

\begin{proof}
Let $E_N$ be the event in \eqref{eq:coldf-entrance}. It belongs to $\mathcal F_{T_{\mathrm e}}$, has probability at least $1-\min\{\eps,1/2\}-o(1)>0$ for large $N$, and implies $X_1(T_{\mathrm e})^2\ge h^2N/M=\sigma N$. Apply \cref{prop:coldf-residence} to the original uniform-start path with $T=T_{\mathrm e}$ and $w=\ind_{E_N}$. Under the resulting conditional law, the probability of leaving $\{x_1^2\ge\sigma'N\}$ before $T_{\mathrm e}+U_K$ is at most $\eps_N=o(1)$. Multiplying by $\PP(E_N)$ proves the claim. Continuity and the positive lower bound on $X_1^2$ give sign constancy.
\end{proof}
\section{Retained coordinates and a logarithmic Sobolev inequality on a cap}\label{sec:coldf-bounded-head}

In this section, we prove two deterministic facts that \cref{sec:coldf-selected,sec:coldf-entropy} use. For every fixed $A>1$, the number of coordinates $x_j$ with $a_j\le Aa_2$ is bounded under $\mathrm H(\theta_*,B,c_*,C_*)$ (\cref{lem:coldf-bounded-head}). We retain these coordinates. For a density that vanishes outside a cap, the conditional relative entropy of the other coordinates given the retained ones is at most a constant of order $(Aa_2)^{-1}$ times its dissipation (\cref{lem:coldf-coarse-lsi}).

\begin{lemma}[The number of retained coordinates]\label{lem:coldf-bounded-head}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$ and fix $A>1$. There exists an integer $M_A\ge2$, depending only on $A,c_*,C_*$, such that $c_*M_A^{2/3}>AC_*$. Fix one such $M_A$.
For $N>M_A$, let $J_N=\max\{j\in\{2,\dots,N\}:a_j\le Aa_2\}$. Then
\begin{equation}\label{eq:coldf-bounded-head}
2\le J_N\le M_A<N,\qquad a_j\le Aa_2\quad(2\le j\le J_N),\qquad a_j>Aa_2\quad(J_N<j\le N).
\end{equation}
\end{lemma}

\begin{proof}
The set in the definition of $J_N$ contains $2$, because $A>1$. Choose $M_A$ large enough that $c_*M_A^{2/3}>AC_*$. Such an integer exists because $m^{2/3}\to\infty$. With this choice, the hypotheses $\mathrm H(\theta_*,B,c_*,C_*)$ give
\[
a_{M_A+1}\ge c_*M_A^{2/3}N^{-2/3}>AC_*N^{-2/3}\ge Aa_2 .
\]
As the $a_j$ are nondecreasing in $j$, every $j>M_A$ satisfies $a_j>Aa_2$, which implies $J_N\le M_A$. The other claims follow from the monotonicity of the $a_j$ and the definition of $J_N$.
\end{proof}

Every coordinate with $a_j=Aa_2$ is retained, and equal values among the $a_j$ cause no difficulty. The index $J_N$ may depend on $N$ and on the spectrum, and we use only the bound $J_N\le M_A$. The hypotheses $\mathrm H(\theta_*,B,c_*,C_*)$ give no upper bound on $a_k/a_2$ for a prescribed index $k\ge3$, and we do not assume one.

For the second fact, we use the notation of \cref{sec:score-coordinates,sec:score-chart-entropy}. Let $2\le J<N$. As in \cref{sec:scores}, $J$ is the index of the last retained coordinate, and it is unrelated to the function $J(y)$ of \cref{sec:coldf-stationary}, which does not appear in the rest of this section. We split $y=(z,v)$ with $z=(y_2,\dots,y_J)$ and $v=(y_{J+1},\dots,y_N)$, so that $G=G_{\mathrm z}\otimes G_{\mathrm v}$. We use the matrix $\mathfrak g(y)=I-yy^\T/N$, the carr\'e du champ $\Gamma(\varphi)=\nabla\varphi^\T\mathfrak g\nabla\varphi$, and the quantities $\mathscr H(w)$ and $\mathscr D(w)$ of \eqref{eq:score-H-D}. In graph coordinates, we write $q(y)=1-|y|^2/N$, so that $q=x_1^2/N$.

\begin{lemma}[A logarithmic Sobolev inequality on a cap]\label{lem:coldf-coarse-lsi}
Let $2\le J<N$, $A>0$, and $r_0\in(0,1)$, and suppose that $a_j\ge Aa_2$ for $J<j\le N$. Let $w\in C^2(\R^{N-1})$ be a nonnegative probability density with respect to $G$, with compact support in $U_N$, such that $w=0$ on $\{q<r_0\}$. Then $\mathscr H(w)$ and $\mathscr D(w)$, the latter defined by the formula in \eqref{eq:score-H-D}, are finite, and
\begin{equation}\label{eq:coldf-coarse-lsi}
0\le\mathscr H(w)\le\frac1{4Aa_2r_0}\,\mathscr D(w).
\end{equation}
\end{lemma}

\begin{proof}
Let $y\in U_N$ and $b\in\R^{N-1}$. The Cauchy--Schwarz inequality $(y\cdot b)^2\le|y|^2|b|^2$ implies that
\begin{equation}\label{eq:coldf-metric-cap}
b^\T\mathfrak g(y)b=|b|^2-\frac{(y\cdot b)^2}N\ge\Big(1-\frac{|y|^2}N\Big)|b|^2=q(y)|b|^2 .
\end{equation}

We first show that both quantities are finite. Since $\int w\,dG=1$ and $w$ is bounded, we have $\KL(wG\Vert G)=\int w\log w\,dG\le\log\sup w<\infty$, and the chain rule \eqref{eq:score-chain-rule} gives $0\le\mathscr H(w)\le\KL(wG\Vert G)$. Let $M_2>0$ be an upper bound for the operator norm of the Hessian of $w$. For $y$ with $\nabla w(y)\ne0$ and $s\ge0$, Taylor's formula along $-\nabla w(y)/|\nabla w(y)|$ gives $0\le w(y)-s|\nabla w(y)|+M_2s^2/2$, and the choice $s=|\nabla w(y)|/M_2$ yields $|\nabla w|^2\le2M_2w$. In particular, $\nabla w=0$ where $w=0$, and $\int|\nabla w|^2/w\,dG\le2M_2$, where the integrand is zero on $\{w=0\}$. The function $\bar w(z)=\int w(z,v)\,G_{\mathrm v}(dv)$ is of class $C^2$, since $w$ has compact support and we may differentiate under the integral, and the Cauchy--Schwarz inequality on each fiber gives $|\nabla\bar w|^2\le\bar w\int|\nabla_zw|^2/w\,dG_{\mathrm v}$. As $\mathfrak g\le I$ and $\nabla g=\nabla w/w-\nabla\bar w/\bar w$ on $\{w>0\}$, we obtain
\[
\mathscr D(w)\le2\int\frac{|\nabla w|^2}w\,dG+2\int\frac{|\nabla\bar w|^2}{\bar w}\,dG_{\mathrm z}\le4\int\frac{|\nabla w|^2}w\,dG<\infty .
\]

Now fix $z$ with $\bar w(z)>0$, and let $\phi=w(z,\cdot)/\bar w(z)$, a nonnegative function of class $C^2$ with compact support and $\int\phi\,dG_{\mathrm v}=1$. Then $\int\phi\log\phi\,dG_{\mathrm v}=\E_w[g\mid z]$, and $(\partial_{v_i}\phi)^2/\phi=\phi\,(\partial_{v_i}g)^2$ on $\{\phi>0\}$. By the Gaussian logarithmic Sobolev inequality \cite{Gross1975}, in the form \eqref{eq:score-anisotropic-lsi}, which follows from \eqref{eq:score-gaussian-lsi} \cite[Theorem~2.7, (9), p.~5]{Gentil2010} after a diagonal change of variables, and by $a_i\ge Aa_2$ for $i>J$, we have
\[
\E_w[g\mid z]\le\sum_{i>J}\frac1{4a_i}\E_w\big[(\partial_{v_i}g)^2\mid z\big]\le\frac1{4Aa_2}\E_w\big[|\nabla_vg|^2\mid z\big].
\]
Here \eqref{eq:score-anisotropic-lsi}, which is stated for smooth functions, applies to $\phi$, with the integrand on the right set to zero on $\{\phi=0\}$. Indeed, the mollifications $\phi_n$ of $\phi$ are smooth and nonnegative, they are supported in a common compact set, and they converge to $\phi$ in $C^2$, because the derivatives of $\phi$ up to order two are uniformly continuous. Fix $e>0$, and apply \eqref{eq:score-anisotropic-lsi} to $(\phi_n+e)/(c_n+e)$, where $c_n=\int\phi_n\,dG_{\mathrm v}\to1$. The integrands on both sides are bounded uniformly in $n$, since $\phi_n+e\ge e$, and they are constant outside the common support. Dominated convergence then gives the inequality for $(\phi+e)/(1+e)$. As $e\downarrow0$, the left side converges by bounded convergence, because $\phi$ is bounded and $s\log s$ is continuous on $[0,\infty)$. The right side is $\sum_{i>J}(4a_i)^{-1}\int(\partial_{v_i}\phi)^2/((\phi+e)(1+e))\,dG_{\mathrm v}$, and it converges by monotone convergence, since $\nabla\phi=0$ on $\{\phi=0\}$. Since $w=0$ on $\{q<r_0\}$, we have $|\nabla_vg|^2\le|\nabla g|^2\le\Gamma(g)/r_0$ on $\{w>0\}$, by \eqref{eq:coldf-metric-cap}. Multiplying by $\bar w(z)$ and integrating with respect to $G_{\mathrm z}$, we obtain \eqref{eq:coldf-coarse-lsi}. Fibers with $\bar w(z)=0$ contribute zero to both sides.
\end{proof}

For $r\in(0,1)$, we use \cref{lem:coldf-coarse-lsi} with $r_0=r/2$ and $A=16/r$. Then $4Aa_2r_0=32a_2$, and under the hypotheses of the lemma,
\begin{equation}\label{eq:coldf-lsi}
\mathscr H(w)\le\frac1{32a_2}\,\mathscr D(w).
\end{equation}
The constant in \eqref{eq:coldf-coarse-lsi} is cruder than that of \cref{prop:score-conditional-lsi}, which keeps the matrix $A_v$ of \cref{lem:score-schur}. Here we bound $\mathfrak g$ from below by $q$ times the identity, which loses the fixed factor $r_0$, and we compensate for this loss by retaining all coordinates with $a_j\le Aa_2$ for a large $A$. The dissipation $\mathscr D(w)$ contains the derivatives in all variables, and the proof uses only its part in $v$.
\section{Gaussian approximation of the coordinates at time \texorpdfstring{$t_{\mathrm p}$}{tp}}\label{sec:coldf-gaussian}\label{cq-gs}

In this section, we show that at the time
\[
t_{\mathrm p}=\frac{\log N}{100},
\]
finitely many coordinates of the unnormalized process are close in total variation to independent centered Gaussian variables with variance $\theta^{-1}$, and that the first coordinate barely moves afterwards. These Gaussian variables are not the initial coordinates of $X_0$; they are built from $X_0$ and the noise up to time $t_{\mathrm p}$. The proof conditions on the matrix $\Phi$, which is independent of $X_0$, and uses the rotation invariance of $\nu_N$; it does not use a central limit theorem for martingales.

\begin{proposition}[Gaussian approximation at time $t_{\mathrm p}$]\label{prop:coldf-initial-coordinates}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$.
\begin{enumerate}
\item Suppose that $\theta_N\to\theta\in(0,1)$, and let $k\ge1$ be fixed with $a_k\log N\to0$. Then
\begin{equation}\label{eq:coldf-GS}
\big\|\Law\big(Y_1(t_{\mathrm p}),\dots,Y_k(t_{\mathrm p})\big)-\cN(0,\theta^{-1}I_k)\big\|_{\TV}\longrightarrow0 .
\end{equation}
\item For every $K\ge1$ and every $U$ with $t_{\mathrm p}\le U\le U_K$,
\begin{equation}\label{eq:coldf-TS}
\E\sup_{t_{\mathrm p}\le t\le U}\big|Y_1(t)-Y_1(t_{\mathrm p})\big|^2\le C\Big[\frac UN+(1+t_{\mathrm p})^{-1/2}\Big],
\end{equation}
where $C$ depends only on $\theta_*$, $c_*$, and $C_*$. The right side tends to zero.
\end{enumerate}
\end{proposition}

The hypothesis $a_k\log N\to0$ holds for $k\le2$ under $\mathrm H(\theta_*,B,c_*,C_*)$, and for every fixed $k$ under $\mathrm H_k(\varpi,D_k)$ as well, since then $a_k\le D_kC_*N^{-2/3}$. The second part, combined with the first, gives the weak limit $\cN(0,\theta^{-1})$ for $Y_1(t)$ at every later time $t\le U_K$, jointly with $(Y_1(t_{\mathrm p}),\dots,Y_k(t_{\mathrm p}))$. It does not give convergence in total variation at later times.

\begin{proof}[Proof of the first part]
Let $\Sigma=\Phi\Phi^\T$, with entries $\Sigma_{ij}$. As $X_0$ is independent of $\Phi$ and $\E X_0X_0^\T=I$, we have $\E[Y(t)Y(t)^\T\mid\Phi]=\Sigma(t)$, and in particular $\E\Sigma_{ii}(t)=h_i(t)$. The proof has three steps: we bound the second moments of $\Sigma$, show that the first $k\times k$ block of $\Sigma(t_{\mathrm p})$ concentrates around $\theta^{-1}I_k$, and then use the conditional law of $Y$ given $\Phi$.

\paragraph{Step 1: second moments of $\Sigma$.} By Itô's formula and \eqref{eq:coldf-matrix-flow},
\begin{equation}\label{eq:coldf-Sigma}
d\Sigma=\Big[-2(D_a\Sigma+\Sigma D_a)+\frac2N\Tr(\Sigma)I\Big]dt+\sqrt{\frac2N}\big(d\mathcal B\,\Sigma+\Sigma\,d\mathcal B^\T\big).
\end{equation}
The drift term $(2/N)\Tr(\Sigma)I$ comes from $(d\mathcal B\,\Sigma\,d\mathcal B^\T)_{ik}=\delta_{ik}\Tr(\Sigma)\,dt$. Expanding $d\Tr(\Sigma^2)=2\Tr(\Sigma\,d\Sigma)+\Tr(d\Sigma\,d\Sigma)$, the noise contributes four contractions: $\Tr(d\mathcal B\,\Sigma\,d\mathcal B\,\Sigma)$ and $\Tr(\Sigma\,d\mathcal B^\T\Sigma\,d\mathcal B^\T)$ each equal $\Tr(\Sigma^2)\,dt$, and $\Tr(d\mathcal B\,\Sigma^2d\mathcal B^\T)$ and $\Tr(\Sigma\,d\mathcal B^\T d\mathcal B\,\Sigma)$ each equal $N\Tr(\Sigma^2)\,dt$. We obtain
\[
\frac{d}{dt}\E\Tr(\Sigma^2)=\E\Big[-8\Tr(D_a\Sigma^2)+\frac4N(\Tr\Sigma)^2+\Big(4+\frac4N\Big)\Tr(\Sigma^2)\Big].
\]
Note that $\Tr(D_a\Sigma^2)\ge0$, because $a_i\ge0$, and that $(\Tr\Sigma)^2\le N\Tr(\Sigma^2)$. With $\Tr(\Sigma(0)^2)=N$ and $N\ge2$, we obtain $\E\Tr(\Sigma(t)^2)\le Ne^{12t}$. Next, let $\mathcal U_i=(\Sigma^2)_{ii}=\sum_j\Sigma_{ij}^2$. We see from \eqref{eq:coldf-Sigma} that the quadratic variation of $\Sigma_{ij}$ grows at rate $(2/N)(\mathcal U_i+\mathcal U_j)$ for $i\ne j$, and that of $\Sigma_{ii}$ at rate $(8/N)\mathcal U_i$. Summing $d(\Sigma_{ij}^2)$ over $j$, we find that the drift of $\mathcal U_i$ is
\[
-4a_i\mathcal U_i-4\sum_ja_j\Sigma_{ij}^2+\frac4N\Tr(\Sigma)\Sigma_{ii}+\frac2N\big[(N+2)\mathcal U_i+\Tr(\Sigma^2)\big].
\]
Using $\Sigma_{ii}\le\mathcal U_i^{1/2}$, we find $(4/N)\Tr(\Sigma)\Sigma_{ii}\le4(\Tr(\Sigma^2)/N)^{1/2}\mathcal U_i^{1/2}\le2\mathcal U_i+2\Tr(\Sigma^2)/N$. Then $u_i=\E\,\mathcal U_i$ satisfies $u_i'\le6u_i+4e^{12t}$ and $u_i(0)=1$, and Gronwall's inequality gives
\begin{equation}\label{eq:coldf-Ui}
\E\,\mathcal U_i(t)\le2e^{12t}.
\end{equation}
These computations are justified first up to bounded stopping times and then by monotone convergence, since all moments of $\Phi(t)$ are finite.

