\chapter{Score bounds, smoothing, and conditional relative entropy}\label{sec:scores}

In this section, we collect the analytic tools that we use to prove upper bounds on the total variation distance at low temperature, in \Cref{sec:cold-fixed} (fixed $c>1$) and \Cref{sec:cold-meso} (the approach to the critical temperature from below). In both regimes, we compare the time-$t$ law with the product of a law of finitely many eigencoordinates and a Gaussian law of the remaining coordinates. The results below control the two factors and their combination.

In \Cref{sec:score-setting}, we define weak scores along rotations and record three elementary facts: the effect of a cutoff function on a score (\Cref{csh-lem:cut-score}), the total variation cost of a path weight (\Cref{lem:score-domination}), and the smoothness of weighted and killed endpoint measures (\Cref{lem:score-regularity}). \Cref{sec:score-bismut} proves a Bismut-type formula for the score over a final time interval: \Cref{csr-thm:reset} bounds the moments of the derivative along a rotation of the logarithm of the density of a weighted endpoint law, using only the final time interval, and \Cref{csr-cor:cold-scale} specializes it to the time scale $a_2^{-1}$. In \Cref{sec:score-smoothing}, we show that weak convergence of finite-dimensional laws implies convergence in total variation when nearby laws have bounded Fisher information (\Cref{csh-thm:weak-to-tv}). \Cref{sec:score-coordinates} combines these results and proves total variation convergence of finitely many standardized eigencoordinates (\Cref{csh-cor:cold-head}). \Cref{sec:score-entropy} concerns the conditional relative entropy of the remaining coordinates given the first ones. It contains an exact identity for its time derivative (\Cref{prop:score-entropy-identity}), a logarithmic Sobolev inequality for it (\Cref{prop:score-conditional-lsi}), and an entropy dissipation inequality under a cutoff function adapted to the approach to the critical temperature (\Cref{cce-thm:consumer}). Finally, \Cref{ccf-lem:finite-product-closure} in \Cref{sec:score-product} combines the two factors in total variation.

Besides the conventions of \Cref{sec:model}, in particular \Cref{lem:model-tv,lem:model-symmetry}, we use only classical tools: Itô's formula, Lévy's characterization of Brownian motion, Hörmander's hypoellipticity theorem, and the Gaussian logarithmic Sobolev inequality. The special case of Girsanov's theorem that we need is proved in \Cref{sec:score-bismut}. All statements are deterministic in the coefficients of the potential. Probabilities and expectations refer only to the initial point and the driving Brownian motion. \Cref{sec:cold-fixed} uses \Cref{csr-thm:reset}, \Cref{csh-thm:weak-to-tv,csh-lem:convolution-fisher}, the results of \Cref{sec:score-coordinates,sec:score-chart-entropy,sec:score-weighted-logs,sec:score-entropy-identity}, and \Cref{ccf-lem:finite-product-closure}. \Cref{sec:cold-meso} uses \Cref{csh-cor:cold-head}, \Cref{cce-thm:consumer}, and \Cref{ccf-lem:finite-product-closure}; the hypotheses of these statements are verified there.

\section{Setting, weak scores, and weighted endpoint measures}\label{sec:score-setting}

\paragraph{The diffusion.}
Throughout this section, $a_1,\dots,a_N$ are real numbers, coordinates refer to a fixed orthonormal basis of $\R^N$, and
\begin{equation}\label{eq:score-potential}
F(x)=\sum_{k=1}^Na_kx_k^2,\qquad L=\Delta_{S_N}-\nabla_{S_N}F\cdot\nabla_{S_N}.
\end{equation}
For the model at low temperature, the coordinates are the eigencoordinates and the $a_k=c(\lambda_1-\lambda_k)/2$ are the scaled eigenvalue gaps of \eqref{eq:model-raw}. More generally, by \eqref{eq:model-shift}, the generator $L_A$ of \eqref{eq:model-general} is of the form \eqref{eq:score-potential} with $a_k=(\alpha_1-\alpha_k)/2$ in an eigenbasis of $A$. As in \eqref{eq:model-sde}, the diffusion generated by $L$ solves
\begin{equation}\label{mc-eq:sr1}
dX_s=\sqrt2\,\Pi(X_s)\,dB_s+\mathbf b(X_s)\,ds,\qquad
\mathbf b(x)=-\frac{N-1}{N}\,x-\nabla_{S_N}F(x),\qquad
\Pi(x)=I-\frac{xx^\T}{N},
\end{equation}
where $\nabla_{S_N}F(x)=\Pi(x)\nabla F(x)$. We always work on a filtered probability space with a filtration $(\mathcal F_s)_{s\ge0}$, on which $B$ is an $(\mathcal F_s)$-Brownian motion in $\R^N$ and $X$ is $(\mathcal F_s)$-adapted, with $X_0\sim\nu_N$. All process-specific statements in this chapter concern this original uniformly initialized diffusion; the statements about probability measures and weak scores are independent of a diffusion. Solutions of \eqref{mc-eq:sr1} remain on $S_N$, by the computation following \eqref{eq:model-sde}. The coefficients are smooth and may be modified outside a neighborhood of $S_N$ to become globally Lipschitz. This implies pathwise uniqueness, and that, for $u\ge0$, the path $(X_s)_{s\ge u}$ is a measurable function of $X_u$ and of the increments of $B$ after time $u$.

\paragraph{Rotations and reflections.}
For $i\ne j$, let $\Omega_{ij}=e_je_i^\T-e_ie_j^\T$ and $\mathcal R^{ij}_\vartheta=e^{\vartheta\Omega_{ij}}$, the rotation by the angle $\vartheta$ in the $(i,j)$-plane, and let $K_{ij}$ be the vector field
\[
K_{ij}\varphi(x)=\frac{d}{d\vartheta}\varphi(\mathcal R^{ij}_\vartheta x)\Big|_{\vartheta=0}=x_i\partial_j\varphi(x)-x_j\partial_i\varphi(x).
\]
It is tangent to $S_N$, and $K_{ij}x=\Omega_{ij}x$. Since each $\mathcal R^{ij}_\vartheta$ preserves $\nu_N$, differentiating $\int\varphi\circ\mathcal R^{ij}_\vartheta\,d\nu_N=\int\varphi\,d\nu_N$ at $\vartheta=0$, we obtain
\begin{equation}\label{eq:score-divergence-free}
\int_{S_N}K_{ij}\varphi\,d\nu_N=0\qquad\text{for every }\varphi\in C^1(S_N).
\end{equation}
We write $R_1(x)=(-x_1,x_2,\dots,x_N)$ for the reflection of the first coordinate, as in \Cref{lem:model-symmetry}. The equation \eqref{mc-eq:sr1} is invariant under $R_1$, because $F$ is even in $x_1$: if $X$ solves it with Brownian motion $B$, then $R_1X$ solves it with Brownian motion $R_1B$.

\paragraph{Weak scores.}
The following definition makes sense of a logarithmic derivative along a rotation without assuming that a density is differentiable or positive.

\begin{definition}[Weak score]\label{def:score-weak-score}
Let $\rho$ be a probability measure on $S_N$ with $\rho\ll\nu_N$, and let $K=K_{ij}$. A function $s\in L^1(\rho)$ is a \emph{weak $K$-score} of $\rho$ if
\begin{equation}\label{eq:score-weak-score}
\int K\varphi\,d\rho=-\int\varphi\,s\,d\rho\qquad\text{for every }\varphi\in C^\infty(S_N).
\end{equation}
We then write $s=K\log(d\rho/d\nu_N)$.
\end{definition}

A weak $K$-score is unique $\rho$-almost everywhere, because \eqref{eq:score-weak-score} determines the signed measure $s\rho$ on the smooth functions, which are dense in $C(S_N)$. Approximating a function of class $C^1$ uniformly, together with its $K$-derivative, by smooth functions, we see that \eqref{eq:score-weak-score} also holds for $\varphi\in C^1(S_N)$. If $d\rho/d\nu_N=f$ is of class $C^1$ and positive on an open set $U$, then $s=Kf/f$ almost everywhere on $U$. This follows from \eqref{eq:score-weak-score} with $\varphi$ supported in $U$ and \eqref{eq:score-divergence-free} applied to $\varphi f$. In particular, the weak score is the usual score where the latter is defined. We note that \eqref{eq:score-weak-score} is a global identity. A jump of nonzero size across a smooth hypersurface transverse to $K$ contributes a singular part to $Kf$, and therefore prevents a weak $K$-score. For this reason, the cutoff functions below are smooth.

The first lemma computes the score after multiplication by a cutoff function. For $p=1$, its hypothesis is $|K\eta|\le B\eta^{1/2}$.

\begin{lemma}[Multiplying by a cutoff function]\label{csh-lem:cut-score}
Let $\rho$ be a probability measure on $S_N$ with a weak $K$-score $s$, where $K=K_{ij}$, and let $p\ge1$ be an integer with $s\in L^{2p}(\rho)$. Let $\eta\in C^1(S_N)$ take values in $[0,1]$, with $Z_\eta=\int\eta\,d\rho>0$. Suppose that there exists $B\ge0$ such that
\begin{equation}\label{mc-eq:hv3}
|K\eta|\le B\,\eta^{1-1/(2p)}\qquad\rho\text{-almost everywhere}.
\end{equation}
Then the normalized cutoff-weighted law $\widehat\rho=\eta\rho/Z_\eta$, with normalizing mass $Z_\eta$, has the weak $K$-score $\widehat s=s+\ind_{\{\eta>0\}}K\eta/\eta$, and
\begin{equation}\label{mc-eq:hv4}
\E_{\widehat\rho}|\widehat s|^{2p}\le\frac{2^{2p-1}}{Z_\eta}\big(\E_\rho|s|^{2p}+B^{2p}\big).
\end{equation}
\end{lemma}

\begin{proof}
We have $K\eta=0$ $\rho$-almost everywhere on $\{\eta=0\}$, by \eqref{mc-eq:hv3}. Let $\varphi\in C^\infty(S_N)$. Applying the identity \eqref{eq:score-weak-score} to $\eta\varphi\in C^1(S_N)$, we find
\[
\int K\varphi\,d\widehat\rho=\frac1{Z_\eta}\int\big(K(\eta\varphi)-\varphi K\eta\big)\,d\rho
=-\frac1{Z_\eta}\int\varphi\,(\eta s+K\eta)\,d\rho
=-\int\varphi\,\widehat s\,d\widehat\rho .
\]
This proves the formula for the score, once we know that $\widehat s\in L^1(\widehat\rho)$, which follows from the moment bound below. By convexity, $|\widehat s|^{2p}\le2^{2p-1}(|s|^{2p}+\ind_{\{\eta>0\}}|K\eta/\eta|^{2p})$. We have $\E_{\widehat\rho}|s|^{2p}=Z_\eta^{-1}\int\eta|s|^{2p}\,d\rho\le Z_\eta^{-1}\E_\rho|s|^{2p}$, and, by \eqref{mc-eq:hv3},
\[
\E_{\widehat\rho}\Big[\ind_{\{\eta>0\}}\Big|\frac{K\eta}{\eta}\Big|^{2p}\Big]
=\frac1{Z_\eta}\int_{\{\eta>0\}}|K\eta|^{2p}\eta^{1-2p}\,d\rho\le\frac{B^{2p}}{Z_\eta}.
\]
Combining the two bounds, we obtain \eqref{mc-eq:hv4}.
\end{proof}

The second lemma bounds the total variation cost of a path weight. We use it in the following form: if $0\le U\le1$ is a random variable with $\E U>0$ and $Y$ is a random element, then the laws of $Y$ under $\PP$ and under $(U/\E U)\PP$ are within $1-\E U$ in total variation. This is the case $\sigma(A)=\E[U\ind_{\{Y\in A\}}]$ and $\rho=\Law(Y)$ below.

\begin{lemma}[Removing a path weight]\label{lem:score-domination}
Let $\rho$ be a probability measure and $\sigma\le\rho$ a measure on the same space with total mass $Z>0$. Then $\|\rho-\sigma/Z\|_{\TV}\le1-Z$.
\end{lemma}

\begin{proof}
For both probability measures, $\|\rho-\sigma/Z\|_{\TV}$ is the supremum of $\sigma(A)/Z-\rho(A)$ over measurable sets $A$, by the first part of \Cref{lem:model-tv}. Since $\rho(A)\ge\sigma(A)$, we have $\sigma(A)/Z-\rho(A)\le(Z^{-1}-1)\sigma(A)\le1-Z$.
\end{proof}

The third lemma records the forward equation and the smoothness of the endpoint measures that appear later: the path is weighted by a variable known at a time $T$, and possibly killed after $T$ when it leaves an open set.

\begin{lemma}[Weighted and killed endpoint measures]\label{lem:score-regularity}
Let $X$ solve \eqref{mc-eq:sr1} with $X_0\sim\nu_N$, let $T\ge0$, let $\Psi$ be an $\mathcal F_T$-measurable random variable with values in $[0,1]$, and let $D\subset S_N$ be open (possibly $D=S_N$). Let $\tau=\inf\{s\ge T:X_s\notin D\}$, with $\inf\emptyset=\infty$, and for $t\ge T$ define the endpoint subprobability measure
\begin{equation}\label{eq:score-sigma}
\sigma_t(A)=\E\big[\Psi\,\ind_{\{\tau>t\}}\ind_{\{X_t\in A\}}\big].
\end{equation}
\begin{enumerate}
\item For every $\varphi\in C^\infty(S_N)$ with compact support in $D$ and all $T\le s\le t$,
\begin{equation}\label{eq:score-forward}
\sigma_t(\varphi)-\sigma_s(\varphi)=\int_s^t\sigma_r(L\varphi)\,dr .
\end{equation}
\item There exists $u\in C^\infty((T,\infty)\times D)$ such that $\sigma_t(dx)=u(t,x)\,\nu_N(dx)$ for every $t>T$.
\item If $D$ and $\sigma_T$ are invariant under $R_1$, then $\sigma_t$ is invariant under $R_1$ for every $t\ge T$.
\end{enumerate}
\end{lemma}

\begin{proof}
For the first part, note that $X_\tau\notin D$ on $\{\tau<\infty\}$, because $D$ is open and the path is continuous. Let $\chi\in C^\infty(\R\times S_N)$ be such that each $\chi(r,\cdot)$ has its support in a fixed compact subset of $D$. By Itô's formula,
\[
\chi(t\wedge\tau,X_{t\wedge\tau})-\chi(s\wedge\tau,X_{s\wedge\tau})
=\int_{s\wedge\tau}^{t\wedge\tau}(\partial_r+L)\chi(r,X_r)\,dr
+\sqrt2\int_{s\wedge\tau}^{t\wedge\tau}\nabla_{S_N}\chi(r,X_r)\cdot dB_r .
\]
On $\{\tau\le t\}$ we have $\chi(\tau,X_\tau)=0$, and the same holds with $s$ in place of $t$. We multiply by $\Psi$ and take expectations. The stochastic integral has a bounded integrand and starts after time $T$, and its product with the $\mathcal F_T$-measurable variable $\Psi$ has mean zero. Using Fubini's theorem, we obtain
\begin{equation}\label{eq:score-forward-time}
\sigma_t(\chi(t,\cdot))-\sigma_s(\chi(s,\cdot))=\int_s^t\sigma_r\big((\partial_r+L)\chi(r,\cdot)\big)\,dr ,
\end{equation}
and \eqref{eq:score-forward} is the case of a $\chi$ that does not depend on $r$.