\paragraph{Step 2: concentration of the covariance.} The diagonal of \eqref{eq:coldf-Sigma} reads
\[
d(\Sigma_{11},\dots,\Sigma_{NN})^\T=\mathsf A(\Sigma_{11},\dots,\Sigma_{NN})^\T dt+d\mathcal M,\qquad \mathsf A=-4D_a+\frac2N\mathbf1\mathbf1^\T,
\]
where $d\mathcal M_i=2\sqrt{2/N}\sum_k\Sigma_{ki}\,d\mathcal B_{ik}$. Since the martingale $\mathcal M_i$ uses only the $i$-th row of $\mathcal B$, we have $\langle\mathcal M_i,\mathcal M_j\rangle=0$ for $i\ne j$, and $d\langle\mathcal M_i\rangle=(8/N)\mathcal U_i\,dt$. The vector $h=(h_i)$ solves $h'=\mathsf Ah$ with $h(0)=\mathbf1$. The off-diagonal entries of $\mathsf A$ are nonnegative, so $e^{t\mathsf A}$ has nonnegative entries; together with $\mathsf A\mathbf1\le2\cdot\mathbf1$ entrywise, this shows that its row sums are at most $e^{2t}$. By variation of constants, the orthogonality of the $\mathcal M_i$, and \eqref{eq:coldf-Ui},
\[
\E|\Sigma_{ii}(t)-h_i(t)|^2=\frac8N\int_0^t\sum_j\big(e^{(t-s)\mathsf A}\big)_{ij}^2\,\E\,\mathcal U_j(s)\,ds\le\frac{16}N\int_0^te^{4(t-s)+12s}\,ds\le\frac{2e^{12t}}N .
\]
For $i\ne j$, the entry $\Sigma_{ij}$ starts at zero and has drift $-2(a_i+a_j)\Sigma_{ij}$, so that
\[
\E\,\Sigma_{ij}(t)^2\le\frac2N\int_0^t\E\big(\mathcal U_i(s)+\mathcal U_j(s)\big)\,ds\le\frac{2e^{12t}}{3N}.
\]
At $t=t_{\mathrm p}$, we have $e^{12t_{\mathrm p}}/N=N^{-22/25}$. Recall from the third part of \cref{lem:coldf-renewal} that $|h_1(t_{\mathrm p})-\theta_N^{-1}|\le C[(1+t_{\mathrm p})^{-1/2}+t_{\mathrm p}/N]$ and $|h_1(t_{\mathrm p})-h_i(t_{\mathrm p})|\le4a_kt_{\mathrm p}h_1(t_{\mathrm p})$ for $i\le k$. Using $\theta_N\to\theta$ and $a_kt_{\mathrm p}\to0$, we conclude that the block $\Sigma_{[k]}=(\Sigma_{ij}(t_{\mathrm p}))_{i,j\le k}$ satisfies
\begin{equation}\label{eq:coldf-block}
\E\big\|\Sigma_{[k]}-\theta^{-1}I_k\big\|_{\mathrm{HS}}^2\longrightarrow0 .
\end{equation}

\paragraph{Step 3: the conditional law given $\Phi$.} Let $\Phi_{[k]}$ be the $k\times N$ matrix formed by the first $k$ rows of $\Phi(t_{\mathrm p})$. As $\Phi(t_{\mathrm p})$ is invertible, the matrix $\Sigma_{[k]}=\Phi_{[k]}\Phi_{[k]}^\T$ is positive definite, and $O=\Sigma_{[k]}^{-1/2}\Phi_{[k]}$ has orthonormal rows. We complete $O$ to an orthogonal matrix. By the rotation invariance of $\nu_N$ and the independence of $X_0$ and $\Phi$, conditionally on $\Phi$ the vector $OX_0$ has the law of $Z_{N,k}=(x_1,\dots,x_k)$ under $\nu_N$. It follows that $\Law(Y_{[k]}(t_{\mathrm p})\mid\Phi)=\Law(\Sigma_{[k]}^{1/2}Z_{N,k})$. Since an invertible linear map preserves total variation, we obtain
\[
\big\|\Law(Y_{[k]}(t_{\mathrm p})\mid\Phi)-\cN(0,\Sigma_{[k]})\big\|_{\TV}=\eps_{N,k},\qquad \eps_{N,k}=\|\Law(Z_{N,k})-\cN(0,I_k)\|_{\TV}.
\]
For $N>k+2$, the law of $Z_{N,k}$ has the density
\[
\frac{\Gamma(N/2)}{\Gamma((N-k)/2)(\pi N)^{k/2}}\Big(1-\frac{|z|^2}N\Big)_+^{(N-k-2)/2},
\]
which converges pointwise to the standard Gaussian density on $\R^k$, since $\Gamma(N/2)/\Gamma((N-k)/2)\sim(N/2)^{k/2}$. Scheffé's lemma then implies $\eps_{N,k}\to0$. This is the classical Gaussian approximation of the coordinates of a uniform point on a high-dimensional sphere; quantitative bounds are in \cite[(1)]{DiaconisFreedman1987}.
Further, for positive definite $\Sigma_1,\Sigma_0$, Pinsker's inequality and the formula for the relative entropy of Gaussian laws give
\[
\|\cN(0,\Sigma_1)-\cN(0,\Sigma_0)\|_{\TV}^2\le\frac14\Big[\Tr(\Sigma_0^{-1}\Sigma_1)-k-\log\det(\Sigma_0^{-1}\Sigma_1)\Big],
\]
and the right side tends to zero as $\Sigma_1\to\Sigma_0$. Using \eqref{eq:coldf-block} and bounded convergence, we find $\E\|\cN(0,\Sigma_{[k]})-\cN(0,\theta^{-1}I_k)\|_{\TV}\to0$. By the convexity of total variation under mixtures, we obtain
\[
\big\|\Law(Y_{[k]}(t_{\mathrm p}))-\cN(0,\theta^{-1}I_k)\big\|_{\TV}\le\eps_{N,k}+\E\big\|\cN(0,\Sigma_{[k]})-\cN(0,\theta^{-1}I_k)\big\|_{\TV}\longrightarrow0,
\]
which proves \eqref{eq:coldf-GS}.
\end{proof}

\begin{proof}[Proof of the second part]
Recall that $Y_1$ is a martingale with $d\langle Y_1\rangle=(2/N)|Y|^2\,dt$. By Doob's inequality in $L^2$ and \eqref{eq:coldf-v}, we have, for $t_{\mathrm p}\le U\le U_K\le N$,
\[
\begin{split}
\E\sup_{t_{\mathrm p}\le t\le U}|Y_1(t)-Y_1(t_{\mathrm p})|^2&\le\frac8N\int_{t_{\mathrm p}}^U\mathsf v(s)\,ds\\
&\le\frac CN\int_{t_{\mathrm p}}^U\big[1+N(1+s)^{-3/2}\big]ds\le C\Big[\frac UN+(1+t_{\mathrm p})^{-1/2}\Big].
\end{split}
\]
The right side tends to zero, since $U_K/N=K\log R_N/R_N\to0$ by \cref{lem:coldf-consequences}.
\end{proof}
\section{The late-time squared norm and finitely many coordinates}\label{sec:coldf-finite}\label{cq-ph}

In this section, we fix
\[
\chi_N=\sqrt{\log\log N},\qquad T_{\mathrm e}=\frac{\chi_N}{a_2},\qquad Z_N=Y_1(t_{\mathrm p}).
\]
Under $\mathrm H(\theta_*,B,c_*,C_*)$, we have $\chi_N=o(\log R_N)$, and \cref{prop:coldf-entrance} applies at the time $T_{\mathrm e}$. Further, $t_{\mathrm p}<T_{\mathrm e}<t_u$ for large $N$, and
\begin{equation}\label{eq:coldf-lags}
a_2(t_u-T_{\mathrm e})=\frac{\log R_N}4+u-\chi_N\longrightarrow\infty,\qquad t_u-T_{\mathrm e}\le U_1
\end{equation}
for every fixed $u$ and large $N$. In this section, ``in probability'' refers to the randomness of the initial point and the noise; the spectra are deterministic.

\begin{proposition}[The squared norm at time $t_u$]\label{prop:coldf-normalization}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$, and fix $u\in\R$. Then
\begin{equation}\label{eq:coldf-PN}
|Y(t_u)|^2-\frac{Z_N^2}{\theta_N}\longrightarrow0,\qquad Y_1(t_u)-Z_N\longrightarrow0
\end{equation}
in probability. If, in addition, $\theta_N\to\theta\in(0,1)$, then $X_1(t_u)^2/N\to\theta$ in probability.
\end{proposition}

The first convergence says that the squared norm of the unnormalized process at time $t_u$ is determined by its first coordinate: the remaining coordinates have relaxed, and their total squared norm is the fraction $(1-\theta_N)/\theta_N$ of $Y_1^2$. The proof conditions the path law on the event of \cref{prop:coldf-entrance}, stops the process when it leaves a cap, and solves the renewal equation for $|Y|^2$ along the stopped process.

\begin{proof}
Fix $\eps\in(0,1/4)$, let $h,M$ be as in \cref{prop:coldf-entrance}, let $E_N=\{|Y_1(T_{\mathrm e})|\ge h,\ |Y(T_{\mathrm e})|^2\le M\}$, and choose $\sigma'>0$ so that $2\sigma'$ satisfies \eqref{eq:coldf-sigma-prime} with $\sigma=h^2/M$. Let $U=U_1$. Throughout the proof, $\PP_E$ and $\E_E$ denote probability and expectation conditionally on $E_N$, and $C$ denotes constants depending only on $\eps$ and the spectral constants.

\paragraph{Step 1: the stopped process.} Let $s_N=\operatorname{sgn}Y_1(T_{\mathrm e})$. Since $E_N\in\mathcal F_{T_{\mathrm e}}$ and the increments of $B$ after $T_{\mathrm e}$ are independent of $\mathcal F_{T_{\mathrm e}}$, the process $r\mapsto(s_NY_1,Y_2,\dots,Y_N)(T_{\mathrm e}+r)$ solves \eqref{eq:coldf-Y} under $\PP_E$ with the Brownian motion $r\mapsto(s_NB_1,B_2,\dots,B_N)(T_{\mathrm e}+r)-(s_NB_1,B_2,\dots,B_N)(T_{\mathrm e})$, which we denote by $\bar B$. Let $\tau=\inf\{r\ge0:s_NX_1(T_{\mathrm e}+r)\le\sqrt{\sigma'N}\}$. On $E_N$, the first squared coordinate at $T_{\mathrm e}$ is at least $\sigma N$. Apply \cref{prop:coldf-residence} directly to the original uniform-start path with $T=T_{\mathrm e}$, $w=\ind_{E_N}$, and residence threshold $2\sigma'$. Since hitting the stopping threshold $\sigma'$ entails falling below $2\sigma'$, we obtain
\[
p_N=\PP_E(\tau\le U)\le\eps_N\longrightarrow0 .
\]
We define $\bar Y(r)=(s_NY_1,Y_2,\dots,Y_N)(T_{\mathrm e}+r)$ for $r\le\tau$; for $r>\tau$, let $\bar Y_1(r)=\bar Y_1(\tau)$ and $\bar Y_j(r)=e^{-2a_j(r-\tau)}\bar Y_j(\tau)$ for $j\ge2$. In other words, $\bar Y$ follows the unnormalized process until $\tau$, and afterwards its noise is removed and its first coordinate is frozen. Let $\bar A=|\bar Y|^2$ and $y=\bar Y_1(0)=|Y_1(T_{\mathrm e})|$, so that $h\le y\le\sqrt M$ on $E_N$. Then
\begin{equation}\label{eq:coldf-stopped}
d\bar Y_1=\sqrt{2/N}\,\bar A^{1/2}\ind_{\{r<\tau\}}\,d\bar B_1,\qquad
d\bar Y_j=-2a_j\bar Y_j\,dr+\sqrt{2/N}\,\bar A^{1/2}\ind_{\{r<\tau\}}\,d\bar B_j\quad(j\ge2).
\end{equation}
Before $\tau$, we have $\bar Y_1^2\ge\sigma'\bar A$. After $\tau$, the first coordinate is frozen and the others decrease, which gives $\bar A(r)\le\bar A(\tau)\le\bar Y_1(\tau)^2/\sigma'$. We conclude that
\begin{equation}\label{eq:coldf-stopped-norm}
\bar A(r)\le\frac{\bar Y_1(r)^2}{\sigma'}\qquad(r\ge0).
\end{equation}

\paragraph{Step 2: moments of the first coordinate.} For an integer $\ell\ge1$, Itô's formula, \eqref{eq:coldf-stopped}, and \eqref{eq:coldf-stopped-norm} show that the drift of $\bar Y_1^{2\ell}$ is at most $2\ell(2\ell-1)(\sigma'N)^{-1}\bar Y_1^{2\ell}$. We then have $\E_E\bar Y_1(r)^{2\ell}\le M^\ell e^{2\ell(2\ell-1)r/(\sigma'N)}$ by Gronwall's inequality. As $U/N\to0$, the moments of order up to eight of $\bar Y_1(r)$ and, by \eqref{eq:coldf-stopped-norm}, of $\bar A(r)^{1/2}$ are bounded uniformly for $r\le U$. The process $\bar Y_1^2-y^2$ has drift $(2/N)\bar A\ind_{\{r<\tau\}}$ and a martingale part with quadratic variation $(8/N)\bar A\bar Y_1^2\ind_{\{r<\tau\}}\,dr$. By Doob's inequality in $L^2$ and Minkowski's inequality, we obtain
\begin{equation}\label{eq:coldf-top-stable}
\Big\|\sup_{r\le U}\big|\bar Y_1(r)^2-y^2\big|\Big\|_{L^2(\PP_E)}\le C\Big(\sqrt{\frac UN}+\frac UN\Big).
\end{equation}

\paragraph{Step 3: the renewal equation for the stopped process.} For $j\ge2$, Itô's formula and variation of constants give
\[
\begin{split}
\bar Y_j(r)^2={}&e^{-4a_jr}\bar Y_j(0)^2+\frac2N\int_0^re^{-4a_j(r-w)}\bar A(w)\ind_{\{w<\tau\}}\,dw\\
&+\sqrt{\frac8N}\int_0^re^{-4a_j(r-w)}\bar A(w)^{1/2}\bar Y_j(w)\ind_{\{w<\tau\}}\,d\bar B_j(w).
\end{split}
\]
Summing over $j\ge2$, we obtain
\begin{equation}\label{eq:coldf-stopped-renewal}
\bar A=\bar Y_1^2+\ell_0+k*\bar A+\mathsf n-\mathsf e,
\end{equation}
where $k$ is the kernel of \cref{lem:coldf-renewal} and
\[
\ell_0(r)=\sum_{j\ge2}e^{-4a_jr}\bar Y_j(0)^2\le Me^{-4a_2r},\qquad
\mathsf e(r)=\int_0^rk(r-w)\bar A(w)\ind_{\{\tau\le w\}}\,dw,
\]
and $\mathsf n(r)$ is the sum of the stochastic integrals. The term $\mathsf e$ is present because the noise was removed after $\tau$. Using the independence of the $\bar B_j$ and Step 2, we find
\[
\E_E\,\mathsf n(r)^2=\frac8N\sum_{j\ge2}\int_0^re^{-8a_j(r-w)}\E_E\big[\bar A\bar Y_j^2\ind_{\{w<\tau\}}\big]\,dw\le\frac8N\int_0^r\E_E\bar A(w)^2\,dw\le\frac{CU}N .
\]
Using Hölder's inequality, we find $\|\bar A(w)\ind_{\{\tau\le w\}}\|_{L^2(\PP_E)}\le(\E_E\bar A(w)^4)^{1/4}p_N^{1/4}\le Cp_N^{1/4}$, and Minkowski's inequality and $\int k\le1$ then give $\|\mathsf e(r)\|_{L^2(\PP_E)}\le Cp_N^{1/4}$ for $r\le U$.