For the second part, we take $\chi\in C^\infty_c((T,\infty)\times D)$, $s=T$, and $t$ larger than every time in the support of $\chi$. Then \eqref{eq:score-forward-time} says that the measure $\sigma_r(dx)\,dr$ on $(T,\infty)\times D$ is annihilated, in the sense of distributions, by the formal adjoint of $\partial_r+L$. In local coordinates on $D$, this adjoint is $P=-\partial_r+L^*$, where $L^*$ is the formal adjoint of $L$ with respect to the coordinate Lebesgue measure. Its second-order part is that of $L$, which is $\sum_{k,l}\mathsf a^{kl}\partial_k\partial_l$ with a smooth positive definite matrix $\mathsf a$. Writing $\mathsf a=\mathsf s\mathsf s^\T$ with $\mathsf s=\mathsf a^{1/2}$ smooth, we have $P=\sum_kV_k^2+V_0+c_0$, where $V_k=\sum_l\mathsf s_{lk}\partial_l$, $c_0$ is a smooth function, and $V_0=-\partial_r+V_0'$ with $V_0'$ a first-order operator in the space variables. The fields $V_1,\dots,V_{N-1}$ span all space directions, and $V_0$ has a nonzero time component. This implies that $V_0,\dots,V_{N-1}$ are linearly independent at every point. By Hörmander's theorem \cite[Theorem~1.1]{Hormander1967}, $P$ is hypoelliptic, and the measure $\sigma_r(dx)\,dr$ is a smooth function times $dx\,dr$.
In other words, there exists $u\in C^\infty((T,\infty)\times D)$ such that $\sigma_r(dx)\,dr=u(r,x)\,\nu_N(dx)\,dr$. For $\varphi\in C^\infty_c(D)$, both $r\mapsto\sigma_r(\varphi)$ and $r\mapsto\int\varphi\,u(r,\cdot)\,d\nu_N$ are continuous on $(T,\infty)$, the first by \eqref{eq:score-forward}. They agree for almost every $r$, hence for every $r>T$. We conclude that $\sigma_r=u(r,\cdot)\nu_N$ on $D$. Since $\sigma_r(D^c)=0$ for $r\ge T$, the second part follows.

For the third part, let $\mathsf k_v(x,A)$ be the probability that the solution of \eqref{mc-eq:sr1} started at $x$ stays in $D$ during $[0,v]$ and lies in $A$ at time $v$. By the Markov property at the time $T$, we have $\sigma_t(A)=\int\sigma_T(dx)\,\mathsf k_{t-T}(x,A)$. Since the equation and $D$ are invariant under $R_1$, we have $\mathsf k_v(R_1x,R_1A)=\mathsf k_v(x,A)$. Together with the invariance of $\sigma_T$, this gives $\sigma_t(R_1A)=\sigma_t(A)$.
\end{proof}

\paragraph{Conditioned, weighted, and killed measures.}
We use the following names for the measures that appear here and in \Cref{sec:cold-fixed,sec:cold-meso}. For an event $E$ with $\PP(E)>0$, the law conditioned on $E$ is $\PP(\cdot\mid E)=\PP(\cdot\cap E)/\PP(E)$, with normalizing mass $\PP(E)$. The measure $\sigma_t$ of \eqref{eq:score-sigma} is the endpoint subprobability measure weighted by $\Psi$ and killed on leaving $D$. It is bounded by the law of $X_t$, and its mass $\sigma_t(S_N)\le1$ need not be one. For a cutoff function $\eta:S_N\to[0,1]$ with $\int\eta\,d\sigma_t>0$, the normalized cutoff-weighted law is the probability measure $\eta\sigma_t/\int\eta\,d\sigma_t$, with normalizing mass $\int\eta\,d\sigma_t$. If $D=S_N$ and $\Psi=\ind_E$ for an event $E\in\mathcal F_T$, then $\sigma_t(S_N)=\PP(E)$, and $\sigma_t/\sigma_t(S_N)$ is the law of $X_t$ under $\PP(\cdot\mid E)$.

\section{A Bismut-type formula for the score over a final time interval}\label{sec:score-bismut}\label{pr-sec:scores-smoothing}\label{csr-sec:reset}

Formulas that express the derivative of a diffusion semigroup through a stochastic integral, without differentiating the initial data, go back to Bismut \cite{Bismut1984} and Elworthy and Li \cite[Theorems~2.1 and~3.1 and Corollary~3.2]{ElworthyLi1994}; see also \cite[Section~8.3, Theorems~8.3.3--8.3.4]{Hsu2002}.
\Cref{csr-thm:reset} below is a version of this idea for the logarithmic derivative of an endpoint density along a rotation. Three features are needed in the applications. The derivative is that of the density of an endpoint law weighted by a functional of the path and normalized. The paths are restricted to a rotation-invariant open set on which the change of drift is small. The bound uses only a final time interval of the original uniformly initialized diffusion. It requires no score bound before that interval and no new initialization at its left endpoint; the law there may be the evolved and prefix-weighted law. The proof differentiates a Girsanov density with respect to a rotation of the path during the final interval.

\paragraph{Setting.}
Fix $i\ne j$ and write $K=K_{ij}$, $\Omega=\Omega_{ij}$, $\mathcal R_\vartheta=\mathcal R^{ij}_\vartheta$, and $\Delta=|a_j-a_i|$. Let $D\subset S_N$ be an open set with $\mathcal R_\vartheta D=D$ for every $\vartheta\in\R$, and let $H<\infty$ satisfy
\begin{equation}\label{mc-eq:sr2}
x_i^2+x_j^2\le H\qquad\text{for }x\in D .
\end{equation}
Fix $t>0$ and $\ell\in(0,t]$, let $I=[t-\ell,t]$, and let $\kappa:I\to\R$ be absolutely continuous with $\kappa(t-\ell)=0$, $\kappa(t)=1$, and $\int_I\kappa'(s)^2\,ds<\infty$. For $\varepsilon\in\R$, let $\Theta_\varepsilon$ act on paths $\omega\in C(I;S_N)$ by $(\Theta_\varepsilon\omega)_s=\mathcal R_{\varepsilon\kappa(s)}\omega_s$. Let $\xi$ be an $\mathcal F_{t-\ell}$-measurable random element of a measurable space $E$, and let $\Phi:E\times C(I;S_N)\to[0,1]$ be measurable with
\begin{equation}\label{mc-eq:sr3}
\Phi(e,\omega)=0\ \text{ unless }\omega_s\in D\text{ for all }s\in I,\qquad
\Phi(e,\Theta_\varepsilon\omega)=\Phi(e,\omega)\ \text{ for all }\varepsilon\in\R .
\end{equation}
We call $W=\Phi(\xi,X|_I)$ a path weight for the final interval $I$. For example, $\xi$ may be the path up to time $t-\ell$, together with any other $\mathcal F_{t-\ell}$-measurable variable. We assume $Z=\E W>0$ and define the weighted endpoint law $\rho^W(A)=Z^{-1}\E[W\ind_{\{X_t\in A\}}]$, whose normalizing mass is $Z$.

\begin{theorem}[A Bismut-type formula for the score over a final time interval]\label{csr-thm:reset}
For the uniformly initialized diffusion in this setting, the law $\rho^W$ has a weak $K$-score, and for every integer $p\ge1$,
\begin{equation}\label{mc-eq:sr4}
\E_{\rho^W}\Big|K\log\frac{d\rho^W}{d\nu_N}\Big|^{2p}
\le\frac{C_p}{Z}\Big[H\int_I\big(\kappa'(s)^2+\Delta^2\kappa(s)^2\big)\,ds\Big]^p,
\end{equation}
where $C_p$ depends only on $p$. For the linear ramp $\kappa(s)=(s-t+\ell)/\ell$,
\begin{equation}\label{mc-eq:sr5}
\E_{\rho^W}\Big|K\log\frac{d\rho^W}{d\nu_N}\Big|^{2p}
\le\frac{C_p}{Z}\big[H(\ell^{-1}+\Delta^2\ell)\big]^p .
\end{equation}
\end{theorem}

The proof has four steps. We first show that rotating the path during $I$ changes its law by an explicit Girsanov density. Using the invariance of $W$, we deduce the identity \eqref{mc-eq:sr12}. Differentiating it at $\varepsilon=0$ represents $\int K\varphi\,d\rho^W$ through a stochastic integral $M$. Finally, the moments of $M$ are bounded by its quadratic variation.

\begin{proof}
Let $\tau=\inf\{s\ge t-\ell:X_s\notin D\}$. All integrals below may be restricted to $\{\tau>t\}$, because $W=0$ unless $\tau>t$.

\paragraph{An exponential bound.}
We use the following elementary bound twice. Let $M'$ be a continuous local martingale with $M'_0=0$ and $\langle M'\rangle_t\le V$ for a deterministic constant $V\ge0$. If $V=0$, then $M'_s=0$ almost surely for all $s\le t$, so its exponential moments equal one and all its positive moments and tails vanish. Suppose now that $V>0$. For $\lambda\in\R$, the process $\exp(\lambda M'_s-\lambda^2\langle M'\rangle_s/2)$ is a nonnegative local martingale, hence a supermartingale with expectation at most one. Therefore
\begin{equation}\label{eq:score-subgaussian}
\E e^{\lambda M'_t}\le e^{\lambda^2V/2},\qquad\PP(|M'_t|>r)\le2e^{-r^2/(2V)},\qquad\E|M'_t|^{2p}\le2\,p!\,(2V)^p
\end{equation}
for $\lambda\in\R$, $r>0$, and integers $p\ge1$. The second bound follows from the first by Markov's inequality with $\lambda=\pm r/V$, and the third from the second, since $\E|M'_t|^{2p}=\int_0^\infty2pr^{2p-1}\PP(|M'_t|>r)\,dr$.

\paragraph{Step 1: the change of drift.}
For $\vartheta\in\R$, the function $y\mapsto F(y)-F(\mathcal R_{-\vartheta}y)$ is the quadratic form of a symmetric matrix $Q_\vartheta$ that acts only on the $(i,j)$-plane. In that plane, we write $\diag(a_i,a_j)=\frac{a_i+a_j}2I+\frac{a_i-a_j}2E$ with $E=\diag(1,-1)$. Then $Q_\vartheta=\frac{a_i-a_j}2(E-\mathcal R_\vartheta E\mathcal R_\vartheta^\T)$, and $\mathcal R_\vartheta E\mathcal R_\vartheta^\T$ is the reflection matrix with entries $\cos2\vartheta,\sin2\vartheta,\sin2\vartheta,-\cos2\vartheta$. We have
\begin{equation}\label{eq:score-Q-norms}
\|Q_\vartheta\|=\Delta|\sin\vartheta|,\qquad\|\partial_\vartheta Q_\vartheta\|=\Delta,\qquad\|\partial^2_\vartheta Q_\vartheta\|=2\Delta,
\end{equation}
since $E-\mathcal R_\vartheta E\mathcal R_\vartheta^\T$ has eigenvalues $\pm2|\sin\vartheta|$, and the first two derivatives of the reflection matrix are $2$ and $4$ times symmetric orthogonal matrices. For $s\in I$ and $y\in S_N$, we define the tangent vector field
\begin{equation}\label{mc-eq:sr6}
u_\varepsilon(s,y)=\varepsilon\kappa'(s)\,\Omega y+\nabla_{S_N}\big\{F-F\circ\mathcal R_{-\varepsilon\kappa(s)}\big\}(y).
\end{equation}
We have $|\Omega y|=(y_i^2+y_j^2)^{1/2}$ and $|\nabla(y^\T Q_\vartheta y)|=2|Q_\vartheta y|\le2\|Q_\vartheta\|(y_i^2+y_j^2)^{1/2}$, and the tangent projection does not increase lengths. Using \eqref{mc-eq:sr2} and \eqref{eq:score-Q-norms}, we obtain
\begin{equation}\label{mc-eq:sr7}
|u_\varepsilon(s,y)|\le|\varepsilon|\sqrt H\big(|\kappa'(s)|+2\Delta|\kappa(s)|\big)\qquad(y\in D).
\end{equation}
We set
\begin{equation}\label{mc-eq:sr8}
\Lambda_\varepsilon=\exp\Big\{\frac1{\sqrt2}\int_{t-\ell}^{\tau\wedge t}u_\varepsilon(s,X_s)\cdot dB_s-\frac14\int_{t-\ell}^{\tau\wedge t}|u_\varepsilon(s,X_s)|^2\,ds\Big\}.
\end{equation}
The integrand is bounded by a deterministic function in $L^2(I)$, by \eqref{mc-eq:sr7}. The following special case of Girsanov's theorem (see \cite[Chapter~VIII, Section~1, Theorem~(1.4)]{RevuzYor1999}) is then elementary. We include the argument. Let $v_s=2^{-1/2}\ind_{\{s<\tau\}}u_\varepsilon(s,X_s)$ for $s\in I$ and $v_s=0$ for $s<t-\ell$, and let $\Lambda_s=\exp(\int_0^sv_r\cdot dB_r-\frac12\int_0^s|v_r|^2\,dr)$, so that $\Lambda_\varepsilon=\Lambda_t$. The quadratic variation $\int_0^t|v_r|^2\,dr$ is bounded by a deterministic constant $V_\varepsilon$. For $q>1$ and every stopping time $\sigma\le t$, we have $\Lambda_\sigma^q=\mathcal E_\sigma\exp\{(q^2-q)\int_0^\sigma|v_r|^2\,dr/2\}$, where $\mathcal E$ is the stochastic exponential of $q\int v\cdot dB$, a nonnegative local martingale and hence a supermartingale. Thus $\E\Lambda_\sigma^q\le e^{(q^2-q)V_\varepsilon/2}$, the local martingale $\Lambda$ is of class (D) on $[0,t]$, and it is a martingale. In particular, $\mathbb Q_\varepsilon=\Lambda_\varepsilon\PP$ is a probability measure that agrees with $\PP$ on $\mathcal F_{t-\ell}$. Using Itô's product formula, we see that each component of $\Lambda_s(B_s-\int_0^sv_r\,dr)$ is a local $\PP$-martingale. By Bayes' formula, after localization, $\widehat B_s=B_s-\int_0^sv_r\,dr$ is a continuous local $\mathbb Q_\varepsilon$-martingale with quadratic covariation $sI_N$, and Lévy's characterization \cite[Theorem~IV.3.6]{RevuzYor1999} shows that $\widehat B$ is a $\mathbb Q_\varepsilon$-Brownian motion. For $s\in I$, we have $\widehat B_s=B_s-2^{-1/2}\int_{t-\ell}^{s\wedge\tau}u_\varepsilon(r,X_r)\,dr$.
Because $u_\varepsilon$ is tangent, we have $\Pi(X_s)u_\varepsilon=u_\varepsilon$, and under $\mathbb Q_\varepsilon$ the process $X$ solves $dX_s=\sqrt2\,\Pi(X_s)\,d\widehat B_s+(\mathbf b+u_\varepsilon)(s,X_s)\,ds$ for $s\in[t-\ell,\tau\wedge t]$.

We define $\widehat X_s=\mathcal R_{-\varepsilon\kappa(s)}X_s$ for $s\in I$. The map $s\mapsto\mathcal R_{-\varepsilon\kappa(s)}$ is absolutely continuous and $\Omega$ commutes with every $\mathcal R_\vartheta$, and we find, for $s\le\tau\wedge t$,
\[
d\widehat X_s=-\varepsilon\kappa'(s)\,\Omega\widehat X_s\,ds+\mathcal R_{-\varepsilon\kappa(s)}\big[\sqrt2\,\Pi(X_s)\,d\widehat B_s+(\mathbf b+u_\varepsilon)(s,X_s)\,ds\big].
\]
For an orthogonal matrix $\mathcal R$, we have $\mathcal R^\T\Pi(\mathcal Rx)\mathcal R=\Pi(x)$ and $\mathcal R^\T\nabla_{S_N}\varphi(\mathcal Rx)=\nabla_{S_N}(\varphi\circ\mathcal R)(x)$ for every smooth $\varphi$. With $\mathcal R=\mathcal R_{\varepsilon\kappa(s)}$, we deduce from these identities that $\mathcal R^\T\mathbf b(\mathcal R\widehat x)=-\frac{N-1}N\widehat x-\nabla_{S_N}(F\circ\mathcal R)(\widehat x)$ and $\mathcal R^\T u_\varepsilon(s,\mathcal R\widehat x)=\varepsilon\kappa'(s)\Omega\widehat x+\nabla_{S_N}(F\circ\mathcal R-F)(\widehat x)$. The terms with $\Omega$ and with $F\circ\mathcal R$ cancel, which gives
\[
d\widehat X_s=\sqrt2\,\Pi(\widehat X_s)\,d\check B_s+\mathbf b(\widehat X_s)\,ds,\qquad
\check B_s=\int_{t-\ell}^s\mathcal R_{-\varepsilon\kappa(r)}\,d\widehat B_r ,
\]
for $s\in[t-\ell,\tau\wedge t]$. By Lévy's characterization, $\check B$ is a $\mathbb Q_\varepsilon$-Brownian motion. Moreover, $\widehat X_{t-\ell}=X_{t-\ell}$, and the exit time of $\widehat X$ from $D$ is $\tau$, because $D$ is rotation invariant. Thus, under $\mathbb Q_\varepsilon$, the path $\widehat X$ stopped at its exit time solves \eqref{mc-eq:sr1} with the same $\mathcal F_{t-\ell}$-measurable data $(\xi,X_{t-\ell})$, whose joint law is the same under $\mathbb Q_\varepsilon$ and $\PP$. Pathwise uniqueness implies that the joint law of $\xi$ and the stopped path $\widehat X$ under $\mathbb Q_\varepsilon$ equals the joint law of $\xi$ and the stopped path $X$ under $\PP$.

Let $\varphi$ be bounded and measurable on $S_N$. The function $(e,\omega)\mapsto\Phi(e,\omega)\varphi(\mathcal R_\varepsilon\omega_t)$ vanishes unless $\omega$ stays in $D$, by \eqref{mc-eq:sr3}. In particular, it depends only on $e$ and on the path stopped at its exit time. Using the equality of laws, then the invariance in \eqref{mc-eq:sr3}, and finally $X=\Theta_\varepsilon\widehat X$ and $\kappa(t)=1$, we find
\[
\begin{aligned}
\E\big[\Phi(\xi,X|_I)\varphi(\mathcal R_\varepsilon X_t)\big]
&=\E_{\mathbb Q_\varepsilon}\big[\Phi(\xi,\widehat X|_I)\varphi(\mathcal R_\varepsilon\widehat X_t)\big]\\
&=\E_{\mathbb Q_\varepsilon}\big[\Phi(\xi,\Theta_\varepsilon\widehat X|_I)\varphi(\mathcal R_\varepsilon\widehat X_t)\big]
=\E_{\mathbb Q_\varepsilon}\big[W\varphi(X_t)\big].
\end{aligned}
\]
That is,
\begin{equation}\label{mc-eq:sr12}
\E\big[W\varphi(\mathcal R_\varepsilon X_t)\big]=\E\big[W\Lambda_\varepsilon\,\varphi(X_t)\big]\qquad(\varepsilon\in\R).
\end{equation}
For $\varphi=1$, this shows that $\E[W\Lambda_\varepsilon]=Z$ for every $\varepsilon$.

\paragraph{Step 2: differentiation at $\varepsilon=0$.}
Let
\begin{equation}\label{mc-eq:sr9}
M=\frac1{\sqrt2}\int_{t-\ell}^{\tau\wedge t}\big\{\kappa'(s)\,\Omega X_s+\kappa(s)\nabla_{S_N}(KF)(X_s)\big\}\cdot dB_s .
\end{equation}
The integrand is $\partial_\varepsilon u_\varepsilon(s,X_s)$ at $\varepsilon=0$, because $\partial_\varepsilon[F(y)-F(\mathcal R_{-\varepsilon\kappa}y)]=\kappa\,(KF)(\mathcal R_{-\varepsilon\kappa}y)$. We claim that $(\Lambda_\varepsilon-1)/\varepsilon\to M$ in $L^1(\PP)$ as $\varepsilon\to0$. We write $\Lambda_\varepsilon=\exp(N_\varepsilon-\frac12\langle N_\varepsilon\rangle)$, where $N_\varepsilon$ is the stochastic integral in \eqref{mc-eq:sr8}. By \eqref{mc-eq:sr7}, we have $\langle N_\varepsilon\rangle\le\varepsilon^2V_1$ with the deterministic constant $V_1=\frac H2\int_I(|\kappa'|+2\Delta|\kappa|)^2\,ds$. The second derivative of $u_\varepsilon$ in $\varepsilon$ is bounded by $4\Delta\kappa^2\sqrt H$ on $D$, by \eqref{eq:score-Q-norms}. The integrand of $N_\varepsilon-\varepsilon M$ is therefore bounded by $2\varepsilon^2\Delta\kappa(s)^2\sqrt H$ before $\tau$, and $\E(N_\varepsilon-\varepsilon M)^2=O(\varepsilon^4)$, since $\kappa$ is bounded on $I$. Let $x_\varepsilon=N_\varepsilon-\frac12\langle N_\varepsilon\rangle$. By \eqref{eq:score-subgaussian}, we have $\E e^{\pm2N_\varepsilon}\le e^{2\varepsilon^2V_1}$ and $\E N_\varepsilon^4\le16\varepsilon^4V_1^2$, and therefore $\E x_\varepsilon^4=O(\varepsilon^4)$. Using also $|e^x-1-x|\le\frac12x^2e^{|x|}$ and the Cauchy--Schwarz inequality, we obtain $\E|e^{x_\varepsilon}-1-x_\varepsilon|=O(\varepsilon^2)$. Since $x_\varepsilon-\varepsilon M=(N_\varepsilon-\varepsilon M)-\frac12\langle N_\varepsilon\rangle$, we conclude that $\E|\Lambda_\varepsilon-1-\varepsilon M|=O(\varepsilon^2)$, as claimed.

Now let $\varphi\in C^\infty(S_N)$. The left side of \eqref{mc-eq:sr12} has derivative $\E[W(K\varphi)(X_t)]$ at $\varepsilon=0$, by dominated convergence, and the claim shows that the right side has derivative $\E[WM\varphi(X_t)]$. Equating the two derivatives, we find
\begin{equation}\label{mc-eq:sr13}
\int K\varphi\,d\rho^W=Z^{-1}\E\big[WM\varphi(X_t)\big].
\end{equation}

\paragraph{Step 3: identification of the score.}
Let $\mathbb Q=(W/Z)\PP$. Since $\E[W|M|]\le\E|M|<\infty$, we may define $\mathsf m(x)=\E_{\mathbb Q}[M\mid X_t=x]$. Then \eqref{mc-eq:sr13} reads $\int K\varphi\,d\rho^W=\int\mathsf m\varphi\,d\rho^W$. The law $\rho^W$ is absolutely continuous with respect to $\nu_N$: it is bounded by $Z^{-1}$ times the law of $X_t$, which has a density by the second part of \Cref{lem:score-regularity} applied with $T=0$, $\Psi=1$, and $D=S_N$. We conclude that $-\mathsf m$ is a weak $K$-score of $\rho^W$, provided that it lies in $L^1(\rho^W)$, which we show in the next step.