\paragraph{Step 4: solving the renewal equation.} Convolving \eqref{eq:coldf-stopped-renewal} with $\delta_0+\Upsilon$, which has total mass $1/\theta_N$, we obtain $\bar A=(\delta_0+\Upsilon)*(\bar Y_1^2+\ell_0+\mathsf n-\mathsf e)$. Recall that $(\delta_0+\Upsilon)*1=H$ and that $\theta_N^{-1}-H(t)=\int_t^\infty\Upsilon\le C(1+t)^{-1/2}$ by \cref{lem:coldf-renewal}. We obtain
\[
\Big\|(\delta_0+\Upsilon)*\bar Y_1^2(t)-\frac{y^2}{\theta_N}\Big\|_{L^2(\PP_E)}\le\frac1{\theta_*}\Big\|\sup_{r\le U}|\bar Y_1(r)^2-y^2|\Big\|_{L^2(\PP_E)}+CM(1+t)^{-1/2}.
\]
Splitting the convolution at $t/2$, we find
\[
(\delta_0+\Upsilon)*\ell_0(t)\le Me^{-4a_2t}+M\int_{t/2}^\infty\Upsilon+\frac M{\theta_*}e^{-2a_2t}\le C\big[e^{-2a_2t}+(1+t)^{-1/2}\big].
\]
The convolutions of $\mathsf n$ and $\mathsf e$ with $\delta_0+\Upsilon$ are bounded in $L^2(\PP_E)$, uniformly for $t\le U$, by $\theta_*^{-1}$ times the bounds of Step 3. Combining these bounds with \eqref{eq:coldf-top-stable}, we obtain, for every $0\le v_N\le U$,
\begin{equation}\label{eq:coldf-PN-quant}
\sup_{v_N\le t\le U}\Big\|\bar A(t)-\frac{y^2}{\theta_N}\Big\|_{L^2(\PP_E)}\le C\Big[\sqrt{\frac UN}+\frac UN+p_N^{1/4}+e^{-2a_2v_N}+(1+v_N)^{-1/2}\Big].
\end{equation}
The supremum is over the values of the norms at fixed times; this is not a bound on the norm of a supremum. If $a_2v_N\to\infty$, then also $v_N\to\infty$, and the right side tends to zero.

\paragraph{Step 5: conclusion.} By \eqref{eq:coldf-lags}, we may take $t=t_u-T_{\mathrm e}$ and $v_N=t$ in \eqref{eq:coldf-PN-quant}. On $E_N\cap\{\tau>U\}$, we have $\bar A(t_u-T_{\mathrm e})=|Y(t_u)|^2$ and $y^2=Y_1(T_{\mathrm e})^2$. Then, for every $\gamma>0$, Markov's inequality and Cauchy--Schwarz imply
\[
\PP\big(E_N\cap\{||Y(t_u)|^2-Y_1(T_{\mathrm e})^2/\theta_N|>\gamma\}\big)\le p_N+\gamma^{-1}\Big\|\bar A(t_u-T_{\mathrm e})-\frac{y^2}{\theta_N}\Big\|_{L^2(\PP_E)}\longrightarrow0 .
\]
Further, \eqref{eq:coldf-TS} gives $\E|Y_1(T_{\mathrm e})-Z_N|^2\to0$ and $\E|Y_1(t_u)-Z_N|^2\to0$, and $\E Y_1(t)^2=h_1(t)\le C$ for $t\le N$ by \cref{lem:coldf-renewal}. It follows that $Y_1(T_{\mathrm e})^2-Z_N^2\to0$ in probability, and
\[
\limsup_N\PP\Big(\Big||Y(t_u)|^2-\frac{Z_N^2}{\theta_N}\Big|>2\gamma\Big)\le\limsup_N\PP(E_N^c)\le\eps .
\]
As $\eps$ is arbitrary, \eqref{eq:coldf-PN} follows. Finally, assume $\theta_N\to\theta$. We know from \cref{prop:coldf-initial-coordinates} with $k=1$ that $Z_N$ converges in distribution to $\cN(0,\theta^{-1})$. For $r_0>0$, on $\{|Z_N|\ge r_0\}$ the ratio $X_1(t_u)^2/N=Y_1(t_u)^2/|Y(t_u)|^2$ differs from $\theta_N$ by a quantity tending to zero in probability, by \eqref{eq:coldf-PN}, and $\limsup_N\PP(|Z_N|<r_0)\le\PP(|\cN(0,\theta^{-1})|\le r_0)$, which tends to zero as $r_0\to0$.
\end{proof}

\begin{theorem}[Finitely many coordinates]\label{thm:coldf-finite-coordinates}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$ and $\theta_N\to\theta\in(0,1)$. Fix $k\ge2$, and if $k\ge3$ assume also $\mathrm H_k(\varpi,D_k)$. Let $G_2,\dots,G_k$ be independent standard Gaussian variables and $C$ an independent standard Cauchy variable. Then for every fixed $u\in\R$,
\begin{equation}\label{eq:coldf-PH}
\big(\sqrt{2a_j}\,X_j(t_u)\big)_{j=2}^k\Longrightarrow\big(G_2+\sqrt{2\theta}\,e^{-2u}C,\,G_3,\dots,G_k\big).
\end{equation}
\end{theorem}

If, in addition, $a_3/a_2$ is bounded uniformly in $N$, then \cref{mesocold-prop:stationary-graph} shows that the same vector under $\pi$ converges in total variation to $(G_2,\dots,G_k)$. This upper bound on $a_3/a_2$ is included in $\mathrm H_3(\varpi,D_3)$ in the main deterministic theorem. Under these hypotheses, since the total-variation distance is at least half the difference of expectations of a function with values in $[-1,1]$, \eqref{eq:coldf-PH} for $k=2$ gives $\liminf_ND_N(t_u)\ge\cV_\theta(u)$. The matching upper bound requires control of all $N$ coordinates, which is the content of \cref{sec:coldf-selected,sec:coldf-entropy,sec:coldf-head,sec:coldf-assembly}.

\begin{proof}
We write $\varkappa_j=a_j/a_2$ for $2\le j\le k$, so that $1\le\varkappa_j\le D_k$ and $\varkappa_j\ge1+\varpi$ for $j\ge3$; for $k=2$, set $D_2=1$. Note that $a_kt_{\mathrm p}\le D_kC_*N^{-2/3}t_{\mathrm p}\to0$. By \cref{prop:coldf-initial-coordinates}, we have
\begin{equation}\label{eq:coldf-seeds}
\big\|\Law\big(Y_1(t_{\mathrm p}),\dots,Y_k(t_{\mathrm p})\big)-\Law\big(\theta^{-1/2}(H_1,\dots,H_k)\big)\big\|_{\TV}\to0,
\end{equation}
where $H_1,\dots,H_k$ are independent standard Gaussian variables. Let $\Lambda_N=\sqrt{\theta_N}\,|Y(t_u)|/|Z_N|$. Then \cref{prop:coldf-normalization} implies that, for every $r_0>0$, $\Lambda_N\to1$ in probability on the event $\{|Z_N|\ge r_0\}$.

\paragraph{Step 1: decomposition.} For $2\le j\le k$, variation of constants in \eqref{eq:coldf-Y} from time $t_{\mathrm p}$ gives
\[
\sqrt{2a_j}\,X_j(t_u)=\frac{\sqrt{2a_jN}\,Y_j(t_u)}{|Y(t_u)|}=\Lambda_N^{-1}\Big[\mathsf S_{j,N}+\frac{\sqrt{\theta_N}}{|Z_N|}\big(I_j^{(1)}+I_j^{(2)}+I_j^{(3)}\big)\Big],
\]
where
\[
\mathsf S_{j,N}=\frac{\sqrt{2a_jN\theta_N}\,e^{-2a_j(t_u-t_{\mathrm p})}\,Y_j(t_{\mathrm p})}{|Z_N|}
=\sqrt{2\theta_N\varkappa_j}\,R_N^{(1-\varkappa_j)/2}e^{-2\varkappa_ju}\,e^{2a_jt_{\mathrm p}}\,\frac{Y_j(t_{\mathrm p})}{|Z_N|},
\]
and $I_j^{(1)},I_j^{(2)},I_j^{(3)}$ are the integrals $\sqrt{4a_j}\int e^{-2a_j(t_u-r)}|Y(r)|\,dB_j(r)$ over the intervals $[t_{\mathrm p},T_{\mathrm e}]$, $[T_{\mathrm e},l_N]$, and $[l_N,t_u]$, with $l_N=t_u-M_*/a_2$ for a fixed $M_*>0$. In the formula for $\mathsf S_{j,N}$, we used $e^{-2a_jt_u}=R_N^{-\varkappa_j/2}e^{-2\varkappa_ju}$ and $\sqrt{2a_jN\theta_N}=\sqrt{2\theta_N\varkappa_j}\,R_N^{1/2}$.

\paragraph{Step 2: the initial terms.} For $j=2$, we have $\mathsf S_{2,N}=\sqrt{2\theta_N}\,e^{-2u}e^{2a_2t_{\mathrm p}}Y_2(t_{\mathrm p})/|Z_N|$. For $j\ge3$, we have $R_N^{(1-\varkappa_j)/2}\le R_N^{-\varpi/2}\to0$, while $e^{-2\varkappa_ju}e^{2a_jt_{\mathrm p}}\le e^{2D_k|u|+1}$ for large $N$. Since $Y_j(t_{\mathrm p})$ is tight by \eqref{eq:coldf-seeds}, we conclude that $\mathsf S_{j,N}\to0$ in probability on $\{|Z_N|\ge r_0\}$ for $j\ge3$.

\paragraph{Step 3: the early noise.} By Itô's isometry and \eqref{eq:coldf-v},
\[
\E|I_j^{(1)}|^2=4a_j\int_{t_{\mathrm p}}^{T_{\mathrm e}}e^{-4a_j(t_u-r)}\mathsf v(r)\,dr\le4a_je^{-4a_j(t_u-T_{\mathrm e})}\,C\big[T_{\mathrm e}+N(1+t_{\mathrm p})^{-1/2}\big].
\]
Recall from \eqref{eq:coldf-lags} that $e^{-4a_j(t_u-T_{\mathrm e})}=R_N^{-\varkappa_j}e^{-4\varkappa_j(u-\chi_N)}$. Using $a_jT_{\mathrm e}=\varkappa_j\chi_N$ and $R_N^{1-\varkappa_j}\le1$, we obtain
\[
\E|I_j^{(1)}|^2\le CD_ke^{4D_k(|u|+\chi_N)}\Big[\frac{\chi_N}{R_N}+(1+t_{\mathrm p})^{-1/2}\Big]\longrightarrow0,
\]
because $e^{4D_k\chi_N}=(\log N)^{4D_k/\chi_N}$ grows more slowly than every positive power of $\log N$. For $T_{\mathrm e}\le r\le T_{\mathrm e}+U_1$, we have $\mathsf v(r)\le C$ by \eqref{eq:coldf-v}, because $N(1+T_{\mathrm e})^{-3/2}\le C$. It follows that
\[
\E|I_j^{(2)}|^2\le4a_jC\int_{-\infty}^{l_N}e^{-4a_j(t_u-r)}\,dr=Ce^{-4\varkappa_jM_*}\le Ce^{-4M_*}.
\]

\paragraph{Step 4: the late noise.} Let $\mathcal G_{j,N}=\sqrt{4a_j}\int_{l_N}^{t_u}e^{-2a_j(t_u-r)}\,dB_j(r)$. These variables are independent centered Gaussian variables with variances $1-e^{-4\varkappa_jM_*}$, and they are independent of $\mathcal F_{l_N}$, in particular of $\mathcal F_{T_{\mathrm e}}$ and of $(Y_1,\dots,Y_k)(t_{\mathrm p})$. We compare $I_j^{(3)}$ with $|Z_N|\theta_N^{-1/2}\mathcal G_{j,N}$. Let $E_N$, $\PP_E$, $\tau$, $\bar A$, $y$, and $U=U_1$ be as in the proof of \cref{prop:coldf-normalization}, for a fixed $\eps$. On $E_N\cap\{\tau>U\}$, we have $I_j^{(3)}=\tilde I_j$, where
\[
\tilde I_j=\sqrt{4a_j}\int_{l_N}^{t_u}e^{-2a_j(t_u-r)}\,\bar A(r-T_{\mathrm e})^{1/2}\ind_{\{r-T_{\mathrm e}<\tau\}}\,dB_j(r)
\]
is the integral with the stopped amplitude. Let $c_{\mathrm e}=y/\sqrt{\theta_N}$, which is $\mathcal F_{T_{\mathrm e}}$-measurable and at most $(M/\theta_*)^{1/2}$ on $E_N$. By Itô's isometry and $4a_j\int_{l_N}^{t_u}e^{-4a_j(t_u-r)}\,dr\le1$,
\[
\begin{split}
\E_E\big|\tilde I_j-c_{\mathrm e}\mathcal G_{j,N}\big|^2&\le\sup_{l_N\le r\le t_u}\E_E\big|\bar A(r-T_{\mathrm e})^{1/2}\ind_{\{r-T_{\mathrm e}<\tau\}}-c_{\mathrm e}\big|^2\\
&\le2\sup_{l_N\le r\le t_u}\E_E\Big|\bar A(r-T_{\mathrm e})-\frac{y^2}{\theta_N}\Big|+\frac{2M}{\theta_*}p_N,
\end{split}
\]
where we used $|\sqrt a-\sqrt b|^2\le|a-b|$ for $a,b\ge0$. As $a_2(l_N-T_{\mathrm e})\to\infty$, the right side tends to zero by \eqref{eq:coldf-PN-quant}. Further, $\mathcal G_{j,N}$ is independent of $\mathcal F_{T_{\mathrm e}}$ and has variance at most one, and we have
\[
\E_E\Big|\Big(c_{\mathrm e}-\frac{|Z_N|}{\sqrt{\theta_N}}\Big)\mathcal G_{j,N}\Big|^2\le\frac{\E|Y_1(T_{\mathrm e})-Z_N|^2}{\theta_*\PP(E_N)}\longrightarrow0,
\]
by \eqref{eq:coldf-TS}. We conclude that $\sqrt{\theta_N}|Z_N|^{-1}I^{(3)}_j-\mathcal G_{j,N}\to0$ in probability on $E_N\cap\{|Z_N|\ge r_0\}$.