\paragraph{Step 4: moments.}
For the quadratic potential, we have
\begin{equation}\label{mc-eq:sr10}
KF=2(a_j-a_i)x_ix_j,\qquad|\nabla_{S_N}(KF)|^2\le|\nabla(KF)|^2=4\Delta^2(x_i^2+x_j^2).
\end{equation}
Together with $|\Omega X_s|=(X_{s,i}^2+X_{s,j}^2)^{1/2}$ and \eqref{mc-eq:sr2}, this bounds the integrand in \eqref{mc-eq:sr9} by $\sqrt H(|\kappa'(s)|+2\Delta|\kappa(s)|)$ before $\tau$. The quadratic variation of $M$ therefore satisfies the deterministic bound
\begin{equation}\label{mc-eq:sr11}
\langle M\rangle\le\frac H2\int_I\big(|\kappa'|+2\Delta|\kappa|\big)^2\,ds\le4H\int_I\big(\kappa'(s)^2+\Delta^2\kappa(s)^2\big)\,ds=V_* .
\end{equation}
By \eqref{eq:score-subgaussian}, we have $\E|M|^{2p}\le2\,p!\,(2V_*)^p$.
Applying Jensen's inequality for conditional expectations, we see that
\[
\E_{\rho^W}|\mathsf m|^{2p}\le\E_{\mathbb Q}|M|^{2p}=Z^{-1}\E\big[W|M|^{2p}\big]\le Z^{-1}\E|M|^{2p}\le2\,p!\,Z^{-1}(2V_*)^p ,
\]
which yields \eqref{mc-eq:sr4} with $C_p=2\cdot8^p\,p!$. For the linear ramp, $\int_I\kappa'^2=\ell^{-1}$ and $\int_I\kappa^2=\ell/3$, and \eqref{mc-eq:sr5} follows.
\end{proof}

No derivative of $W$ is taken in the proof. For this reason, the path weight may be any measurable functional satisfying \eqref{mc-eq:sr3}. We now specialize to the low-temperature time scale. Recall that the first coordinate is singled out by $a_1=0$, and that in the applications the parameter $\theta$ below is the top fraction $\theta_N$ of \eqref{eq:model-cold-parameters}.

\begin{corollary}[Scores on the time scale $a_2^{-1}$]\label{csr-cor:cold-scale}
Let $J\ge2$, $C_J\ge1$, $L>0$, $z_0>0$, and $\theta>0$. Suppose that $0=a_1<a=a_2\le a_3\le\cdots\le a_J\le C_Ja$, and set $m=\theta Na$. Let $D\subset S_N$ be an open set that is invariant under all rotations of the coordinates $x_1,\dots,x_J$ and contained in
\begin{equation}\label{mc-eq:sr14}
\Big\{x\in S_N:\sum_{k=1}^Jx_k^2<L\theta N\Big\}.
\end{equation}
Let $t\ge1/a$, and let $W$ be a path weight for the final interval $I=[t-1/a,t]$ and the set $D$, as in \eqref{mc-eq:sr3}, which is invariant under the rotations $\Theta_\varepsilon$ in the $(1,j)$-plane with the linear ramp, for every $2\le j\le J$ and $\varepsilon\in\R$. Suppose that $Z=\E W\ge z_0$. Then, for every $2\le j\le J$ and every integer $p\ge1$,
\begin{equation}\label{mc-eq:sr15}
\E_{\rho^W}\Big|K_{1j}\log\frac{d\rho^W}{d\nu_N}\Big|^{2p}\le C\,m^p,
\end{equation}
where $C$ depends only on $p$, $z_0$, $L$, and $C_J$.
\end{corollary}

\begin{proof}
We use \Cref{csr-thm:reset} with $(i,j)$ replaced by $(1,j)$, $\ell=1/a$, and $H=L\theta N$. The set $D$ is invariant under the rotations of the $(1,j)$-plane, and \eqref{mc-eq:sr2} holds because $x_1^2+x_j^2\le\sum_{k\le J}x_k^2$. With $\Delta=a_j\le C_Ja$, the bound \eqref{mc-eq:sr5} becomes
\[
\E_{\rho^W}\Big|K_{1j}\log\frac{d\rho^W}{d\nu_N}\Big|^{2p}\le\frac{C_p}{z_0}\big[L\theta N(a+C_J^2a)\big]^p=\frac{C_p}{z_0}\big[L(1+C_J^2)m\big]^p .
\]
This is \eqref{mc-eq:sr15}.
\end{proof}

A weight that depends on the path only through its values before $t-1/a$, multiplied by the indicator that the path stays in $D$ from some time $T\le t-1/a$ until $t$, satisfies the hypotheses: the rotations act only on $I$ and preserve $D$.

\section{Weak convergence to total variation convergence by Gaussian smoothing}\label{sec:score-smoothing}\label{csh-sec:pure-smoothing}

In this section $d\ge1$ is a fixed integer and densities are taken with respect to Lebesgue measure on $\R^d$. For a probability density $p$ on $\R^d$ that belongs to $W^{1,1}_{\mathrm{loc}}(\R^d)$, its \emph{Fisher information} is
\[
\mathrm{FI}(p)=\int_{\R^d}\frac{|\nabla p|^2}{p}\,dz,
\]
where the integrand is $0$ where $p=0$ and $\nabla p=0$, and $+\infty$ where $p=0$ and $\nabla p\ne0$. For other densities, $\mathrm{FI}(p)=\infty$. We write $\mathrm{FI}(\mu)=\mathrm{FI}(p)$ for a probability measure $\mu$ with density $p$, and $\mathrm{FI}(\mu)=\infty$ if $\mu$ has no density. If $\mathrm{FI}(p)<\infty$, then by the Cauchy--Schwarz inequality, we have
\begin{equation}\label{eq:score-fisher-l1}
\int_{\R^d}|\nabla p|\,dz\le\Big(\int_{\{p>0\}}\frac{|\nabla p|^2}{p}\,dz\Big)^{1/2}\Big(\int p\,dz\Big)^{1/2}=\mathrm{FI}(p)^{1/2},
\end{equation}
so $p\in W^{1,1}(\R^d)$. The first lemma is the classical Fisher information bound for a Gaussian convolution, related to the convolution inequality of Stam \cite{Stam1959}. Its proof uses the conditional-score and Cauchy--Schwarz argument of Blachman \cite{Blachman1965}. We include it because the input law is arbitrary and may, for instance, have no moments.

\begin{lemma}[Fisher information of a Gaussian convolution]\label{csh-lem:convolution-fisher}
For every probability measure $\lambda$ on $\R^d$, the convolution $\mu=\lambda*\cN(0,I_d)$ has a smooth positive density $p$ with $\mathrm{FI}(p)\le d$.
\end{lemma}

\begin{proof}
Let $\phi_d$ be the standard Gaussian density on $\R^d$. The density of $\mu$ and its gradient are
\[
p(y)=\int\phi_d(y-v)\,\lambda(dv)>0,\qquad\nabla p(y)=-\int(y-v)\phi_d(y-v)\,\lambda(dv).
\]
Every derivative of $\phi_d$ is bounded, and dominated convergence justifies differentiation under the integral. In particular, $p$ is smooth. Applying the Cauchy--Schwarz inequality with respect to the finite measure $\phi_d(y-v)\lambda(dv)$, we find
\[
\frac{|\nabla p(y)|^2}{p(y)}\le\int|y-v|^2\phi_d(y-v)\,\lambda(dv).
\]
We integrate in $y$ and use Tonelli's theorem and $\int|y-v|^2\phi_d(y-v)\,dy=d$ for every $v$. This gives $\mathrm{FI}(p)\le d$.
\end{proof}

For probability measures $\mu,\mu'$ on $\R^d$, let
\begin{equation}\label{mc-eq:hv10}
d_{\mathrm{BL}}(\mu,\mu')=\sup\Big\{\Big|\int\varphi\,d\mu-\int\varphi\,d\mu'\Big|:\|\varphi\|_\infty\le1,\ \Lip(\varphi)\le1\Big\}
\end{equation}
be the bounded Lipschitz distance. The next theorem shows that weak convergence implies convergence in total variation when nearby laws have bounded Fisher information. We smooth with a Gaussian kernel. The Fisher information is required only of approximating laws that are close in total variation. In the applications, these are obtained from the laws of interest by path weights and cutoff functions.

\begin{theorem}[From weak convergence to total variation]\label{csh-thm:weak-to-tv}
Let $\mu_N,\mu,\widehat\mu_N$ be probability measures on $\R^d$. Suppose that $\mathrm{FI}(\mu)\le I_*$, $\mathrm{FI}(\widehat\mu_N)\le C$, and $\|\mu_N-\widehat\mu_N\|_{\TV}\le\delta_N$. Then for every $0<\varepsilon\le1$,
\begin{equation}\label{mc-eq:hv11}
\|\mu_N-\mu\|_{\TV}\le2\delta_N+C_d\,\varepsilon\big(\sqrt C+\sqrt{I_*}\big)+\big(1+C_d/\varepsilon\big)\,d_{\mathrm{BL}}(\mu_N,\mu),
\end{equation}
with $C_d=\sqrt d$. In particular, suppose that $\mu_N\Rightarrow\mu$, that $\mathrm{FI}(\mu)<\infty$, and that for every $\delta>0$ there exist $C_\delta<\infty$ and probability measures $\widehat\mu_N^\delta$ with $\mathrm{FI}(\widehat\mu_N^\delta)\le C_\delta$ for all large $N$ and $\limsup_N\|\mu_N-\widehat\mu_N^\delta\|_{\TV}\le\delta$. Then $\|\mu_N-\mu\|_{\TV}\to0$.
\end{theorem}

\begin{proof}
We begin with two estimates for a density $q$ with $\mathrm{FI}(q)<\infty$. Note that $q\in W^{1,1}(\R^d)$, by \eqref{eq:score-fisher-l1}. For $q\in C^\infty_c(\R^d)$, we see that $\|q(\cdot+h)-q\|_{L^1}\le|h|\,\|\nabla q\|_{L^1}$ by the fundamental theorem of calculus along segments. Smooth compactly supported functions are dense in $W^{1,1}(\R^d)$ \cite[Theorem~9.2, p.~265]{Brezis2011}, and the same inequality holds for our $q$. Together with \eqref{eq:score-fisher-l1}, this shows that
\begin{equation}\label{eq:score-translation}
\|q(\cdot+h)-q\|_{L^1}\le|h|\,\mathrm{FI}(q)^{1/2}\qquad(h\in\R^d).
\end{equation}
Let $\phi_\varepsilon$ be the density of $\cN(0,\varepsilon^2I_d)$. For a Borel function $f:\R^d\to[0,1]$, let $f_\varepsilon=f*\phi_\varepsilon$. Then $0\le f_\varepsilon\le1$, and $\Lip(f_\varepsilon)\le\int|\nabla\phi_\varepsilon|=\varepsilon^{-1}\E|\mathsf g|\le\sqrt d/\varepsilon$, where $\mathsf g$ is a standard Gaussian vector. The density $\phi_\varepsilon$ is even, so that $\int f_\varepsilon q=\int f\,(q*\phi_\varepsilon)$. Using \eqref{eq:score-translation}, we deduce that
\[
\Big|\int(f-f_\varepsilon)\,q\Big|\le\int\phi_\varepsilon(h)\,\|q-q(\cdot-h)\|_{L^1}\,dh\le\mathrm{FI}(q)^{1/2}\int|h|\phi_\varepsilon(h)\,dh\le\sqrt d\,\varepsilon\,\mathrm{FI}(q)^{1/2}.
\]
We now decompose
\[
\mu_N(f)-\mu(f)=(\mu_N-\widehat\mu_N)(f)+\widehat\mu_N(f-f_\varepsilon)+(\widehat\mu_N-\mu_N)(f_\varepsilon)+(\mu_N-\mu)(f_\varepsilon)+\mu(f_\varepsilon-f).
\]
The first and third terms are at most $\delta_N$ in absolute value, by \Cref{lem:model-tv}, because $f$ and $f_\varepsilon$ take values in $[0,1]$. The second and fifth terms are at most $\sqrt d\,\varepsilon\sqrt C$ and $\sqrt d\,\varepsilon\sqrt{I_*}$. The function $f_\varepsilon/(1+\sqrt d/\varepsilon)$ is admissible in \eqref{mc-eq:hv10}, and the fourth term is at most $(1+\sqrt d/\varepsilon)\,d_{\mathrm{BL}}(\mu_N,\mu)$. Taking the supremum over $f$ proves \eqref{mc-eq:hv11}.

For the second statement, we first show that $d_{\mathrm{BL}}(\mu_N,\mu)\to0$. This is classical, but we include the argument. Fix $\kappa_0>0$, let $R$ satisfy $\mu(|z|\le R)\ge1-\kappa_0$, and let $\psi:\R^d\to[0,1]$ be continuous, equal to one on $\{|z|\le R\}$ and to zero outside $\{|z|\le R+1\}$. By weak convergence, $\mu_N(|z|\le R+1)\ge\mu_N(\psi)\ge1-2\kappa_0$ for large $N$. The Arzelà--Ascoli theorem shows that the restrictions to $\{|z|\le R+1\}$ of the functions admissible in \eqref{mc-eq:hv10} have a finite $\kappa_0$-net $\varphi_1,\dots,\varphi_n$ in the uniform norm, which we may take admissible. For an admissible $\varphi$ and $\varphi_k$ within $\kappa_0$ of it on $\{|z|\le R+1\}$, we have
\[
|\mu_N(\varphi)-\mu(\varphi)|\le|\mu_N(\varphi_k)-\mu(\varphi_k)|+2\kappa_0+2\big(\mu_N(|z|>R+1)+\mu(|z|>R+1)\big).
\]
This gives $\limsup_Nd_{\mathrm{BL}}(\mu_N,\mu)\le8\kappa_0$, and $\kappa_0$ is arbitrary. Now we fix $\delta>0$ and use \eqref{mc-eq:hv11} with $\widehat\mu_N=\widehat\mu_N^\delta$. Letting $N\to\infty$ and then $\varepsilon\to0$, we find $\limsup_N\|\mu_N-\mu\|_{\TV}\le2\delta$. Since $\delta$ is arbitrary, this completes the proof.
\end{proof}

In the second statement, the approximating laws $\widehat\mu_N^\delta$ need not converge weakly, and their limit points need not be $\mu$. Weak convergence is required only of $\mu_N$. The order of the limits is: $\delta$ and the approximating laws are fixed, then $N\to\infty$, then $\varepsilon\to0$, then $\delta\to0$.

\section{Fisher information of finitely many coordinates}\label{sec:score-coordinates}\label{csh-sec:head-tv}

We now relate scores along the rotations $K_{1j}$ to derivatives in the coordinates $x_2,\dots,x_N$, viewed as coordinates on the hemisphere $\{x_1>0\}$.