\paragraph{Step 5: conclusion.} Combining Steps 1--4, for fixed $\eps$, $r_0$, $M_*$, and $\gamma>0$,
\[
\begin{split}
&\limsup_N\PP\Big(\max_{2\le j\le k}\big|\sqrt{2a_j}X_j(t_u)-\big(\mathsf S_{j,N}\ind_{\{j=2\}}+\mathcal G_{j,N}\big)\big|>\gamma\Big)\\
&\qquad\le\eps+\PP\big(|H_1|\le\sqrt\theta r_0\big)+\frac{Ck}{\gamma^2r_0^2}e^{-4M_*},
\end{split}
\]
where we used that $\mathsf S_{j,N}$ and $\mathcal G_{j,N}$ are tight, Chebyshev's inequality for $I_j^{(2)}$, and the factor $\sqrt{\theta_N}/|Z_N|\le r_0^{-1}$. The vector $(\mathcal G_{j,N})_{2\le j\le k}$ is independent of $(Y_1,Y_2)(t_{\mathrm p})$, and its law is within total-variation distance $Cke^{-4M_*}$ of the standard Gaussian law on $\R^{k-1}$. Using \eqref{eq:coldf-seeds}, $e^{2a_2t_{\mathrm p}}\to1$, and $\theta_N\to\theta$, we see that the variable $\mathsf S_{2,N}$ converges in distribution to $\sqrt{2\theta}\,e^{-2u}H_2/|H_1|$. Testing against bounded Lipschitz functions, letting first $N\to\infty$, then $M_*\to\infty$, then $\gamma\downarrow0$ and $r_0\downarrow0$, and finally $\eps\downarrow0$, we obtain that $(\sqrt{2a_j}X_j(t_u))_{j=2}^k$ converges in distribution to $(G_2+\sqrt{2\theta}e^{-2u}H_2/|H_1|,G_3,\dots,G_k)$, where $G_2,\dots,G_k$ are independent standard Gaussian variables, independent of $(H_1,H_2)$. Finally, the density of $H_2/|H_1|$ at $w\in\R$ is
\[
2\int_0^\infty s\,\phi(s)\phi(ws)\,ds=\frac1\pi\int_0^\infty se^{-(1+w^2)s^2/2}\,ds=\frac1{\pi(1+w^2)},
\]
and $H_2/|H_1|$ has the standard Cauchy law. This proves \eqref{eq:coldf-PH}.
\end{proof}

The Cauchy variable in \eqref{eq:coldf-PH} is the ratio $H_2/|H_1|$ of the Gaussian approximations of the first two coordinates at time $t_{\mathrm p}$, and the Gaussian variable $G_2$ is produced by the noise in the second coordinate during the last time interval $[t_u-M_*/a_2,t_u]$. They are independent because the Brownian increments after time $l_N$ are independent of $\mathcal F_{l_N}$. We never assign a Gaussian law to the stochastic integrals with the random amplitude $|Y(r)|$; they are compared with integrals of deterministic functions after the amplitude has been replaced by an $\mathcal F_{T_{\mathrm e}}$-measurable variable.
\section{The weighted law, the cutoff function, and scores}\label{sec:coldf-selected}\label{cq-sc}

In this section, we weight the path law by a smooth function of the position at the time $T_{\mathrm e}$, introduce a cutoff function, and prove the estimates that \cref{sec:coldf-entropy,sec:coldf-head} use: a formula for the density of the weighted law and its forward equation (\cref{lem:coldf-selected-density}), the probability that the cutoff differs from one and the fourth moments of the coordinates (\cref{lem:coldf-collar}), bounds on the derivatives of the cutoff (\cref{lem:coldf-cutoff}), bounds on the scores (\cref{lem:coldf-scores,lem:coldf-marginal-score}), and an upper bound on the density (\cref{lem:coldf-density}). Throughout the section, we assume $\mathrm H(\theta_*,B,c_*,C_*)$; no bound on $a_3/a_2$ is used.

\paragraph{Times and the weight.}
Fix $\eps\in(0,1/8)$. With $\chi_N=\sqrt{\log\log N}$ and $T_{\mathrm e}=\chi_N/a_2$ as in \cref{sec:coldf-finite}, let
\begin{equation}\label{eq:coldf-times}
t_0=T_{\mathrm e}+\frac1{a_2},\qquad t_1=T_{\mathrm e}+\frac{\log R_N}{a_2},
\end{equation}
and recall that $U_2=2\log R_N/a_2$. For every fixed $u\in\R$, we have $t_0\le t_u\le t_1$ for large $N$, by \eqref{eq:coldf-lags}, and $t_1<T_{\mathrm e}+U_2$. Fix once and for all the smooth transition $\psi(s)=b(s)/(b(s)+b(1-s))$, where $b(s)=e^{-1/s}$ for $s>0$ and $b(s)=0$ for $s\le0$. It satisfies $\psi=0$ on $(-\infty,0]$ and $\psi=1$ on $[1,\infty)$. Only bounds on $\psi'$ and $\psi''$ enter the estimates below.

Let $h,M$ be given by \cref{prop:coldf-entrance} with $\eps/2$ in place of $\eps$, so that they depend only on $\eps$, $\theta_*$, $c_*$, and $C_*$. Let $\sigma_{\mathrm e}=\min\{h^2/M,1/2\}$ and $\sigma_0=\sigma_{\mathrm e}/2$, and write $q(x)=x_1^2/N$. By \cref{prop:coldf-entrance}, we have $\PP(q(X_{T_{\mathrm e}})\ge\sigma_{\mathrm e})\ge1-\eps/2-o(1)$. We define
\begin{equation}\label{eq:coldf-weight}
\Psi_0(x)=\psi\Big(\frac{q(x)}{\sigma_0}-1\Big),\qquad\Psi=\Psi_0(X_{T_{\mathrm e}}),\qquad Z_\Psi=\E\Psi .
\end{equation}
The function $\Psi_0$ is smooth on $S_N$, invariant under $R_1$, equal to one on $\{q\ge\sigma_{\mathrm e}\}$, and equal to zero on $\{q\le\sigma_0\}$. It follows that $Z_\Psi\ge\PP(q(X_{T_{\mathrm e}})\ge\sigma_{\mathrm e})\ge1-\eps$ for large $N$. We write $P$ for the law of the whole process $(X_0,B)$, as in \cref{sec:score-setting}, and we call
\[
Q=\frac{\Psi}{Z_\Psi}\,P
\]
the weighted law. For $t\ge0$, we let $Q_t$ be the law of $X_t$ under $Q$.

The variable $\Psi$ is $\mathcal F_{T_{\mathrm e}}$-measurable. By the Markov property of \eqref{eq:coldf-X}, conditionally on $\mathcal F_{T_{\mathrm e}}$, the process $(X_{T_{\mathrm e}+s})_{s\ge0}$ is the diffusion generated by $L$ started from $X_{T_{\mathrm e}}$, and the increments of $B$ after $T_{\mathrm e}$ are independent of $\mathcal F_{T_{\mathrm e}}$. Both statements remain true under $Q$, because $dQ/dP=\Psi/Z_\Psi$ is $\mathcal F_{T_{\mathrm e}}$-measurable. In particular, $Q_t=Q_{T_{\mathrm e}}P_{t-T_{\mathrm e}}$ for $t\ge T_{\mathrm e}$, and $Q_{T_{\mathrm e}}$ is concentrated on $\{q>\sigma_0\}$. We make no claim about the law of $B$ before $T_{\mathrm e}$ under $Q$. The law of $(X_0,B)$ is invariant under the reflection of the first coordinates of $X_0$ and $B$, which maps $X$ to $R_1X$ and leaves $\Psi$ unchanged. It follows that every $Q_t$ is invariant under $R_1$. By \cref{lem:score-domination}, applied to the law of $X_t$ and the measure $A\mapsto\E[\Psi\ind_{\{X_t\in A\}}]$, and since $0\le\Psi\le1$, we have
\begin{equation}\label{eq:coldf-restriction-loss}
Q_t\le\frac{\nu_NP_t}{Z_\Psi},\qquad\|Q_t-\nu_NP_t\|_{\TV}\le1-Z_\Psi\le\eps .
\end{equation}

\begin{lemma}[The density of the weighted law]\label{lem:coldf-selected-density}
For $t>T_{\mathrm e}$, the law $Q_t$ has a density $\rho_t=dQ_t/d\nu_N$ which is twice continuously differentiable in space and once continuously differentiable in time. The density, its first and second spatial derivatives, and its time derivative are jointly continuous on $(T_{\mathrm e},\infty)\times S_N$. Let $Z_\pi=\int e^{-F}\,d\nu_N$. For $s\ge0$,
\begin{equation}\label{eq:coldf-selected-density}
\frac{dQ_{T_{\mathrm e}}}{d\pi}=\frac{\Psi_0}{Z_\Psi}\,P_{T_{\mathrm e}}\big(Z_\pi e^F\big),\qquad
\frac{dQ_{T_{\mathrm e}+s}}{d\pi}=P_s\Big(\frac{dQ_{T_{\mathrm e}}}{d\pi}\Big).
\end{equation}
In graph coordinates, a nonnegative representative $f_t=dQ_t^+/dG$ has the same $C^2$ spatial and $C^1$ time regularity, with $f_t$, $\nabla f_t$, $\nabla^2f_t$, and $\partial_tf_t$ jointly continuous, hence bounded on compact cylinders in $(T_{\mathrm e},\infty)\times U_N$, and satisfies
\begin{equation}\label{eq:coldf-forward}
\partial_tf_t=L_Gf_t-\operatorname{div}_G\Big(\frac yN\,f_t\Big),\qquad L_G=\operatorname{div}_G(\mathfrak g\nabla),
\end{equation}
with $\mathfrak g$ and $\operatorname{div}_G$ as in \cref{sec:score-chart-entropy}.
\end{lemma}

\begin{proof}
The density of $\nu_N$ with respect to $\pi$ is $Z_\pi e^F$, which is bounded. For bounded measurable $\varphi$, the reversibility of $\pi$ gives $\int\varphi\,d(\nu_NP_{T_{\mathrm e}})=\int P_{T_{\mathrm e}}\varphi\,Z_\pi e^F\,d\pi=\int\varphi\,P_{T_{\mathrm e}}(Z_\pi e^F)\,d\pi$, so that $d(\nu_NP_{T_{\mathrm e}})/d\pi=P_{T_{\mathrm e}}(Z_\pi e^F)$. As $Q_{T_{\mathrm e}}(A)=Z_\Psi^{-1}\E[\Psi_0(X_{T_{\mathrm e}})\ind_{\{X_{T_{\mathrm e}}\in A\}}]$, we obtain the first formula in \eqref{eq:coldf-selected-density}. For $s\ge0$, the identity $Q_{T_{\mathrm e}+s}=Q_{T_{\mathrm e}}P_s$ and reversibility give
\[
\int\varphi\,dQ_{T_{\mathrm e}+s}=\int\frac{dQ_{T_{\mathrm e}}}{d\pi}\,P_s\varphi\,d\pi=\int\varphi\,P_s\Big(\frac{dQ_{T_{\mathrm e}}}{d\pi}\Big)\,d\pi ,
\]
which is the second formula.

For the regularity, let $\sigma_t(A)=\E[\Psi\ind_{\{X_t\in A\}}]=Z_\Psi Q_t(A)$ be the endpoint measure of \cref{lem:score-regularity} with $T=T_{\mathrm e}$, the weight $\Psi$, and $D=S_N$. Apply that lemma to the original uniformly initialized diffusion and choose its density representative $u$, retaining its first time derivative and first two spatial derivatives. Set $\rho_t=u(t,\cdot)/Z_\Psi$. The graph formula in \cref{lem:score-chart} gives $f_t(y)=2\omega_N\rho_t(x^+(y))/(x_1(y)\gamma(y))$ on $U_N$, where $\gamma$ is the density \eqref{eq:score-gaussian} of $G$. The chart map and its prefactor are smooth on $U_N$, so the chain rule gives the asserted spatial derivatives and their joint continuity; the same holds for the time derivative. Let $\varphi\in C^\infty_c(U_N)$. By \eqref{eq:score-forward-chart}, divided by $Z_\Psi$,
\[
\frac d{dt}\int\varphi f_t\,dG=\int f_t\Big(L_G\varphi+\frac yN\cdot\nabla\varphi\Big)\,dG=\int\varphi\Big(L_Gf_t-\operatorname{div}_G\Big(\frac yN\,f_t\Big)\Big)\,dG .
\]
In the second step, we used \eqref{eq:score-chart-ibp} and the divergence theorem for the weighted divergence $\operatorname{div}_G$. Differentiating under the integral on the left, which is allowed by the joint continuity of $\partial_tf_t$ and the compact support of $\varphi$, and using that $\varphi$ is arbitrary, we obtain \eqref{eq:coldf-forward}.
\end{proof}

We work with this representative of the density throughout. The estimates below use only this finite regularity near the support of the cutoff function, and no lower bound on it.

\paragraph{The cap of the cutoff and the residence error.}
Let
\begin{equation}\label{eq:coldf-r}
r=\frac14\min\Big\{\sigma_0,\ \frac{\theta_*}8,\ \frac{\theta_*\sigma_0}{16(1+32/\theta_*)}\Big\},\qquad A=\frac{16}r,
\end{equation}
and let $M_A$ and $J_N$ be as in \cref{lem:coldf-bounded-head}, for $N>M_A$. Then $\sigma'=2r$ satisfies \eqref{eq:coldf-sigma-prime} with $\sigma=\sigma_0$, with the strict inequality $2r<\sigma_0$. Let $c_0>0$ be small enough that the bounds in \cref{prop:coldf-residence} and its ratio construction hold with $\sigma=\sigma_0$, $\sigma'=2r$, for both $K=1$ and $K=2$; it depends only on $r$, $\theta_*$, and $B$. We set
\begin{equation}\label{eq:coldf-pN}
p_N=2\exp\Big(-\frac{c_0c_*N^{1/3}}{\log(C_*N^{1/3})}\Big)+(N^3+2)\,e^{-3\theta_*^2c_*N^{1/3}/1024}+2(N^3+2)\,e^{-c_0N^3}.
\end{equation}
By \cref{lem:coldf-consequences}, we have $c_*N^{1/3}\le R_N\le C_*N^{1/3}$, and therefore, for large $N$, the bound $\eps_N$ of \eqref{eq:coldf-residence} with these parameters satisfies $\eps_N\le p_N$. The sequence $p_N$ does not depend on the spectrum, and $N^bp_N\to0$ for every $b>0$: the first exponent is of order $N^{1/3}/\log N$, and the other two exponentials decay faster than every power of $N$. Here and below, ``for large $N$'' means for $N$ larger than a constant depending only on $\eps$, the spectral constants, and $\psi$.