\paragraph{The chart.}
For $x\in S_N$, let $y(x)=(x_2,\dots,x_N)\in\R^{N-1}$. Let $U_N=\{y\in\R^{N-1}:|y|^2<N\}$, and for $y\in U_N$ let $x_1(y)=(N-|y|^2)^{1/2}$ and $x^+(y)=(x_1(y),y)\in S_N$. For a measure $\rho$ on $S_N$, let $\rho^+$ be its image under $x\mapsto y(x)$. If $\rho$ is invariant under $R_1$, then $\rho^+$ determines $\rho$, and by the second part of \Cref{lem:model-symmetry}, for two such probability measures $\rho,\rho'$,
\begin{equation}\label{eq:score-fold-tv}
\|\rho-\rho'\|_{\TV}=\|\rho^+-\rho'^+\|_{\TV}.
\end{equation}
Indeed, $\rho^+$ is the image of the law of $(|x_1|,x_2,\dots,x_N)$ under a bijection. We write $|S_N|$ for the surface area of $S_N$ and $\omega_N=\sqrt N/|S_N|$.

\begin{lemma}[Densities and scores in the chart]\label{lem:score-chart}
Let $\rho=f\nu_N$ be a probability measure on $S_N$ that is invariant under $R_1$.
\begin{enumerate}
\item The measure $\rho^+$ has the Lebesgue density $f^+(y)=2\omega_Nf(x^+(y))/x_1(y)$ on $U_N$, and $\rho^+(\R^{N-1}\setminus U_N)=0$.
\item Suppose that $\rho$ has a weak $K_{1j}$-score $s_j$ for some $2\le j\le N$, and write $s_j^+=s_j\circ x^+$. Then the distributional derivative $\partial_{y_j}f^+$ belongs to $L^1_{\mathrm{loc}}(U_N)$, and
\begin{equation}\label{eq:score-jacobian}
\partial_{y_j}f^+=f^+\Big(\frac{s_j^+}{x_1}+\frac{y_j}{x_1^2}\Big)\qquad\text{almost everywhere on }U_N .
\end{equation}
\item If, in addition, $\rho(\{x_1^2<r_0N\})=0$ for some $r_0\in(0,1]$, then the extension of $f^+$ by zero has a weak derivative in $y_j$ on $\R^{N-1}$ which belongs to $L^1(\R^{N-1})$ and is given by \eqref{eq:score-jacobian} on $U_N$ and by zero outside $U_N$.
\end{enumerate}
\end{lemma}

The term $y_j/x_1^2$ in \eqref{eq:score-jacobian} comes from the surface element $x_1^{-1}dy$ of the chart, and it has a positive sign. In the situation of the second part, the score is odd under $R_1$: since $K_{1j}(\varphi\circ R_1)=-(K_{1j}\varphi)\circ R_1$ and $\rho$ is invariant, the function $-s_j\circ R_1$ also satisfies \eqref{eq:score-weak-score}, so $s_j\circ R_1=-s_j$ $\rho$-almost everywhere by uniqueness. In particular $|s_j|$ is invariant under $R_1$, and $\int|s_j^+|^qf^+\,dy=\E_\rho|s_j|^q$ for every $q\ge1$.

\begin{proof}
The upper hemisphere is the graph of $y\mapsto x_1(y)$ over $U_N$. We have $\nabla x_1=-y/x_1$, so its surface element is $(1+|y|^2/x_1^2)^{1/2}dy=\sqrt N\,x_1^{-1}dy$, and we have $\int_{\{x_1>0\}}\varphi\,d\nu_N=\omega_N\int_{U_N}\varphi(x^+(y))\,x_1(y)^{-1}dy$. The equator has $\nu_N$-measure zero, and by the invariance of $\rho$ we have $\rho^+(A)=2\rho(\{y(x)\in A,\,x_1>0\})$. The first part follows.

For the second part, let $\psi\in C^\infty_c(U_N)$, and let $\varphi(x)=\psi(y(x))$ if $x_1>0$ and $\varphi(x)=0$ otherwise. Since $\psi$ vanishes near $\partial U_N$, the function $\varphi$ vanishes near the equator and is smooth on $S_N$, with $K_{1j}\varphi(x)=x_1(\partial_j\psi)(y(x))$ for $x_1>0$. Inserting $\varphi$ in \eqref{eq:score-weak-score} and using the first part, we find
\[
\int_{U_N}x_1\,\partial_j\psi\,f^+\,dy=-\int_{U_N}\psi\,s_j^+f^+\,dy .
\]
In other words, the distributional derivative of $x_1f^+$ in $y_j$ is $s_j^+f^+$, which lies in $L^1_{\mathrm{loc}}(U_N)$ because $s_j\in L^1(\rho)$ and $x_1^{-1}$ is locally bounded on $U_N$. The function $x_1^{-1}$ is smooth on $U_N$ with $\partial_j(x_1^{-1})=y_j/x_1^3$, and by the product rule we have $\partial_jf^+=\partial_j(x_1f^+\cdot x_1^{-1})=s_j^+f^+/x_1+f^+y_j/x_1^2$.

For the third part, $f^+=0$ almost everywhere on $\{|y|^2>(1-r_0)N\}$. Let $\chi\in C^\infty_c(U_N)$ equal one on $\{|y|^2\le(1-r_0)N\}$. For $\psi\in C^\infty_c(\R^{N-1})$, we have $\int\partial_j\psi\,f^+=\int\partial_j(\psi\chi)f^+-\int\psi\,\partial_j\chi\,f^+$, and the last integral vanishes because $\partial_j\chi=0$ wherever $f^+\ne0$. The claim follows by applying the second part to $\psi\chi$. The derivative lies in $L^1$ because $x_1^2\ge r_0N$ on the support of $f^+$ and $s_j\in L^1(\rho)$.
\end{proof}

We next record the score of a marginal. Let $a_2,\dots,a_N>0$ and let $G$ be the centered Gaussian product law on $\R^{N-1}$ whose coordinates $y_i$ have variance $(2a_i)^{-1}$, $2\le i\le N$. Its density is
\begin{equation}\label{eq:score-gaussian}
\gamma(y)=\prod_{i=2}^N\sqrt{a_i/\pi}\,e^{-a_iy_i^2}.
\end{equation}
Fix $2\le J\le N-1$ and split $y=(z,v)$ with $z=(y_2,\dots,y_J)$ and $v=(y_{J+1},\dots,y_N)$, the remaining coordinates. Then $G=G_{\mathrm z}\otimes G_{\mathrm v}$, the product of the corresponding Gaussian laws on $\R^{J-1}$ and $\R^{N-J}$. For a probability density $w$ with respect to $G$, let $\bar w(z)=\int w(z,v)\,G_{\mathrm v}(dv)$ be the density of its $z$-marginal with respect to $G_{\mathrm z}$, and, for $\bar w(z)>0$, let $\E_w[\,\cdot\mid z]=\bar w(z)^{-1}\int\cdot\ w(z,v)\,G_{\mathrm v}(dv)$ be the conditional expectation given $z$.

\begin{lemma}[Marginal score]\label{lem:score-marginal-score}
Let $\widehat\rho$ be a probability measure on $S_N$, invariant under $R_1$, with $\widehat\rho(\{x_1^2<r_0N\})=0$ for some $r_0\in(0,1]$ and with weak $K_{1j}$-scores $\vartheta_j$ for $2\le j\le J$. Let $w=d\widehat\rho^+/dG$. Then $\bar w\in W^{1,1}_{\mathrm{loc}}(\R^{J-1})$ and, almost everywhere on $\{\bar w>0\}$, for $2\le j\le J$,
\begin{equation}\label{eq:score-marginal}
\partial_{z_j}\log\bar w=\E_w\Big[\frac{\vartheta_j\circ x^+}{x_1}+\frac{z_j}{x_1^2}\,\Big|\,z\Big]+2a_jz_j .
\end{equation}
In particular, with $h=\log\bar w$,
\begin{equation}\label{mc-eq:te14}
z_j\partial_jh=\E_w\Big[\frac{z_j}{x_1}\,\vartheta_j\circ x^++\frac{z_j^2}{x_1^2}\,\Big|\,z\Big]+2a_jz_j^2 .
\end{equation}
\end{lemma}

\begin{proof}
Let $f^+=w\gamma$ be the Lebesgue density of $\widehat\rho^+$, and let $p(z)=\int f^+(z,v)\,dv$. The third part of \Cref{lem:score-chart} implies that $\partial_jf^+$ exists in $L^1(\R^{N-1})$ and is given by \eqref{eq:score-jacobian}. For $\psi\in C^\infty_c(\R^{J-1})$, we use the definition of the weak derivative with the test function $\psi(z)\chi_n(v)$, where $\chi_n\in C_c^\infty(\R^{N-J})$ take values in $[0,1]$ and increase to one, and let $n\to\infty$ by dominated convergence. This shows that $p\in W^{1,1}(\R^{J-1})$ with $\partial_jp(z)=\int\partial_jf^+(z,v)\,dv$, that is, $\partial_jp=p\,\E_w[\vartheta_j\circ x^+/x_1+z_j/x_1^2\mid z]$ on $\{p>0\}$. Here the conditional law of $v$ given $z$ is the same under $f^+(z,v)\,dv/p(z)$ and under $w(z,v)\,G_{\mathrm v}(dv)/\bar w(z)$. Finally $\bar w=p/\gamma_{\mathrm z}$, where $\gamma_{\mathrm z}(z)=\prod_{j=2}^J\sqrt{a_j/\pi}e^{-a_jz_j^2}$ is smooth and positive, and $\partial_j\log\gamma_{\mathrm z}=-2a_jz_j$. We obtain \eqref{eq:score-marginal}, and \eqref{mc-eq:te14} follows on multiplying by $z_j$.
\end{proof}

\paragraph{Standardized coordinates.}
For the rest of this section we assume
\begin{equation}\label{mc-eq:hv1}
0=a_1<a=a_2\le a_3\le\cdots\le a_J\le C_Ja,\qquad\theta>0,\qquad m=\theta Na ,
\end{equation}
for a fixed integer $J\ge2$ and a constant $C_J\ge1$; the coefficients $a_k$ with $k>J$ are arbitrary real numbers. In the applications, $\theta$ is the top fraction $\theta_N$ and $m=m_N$, as in \eqref{eq:model-cold-parameters}. We write
\begin{equation}\label{mc-eq:hv2}
\mathbf z(x)=\big(\sqrt{2a_2}\,x_2,\dots,\sqrt{2a_J}\,x_J\big)\in\R^{J-1}
\end{equation}
for the standardized coordinates. This scaling sends $\bigotimes_{j=2}^J\cN(0,(2a_j)^{-1})$ to $\cN(0,I_{J-1})$.

\begin{lemma}[Fisher information of the standardized coordinates]\label{csh-lem:graph-fisher}
Assume \eqref{mc-eq:hv1} with $m\ge1$, and let $s,K_0,C_0>0$. Let $\widehat\rho$ be a probability measure on $S_N$ that is invariant under $R_1$ and satisfies
\begin{equation}\label{mc-eq:hv5}
\widehat\rho\big(\{x_1^2<s\theta N\}\cup\{|\mathbf z(x)|>K_0\}\big)=0 .
\end{equation}
Suppose that $\widehat\rho$ has weak $K_{1j}$-scores $\vartheta_j$ with
\begin{equation}\label{mc-eq:hv6}
\E_{\widehat\rho}|\vartheta_j|^2\le C_0\,m,\qquad2\le j\le J .
\end{equation}
Then the law of $\mathbf z(x)$ under $\widehat\rho$ has a density $p$ with
\begin{equation}\label{mc-eq:hv7}
\mathrm{FI}(p)\le\frac{(J-1)C_0}{s}+\frac{K_0^2}{2s^2},\qquad\supp p\subset\{|\mathbf z|\le K_0\}.
\end{equation}
\end{lemma}

\begin{proof}
Let $f^+$ be the Lebesgue density of $\widehat\rho^+$, and change variables to $(\mathbf z,v)$ with $\mathbf z_j=\sqrt{2a_j}\,y_j$ for $2\le j\le J$. The joint density of $(\mathbf z,v)$ is $Q(\mathbf z,v)=f^+(y)\prod_{j=2}^J(2a_j)^{-1/2}$. We have $\partial_{\mathbf z_j}=(2a_j)^{-1/2}\partial_{y_j}$ and $y_j=\mathbf z_j/\sqrt{2a_j}$, and by the third part of \Cref{lem:score-chart},
\begin{equation}\label{mc-eq:hv8}
\partial_{\mathbf z_j}\log Q=\frac{\vartheta_j\circ x^+}{\sqrt{2a_j}\,x_1}+\frac{\mathbf z_j}{2a_jx_1^2}
\end{equation}
almost everywhere on $\{Q>0\}$, and $\partial_{\mathbf z_j}Q=0$ almost everywhere on $\{Q=0\}$. On the support of $\widehat\rho$, we have $a_jx_1^2\ge a\,s\theta N=sm$, because $a_j\ge a$. Using $(\alpha+\beta)^2\le2\alpha^2+2\beta^2$, \eqref{mc-eq:hv6}, $|\mathbf z|\le K_0$, and $m\ge1$, we have
\begin{equation}\label{mc-eq:hv9}
\sum_{j=2}^J\int Q\,|\partial_{\mathbf z_j}\log Q|^2\,d\mathbf z\,dv
\le\sum_{j=2}^J\Big(\frac{\E_{\widehat\rho}|\vartheta_j|^2}{sm}+\frac{\E_{\widehat\rho}\mathbf z_j^2}{2s^2m^2}\Big)
\le\frac{(J-1)C_0}{s}+\frac{K_0^2}{2s^2}.
\end{equation}
Let $p(\mathbf z)=\int Q(\mathbf z,v)\,dv$. As in the proof of \Cref{lem:score-marginal-score}, $\partial_jp(\mathbf z)=\int\partial_{\mathbf z_j}Q(\mathbf z,v)\,dv$. By the Cauchy--Schwarz inequality on each fiber, we have $|\partial_jp|^2\le p\int Q\,|\partial_{\mathbf z_j}\log Q|^2\,dv$, and in particular $\partial_jp=0$ where $p=0$. Dividing by $p$, integrating in $\mathbf z$, and summing over $j$, we deduce \eqref{mc-eq:hv7} from \eqref{mc-eq:hv9}.
\end{proof}