\paragraph{The cutoff function.}
Let
\begin{equation}\label{eq:coldf-eta}
\chi(s)=\psi\Big(\frac{2s}r-1\Big),\qquad\zeta=\chi\circ q,\qquad\eta=\zeta^8 .
\end{equation}
Then $\chi=0$ on $(-\infty,r/2]$ and $\chi=1$ on $[r,\infty)$, and we have $\|\chi'\|_\infty\le2\|\psi'\|_\infty/r$ and $\|\chi''\|_\infty\le4\|\psi''\|_\infty/r^2$. The functions $\zeta$ and $\eta$ are smooth on $S_N$, invariant under every reflection $R_j$, equal to one on $\{q\ge r\}$, and equal to zero on $\{q\le r/2\}$. They depend only on $x_1^2$, and they do not restrict the other coordinates. Let
\[
\mathcal K=\{q<r\},\qquad M_t=\E_Q\,\eta(X_t),\qquad Q_t^\eta=\frac{\eta\,Q_t}{M_t}.
\]
Outside $\mathcal K$, we have $\eta=1$, and all derivatives of $\eta$ vanish. The probability measure $Q_t^\eta$ is the normalized cutoff-weighted law of \cref{sec:score-setting}, with normalizing mass $M_t$.

\begin{lemma}[The region outside which the cutoff equals one]\label{lem:coldf-collar}
There exists $C>0$, depending only on $\eps$ and the spectral constants, such that for large $N$ and all $t\in[T_{\mathrm e},T_{\mathrm e}+U_2]$,
\begin{equation}\label{eq:coldf-collar}
Q\big(q(X_t)<2r\big)\le p_N,\qquad Q(X_t\in\mathcal K)\le p_N,\qquad1-M_t\le p_N,\qquad Q\big(\eta(X_t)\ne1\big)\le p_N,
\end{equation}
and, for $t\in[T_{\mathrm e},t_1]$ with $s=t-T_{\mathrm e}$, for every $2\le j\le N$,
\begin{equation}\label{eq:coldf-X2-moment}
\E_QX_j(t)^4\le CN^2\big[e^{-8a_js}+(Na_j)^{-2}+p_N\big].
\end{equation}
\end{lemma}

The constant in \eqref{eq:coldf-X2-moment} does not depend on $j$, $N$, or $t$. The bounds \eqref{eq:coldf-collar} hold at each fixed time through $T_{\mathrm e}+U_2$; the fourth-moment bound is asserted only through $t_1=T_{\mathrm e}+\log R_N/a_2$. This shorter interval contains every fixed-offset profile time $t_u$ for all sufficiently large $N$.

\begin{proof}
Apply \cref{prop:coldf-residence} to the original uniform-start path with $T=T_{\mathrm e}$, $w=\Psi$, $\sigma=\sigma_0$, $\sigma'=2r$, and $K=2$. Its support hypothesis holds because $\Psi\ne0$ implies $q(X_{T_{\mathrm e}})>\sigma_0$. The resulting weighted law is exactly $Q$, so the first estimate in \eqref{eq:coldf-collar} follows, with error at most $\eps_N\le p_N$. For the ratio and moment estimates, use the same weighted-path construction over the shorter interval $0\le s\le\log R_N/a_2$. Reflecting the first coordinate by its $\mathcal F_{T_{\mathrm e}}$-measurable sign leaves the later Brownian increments Brownian and changes neither $X_j$ for $j\ge2$ nor any cap event. The construction in the proof of \cref{prop:coldf-residence}, now with $K=1$, supplies an event $\mathcal A$ with $Q(\mathcal A^c)\le p_N$, on which $X_1(T_{\mathrm e}+s)^2\ge2rN$, $|\log\cE_s|\le\gamma$ with $\gamma=\theta_*/1024$, and
\begin{equation}\label{eq:coldf-ratio-after}
\frac{X_j(T_{\mathrm e}+s)}{|X_1(T_{\mathrm e}+s)|}=\cE_s\Big[e^{-2a_js}\,\frac{X_j(T_{\mathrm e})}{|X_1(T_{\mathrm e})|}+M_j(s)\Big],\qquad j\ge2,
\end{equation}
where $\cE$ and the stopped integrals $M_j$ are those of \eqref{eq:coldf-ratio-formula} for the reflected process.

The other three bounds in \eqref{eq:coldf-collar} follow from $\mathcal K\subset\{q<2r\}$, $\{\eta\ne1\}\subset\mathcal K$, and $1-M_t\le Q(\eta(X_t)\ne1)$, which holds because $0\le\eta\le1$.

For the fourth moments, fix $0\le s\le\log R_N/a_2$ and $j\ge2$. As in Step 3 of the proof of \cref{prop:coldf-residence}, with $\sigma'=2r$, the variable $M_j(s)$ is the terminal value of a continuous martingale whose quadratic variation is at most $d_j=e^{2\gamma}/(4rNa_j)$. Under $Q$, the Brownian increments after $T_{\mathrm e}$ remain Brownian, so the third bound in \eqref{eq:score-subgaussian} with $p=2$ directly gives $\E_QM_j(s)^4\le16d_j^2$. We use only this moment bound, and we make no claim about the conditional law of $M_j(s)$. On $\mathcal A$, we have $X_1^2\le N$, $X_j(T_{\mathrm e})^2/X_1(T_{\mathrm e})^2\le\sigma_0^{-1}$, and $e^{4\gamma}\le2$. Using \eqref{eq:coldf-ratio-after} and $(b_1+b_2)^4\le8(b_1^4+b_2^4)$, we find that on $\mathcal A$,
\[
X_j(T_{\mathrm e}+s)^4\le N^2\,\frac{X_j(T_{\mathrm e}+s)^4}{X_1(T_{\mathrm e}+s)^4}\le16N^2\big[\sigma_0^{-2}e^{-8a_js}+M_j(s)^4\big].
\]
On $\mathcal A^c$, we use $X_j^4\le N^2$. Combining these bounds, we obtain
\[
\E_QX_j(T_{\mathrm e}+s)^4\le16N^2\big[\sigma_0^{-2}e^{-8a_js}+16d_j^2\big]+N^2p_N ,
\]
and $16d_j^2=e^{4\gamma}r^{-2}(Na_j)^{-2}\le2r^{-2}(Na_j)^{-2}$. This proves \eqref{eq:coldf-X2-moment}.
\end{proof}

\begin{lemma}[Derivatives of the cutoff]\label{lem:coldf-cutoff}
There exists $C>0$, depending only on $r$, $B$, and $\psi$, such that on $S_N$, for $2\le j\le N$,
\begin{equation}\label{eq:coldf-cutoff-derivatives}
\frac{\Gamma(\eta)}{\eta}\le\frac CN\,\eta^{3/4}\ind_{\mathcal K},\qquad
|L\eta|\le C\eta^{3/4}\ind_{\mathcal K},\qquad
|K_{1j}\eta|\le C\eta^{7/8}\le C\eta^{1/2},
\end{equation}
where the ratio is zero where $\eta=0$, and $K_{1j}=x_1\partial_j-x_j\partial_1$.
\end{lemma}

\begin{proof}
Recall from \eqref{eq:model-sde} that, for a function $f$ smooth near $S_N$, we have $\Gamma(f)=|\Pi(x)\nabla f|^2$ and $Lf=\Tr(\Pi(x)\nabla^2f)-(1-1/N)\ip{x}{\nabla f}-\ip{\Pi(x)\nabla F}{\nabla f}$. For $q$, a direct computation gives
\[
\Gamma(q)=\frac{4q(1-q)}N\le\frac1N,\qquad Lq=\frac2N-2q+\frac{4qF}N,\qquad|Lq|\le2+4B,
\]
where we used $\Pi(x)\nabla F\cdot\nabla q=-4qF/N$, which holds because $(\nabla F)_1=0$ and $x\cdot\nabla F=2F$, and $0\le F\le BN$. Further, $K_{1j}q=-2x_1x_j/N$, so that $|K_{1j}q|\le(x_1^2+x_j^2)/N\le1$. By the chain rule, with $\chi$ and its derivatives evaluated at $q$,
\[
\nabla\eta=8\chi^7\chi'\,\nabla q,\qquad
L\eta=8\chi^7\chi'\,Lq+\big(56\chi^6(\chi')^2+8\chi^7\chi''\big)\Gamma(q),\qquad
K_{1j}\eta=8\chi^7\chi'\,K_{1j}q .
\]
Since $\eta^{3/4}=\chi^6$ and $\eta^{7/8}=\chi^7$, we obtain $\Gamma(\eta)/\eta=64\chi^6(\chi')^2\Gamma(q)\le64\|\chi'\|_\infty^2N^{-1}\eta^{3/4}$,
\[
|L\eta|\le\big(8\|\chi'\|_\infty(2+4B)+56\|\chi'\|_\infty^2+8\|\chi''\|_\infty\big)\eta^{3/4},
\]
and $|K_{1j}\eta|\le8\|\chi'\|_\infty\eta^{7/8}$. In the bound on $L\eta$, we used $\chi^7\le\chi^6$ and $\Gamma(q)\le1$. All these derivatives vanish unless $\chi'(q)\ne0$ or $\chi''(q)\ne0$, which happens only in $\{r/2<q<r\}\subset\mathcal K$. The bounds on $\chi'$ and $\chi''$ stated after \eqref{eq:coldf-eta} complete the proof.
\end{proof}

\begin{lemma}[Scores of the weighted law]\label{lem:coldf-scores}
There exists a numerical constant $C>0$ such that for large $N$, $t\in[t_0,t_1]$, and $1\le i<j\le N$, the law $Q_t$ has a weak $K_{ij}$-score $K_{ij}\log\rho_t$, and
\begin{equation}\label{cq-sc-reset-scores}
\E_Q\big|K_{ij}\log\rho_t(X_t)\big|^4\le CN^2a_j^2 .
\end{equation}
In particular, $\E_Q|K_{12}\log\rho_t(X_t)|^2\le CR_N$.
\end{lemma}

\begin{proof}
We use \cref{csr-thm:reset} with $p=2$, the domain $D=S_N$ (so that there is no killing), $H=N$, the Killing field $K_{ij}$, the final interval $[t-\ell,t]$ with $\ell=1/a_j$, the linear ramp, and the weight $W=\Psi$. We check its hypotheses. The domain $S_N$ is invariant under all rotations, and $x_i^2+x_j^2\le N$ on $S_N$. The process $X$ solves \eqref{eq:coldf-X}, which is \eqref{mc-eq:sr1}, and it is adapted to the filtration $(\mathcal F_s)$ generated by $X_0$ and $\mathcal B$, for which $B$ is a Brownian motion (\cref{sec:coldf-process}). Since $\ell\le1/a_2$ and $t\ge t_0=T_{\mathrm e}+1/a_2$, we have $t-\ell\ge T_{\mathrm e}$. In the notation of \cref{csr-thm:reset}, the weight is $W=\Phi(\xi,X|_{[t-\ell,t]})$ with the $\mathcal F_{t-\ell}$-measurable variable $\xi=\Psi$ and $\Phi(e,\omega)=e$; it takes values in $[0,1]$, and it is unchanged by the rotations, which act only on $[t-\ell,t]$. The normalizing constant is $\E W=Z_\Psi\ge1/2$, and the weighted endpoint law is $Q_t$. With $\Delta=a_j-a_i\le a_j$, \eqref{mc-eq:sr5} implies
\[
\E_Q|K_{ij}\log\rho_t(X_t)|^4\le\frac{C_2}{Z_\Psi}\big[N(\ell^{-1}+\Delta^2\ell)\big]^2\le8C_2N^2a_j^2 ,
\]
where $C_2$ is the constant of \cref{csr-thm:reset} with $p=2$. For $(i,j)=(1,2)$, this is $8C_2R_N^2$, and the Cauchy--Schwarz inequality gives the last claim.
\end{proof}

The bound \eqref{cq-sc-reset-scores} requires no bound on the scores of the law at earlier times: the rotation acts only on the last interval of length $1/a_j$, and it starts from the identity, also when $t-\ell=T_{\mathrm e}$.

\paragraph{Graph coordinates for the weighted law.}
We use the graph coordinates $y=(x_2,\dots,x_N)$ of \cref{sec:coldf-stationary} and the splitting $y=(z,v)$ of \cref{sec:score-coordinates} with $J=J_N$. Thus $z=(x_2,\dots,x_{J_N})$ are the retained coordinates and $v=(x_{J_N+1},\dots,x_N)$ the remaining ones, and $G=G_{\mathrm z}\otimes G_{\mathrm v}$. Recall that $J_N$ is the index of \cref{lem:coldf-bounded-head} with $A=16/r$; it may depend on $N$ and on the spectrum, and $2\le J_N\le M_A$. Recall also from \cref{lem:coldf-selected-density} that $f_t=dQ_t^+/dG$ for $t>T_{\mathrm e}$. Viewed as a function of $y$, $\eta$ is smooth with compact support in $U_N$, since $\eta(y)\ne0$ only if $|y|^2<(1-r/2)N$. Let
\begin{equation}\label{eq:coldf-w}
w_t=\frac{\eta f_t}{M_t},\qquad\bar w_t(z)=\int w_t(z,v)\,G_{\mathrm v}(dv),
\end{equation}
extended by zero outside $U_N$. Then $w_tG=(Q_t^\eta)^+$, and $\bar w_t$ is the density of the law of $z$ under $Q_t^\eta$ with respect to $G_{\mathrm z}$. The function $w_t$ vanishes on $\{q\le r/2\}$.

\begin{lemma}[An upper bound on the density]\label{lem:coldf-density}
For $t\ge T_{\mathrm e}$, we have $dQ_t/d\pi\le e^{BN}/Z_\Psi$ on $S_N$. For $t>T_{\mathrm e}$ and large $N$,
\begin{equation}\label{eq:coldf-density}
f_t\le\frac{e^{BN}}{Z_\Psi\,\theta_*\,(r/2)^{1/2}}\le e^{CN}\qquad\text{on }D_{r/2}=\{y:|y|^2\le(1-r/2)N\},
\end{equation}
with $C$ depending only on $B$, $\theta_*$, and $r$.
\end{lemma}

\begin{proof}
The density of $\nu_N$ with respect to $\pi$ is $Z_\pi e^F$, with $Z_\pi\le1$, and it is at most $e^{BN}$ by \cref{lem:coldf-consequences}. By the reversibility of $\pi$, the density of $\nu_NP_t$ with respect to $\pi$ is $P_t(Z_\pi e^F)\le e^{BN}$, by the positivity of $P_t$ and $P_t1=1$. Then \eqref{eq:coldf-restriction-loss} implies $dQ_t/d\pi\le e^{BN}/Z_\Psi$. Both $Q_t$ and $\pi$ are invariant under $R_1$, and the same bound holds for $dQ_t^+/d\pi^+$. Finally, $r/2\le\theta_*/2$, and \cref{lem:coldf-cap-gibbs} gives $d\pi^+/dG\le(r/2)^{-1/2}\theta_*^{-1}$ on $D_{r/2}$. This proves the first bound in \eqref{eq:coldf-density}, and the second follows from $Z_\Psi\ge1/2$.
\end{proof}

The bound \eqref{eq:coldf-density} is crude, but it holds for $Q_t$ itself, at every time. We use it only through logarithms, where it costs a factor of order $N$.

\paragraph{The score of the retained coordinates.}
For $t\in[t_0,t_1]$, let
\begin{equation}\label{eq:coldf-B}
S_t(z)=z\cdot\nabla_z\log\bar w_t(z),\qquad\mathsf B(t)=\int w_t\,S_t^2\,dG ,
\end{equation}
where $S_t$ is defined on $\{\bar w_t>0\}$. The function $\bar w_t$ is $C^2$, since $w_t$ is $C^2$ with compact support. The quantity $\mathsf B(t)$ is the second moment, under $Q_t^\eta$, of the derivative of $\log\bar w_t$ along the radial vector field $z\cdot\nabla_z$ of the retained coordinates.

\begin{lemma}[The score of the retained coordinates]\label{lem:coldf-marginal-score}
There exists $C>0$, depending only on $\eps$, $\theta_*$, $B$, $c_*$, $C_*$, and $\psi$, such that for large $N$ and all $t\in[t_0,t_1]$,
\begin{equation}\label{cq-sc-reset-marginal-score}
\mathsf B(t)\le C\Big\{1+R_Ne^{-4a_2(t-T_{\mathrm e})}+R_N^2e^{-8a_2(t-T_{\mathrm e})}\Big\}.
\end{equation}
\end{lemma}

The constant depends on the spectral constants also through $A$ and $M_A$, and it is uniform over the index $J_N$ and over the ratios $a_j/a_2$ with $j\le J_N$. We use no convergence of these ratios. The last term in \eqref{cq-sc-reset-marginal-score} comes from the Gaussian reference $G_{\mathrm z}$. It is large at the beginning of the interval, and it is integrated against a contracting factor in \cref{sec:coldf-entropy}.

\begin{proof}
For $2\le j\le N$, let $s_j=K_{1j}\log\rho_t$ be the weak score of \cref{lem:coldf-scores}. The law $Q_t^\eta$ is invariant under $R_1$ and vanishes on $\{x_1^2<rN/2\}$. Since $s_j\in L^2(Q_t)$ and $|K_{1j}\eta|\le C\eta^{1/2}$ (\cref{lem:coldf-cutoff}), \cref{csh-lem:cut-score} with $p=1$ implies that $Q_t^\eta$ has the weak $K_{1j}$-score $s_j+\ind_{\{\eta>0\}}K_{1j}\log\eta$. We apply \eqref{mc-eq:te14} of \cref{lem:score-marginal-score} with $J=J_N$ and $r_0=r/2$. For $2\le j\le J_N$, almost everywhere on $\{\bar w_t>0\}$, we obtain
\begin{equation}\label{eq:coldf-marginal-terms}
z_j\partial_{z_j}\log\bar w_t=\E_w\Big[\frac{z_j}{x_1}\,s_j\circ x^++\frac{z_j}{x_1}\,(K_{1j}\log\eta)\circ x^++\frac{z_j^2}{x_1^2}\,\Big|\,z\Big]+2a_jz_j^2 ,
\end{equation}
where $\E_w[\,\cdot\mid z]$ is the conditional expectation under $w_tG$ and $x_1=x_1(y)$. The term $z_j^2/x_1^2$ comes from the Jacobian $x_1^{-1}$ of the graph coordinates (\cref{lem:score-chart}), and it has a positive sign. The term $2a_jz_j^2$ comes from the Gaussian reference $G_{\mathrm z}$.

Using the conditional Jensen inequality and $(b_1+b_2+b_3+b_4)^2\le4(b_1^2+b_2^2+b_3^2+b_4^2)$, we see that $\int w_t(z_j\partial_{z_j}\log\bar w_t)^2\,dG$ is at most four times the sum of the second moments under $w_tG$ of the four terms. The functions $|s_j|$, $|K_{1j}\eta|$, $x_j^2$, and $x_1^2$ are invariant under $R_1$ (see the discussion after \cref{lem:score-chart}), and these second moments are equal to the corresponding second moments under $Q_t^\eta$. On the support of $\eta$, we have $x_1^2\ge rN/2$, and $Q_t^\eta\le Q_t/M_t\le2Q_t$ for large $N$, by \eqref{eq:coldf-collar}. We write $\E_Q$ for expectations of functions of $X_t$ under $Q$, and recall that $z_j=x_j$.

\emph{The first term.} By the Cauchy--Schwarz inequality, \eqref{eq:coldf-X2-moment}, and \cref{lem:coldf-scores},
\[
\begin{split}
\E_{Q_t^\eta}\frac{x_j^2s_j^2}{x_1^2}&\le\frac4{rN}\big(\E_QX_j^4\big)^{1/2}\big(\E_Q|s_j|^4\big)^{1/2}\\
&\le C\big[Na_je^{-4a_j(t-T_{\mathrm e})}+1+Na_jp_N^{1/2}\big].
\end{split}
\]
\emph{The second term.} Using $K_{1j}q=-2x_1x_j/N$, we find
\[
\frac{z_jK_{1j}\log\eta}{x_1}=-\frac{16z_j^2}N\,\frac{\chi'(q)}{\chi(q)} .
\]
In the second moment under $Q_t^\eta$, the factor $\eta=\chi^8$ cancels the denominator, and this term contributes at most $C\E_QX_j^4/N^2$. \emph{The third term} contributes at most $8\E_QX_j^4/(r^2N^2)$, since $x_1^4\ge r^2N^2/4$ on the support of $\eta$. \emph{The fourth term} contributes at most $8a_j^2\E_QX_j^4$.

For $2\le j\le J_N$, we have $a_2\le a_j\le Aa_2$ by \cref{lem:coldf-bounded-head}, and \eqref{eq:coldf-X2-moment} gives
\[
\frac{\E_QX_j^4}{N^2}+a_j^2\,\E_QX_j^4\le C\big[1+R_N^2e^{-8a_2(t-T_{\mathrm e})}+R_N^2p_N\big]
\]
and
\[
Na_je^{-4a_j(t-T_{\mathrm e})}+Na_jp_N^{1/2}\le A\big[R_Ne^{-4a_2(t-T_{\mathrm e})}+R_Np_N^{1/2}\big].
\]
As $p_N$ tends to zero faster than any power of $N$, the terms $R_Np_N^{1/2}$ and $R_N^2p_N$ are at most one for large $N$. We conclude that, for $2\le j\le J_N$,
\[
\int w_t\big(z_j\partial_{z_j}\log\bar w_t\big)^2\,dG\le C\big\{1+R_Ne^{-4a_2(t-T_{\mathrm e})}+R_N^2e^{-8a_2(t-T_{\mathrm e})}\big\}.
\]
Finally, $S_t=\sum_{j=2}^{J_N}z_j\partial_{z_j}\log\bar w_t$, and the inequality $(\sum_{j=2}^{J_N}b_j)^2\le(J_N-1)\sum_{j=2}^{J_N}b_j^2$ with $J_N\le M_A$ proves \eqref{cq-sc-reset-marginal-score}.
\end{proof}
\section{Conditional relative entropy of the remaining coordinates}\label{sec:coldf-entropy}\label{cq-tl}

In this section, we show that, for the weighted law multiplied by the cutoff and normalized, at the times $t_u$, the conditional law of the remaining coordinates $v=(x_{J_N+1},\dots,x_N)$ given the retained coordinates $z=(x_2,\dots,x_{J_N})$ is close to the Gaussian law $G_{\mathrm v}$ in relative entropy. We use the notation of \cref{sec:coldf-selected}, and for $t\in(T_{\mathrm e},T_{\mathrm e}+U_2)$ we let
\begin{equation}\label{eq:coldf-HD}
g_t=\log w_t-\log\bar w_t,\qquad
\mathscr H(t)=\int w_tg_t\,dG,\qquad
\mathscr D(t)=\int w_t\,\Gamma(g_t)\,dG,
\end{equation}
where, in graph coordinates, $\Gamma(\varphi)=\nabla\varphi^\T(I-yy^\T/N)\nabla\varphi$ is the carré du champ of $L$. Logarithms and their derivatives are evaluated only on $\{w_t>0\}$, and integrands that contain the factor $w_t$ are set to zero elsewhere. These are the quantities \eqref{eq:score-H-D} of \cref{sec:score-entropy}, with $J=J_N$. In particular, $\mathscr H(t)=\KL\big(w_tG\,\big\Vert\,(\bar w_tG_{\mathrm z})\otimes G_{\mathrm v}\big)\in[0,\infty]$ is the conditional relative entropy of the remaining coordinates given $z$, averaged over the law of $z$. Throughout the section, we assume $\mathrm H(\theta_*,B,c_*,C_*)$, and $C$ denotes constants depending only on $\eps$, the spectral constants, and $\psi$.

The argument has three parts. The initial value $\mathscr H(t_0)$ is of order $N$, by the upper bound on the density (\cref{lem:coldf-initial-entropy}). The conditional relative entropy identity of \cref{sec:scores}, together with \cref{lem:coldf-coarse-lsi}, gives a differential inequality with rate $16a_2$. During the time $t_u-t_0$, which is close to $\log R_N/(4a_2)$, this rate multiplies the entropy by about $R_N^{-4}$, and this beats the initial value $N\le c_*^{-3}R_N^3$.

\begin{lemma}[Initial entropy]\label{lem:coldf-initial-entropy}
For large $N$ and every $t\in(T_{\mathrm e},T_{\mathrm e}+U_2)$,
\begin{equation}\label{eq:coldf-initial-entropy}
0\le\mathscr H(t)\le\KL(w_tG\Vert G)\le BN+\log\frac4{\theta_*(r/2)^{1/2}}\le CN .
\end{equation}
In particular, $\mathscr H(t_0)\le CN$.
\end{lemma}

\begin{proof}
We have $M_t\ge1/2$ by \eqref{eq:coldf-collar}, and $\eta$ vanishes outside $D_{r/2}$. Since $Z_\Psi\ge1/2$, \cref{lem:coldf-density} implies that $w_t\le4e^{BN}/(\theta_*(r/2)^{1/2})$. As $\int w_t\,dG=1$, we have $\KL(w_tG\Vert G)=\int w_t\log w_t\,dG\le\log\sup w_t$, which is the third inequality. Here the positive part of $w_t\log w_t$ is integrable because $w_t$ is bounded, and its negative part is bounded by $1/e$. The chain rule \eqref{eq:score-chain-rule} \cite[Theorem~2.5.3 and Section~8.5]{CoverThomas2006} gives $\KL(w_tG\Vert G)=\KL(\bar w_tG_{\mathrm z}\Vert G_{\mathrm z})+\mathscr H(t)$, and both terms on the right are nonnegative. This proves the first two inequalities, and the last follows from $N\ge1$.
\end{proof}

The bound \eqref{eq:coldf-initial-entropy} concerns the weighted law itself, and it uses no comparison with another process. It is of order $N$, which is of order $R_N^3$, and the rate $16a_2$ below compensates for this.

\paragraph{Logarithms.}
Fix $t\in(T_{\mathrm e},T_{\mathrm e}+U_2)$ and let $\bar f=f_t\ind_{D_{r/2}}$. Then $\int\bar f\,dG\le1$, and \cref{lem:coldf-density} gives $\int\bar f^2\,dG\le e^{CN}$. We use \cref{cce-lem:weighted-logs} with $p=2$, its parameter $A$ equal to $N$, $z_0=1/2$, $\alpha_0=1/2$, and the product $G=G_{\mathrm z}\otimes G_{\mathrm v}$. Since $\eta$ vanishes outside $D_{r/2}$, the functions $w_t$, $\bar w_t$, and $g_t$ are the same for $\bar f$ and for $f_t$, and $\int\eta\bar f\,dG=M_t\ge1/2$. We obtain, for $\alpha\in[1/2,1]$,
\begin{equation}\label{eq:coldf-logs}
\int_{\{w_t>0\}}f_t\,\eta^\alpha\big(|g_t|^2+|\log w_t|^2+|\log\bar w_t|^2\big)\,dG\le CN^2 .
\end{equation}
Together with \eqref{eq:coldf-initial-entropy} and $\int f_t\,dG=1$, this gives
\begin{equation}\label{eq:coldf-logs-centered}
\int f_t\,\eta^{1/2}\big(g_t-\mathscr H(t)\big)^2\,dG\le CN^2 .
\end{equation}

\begin{proposition}[Conditional relative entropy at the times $t_u$]\label{prop:coldf-conditional-entropy}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$, fix $\eps\in(0,1/8)$, and let $\mathscr H$ be as in \eqref{eq:coldf-HD}. For every compact set $K\subset\R$, there exist $C_K,N_K<\infty$, depending only on $K$, $\eps$, $\theta_*$, $B$, $c_*$, $C_*$, and $\psi$, such that for $N\ge N_K$ and $u\in K$,
\begin{equation}\label{eq:coldf-conditional-entropy}
0\le\mathscr H(t_u)\le C_K\Big\{R_N^{-1}e^{16\chi_N}+R_N^5p_N^{1/2}\Big\}.
\end{equation}
The right side tends to zero as $N\to\infty$.
\end{proposition}

The bound is uniform over all spectra that satisfy $\mathrm H(\theta_*,B,c_*,C_*)$ with $N\ge N_K$, because $p_N$ does not depend on the spectrum. It concerns the splitting after the coordinate $x_{J_N}$ of \cref{lem:coldf-bounded-head}, and its constants may deteriorate as $\theta_*\to0$.

\begin{proof}
We write $\mathscr H=\mathscr H(t)$, $\mathscr D=\mathscr D(t)$, and $S=S_t$, and we write $\E_w$ for integration against $w_tG$.

\paragraph{Step 1: the entropy identity.} We apply \cref{prop:score-entropy-identity} with $J=J_N$, the domain $D=S_N$ (no killing), the time $T=T_{\mathrm e}$, the weight $\Psi$ of \eqref{eq:coldf-weight}, the function $\zeta=\chi\circ q$ of \eqref{eq:coldf-eta}, $k=8$, and the interval $\mathcal T=(T_{\mathrm e},T_{\mathrm e}+U_2)$. We check its hypotheses. The weight is $\mathcal F_{T_{\mathrm e}}$-measurable with values in $[0,1]$. The endpoint measure at time $T_{\mathrm e}$, $Z_\Psi Q_{T_{\mathrm e}}$, and the domain $S_N$ are invariant under $R_1$. In graph coordinates, $\zeta$ is smooth, takes values in $[0,1]$, and has compact support in $\{|y|^2<N\}=D^+$, since $\zeta(y)\ne0$ only if $|y|^2<(1-r/2)N$. Finally, $M_t\ge1-p_N>0$ on $\mathcal T$, by \eqref{eq:coldf-collar}. The endpoint subprobability measures of \cref{lem:score-regularity} are $Z_\Psi Q_t$, whose densities in graph coordinates are $Z_\Psi f_t$; the quantities $w_t$, $\bar w_t$, $g_t$, $\mathscr H$, $\mathscr D$, and the identity below are unchanged when $f_t$ and $M_t$ are multiplied by the same constant. By \cref{prop:score-entropy-identity}, $\mathscr H(t)$ and $\mathscr D(t)$ are finite for $t\in\mathcal T$, the function $\mathscr H$ is absolutely continuous on $[t_0,t_1]$, and for almost every $t\in\mathcal T$,
\begin{equation}\label{eq:coldf-entropy-identity}
\mathscr H'=-\mathscr D+\frac1N\E_w\big[(1+S)\,v\cdot\nabla_vg\big]+2\E_w\Gamma(g,\log\eta)+\frac1{M_t}\int f_t(g-\mathscr H)\,L\eta\,dG .
\end{equation}
The second term on the right comes from the drift $N^{-1}y\cdot\nabla$, by which the generator in graph coordinates differs from the symmetric operator $L_G$ (see \eqref{mc-eq:te20}); the coefficient one in the factor $1+S$ is the coefficient of this drift. The terms that contain only derivatives in $z$ cancel, because $\int w_t\nabla_zg_t\,dG_{\mathrm v}=0$ for every $z$ with $\bar w_t(z)>0$.