We can now upgrade weak convergence of the standardized coordinates to convergence in total variation. A path weight in the sense of \eqref{mc-eq:sr3} is written $W=\Phi(\xi,X|_I)$, and for $2\le j\le J$ we write $\Theta^{(1j)}_\varepsilon$ for the path rotations in the $(1,j)$-plane with the linear ramp on $I$.

\begin{corollary}[Total variation convergence of finitely many coordinates]\label{csh-cor:cold-head}
For each $N$, let $a_1,\dots,a_N$ and $\theta=\theta_N$ satisfy \eqref{mc-eq:hv1}, with $J$ and $C_J$ independent of $N$ and $m=\theta Na\to\infty$. Let $X$ solve \eqref{mc-eq:sr1} with $X_0\sim\nu_N$, let $t_N\ge1/a$ be deterministic, and let $\mu_N$ be the law of $\mathbf z(X_{t_N})$. Suppose that $\mu_N\Rightarrow\mu$, where $\mu$ is a probability measure on $\R^{J-1}$ with $\mathrm{FI}(\mu)<\infty$. Suppose further that for every $\delta\in(0,1)$ there exist $s,L,B>0$, independent of $N$, and for every $N$ a random variable $W_N$ and a function $\eta_N^0$, with the following properties.
\begin{enumerate}
\item $W_N=\Phi_N(\xi_N,X|_{I_N})$ is a path weight for the final interval $I_N=[t_N-1/a,t_N]$ and an open set $D_N$, as in \eqref{mc-eq:sr3}. The set $D_N$ is invariant under all rotations of $x_1,\dots,x_J$ and contained in $\{\sum_{k\le J}x_k^2<L\theta N\}$. For every $2\le j\le J$ and $\varepsilon\in\R$, the function $\Phi_N$ is invariant under $\Theta^{(1j)}_\varepsilon$. The measure $A\mapsto\E[W_N\ind_{\{X_{t_N}\in A\}}]$ is invariant under $R_1$.
\item $\eta_N^0\in C^1(S_N)$ takes values in $[0,1]$, is invariant under $R_1$, vanishes on $\{x_1^2<s\theta N\}$, and satisfies $|K_{1j}\eta_N^0|\le B(\eta_N^0)^{1/2}$ for $2\le j\le J$.
\item $\liminf_{N\to\infty}\E\big[W_N\,\eta_N^0(X_{t_N})\big]\ge1-\delta$.
\end{enumerate}
Then $\|\mu_N-\mu\|_{\TV}\to0$.
\end{corollary}

\begin{proof}
Fix $\delta\in(0,1/6)$ and the corresponding $s,L,B$, $W_N$, and $\eta^0_N$. We construct approximating laws for \Cref{csh-thm:weak-to-tv}.

\paragraph{A cutoff in the standardized coordinates.}
Let $\psi\in C^\infty(\R;[0,1])$ equal one on $(-\infty,1]$ and zero on $[2,\infty)$, and for $K_0>0$ let $\psi_{K_0}(\mathbf z)=\psi(|\mathbf z|/K_0)^2$. Then $|\nabla\psi_{K_0}|\le(2\|\psi'\|_\infty/K_0)\,\psi_{K_0}^{1/2}$. We fix $K_0$ such that $\mu(|\mathbf z|\ge K_0)\le\delta$. The function $1-\psi_{K_0}$ is bounded, continuous, and vanishes on $\{|\mathbf z|\le K_0\}$, and weak convergence implies that $\limsup_N\E[1-\psi_{K_0}(\mathbf z(X_{t_N}))]\le\delta$. Let $\eta_N=\eta_N^0\,\psi_{K_0}\circ\mathbf z$ and $U_N=W_N\eta_N(X_{t_N})$. From the third hypothesis, we conclude that
\begin{equation}\label{eq:score-head-mass}
\liminf_N\E U_N\ge\liminf_N\E\big[W_N\eta_N^0(X_{t_N})\big]-\limsup_N\E\big[1-\psi_{K_0}(\mathbf z(X_{t_N}))\big]\ge1-2\delta .
\end{equation}

\paragraph{Scores of the weighted endpoint law.}
Let $\rho_N=\rho^{W_N}$ be the endpoint law weighted by $W_N$, with normalizing mass $\E W_N$. We have $\E W_N\ge\E U_N\ge1/2$ for large $N$, and \Cref{csr-cor:cold-scale} with $p=1$ and $z_0=1/2$ gives $\E_{\rho_N}|K_{1j}\log(d\rho_N/d\nu_N)|^2\le Cm$ for $2\le j\le J$, with $C$ depending only on $L$ and $C_J$. The law $\rho_N$ is concentrated on $D_N$, where $x_1^2\le L\theta N$. We have $K_{1j}\mathbf z=\sqrt{2a_j}\,x_1e_j$ and $a_j\le C_Ja$, so that $|K_{1j}\mathbf z|\le(2C_JLm)^{1/2}$ on $D_N$. Using the product rule, we find that, $\rho_N$-almost everywhere,
\begin{equation}\label{mc-eq:hv12}
|K_{1j}\eta_N|\le B(\eta_N^0)^{1/2}\psi_{K_0}+\eta_N^0\,\frac{2\|\psi'\|_\infty}{K_0}\,\psi_{K_0}^{1/2}(2C_JLm)^{1/2}\le B'\sqrt m\,\eta_N^{1/2}
\end{equation}
for $m\ge1$, where $B'$ depends on $B$, $K_0$, $C_J$, and $L$. Here we used $\psi_{K_0}\le\psi_{K_0}^{1/2}$ and $\eta_N^0\le(\eta_N^0)^{1/2}$. The normalizer of $\eta_N\rho_N$ is $\int\eta_N\,d\rho_N=\E U_N/\E W_N\ge\E U_N\ge1/2$ for large $N$. \Cref{csh-lem:cut-score} with $p=1$ implies that the normalized cutoff-weighted law $\widehat\rho_N=\eta_N\rho_N/\int\eta_N\,d\rho_N$ has weak $K_{1j}$-scores with second moments at most $C'm$, where $C'$ depends on $\delta$ through $B,K_0,L$. It is invariant under $R_1$, by the first two hypotheses and because $\mathbf z$ does not involve $x_1$, and it satisfies \eqref{mc-eq:hv5} with $s$ and $2K_0$ in place of $K_0$. Using \Cref{csh-lem:graph-fisher}, we find that the law $\widehat\mu_N$ of $\mathbf z$ under $\widehat\rho_N$ satisfies $\mathrm{FI}(\widehat\mu_N)\le C_\delta$ for large $N$, with $C_\delta$ independent of $N$.

\paragraph{Conclusion.}
The law $\widehat\mu_N$ is the law of $\mathbf z(X_{t_N})$ under $(U_N/\E U_N)\PP$. Combining \Cref{lem:score-domination} and \eqref{eq:score-head-mass}, we find $\limsup_N\|\mu_N-\widehat\mu_N\|_{\TV}\le\limsup_N(1-\E U_N)\le2\delta$. The second statement of \Cref{csh-thm:weak-to-tv}, applied with $2\delta$ in place of $\delta$, completes the proof.
\end{proof}

\begin{remark}[Gaussian convolutions as limits]\label{rem:score-convolution-limits}
By \Cref{csh-lem:convolution-fisher}, every limit of the form $\mu=\lambda*\cN(0,I_{J-1})$ satisfies $\mathrm{FI}(\mu)\le J-1$, for an arbitrary probability measure $\lambda$. In the low-temperature sections, $\mu$ is the law of the sum of a standard Gaussian vector and an independent multiple of a standard Cauchy variable in the first coordinate. Its Fisher information is finite although the Cauchy variable has no moments.
\end{remark}

\section{Conditional relative entropy of the remaining coordinates}\label{sec:score-entropy}

In this section, $a_1=0<a_i$ for $2\le i\le N$, so that $F(x)=\sum_{i\ge2}a_ix_i^2$ and the Gaussian product law $G$ of \eqref{eq:score-gaussian} is defined. We fix $2\le J\le N-1$ and use the splitting $y=(z,v)$ of \Cref{sec:score-coordinates}. We study the relative entropy of the remaining coordinates $v$ given $z$, with respect to their Gaussian law $G_{\mathrm v}$. In \Cref{sec:score-chart-entropy}, we write the generator in the chart and prove a logarithmic Sobolev inequality for this conditional relative entropy. \Cref{sec:score-weighted-logs} contains moment bounds for logarithms of densities. \Cref{sec:score-entropy-identity} computes the time derivative of the conditional relative entropy along the diffusion, and \Cref{sec:score-dissipation} bounds it under a cutoff function adapted to the approach to the critical temperature.

\subsection{The generator in the chart and a logarithmic Sobolev inequality}\label{sec:score-chart-entropy}

For $y\in U_N$, let
\[
\mathfrak g(y)=I-\frac{yy^\T}N,\qquad\Gamma(\varphi,\psi)=\nabla\varphi^\T\mathfrak g\nabla\psi,\qquad\Gamma(\varphi)=\Gamma(\varphi,\varphi).
\]
The matrix $\mathfrak g(y)$ is the inverse of the metric $I+yy^\T/x_1^2$ of $S_N$ in the chart, with eigenvalues $1$ and $x_1^2/N$. If $\varphi$ is a function on $S_N$ with $\varphi(x)=\psi(y(x))$ for $x_1>0$, then $|\nabla_{S_N}\varphi|^2(x^+(y))=\Gamma(\psi)(y)$, so $\Gamma$ is the carré du champ of $L$ in the chart. For a smooth vector field $V$ on $U_N$, let $\operatorname{div}_GV=\gamma^{-1}\operatorname{div}(\gamma V)$, and let $L_G\psi=\operatorname{div}_G(\mathfrak g\nabla\psi)$. By the divergence theorem, for $\varphi,\psi\in C^2(U_N)$, one of which has compact support in $U_N$,
\begin{equation}\label{eq:score-chart-ibp}
\int\varphi\,L_G\psi\,dG=-\int\Gamma(\varphi,\psi)\,dG .
\end{equation}

\begin{lemma}[The generator in the chart]\label{lem:score-chart-generator}
For $\psi\in C^2(\R^{N-1})$, let $\varphi=\psi\circ y$ on $S_N$. Then $L\varphi=(L\psi)\circ y$ on $\{x_1>0\}$, where
\begin{equation}\label{mc-eq:te20}
L\psi=\operatorname{div}_G(\mathfrak g\nabla\psi)+\frac yN\cdot\nabla\psi .
\end{equation}
\end{lemma}

\begin{proof}
Using \eqref{mc-eq:sr1}, we see that the generator acts on $\varphi=\psi(x_2,\dots,x_N)$ as $\sum_{k,l\ge2}\Pi(x)_{kl}\partial_k\partial_l\psi+\sum_{k\ge2}\mathbf b_k(x)\partial_k\psi$. For $k,l\ge2$ we have $\Pi(x)_{kl}=\mathfrak g(y)_{kl}$. Using $\nabla F=2(a_kx_k)_k$ and $x\cdot\nabla F=2F$, we find $\nabla_{S_N}F=\nabla F-(2F/N)x$ and $\mathbf b_k=[-\frac{N-1}N-2a_k+\frac{2F}N]y_k$ for $k\ge2$. On the other hand, $L_G\psi=\mathfrak g:\nabla^2\psi+(\operatorname{div}\mathfrak g+\mathfrak g\nabla\log\gamma)\cdot\nabla\psi$. In dimension $N-1$, we have $(\operatorname{div}\mathfrak g)_k=-N^{-1}\sum_l\partial_l(y_ky_l)=-y_k$, and $\mathfrak g\nabla\log\gamma=-2\diag(a)y+(2F/N)y$ because $y\cdot\diag(a)y=F$. Then the drift of $L_G$ is $[-1-2a_k+2F/N]y_k$, which differs from $\mathbf b_k$ by $y_k/N$. This proves \eqref{mc-eq:te20}. In the uses below, a test in $C^\infty_c(D^+)$ is extended by zero to $\R^{N-1}$, so it belongs to this class; invariance under $R_1$ transfers the resulting integral identity to the folded measure.
\end{proof}

The coefficient one in front of $(y/N)\cdot\nabla\psi$ reflects the dimension $N-1$ of the chart. It produces the term $1$ in the factor $1+S$ in \eqref{mc-eq:te21} below.