The proof of \cref{prop:score-entropy-identity} uses only the stated $C^2$ spatial and $C^1$ time regularity of $f_t$ near the support of $\zeta$, and no lower bound on $f_t$. For completeness, we recall its steps in the present setting. Fix a compact interval $\mathcal I\subset\mathcal T$ and a compact neighborhood $\mathcal O\subset U_N$ of $D_{r/2}$. By \cref{lem:coldf-selected-density}, the functions $f_t$, $\nabla f_t$, $\nabla^2f_t$, and $\partial_tf_t$ are bounded on $\mathcal I\times\mathcal O$, and $M_t$ is bounded below by a positive constant on $\mathcal I$. Let $d_0$ be the distance from $D_{r/2}$ to the complement of $\mathcal O$, and let $M_2>0$ bound $\nabla^2f_t$ on $\mathcal I\times\mathcal O$. On $D_{r/2}$, Taylor's formula along $-\nabla f_t/|\nabla f_t|$ gives $0\le f_t-\varsigma|\nabla f_t|+M_2\varsigma^2/2$ for $0\le\varsigma\le d_0$. The choice $\varsigma=|\nabla f_t|/M_2$ if this is at most $d_0$, and $\varsigma=d_0$ otherwise, yields $|\nabla f_t|^2\le C'f_t$ on $D_{r/2}$. In this paragraph, the constant $C'$ may depend on $N$ and $\mathcal I$; such constants justify the identity, and they do not enter the estimates below. Expanding $w_t=\zeta^8f_t/M_t$, we deduce that, uniformly for $t\in\mathcal I$,
\begin{equation}\label{eq:coldf-fisher-finite}
\int\frac{|\nabla w_t|^2}{w_t}\,dG<\infty,\qquad
\int\frac{|\nabla\bar w_t|^2}{\bar w_t}\,dG_{\mathrm z}\le\int w_t\,|\nabla_z\log w_t|^2\,dG<\infty .
\end{equation}
Here the first integrand is bounded by constants times $f_t\zeta^6|\nabla\zeta|^2$ and $\zeta^8|\nabla f_t|^2/f_t$, and the second inequality is the Cauchy--Schwarz inequality on each fiber. In particular, $\mathscr D(t)$ is finite. For $0<e\le1$, let $\Phi_e(\varsigma)=(\varsigma+e)\log(\varsigma+e)-\varsigma$, $g_e=\log(w_t+e)-\log(\bar w_t+e)$, and $\mathscr H_e(t)=\int\Phi_e(w_t)\,dG-\int\Phi_e(\bar w_t)\,dG_{\mathrm z}$. Then $\mathscr H_e'=\int\partial_tw_t\,g_e\,dG$. Using \eqref{eq:coldf-forward}, integration by parts with the compactly supported function $\eta g_e$, and $M_t'=\int f_tL\eta\,dG$, we find
\begin{equation}\label{eq:coldf-regularized}
\begin{split}
\mathscr H_e'={}&-\int\Gamma(w_t,g_e)\,dG+\int w_t\,\frac yN\cdot\nabla g_e\,dG+\frac1{M_t}\int f_tg_e\,L\eta\,dG\\
&+\frac2{M_t}\int f_t\,\Gamma(\eta,g_e)\,dG-\frac{M_t'}{M_t}\int w_tg_e\,dG .
\end{split}
\end{equation}
As $e\downarrow0$, the integrands converge pointwise, and they are dominated by means of \eqref{eq:coldf-fisher-finite}, \eqref{eq:coldf-logs} with $\alpha=3/4$, the bound $|L\eta|\le C\zeta^6$, and $|\log(\varsigma+e)|\le|\log\varsigma|+\log2$ for $\varsigma>0$. Here we used that the first derivatives of $f_t$, $w_t$, and $\bar w_t$ vanish where these functions vanish, that $f_t=0$ on $\{\eta>0,\ w_t=0\}$, and that $L\eta=0$ where $\zeta=0$. Integrating \eqref{eq:coldf-regularized} over subintervals of $\mathcal I$ and letting $e\downarrow0$ by dominated convergence, we find that $\mathscr H$ is absolutely continuous on $\mathcal I$, and that its derivative is, almost everywhere, the limit of the right side of \eqref{eq:coldf-regularized}. The identity $\int w_t\nabla_zg_t\,dG_{\mathrm v}=0$ and the expansion of $\Gamma$ in the variables $(z,v)$ turn this limit into \eqref{eq:coldf-entropy-identity}. This is the classical entropy dissipation calculation \cite[Section~5.2]{BakryGentilLedoux2014}, with the additional conditional and cutoff terms, which are derived in the proof of \cref{prop:score-entropy-identity}.

\paragraph{Step 2: the drift term.} On the support of $\eta$, we have $q\ge r/2$, and \eqref{eq:coldf-metric-cap} gives $|\nabla g|^2\le2\Gamma(g)/r$. Since $|v|^2\le N$, we have $|v\cdot\nabla_vg|^2\le2N\Gamma(g)/r$ there. The Cauchy--Schwarz inequality and $xy\le x^2/8+2y^2$ give
\begin{equation}\label{eq:coldf-drift-term}
\begin{split}
\frac1N\big|\E_w[(1+S)\,v\cdot\nabla_vg]\big|&\le\mathscr D^{1/2}\Big(\frac{2\E_w(1+S)^2}{rN}\Big)^{1/2}\\
&\le\frac{\mathscr D}8+\frac{4\E_w(1+S)^2}{rN}\le\frac{\mathscr D}8+\frac CN\big(1+\mathsf B(t)\big).
\end{split}
\end{equation}
In the last step, we used $(1+S)^2\le2+2S^2$ and the definition \eqref{eq:coldf-B}.

\paragraph{Step 3: the terms containing the cutoff.} By the Cauchy--Schwarz inequality, $2xy\le x^2/8+8y^2$, \cref{lem:coldf-cutoff}, $M_t\ge1/2$, and \eqref{eq:coldf-collar},
\begin{equation}\label{eq:coldf-cutoff-term}
2\big|\E_w\Gamma(g,\log\eta)\big|\le\frac{\mathscr D}8+8\E_w\Gamma(\log\eta)=\frac{\mathscr D}8+\frac8{M_t}\int f_t\,\frac{\Gamma(\eta)}\eta\,dG\le\frac{\mathscr D}8+\frac CN\,p_N .
\end{equation}
Here we used that $\mathcal K$ is invariant under $R_1$, so that $\int f_t\ind_{\mathcal K}\,dG=Q(X_t\in\mathcal K)\le p_N$. For the last term in \eqref{eq:coldf-entropy-identity}, we write $\eta^{3/4}=\eta^{1/4}\eta^{1/2}$, and we use \cref{lem:coldf-cutoff}, the Cauchy--Schwarz inequality, \eqref{eq:coldf-logs-centered}, and \eqref{eq:coldf-collar}:
\begin{equation}\label{eq:coldf-generator-term}
\begin{split}
\Big|\frac1{M_t}\int f_t(g-\mathscr H)\,L\eta\,dG\Big|&\le C\Big(\int f_t\eta^{1/2}(g-\mathscr H)^2\,dG\Big)^{1/2}\Big(\int f_t\eta\ind_{\mathcal K}\,dG\Big)^{1/2}\\
&\le CN\,p_N^{1/2}.
\end{split}
\end{equation}
All these estimates concern the same weighted law, and no relative entropy is transferred through a total variation bound.

\paragraph{Step 4: the differential inequality.} Combining \eqref{eq:coldf-entropy-identity}, \eqref{eq:coldf-drift-term}, \eqref{eq:coldf-cutoff-term}, \eqref{eq:coldf-generator-term}, and \cref{lem:coldf-marginal-score}, we obtain, for almost every $t\in[t_0,t_1]$,
\begin{equation}\label{eq:coldf-entropy-ineq}
\mathscr H'\le-\frac34\mathscr D+\frac CN\Big\{1+R_Ne^{-4a_2(t-T_{\mathrm e})}+R_N^2e^{-8a_2(t-T_{\mathrm e})}\Big\}+CN\,p_N^{1/2}.
\end{equation}
For each $t\in[t_0,t_1]$, the function $w_t$ is $C^2$, has compact support in $U_N$, and vanishes on $\{q<r/2\}$, and $a_j>Aa_2$ for $j>J_N$ by \cref{lem:coldf-bounded-head}. Then \eqref{eq:coldf-lsi} implies that $\mathscr D\ge32a_2\mathscr H$, and since $\mathscr H\ge0$, we have $-3\mathscr D/4\le-24a_2\mathscr H\le-16a_2\mathscr H$. We conclude that, for almost every $t\in[t_0,t_1]$,
\begin{equation}\label{eq:coldf-entropy-contraction}
\mathscr H'\le-16a_2\mathscr H+\frac CN\Big\{1+R_Ne^{-4a_2(t-T_{\mathrm e})}+R_N^2e^{-8a_2(t-T_{\mathrm e})}\Big\}+CN\,p_N^{1/2}.
\end{equation}
The threshold $A=16/r$ compensates for the factor $r/2$ lost in \eqref{eq:coldf-metric-cap}. No comparison of metrics finer than \eqref{eq:coldf-metric-cap} is used.

\paragraph{Step 5: integration.} Let $K\subset\R$ be compact, let $u\in K$, and let $S_u=a_2(t_u-T_{\mathrm e})=\log R_N/4+u-\chi_N$. For $N$ larger than a constant depending on $K$, we have $1\le S_u\le\log R_N$, that is, $t_0\le t_u\le t_1$. Since $\mathscr H$ is absolutely continuous on $[t_0,t_1]$, so is $e^{16a_2t}\mathscr H(t)$, and by \eqref{eq:coldf-entropy-contraction} its derivative is at most $e^{16a_2t}$ times the last two terms of \eqref{eq:coldf-entropy-contraction}, almost everywhere. Integrating from $t_0$ to $t_u$, changing variables to $\varsigma=a_2(t-T_{\mathrm e})$, and using $1/(Na_2)=1/R_N$ and \cref{lem:coldf-initial-entropy}, we obtain
\begin{equation}\label{eq:coldf-entropy-integrated}
\begin{split}
\mathscr H(t_u)\le{}&CN\,e^{-16(S_u-1)}+\frac C{R_N}\int_1^{S_u}e^{-16(S_u-\varsigma)}\big(1+R_Ne^{-4\varsigma}+R_N^2e^{-8\varsigma}\big)\,d\varsigma\\
&+\frac{CN}{16a_2}\,p_N^{1/2}.
\end{split}
\end{equation}
Only the inequality \eqref{eq:coldf-entropy-contraction} almost everywhere is used here. Factoring out $e^{-16S_u}$, we see that the three parts of the integral are at most $C/(16R_N)$, $Ce^{-4S_u}/12$, and $CR_Ne^{-8S_u}/8$. By \cref{lem:coldf-consequences}, we have $c_*N^{1/3}\le R_N\le C_*N^{1/3}$, and therefore
\begin{equation}\label{eq:coldf-N-R}
N\le c_*^{-3}R_N^3,\qquad\frac N{a_2}=\frac{N^2}{R_N}\le c_*^{-6}R_N^5 .
\end{equation}
Using $e^{-4S_u}=R_N^{-1}e^{4(\chi_N-u)}$, we find that the first term of \eqref{eq:coldf-entropy-integrated} is at most $Cc_*^{-3}e^{16}R_N^{-1}e^{16(\chi_N-u)}$, that the second and third parts of the integral are at most $CR_N^{-1}e^{4(\chi_N-u)}/12$ and $CR_N^{-1}e^{8(\chi_N-u)}/8$, and that the last term is at most $Cc_*^{-6}R_N^5p_N^{1/2}/16$. As $\chi_N\ge0$ and $|u|$ is bounded on $K$, we obtain \eqref{eq:coldf-conditional-entropy} from these bounds.

Finally, $\log R_N\ge(\log N)/3+\log c_*$ and $16\chi_N=16\sqrt{\log\log N}=o(\log N)$, which implies that $R_N^{-1}e^{16\chi_N}\to0$. Further, $R_N^5p_N^{1/2}\to0$, because $p_N$ tends to zero faster than any power of $N$.
\end{proof}

The rate $16a_2$ in \eqref{eq:coldf-entropy-contraction} is four times the rate $4a_2$ that sets the profile. The initial entropy is of order $N\asymp R_N^3$, and it is multiplied by $e^{-16a_2(t_u-t_0)}=e^{16}R_N^{-4}e^{16(\chi_N-u)}$, which leaves a quantity of order $R_N^{-1}e^{16\chi_N}$. This is why we retain all coordinates with $a_j\le16a_2/r$: the inequality \eqref{eq:coldf-lsi} loses the factor $r/2$ from the metric on the cap, and the threshold restores the rate. The lower bound on $a_3/a_2$ in $\mathrm H_3(\varpi,D_3)$ is not used in this section.
\section{Finitely many coordinates in total variation}\label{sec:coldf-head}\label{cq-hc}

In this section, we upgrade the weak convergence of finitely many standardized coordinates in \cref{thm:coldf-finite-coordinates} to convergence in total variation. The tool is \cref{csh-thm:weak-to-tv}: a weak limit with finite Fisher information, together with laws of bounded Fisher information that are close in total variation, gives convergence in total variation. We obtain such laws from the weighted law of \cref{sec:coldf-selected} and the cutoff, with a loss that is sent to zero after $N\to\infty$. Their Fisher information is bounded through the scores of \cref{lem:coldf-scores}, which use only the last time interval of length $a_j^{-1}$ before $t_u$. The weak limit is a Gaussian convolution, although it has no finite variance.

\begin{proposition}[Finitely many coordinates in total variation]\label{prop:coldf-finite-head-tv}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$ and $\theta_N\to\theta\in(0,1)$. Fix $k\ge2$, and if $k\ge3$ assume also $\mathrm H_k(\varpi,D_k)$. Let $G_2,\dots,G_k$ be independent standard Gaussian variables and $C$ an independent standard Cauchy variable. Then for every fixed $u\in\R$,
\begin{equation}\label{eq:coldf-finite-head-tv}
\Big\|\Law\big((\sqrt{2a_j}\,X_j(t_u))_{j=2}^k\big)-\Law\big(G_2+\sqrt{2\theta}\,e^{-2u}C,G_3,\dots,G_k\big)\Big\|_{\TV}\longrightarrow0 .
\end{equation}
\end{proposition}

Like the other results of \cref{sec:cold-fixed}, \cref{prop:coldf-finite-head-tv} holds along every increasing sequence of dimensions, since its proof uses only limits as $N\to\infty$. No convergence of the ratios $a_j/a_2$ is assumed.

\begin{proof}
Let $\mathbf z_N=(\sqrt{2a_j}\,X_j(t_u))_{j=2}^k$, let $\mu_N$ be its law, and let $\mu$ be the law of $(G_2+\sqrt{2\theta}e^{-2u}C,G_3,\dots,G_k)$. \Cref{thm:coldf-finite-coordinates} implies that $\mu_N\Rightarrow\mu$. The law $\mu$ is the convolution of the standard Gaussian law on $\R^{k-1}$ with the law of $(\sqrt{2\theta}e^{-2u}C,0,\dots,0)$, and \cref{csh-lem:convolution-fisher} implies that $\mathrm{FI}(\mu)\le k-1$; no moment of $C$ is needed (see \cref{rem:score-convolution-limits}). This is the classical Fisher information bound for a Gaussian convolution, related to the Stam--Blachman convolution inequality and its score proof \cite{Stam1959,Blachman1965}. By the second statement of \cref{csh-thm:weak-to-tv}, it suffices to construct, for every $\delta\in(0,1/8)$, probability measures $\widehat\mu_N^\delta$ on $\R^{k-1}$ such that $\limsup_N\|\mu_N-\widehat\mu_N^\delta\|_{\TV}\le\delta$ and $\mathrm{FI}(\widehat\mu_N^\delta)\le C_\delta$ for large $N$, where $C_\delta$ does not depend on $N$.

Fix $\delta\in(0,1/8)$. We repeat the construction of the weight and of the cutoff in \cref{sec:coldf-selected} with $\delta$ in place of $\eps$, and we keep $k$ fixed. This gives a weight $\Psi^\delta$ with mean $Z^\delta\ge1-\delta$, the law $Q^\delta$ with the time marginals $Q^\delta_t$ and their densities $\rho^\delta_t=dQ^\delta_t/d\nu_N$, a cap parameter $r^\delta$, a cutoff $\eta^\delta$ that vanishes on $\{q\le r^\delta/2\}$, its mass $M^\delta_t=\E_{Q^\delta}\eta^\delta(X_t)$, and the sequence $p^\delta_N$ of \eqref{eq:coldf-pN}. Below, $t=t_u$, which lies in $[t_0,t_1]$ for large $N$, and $C_\delta$ denotes constants depending only on $\delta$, $k$, and the spectral constants. Let $\widehat\rho=\eta^\delta Q_t^\delta/M_t^\delta$, and let $\widehat\mu_N^\delta$ be the law of $\mathbf z_N$ under $\widehat\rho$.

\paragraph{Total variation.} The law $\widehat\mu_N^\delta$ is the law of $\mathbf z_N$ under $(U/\E U)P$, where $U=\Psi^\delta\eta^\delta(X_t)$ takes values in $[0,1]$ and $\E U=Z^\delta M^\delta_t$. By \cref{lem:score-domination} and \cref{lem:coldf-collar},
\[
\|\mu_N-\widehat\mu_N^\delta\|_{\TV}\le1-Z^\delta M^\delta_t\le(1-Z^\delta)+(1-M^\delta_t)\le\delta+p^\delta_N ,
\]
and $p^\delta_N\to0$. We do not identify a weak limit of $\widehat\mu_N^\delta$.

\paragraph{Scores.} For $2\le j\le k$, let $s_j=K_{1j}\log\rho^\delta_t$. By \cref{lem:coldf-scores} and the Cauchy--Schwarz inequality, $\E_{Q^\delta}s_j^2\le C_\delta Na_j$. By \cref{csh-lem:cut-score} with $p=1$ and \cref{lem:coldf-cutoff}, the law $\widehat\rho$ has the weak $K_{1j}$-score $\vartheta_j=s_j+\ind_{\{\eta^\delta>0\}}K_{1j}\log\eta^\delta$, and
\[
\E_{\widehat\rho}\vartheta_j^2\le\frac2{M^\delta_t}\big(\E_{Q^\delta}s_j^2+C_\delta\big)\le C_\delta(Na_j+1).
\]

\paragraph{Fisher information.} The law $\widehat\rho$ is invariant under $R_1$ and vanishes on $\{x_1^2<r^\delta N/2\}$. Let $f^+$ be the Lebesgue density of $\widehat\rho^+$, extended by zero outside $U_N$. By the third part of \cref{lem:score-chart}, for $2\le j\le k$, the weak derivative of $f^+$ in $y_j$ is $f^+(\vartheta_j\circ x^+/x_1+y_j/x_1^2)$, and it vanishes where $f^+=0$. We change variables to $(\mathbf z,v)$, with $\mathbf z_j=\sqrt{2a_j}\,y_j$ for $2\le j\le k$ and $v=(y_{k+1},\dots,y_N)$, and we let $\tilde f(\mathbf z,v)=f^+(y)\prod_{j=2}^k(2a_j)^{-1/2}$ be the joint density. Then $\partial_{\mathbf z_j}=(2a_j)^{-1/2}\partial_{y_j}$, and on $\{\tilde f>0\}$,
\[
|\partial_{\mathbf z_j}\log\tilde f|^2\le\frac1{a_j}\Big(\frac{(\vartheta_j\circ x^+)^2}{x_1^2}+\frac{y_j^2}{x_1^4}\Big).
\]
On the support of $\widehat\rho$, we have $x_1^2\ge r^\delta N/2$ and $y_j^2\le N$. Integrating, and using that $|\vartheta_j|$ is invariant under $R_1$ (see the discussion after \cref{lem:score-chart}), we find
\[
\int\tilde f\,|\partial_{\mathbf z_j}\log\tilde f|^2\,d\mathbf z\,dv\le\frac{2\E_{\widehat\rho}\vartheta_j^2}{r^\delta Na_j}+\frac4{(r^\delta)^2Na_j}\le C_\delta\Big(1+\frac1{Na_j}\Big)\le C_\delta ,
\]
since $Na_j\ge R_N\ge1$ for large $N$. Let $p(\mathbf z)=\int\tilde f(\mathbf z,v)\,dv$ be the density of $\widehat\mu_N^\delta$. As in the proof of \cref{csh-lem:graph-fisher}, we have $\partial_jp(\mathbf z)=\int\partial_{\mathbf z_j}\tilde f(\mathbf z,v)\,dv$, and the Cauchy--Schwarz inequality on each fiber gives
\[
\frac{|\partial_jp(\mathbf z)|^2}{p(\mathbf z)}\le\int\tilde f(\mathbf z,v)\,|\partial_{\mathbf z_j}\log\tilde f(\mathbf z,v)|^2\,dv
\]
where $p(\mathbf z)>0$, and $\partial_jp=0$ where $p=0$. Integrating in $\mathbf z$ and summing over $j$, we obtain $\mathrm{FI}(\widehat\mu_N^\delta)\le(k-1)C_\delta$ for large $N$.

Applying the second statement of \cref{csh-thm:weak-to-tv}, we obtain \eqref{eq:coldf-finite-head-tv}. In its proof, the Gaussian smoothing scale is sent to zero after $N\to\infty$ and before $\delta\downarrow0$, and the translation estimate \eqref{eq:score-translation} rests on the Sobolev approximation of \cite[Theorem~9.2, p.~265]{Brezis2011}. The constants $C_\delta$ need not be bounded as $\delta\downarrow0$.
\end{proof}

The weighted laws are used only to obtain laws of bounded Fisher information that are close to $\mu_N$ in total variation. The limit in \eqref{eq:coldf-finite-head-tv} concerns the law of the diffusion started from $\nu_N$. The second coordinate is the case $k=2$.

\begin{theorem}[The second coordinate in total variation]\label{cq-hc-head-smoothing}
Assume $\mathrm H(\theta_*,B,c_*,C_*)$ and $\theta_N\to\theta\in(0,1)$. Let $G$ be a standard Gaussian variable and $C$ an independent standard Cauchy variable. Then for every fixed $u\in\R$,
\begin{equation}\label{eq:coldf-head-tv}
\big\|\Law\big(\sqrt{2a_2}\,X_2(t_u)\big)-\Law\big(G+\sqrt{2\theta}\,e^{-2u}C\big)\big\|_{\TV}\longrightarrow0 .
\end{equation}
\end{theorem}

\begin{proof}
This is the case $k=2$ of \cref{prop:coldf-finite-head-tv}, in which $\mathrm H_k(\varpi,D_k)$ is not assumed.
\end{proof}

\Cref{cq-hc-head-smoothing} concerns the law of $X_2(t_u)$ under the whole initial law $\nu_N$. The weight and the cutoff are used only to obtain a score bound for a law that is close to it in total variation; we do not identify a weak limit of the weighted law.

\section{Proof of the deterministic theorem}\label{sec:coldf-assembly}\label{cq-jp}

\begin{proof}[Proof of \cref{thm:coldf-deterministic}]
\emph{Pointwise convergence.} Fix $u\in\R$ and $\eps\in(0,1/8)$, and use the notation of \cref{sec:coldf-selected,sec:coldf-entropy} with this $\eps$. In particular, $J_N\in\{2,\dots,M_A\}$ is the index of the last retained coordinate. We first show that
\begin{equation}\label{eq:coldf-assembly-subsequence}
\limsup_{N\to\infty,\ N\in\mathcal N_J}\big|D_N(t_u)-\cV_\theta(u)\big|\le2\eps
\end{equation}
for every $J$ such that the set $\mathcal N_J=\{N:J_N=J\}$ is infinite. Fix such a $J$. If $J\ge3$, then $a_3/a_2\ge1+\varpi$ by $\mathrm H_3(\varpi,D_3)$ and $a_J/a_2\le A$ by \cref{lem:coldf-bounded-head}, so that the spectra with $N\in\mathcal N_J$ satisfy $\mathrm H_J(\varpi,A)$. For $N\in\mathcal N_J$, we use \cref{ccf-lem:finite-product-closure} on the product space $\R^{J-1}\times\R^{N-J}$ of the coordinates $(z,v)$, where measures on the ball $\{|y|^2\le N\}$ are extended by zero, with the product probability $G=G_{\mathrm z}\otimes G_{\mathrm v}$ and the following choices:
\begin{itemize}
\item $\rho=(\nu_NP_{t_u})^+$, the law of $(X_2,\dots,X_N)(t_u)$;
\item $\sigma(E)=\E_P\big[\Psi\,\eta(X_{t_u})\ind_{\{(X_2,\dots,X_N)(t_u)\in E\}}\big]$ for Borel sets $E$, which satisfies $\sigma\le\rho$ since $0\le\Psi\eta\le1$, and has mass $Z_\Psi M_{t_u}$, so that $\sigma/(Z_\Psi M_{t_u})=(Q^\eta_{t_u})^+=w_{t_u}G$;
\item the target law $\mu$ of $((2a_j)^{-1/2}\Gamma_j)_{j=2}^J$ for the first factor, where $(\Gamma_2,\dots,\Gamma_J)=(G_2+\sqrt{2\theta}e^{-2u}C,G_3,\dots,G_J)$ with $G_j$ and $C$ as in \cref{prop:coldf-finite-head-tv}, and the folded Gibbs measure $\pi^+$ in the role of $\pi$.
\end{itemize}
With these choices, the relative entropy $H_{\mathrm v}$ in \cref{ccf-lem:finite-product-closure} is $\KL(w_{t_u}G\Vert(\bar w_{t_u}G_{\mathrm z})\otimes G_{\mathrm v})=\mathscr H(t_u)$, and the first marginal $\rho_{\mathrm z}$ of $\rho$ is the law of $(X_2,\dots,X_J)(t_u)$ under the original law $P$, without weight. By \eqref{ccf-eq:finite-profile-closure},
\begin{equation}\label{eq:coldf-assembly}
\Big|\|\rho-\pi^+\|_{\TV}-\|\mu-G_{\mathrm z}\|_{\TV}\Big|\le2(1-Z_\Psi M_{t_u})+\sqrt{\mathscr H(t_u)/2}+\|\rho_{\mathrm z}-\mu\|_{\TV}+\|\pi^+-G\|_{\TV}.
\end{equation}
We identify each term. Both $\nu_NP_{t_u}$ and $\pi$ are invariant under $R_1$, and \eqref{eq:coldf-fold} gives $\|\rho-\pi^+\|_{\TV}=D_N(t_u)$. The map $(y_j)_{j=2}^J\mapsto(\sqrt{2a_j}\,y_j)_{j=2}^J$ is a bijection of $\R^{J-1}$ that maps $G_{\mathrm z}$ to the standard Gaussian law and $\mu$ to the law of $(\Gamma_2,\dots,\Gamma_J)$. For probability measures $\alpha,\alpha'$ on $\R$ and a probability measure $\beta$ on $\R^{J-2}$, we have $\|\alpha\otimes\beta-\alpha'\otimes\beta\|_{\TV}=\|\alpha-\alpha'\|_{\TV}$, as in the proof of \cref{ccf-lem:finite-product-closure}. As $\Gamma_3,\dots,\Gamma_J$ are standard Gaussian variables independent of $\Gamma_2$, we obtain
\begin{equation}\label{eq:coldf-assembly-profile}
\|\mu-G_{\mathrm z}\|_{\TV}=\big\|\Law\big(G_2+\sqrt{2\theta}e^{-2u}C\big)-\Law(G_2)\big\|_{\TV}=\cV_\theta(u).
\end{equation}
Using \eqref{eq:coldf-restriction-loss} and \cref{lem:coldf-collar}, we find $1-Z_\Psi M_{t_u}\le(1-Z_\Psi)+(1-M_{t_u})\le\eps+p_N$. \Cref{prop:coldf-conditional-entropy} implies that $\mathscr H(t_u)\to0$. By \cref{prop:coldf-finite-head-tv} with $k=J$, applied along $\mathcal N_J$, and the invariance of total variation under the above bijection, $\|\rho_{\mathrm z}-\mu\|_{\TV}\to0$ along $\mathcal N_J$; for $J\ge3$, this uses $\mathrm H_J(\varpi,A)$, and with it the lower bound on $a_3/a_2$. Finally, $\|\pi^+-G\|_{\TV}\to0$ by \cref{mesocold-prop:stationary-graph} with $D=D_3$, the constant $C_0$ of \cref{lem:coldf-consequences}, and $m_N=\theta_NR_N\ge\theta_*c_*N^{1/3}$; this uses the upper bound $a_3\le D_3a_2$. We obtain \eqref{eq:coldf-assembly-subsequence}.

The index $J_N$ takes at most $M_A-1$ values. If $\limsup_N|D_N(t_u)-\cV_\theta(u)|$ were larger than $2\eps$, there would exist $\kappa>0$ and an infinite set of dimensions $N$ with $|D_N(t_u)-\cV_\theta(u)|\ge2\eps+\kappa$. One of the values of $J_N$ would be taken for infinitely many of these $N$, which contradicts \eqref{eq:coldf-assembly-subsequence}. We conclude that
\[
\limsup_N\big|D_N(t_u)-\cV_\theta(u)\big|\le2\eps .
\]
All the estimates used here hold with constants fixed after $\eps$ and before $N\to\infty$, and we do not assume that $J_N$ converges. Letting $\eps\downarrow0$, we obtain $D_N(t_u)\to\cV_\theta(u)$ for every fixed $u$.

\emph{Uniform convergence and mixing times.} The function $D_N$ is nonincreasing by the second part of \cref{lem:model-tv}, and $D_N(t)\to0$ as $t\to\infty$, by the argument in the proof of \cref{cor:model-profile-cutoff}, which applies to the generator $L$ of this section. For large $N$, we have $a_2T_N''=\log R_N/4\to\infty$ by \cref{lem:coldf-consequences}, and $\cV_\theta$ is continuous and strictly decreasing from one to zero by \cref{cot-profile}. The deterministic version of \cref{hp-monotone-profile-transfer}, with $b_N=T_N''$, $w_N=a_2^{-1}$, and $V=\cV_\theta$, gives the first two parts of \cref{thm:coldf-deterministic}.

\emph{The recentered profile.} For large $N$, we have $\theta_N>0$ and $2\theta_NR_N>1$, so that $T_N=\log(2\theta_NR_N)/(4a_2)$ and $T_N+u/a_2=t_{u+\ell_N}$ with $\ell_N=\log(2\theta_N)/4\to\log(2\theta)/4$. By the first part and the continuity of $\cV_\theta$, $D_N(T_N+u/a_2)\to\cV_\theta(u+\log(2\theta)/4)=\cV(u)$ for every $u$, where the last identity is the third part of \cref{cot-profile}. As $a_2T_N=\log(2\theta_NR_N)/4\to\infty$, \cref{hp-monotone-profile-transfer} with $b_N=T_N$ and $V=\cV$ gives the third part.
\end{proof}

The proof compares four measures on the same space: the law of the diffusion at time $t_u$ in graph coordinates; the measure obtained from it by weighting with $\Psi\eta(X_{t_u})$, normalized; the product of the marginal of this normalized measure on the retained coordinates $(x_2,\dots,x_{J_N})$ with $G_{\mathrm v}$; and the Gibbs measure in graph coordinates. Relative entropy is used only for the normalized weighted law $Q^\eta_{t_u}$, for which it was estimated directly. The weight is removed in total variation, through \eqref{ccf-eq:finite-profile-closure}, and the marginal on $(x_2,\dots,x_{J_N})$ that enters there is that of the diffusion itself.