\paragraph{Conditional relative entropy.}
Let $w$ be a probability density with respect to $G$. With $\bar w$ and $\E_w[\,\cdot\mid z]$ as in \Cref{sec:score-coordinates}, let $h=\log\bar w$ on $\{\bar w>0\}$ and $g=\log w-h$ on $\{w>0\}$. We write $\E_w$ for expectations under the probability measure $wG$, and
\begin{equation}\label{eq:score-H-D}
\mathscr H(w)=\int wg\,dG,\qquad\mathscr D(w)=\int w\,\Gamma(g)\,dG ,
\end{equation}
where $\mathscr D(w)$ is defined when $w\in C^\infty(\R^{N-1})$ has compact support in $U_N$. Then $\mathscr H(w)=\KL(wG\Vert(\bar wG_{\mathrm z})\otimes G_{\mathrm v})\in[0,\infty]$ is the relative entropy of the law $wG$ with respect to the product of its own $z$-marginal and the Gaussian law of the remaining coordinates. Equivalently, it is the average over $z$ of the relative entropy of the conditional law of $v$ given $z$ with respect to $G_{\mathrm v}$. We call it the \emph{conditional relative entropy of the remaining coordinates} with respect to $G_{\mathrm v}$. If $\KL(wG\Vert G)<\infty$, then
\begin{equation}\label{eq:score-chain-rule}
\KL(wG\Vert G)=\KL(\bar wG_{\mathrm z}\Vert G_{\mathrm z})+\mathscr H(w).
\end{equation}
This is the chain rule for relative entropy (see \cite[Theorem~2.5.3 and Section~8.5]{CoverThomas2006}), and we include its short proof. By Jensen's inequality on each fiber, applied to the convex function $x\log x$, we have $\int\bar w\log\bar w\,dG_{\mathrm z}\le\int w\log w\,dG<\infty$; since $x\log x\ge-1/e$, both $w\log w$ and $\bar w\log\bar w$ are integrable. On $\{w>0\}$, we have $\log w=\log\bar w+g$, and Fubini's theorem gives $\int w\log\bar w\,dG=\int\bar w\log\bar w\,dG_{\mathrm z}$. Hence $\int w\log w\,dG=\int\bar w\log\bar w\,dG_{\mathrm z}+\int wg\,dG$, which is \eqref{eq:score-chain-rule}.
The quantity $\mathscr D(w)$ uses the full metric $\mathfrak g$ applied to the gradient of $g$ in all variables. The next lemma separates its part in the $v$ variables.

\begin{lemma}[The metric of the remaining coordinates]\label{lem:score-schur}
Let $y=(z,v)\in U_N$ and $x_1^2=N-|z|^2-|v|^2$.
\begin{enumerate}
\item Let $A_v=I-vv^\T/(N-|z|^2)$. For $\pi=(\pi_z,\pi_v)\in\R^{J-1}\times\R^{N-J}$, let $\pi_z'=\pi_z-z\,(v\cdot\pi_v)/(N-|z|^2)$. Then
\begin{equation}\label{mc-eq:te25}
\pi^\T\mathfrak g\,\pi=\pi_z'^\T\Big(I-\frac{zz^\T}N\Big)\pi_z'+\pi_v^\T A_v\pi_v .
\end{equation}
In particular $\Gamma(\varphi)\ge\nabla_v\varphi^\T A_v\nabla_v\varphi$. Moreover $A_v$ is positive definite and
\[
v^\T A_v^{-1}v=\frac{(N-|z|^2)\,|v|^2}{x_1^2}.
\]
\item Let $b>0$ with $b\le a_i$ for all $i>J$, and suppose that $b\sum_{i>J}v_i^2/a_i\le d\,x_1^2$ for some $d>0$. Then
\begin{equation}\label{mc-eq:te27}
A_v\ge\frac b{1+d}\diag\big(a_i^{-1}\big)_{i>J}.
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We write $\mathfrak g$ in blocks: $\mathfrak g_{zz}=I-zz^\T/N$, $\mathfrak g_{zv}=-zv^\T/N$, and $\mathfrak g_{vv}=I-vv^\T/N$. By the rank-one inverse formula of Bartlett \cite{Bartlett1951}, usually called the Sherman--Morrison formula \cite{ShermanMorrison1950}, $\mathfrak g_{zz}^{-1}=I+zz^\T/(N-|z|^2)$, and a direct computation shows that $\mathfrak g_{zz}^{-1}\mathfrak g_{zv}\pi_v=-z\,(v\cdot\pi_v)/(N-|z|^2)$ and
\[
\mathfrak g_{vv}-\mathfrak g_{zv}^\T\mathfrak g_{zz}^{-1}\mathfrak g_{zv}=I-\frac{vv^\T}N-\frac{|z|^2\,vv^\T}{N(N-|z|^2)}=A_v .
\]
The identity in \eqref{mc-eq:te25} is the completion of the square $\pi^\T\mathfrak g\pi=(\pi_z+\mathfrak g_{zz}^{-1}\mathfrak g_{zv}\pi_v)^\T\mathfrak g_{zz}(\pi_z+\mathfrak g_{zz}^{-1}\mathfrak g_{zv}\pi_v)+\pi_v^\T A_v\pi_v$, and the first term is nonnegative. Using the Sherman--Morrison formula again, we find $A_v^{-1}=I+vv^\T/x_1^2$ and $v^\T A_v^{-1}v=|v|^2+|v|^4/x_1^2=|v|^2(N-|z|^2)/x_1^2$.

For the second part, let $D_0=\diag(1-b/((1+d)a_i))_{i>J}$, a positive definite diagonal matrix. For a positive definite $D_0$ and $s_0>0$, the matrix $D_0-vv^\T/s_0$ is positive semidefinite if and only if $v^\T D_0^{-1}v\le s_0$, as one sees by conjugating with $D_0^{-1/2}$. Since $b\le a_i$, we have $1/(1-b/((1+d)a_i))=1+b/((1+d)a_i-b)\le1+b/(da_i)$, and therefore
\[
v^\T D_0^{-1}v\le|v|^2+\frac bd\sum_{i>J}\frac{v_i^2}{a_i}\le|v|^2+x_1^2=N-|z|^2 .
\]
We conclude that $D_0\ge vv^\T/(N-|z|^2)$, which is equivalent to \eqref{mc-eq:te27}.
\end{proof}

The Gaussian logarithmic Sobolev inequality is due to Gross \cite{Gross1975}. We use it in the following form: if $\mathsf G$ is the standard Gaussian measure on $\R^n$ and $\phi\ge0$ is smooth with $0<\int\phi\,d\mathsf G<\infty$, then
\begin{equation}\label{eq:score-gaussian-lsi}
\int\phi\log\frac{\phi}{\int\phi\,d\mathsf G}\,d\mathsf G\le\frac12\int\frac{|\nabla\phi|^2}{\phi}\,d\mathsf G .
\end{equation}
Here $0\log0=0$, and $|\nabla\phi|^2/\phi$ is set to zero on $\{\phi=0\}$, where $\nabla\phi=0$ because $\phi$ is smooth and nonnegative; the energy integral may equal $+\infty$. The identically zero function has entropy and energy zero. See \cite[Theorem~2.7, (9), p.~5]{Gentil2010}.
Applying \eqref{eq:score-gaussian-lsi} to $\phi(\Sigma^{1/2}\cdot)$ with $\Sigma=\diag((2a_i)^{-1})_{i>J}$, we obtain, for smooth $\phi\ge0$ on $\R^{N-J}$ with $\int\phi\,dG_{\mathrm v}=1$,
\begin{equation}\label{eq:score-anisotropic-lsi}
\int\phi\log\phi\,dG_{\mathrm v}\le\sum_{i>J}\frac1{4a_i}\int\frac{(\partial_{v_i}\phi)^2}{\phi}\,dG_{\mathrm v}.
\end{equation}

\begin{proposition}[A logarithmic Sobolev inequality for the conditional relative entropy]\label{prop:score-conditional-lsi}
Let $w\in C^\infty(\R^{N-1})$ be a nonnegative probability density with respect to $G$, with compact support in $U_N$, and let $b,d$ be as in the second part of \Cref{lem:score-schur}. Suppose that $b\sum_{i>J}v_i^2/a_i\le d\,x_1^2$ at every point of $\{w>0\}$. Then
\begin{equation}\label{eq:score-conditional-lsi}
\mathscr H(w)\le\frac{1+d}{4b}\,\mathscr D(w).
\end{equation}
\end{proposition}

\begin{proof}
Fix $z$ with $\bar w(z)>0$ and let $\phi(v)=w(z,v)/\bar w(z)$, a smooth nonnegative function with $\int\phi\,dG_{\mathrm v}=1$. Then $\int\phi\log\phi\,dG_{\mathrm v}=\E_w[g\mid z]$ and, on $\{\phi>0\}$, $(\partial_{v_i}\phi)^2/\phi=\phi\,(\partial_{v_i}g)^2$. Where $\phi=0$, also $\nabla\phi=0$. By \eqref{eq:score-anisotropic-lsi}, \eqref{mc-eq:te27}, and \eqref{mc-eq:te25}, we have
\[
\begin{aligned}
\E_w[g\mid z]&\le\frac14\E_w\big[\nabla_vg^\T\diag(a_i^{-1})\nabla_vg\mid z\big]\\
&\le\frac{1+d}{4b}\E_w\big[\nabla_vg^\T A_v\nabla_vg\mid z\big]\le\frac{1+d}{4b}\E_w\big[\Gamma(g)\mid z\big].
\end{aligned}
\]
Multiplying by $\bar w(z)$ and integrating with respect to $G_{\mathrm z}$ yields \eqref{eq:score-conditional-lsi}.
\end{proof}

Keeping the matrix $A_v$ leads to the constant $(1+d)/(4b)$, close to the Gaussian value $1/(4b)$ when $d$ is small. Replacing $A_v$ by its smallest eigenvalue $x_1^2/(N-|z|^2)$ would lose a factor of order one, which the applications cannot afford.

\subsection{Weighted logarithmic moments}\label{sec:score-weighted-logs}

The entropy identity below contains logarithms of densities multiplied by powers of a cutoff function. The next lemma bounds them using only an $L^p$ bound on the density, with no lower bound on it. We formulate this measure-theoretic statement on an arbitrary product of probability spaces.

\begin{lemma}[Weighted logarithmic moments]\label{cce-lem:weighted-logs}
Let $G=G_{\mathrm z}\otimes G_{\mathrm v}$ be a product probability measure on a product space $E_{\mathrm z}\times E_{\mathrm v}$. Let $f\ge0$ be measurable with $\int f\,dG\le1$ and
\[
\log\int f^p\,dG\le C_pA
\]
for some $p>1$, $C_p>0$, and $A\ge2$. Let $\eta$ be measurable with values in $[0,1]$, let $Z=\int\eta f\,dG\ge z_0>0$, and let $w=\eta f/Z$ and $\bar w(z)=\int w(z,v)\,G_{\mathrm v}(dv)$. On $E=\{0<w<\infty,\ 0<\bar w<\infty\}$ set $g=\log w-\log\bar w$, and set $g=0$ elsewhere. Then for every $r\in[1,\infty)$ and $\alpha_0\in(0,1]$,
\begin{equation}\label{mc-eq:te10}
\sup_{\alpha\in[\alpha_0,1]}\int_E f\eta^\alpha\big(|g|^r+|\log w|^r+|\log\bar w|^r\big)\,dG\le C_rA^r ,
\end{equation}
where $C_r$ depends only on $r,\alpha_0,p,C_p$, and $z_0$. Moreover $0\le\int wg\,dG\le CA$, with $C$ depending only on $p$, $C_p$, and $z_0$.
\end{lemma}

For every $\alpha>0$, the set $E$ has full $f\eta^\alpha G$-measure on $\{w>0\}$. Indeed, the $L^p$ assumption makes $f$ finite almost everywhere, $\int\bar w\,dG_{\mathrm z}=1$ makes $\bar w$ finite almost everywhere, and Tonelli's theorem gives $w=0$ almost everywhere over the fibers where $\bar w=0$. On the latter fibers this also gives $f\eta^\alpha=0$ almost everywhere. Thus the estimates do not depend on how the logarithms are defined on the exceptional set.

\begin{proof}
Throughout, $C$ denotes a constant depending only on $r,\alpha_0,p,C_p,z_0$. Note that $z_0\le Z\le1$. All density identities and fiber inequalities below are understood almost everywhere. In integrals over the product space, expressions involving $\log w$ or $\log\bar w$ are taken to be zero off $E$.

\paragraph{Step 1: moments slightly above one.}
Hölder's inequality shows that $\log\int f^{1+e}\,dG$ is a convex function of $e\in[0,p-1]$. Since it is at most $0$ at $e=0$ and at most $C_pA$ at $e=p-1$, we have $\int f^{1+e}\,dG\le\exp(C_pAe/(p-1))$. We choose $e=(p-1)/A\le(p-1)/2$, so that $\int f^{1+e}\,dG\le e^{C_p}$. Since $w\le f/z_0$, $z_0\le1$, and $1+e\le(p+1)/2$, we find $\int w^{1+e}\,dG\le z_0^{-(1+e)}\int f^{1+e}\,dG\le z_0^{-(p+1)/2}e^{C_p}$, and by Jensen's inequality on each fiber the functions $\bar w$ and $\bar f(z)=\int f(z,v)\,G_{\mathrm v}(dv)$ satisfy
\begin{equation}\label{eq:score-marginal-moments}
\int\bar w^{1+e}\,dG_{\mathrm z}\le z_0^{-(p+1)/2}e^{C_p},\qquad\int\bar f^{1+e}\,dG_{\mathrm z}\le e^{C_p}.
\end{equation}

\paragraph{Step 2: the logarithms of $f$, $\eta$, and $Z$.}
On $\{f\le1\}$, we have $f|\log f|^r\le(r/e_{\mathrm E})^r$, where $e_{\mathrm E}$ is Euler's number. On $\{f>1\}$, by Markov's inequality, we have $\int_{\{\log f>x\}}f\,dG\le e^{-(p-1)x}\int f^p\,dG\le e^{C_pA-(p-1)x}$, and this integral is also at most $1$. Let $x_0=2C_pA/(p-1)$. For $x\ge x_0$, we have $C_pA-(p-1)x\le-(p-1)x/2$, and therefore
\[
\int f(\log f)_+^r\,dG=\int_0^\infty rx^{r-1}\int_{\{\log f>x\}}f\,dG\,dx\le x_0^r+\int_0^\infty rx^{r-1}e^{-(p-1)x/2}\,dx\le CA^r .
\]
Further, $\eta^\alpha|\log\eta|^r\le(r/(\alpha_0e_{\mathrm E}))^r$ for $\alpha\ge\alpha_0$, and $|\log Z|\le\log(1/z_0)$. We have $\log w=\log f+\log\eta-\log Z$ on $\{w>0\}$, and these bounds prove the part of \eqref{mc-eq:te10} that concerns $\log w$.

\paragraph{Step 3: the logarithm of the marginal.}
Let $k_\alpha(z)=\int f\eta^\alpha\,dG_{\mathrm v}$. By Hölder's inequality on the fiber with exponents $1/\alpha$ and $1/(1-\alpha)$,
\[
k_\alpha=\int(f\eta)^\alpha f^{1-\alpha}\,dG_{\mathrm v}\le(Z\bar w)^\alpha\,\bar f^{1-\alpha},
\]
and in particular $k_\alpha=0$ where $\bar w=0$. On $\{0<\bar w\le1\}$, we have $k_\alpha|\log\bar w|^r\le\bar w^\alpha|\log\bar w|^r\bar f^{1-\alpha}\le C\bar f^{1-\alpha}$, and $\int\bar f^{1-\alpha}\,dG_{\mathrm z}\le1$ by Jensen's inequality. On $\{\bar w>1\}$, the inequality $(\log x)^r\le(r/e_{\mathrm E})^rx$ for $x\ge1$, applied to $x=\bar w^e$, gives $(\log\bar w)^r\le(rA/((p-1)e_{\mathrm E}))^r\bar w^e$. Using Hölder's inequality with exponents $(1+e)/(\alpha+e)$ and $(1+e)/(1-\alpha)$ and \eqref{eq:score-marginal-moments}, we find
\[
\begin{aligned}
\int_{\{\bar w>1\}}k_\alpha(\log\bar w)^r\,dG_{\mathrm z}&\le CA^r\int\bar w^{\alpha+e}\bar f^{1-\alpha}\,dG_{\mathrm z}\\
&\le CA^r\Big(\int\bar w^{1+e}dG_{\mathrm z}\Big)^{\frac{\alpha+e}{1+e}}\Big(\int\bar f^{1+e}dG_{\mathrm z}\Big)^{\frac{1-\alpha}{1+e}}\le CA^r ;
\end{aligned}
\]
for $\alpha=1$, the second factor is absent. We have $\int f\eta^\alpha|\log\bar w|^r\,dG=\int k_\alpha|\log\bar w|^r\,dG_{\mathrm z}$, and this proves the part of \eqref{mc-eq:te10} that concerns $\log\bar w$, and the bound for $g=\log w-\log\bar w$ follows.

\paragraph{Step 4: the entropy.}
The integral $\int wg\,dG$ is the relative entropy $\KL(wG\Vert(\bar wG_{\mathrm z})\otimes G_{\mathrm v})\ge0$. The functions $w\log w$ and $w\log\bar w$ are integrable by Steps~2 and~3 with $\alpha=1$. On $\{w>0\}$, we have $\log w=\log\bar w+g$, and Fubini's theorem gives $\int w\log\bar w\,dG=\int\bar w\log\bar w\,dG_{\mathrm z}\ge0$, by Jensen's inequality for the convex function $x\log x$ and $\int\bar w\,dG_{\mathrm z}=1$. Hence $\int wg\,dG\le\int w\log w\,dG=\KL(wG\Vert G)$. By Jensen's inequality under the probability measure $wG$ and $w\le f/z_0$,
\[
\int w\log w\,dG=\frac{\int w\log w^{p-1}\,dG}{p-1}\le\frac{\log\int w^p\,dG}{p-1}\le\frac{C_pA+p\log(1/z_0)}{p-1}\le CA .
\]
This completes the proof.
\end{proof}

\subsection{The time derivative of the conditional relative entropy}\label{sec:score-entropy-identity}

We adapt the classical entropy dissipation calculation for diffusion semigroups \cite[Proposition~5.2.2, (5.2.3), p.~245]{BakryGentilLedoux2014} to the conditional entropy and cutoff-weighted endpoint law; all additional terms are derived below. Let $T\ge0$, let $\Psi$ be an $\mathcal F_T$-measurable random variable with values in $[0,1]$, let $D\subset S_N$ be open (possibly $D=S_N$), and let $\sigma_t$ be the endpoint subprobability measures \eqref{eq:score-sigma} of \Cref{lem:score-regularity}. We assume that $D$ and $\sigma_T$ are invariant under $R_1$, so that every $\sigma_t$ is. Let
\[
D^+=\{y\in U_N:x^+(y)\in D\}.
\]
For $t>T$, the measure $\sigma_t^+$ vanishes outside $D^+$ up to a null set, and, by \Cref{lem:score-regularity} and the first part of \Cref{lem:score-chart}, it has a density $f_t$ with respect to $G$ such that $(t,y)\mapsto f_t(y)$ is smooth on $(T,\infty)\times D^+$. Applying \eqref{eq:score-forward} to $\varphi=\psi\circ y$ for $\psi\in C^\infty_c(D^+)$, and using \Cref{lem:score-chart-generator} and the invariance under $R_1$, we obtain the weak forward equation
\begin{equation}\label{eq:score-forward-chart}
\frac d{dt}\int\psi f_t\,dG=\int f_t\,L\psi\,dG\qquad(t>T,\ \psi\in C^\infty_c(D^+)),
\end{equation}
where $L$ is the operator \eqref{mc-eq:te20}. Let $\zeta\in C^\infty(\R^{N-1})$ take values in $[0,1]$ and have compact support in $D^+$, let $k\ge3$ be an integer, and let $\eta=\zeta^k$. Let $\mathcal T\subset(T,\infty)$ be an open interval on which $Z_t=\int\eta f_t\,dG>0$. For $t\in\mathcal T$, let
\begin{equation}\label{mc-eq:te5}
w_t=\frac{\eta f_t}{Z_t},\qquad\bar w_t(z)=\int w_t(z,v)\,G_{\mathrm v}(dv),\qquad h_t=\log\bar w_t,\qquad g_t=\log w_t-h_t ,
\end{equation}
and let $\mathscr H(t)=\mathscr H(w_t)$, $\mathscr D(t)=\mathscr D(w_t)$, and $S_t=z\cdot\nabla h_t$. The functions $w_t$ are smooth with compact support in $D^+$, so \eqref{eq:score-H-D} applies. We write $\E_w$ for $\E_{w_t}$ and omit the index $t$ when no confusion arises. Recall that $\Gamma(\varphi,\log\eta)=\Gamma(\varphi,\eta)/\eta$ on $\{\eta>0\}$, and that expectations $\E_w$ only see the set $\{w>0\}\subset\{\eta>0\}$.

\begin{proposition}[Time derivative of the conditional relative entropy]\label{prop:score-entropy-identity}
In this setting, $\mathscr H(t)$ and $\mathscr D(t)$ are finite for $t\in\mathcal T$, the function $\mathscr H$ is absolutely continuous on every compact subinterval of $\mathcal T$, and $Z_t'=\int f_tL\eta\,dG$. For almost every $t\in\mathcal T$,
\begin{equation}\label{mc-eq:te21}
\mathscr H'=-\mathscr D+\frac1N\E_w\big[(1+S)\,v\cdot\nabla_vg\big]+2\E_w\Gamma(g,\log\eta)+\frac1Z\int f(g-\mathscr H)\,L\eta\,dG ,
\end{equation}
and
\begin{equation}\label{mc-eq:te22}
\begin{aligned}
\mathscr H'={}&-\mathscr D+\frac1N\E_w\big[(1+S)\,v\cdot\nabla_vg\big]+\E_w\big[(1-g+\mathscr H)\,\Gamma(g,\log\eta)\big]\\
&-\E_w\big[(g-\mathscr H)\,\Gamma(h,\log\eta)\big]+\E_w\big[(g-\mathscr H)\,\Gamma(\log\eta)\big]+\frac1Z\int f(g-\mathscr H)\,\frac yN\cdot\nabla\eta\,dG .
\end{aligned}
\end{equation}
In the last integrals of \eqref{mc-eq:te21} and \eqref{mc-eq:te22}, the integrand is set to zero where $\eta f=0$.
\end{proposition}

After the second integration by parts, \eqref{mc-eq:te22} contains only first derivatives of $\eta$. The explicit drift contribution involving $\nabla F$ has been absorbed into the carré du champ terms; the remaining first-order term is $(y/N)\cdot\nabla\eta$.

\begin{proof}
We first give the formal computation and then justify it. The justification uses that, at fixed $N$, $f$ is smooth on a neighborhood of the support of $\eta$, and no lower bound on $f$.

\paragraph{Formal computation.}
We have $\int w_t\,dG=1$ and $\int\bar w_t\,dG_{\mathrm z}=1$. Differentiating $\mathscr H=\int w\log w\,dG-\int\bar w\log\bar w\,dG_{\mathrm z}$, we find $\mathscr H'=\int\partial_tw\,g\,dG$. By \eqref{eq:score-forward-chart} with $\psi=\eta$, we have $Z'=\int fL\eta\,dG$, and $\partial_tw=\eta\partial_tf/Z-wZ'/Z$. Applying \eqref{eq:score-forward-chart} with $\psi=\eta g$, we find
\[
\mathscr H'=\frac1Z\int f\,L(\eta g)\,dG-\frac{Z'}Z\mathscr H .
\]
We expand $L(\eta g)=\eta Lg+gL\eta+2\Gamma(\eta,g)$, write $L=L_G+(y/N)\cdot\nabla$, and integrate the term $\int f\eta L_Gg\,dG=Z\int wL_Gg\,dG$ by parts with \eqref{eq:score-chart-ibp}. Using $\nabla w=w\nabla(g+h)$ on $\{w>0\}$, we obtain
\begin{equation}\label{eq:score-identity-pre}
\mathscr H'=-\mathscr D-\E_w\Gamma(h,g)+\frac1N\E_w[y\cdot\nabla g]+2\E_w\Gamma(g,\log\eta)+\frac1Z\int f(g-\mathscr H)\,L\eta\,dG .
\end{equation}
Now we use the conditional structure. On each fiber with $\bar w(z)>0$, we have $\int w\nabla_zg\,dG_{\mathrm v}=\int\nabla_zw\,dG_{\mathrm v}-\bar w\nabla_zh=\nabla_z\bar w-\nabla_z\bar w=0$, that is,
\begin{equation}\label{eq:score-conditional-mean-zero}
\E_w[\nabla_zg\mid z]=0 .
\end{equation}
The function $h$ depends only on $z$. In $\Gamma(h,g)=\nabla_zh\cdot\nabla_zg-(z\cdot\nabla_zh)(z\cdot\nabla_zg)/N-(z\cdot\nabla_zh)(v\cdot\nabla_vg)/N$, the first two terms are functions of $z$ multiplied by components of $\nabla_zg$. Then \eqref{eq:score-conditional-mean-zero} implies that $\E_w\Gamma(h,g)=-N^{-1}\E_w[S\,v\cdot\nabla_vg]$. In the same way, $\E_w[y\cdot\nabla g]=\E_w[v\cdot\nabla_vg]$. Inserting these two identities into \eqref{eq:score-identity-pre} gives \eqref{mc-eq:te21}.

For \eqref{mc-eq:te22}, we write $L\eta=L_G\eta+(y/N)\cdot\nabla\eta$ in the last term of \eqref{mc-eq:te21}, and we integrate the part with $L_G$ by parts once more:
\[
\frac1Z\int f(g-\mathscr H)L_G\eta\,dG=-\frac1Z\int\big[(g-\mathscr H)\Gamma(f,\eta)+f\,\Gamma(g,\eta)\big]\,dG .
\]
On $\{w>0\}$, we have $\log f=g+h-\log\eta+\log Z$ and $\Gamma(f,\eta)=f[\Gamma(g,\eta)+\Gamma(h,\eta)-\Gamma(\eta)/\eta]$. Using $f\Gamma(\cdot,\eta)/Z=w\Gamma(\cdot,\log\eta)$ and $f\Gamma(\eta)/(Z\eta)=w\Gamma(\log\eta)$, we see that the right side equals
\[
\begin{aligned}
&-\E_w\big[(g-\mathscr H)\Gamma(g,\log\eta)\big]-\E_w\big[(g-\mathscr H)\Gamma(h,\log\eta)\big]\\
&\qquad+\E_w\big[(g-\mathscr H)\Gamma(\log\eta)\big]-\E_w\Gamma(g,\log\eta).
\end{aligned}
\]
Together with the term $2\E_w\Gamma(g,\log\eta)$ of \eqref{mc-eq:te21}, the terms containing $\Gamma(g,\log\eta)$ combine to $\E_w[(1-g+\mathscr H)\Gamma(g,\log\eta)]$, and \eqref{mc-eq:te22} follows.

\paragraph{Justification.}
Fix a compact interval $[\alpha,\beta]\subset\mathcal T$. Below, ``finite'' and ``bounded'' refer to bounds that may depend on $N$, $\alpha$, and $\beta$, and that are uniform in $t\in[\alpha,\beta]$. Let $\mathcal K$ be a compact neighborhood of $\supp\zeta$ in $D^+$. On $[\alpha,\beta]\times\mathcal K$, the function $f$ and all its derivatives are bounded, $Z_t$ is bounded below by a positive constant, and $w_t$ and $\bar w_t$ are smooth with bounded derivatives. The function $f\ge0$ is of class $C^2$, and Taylor's formula along $-\nabla f$ shows that $|\nabla f|^2\le C_{\mathcal K}f$ on $\supp\zeta$. Indeed, if $\rho_0$ is the distance from $\supp\zeta$ to the complement of $\mathcal K$ and $M_2$ bounds $\nabla^2f$ on $\mathcal K$, then $0\le f-r|\nabla f|+M_2r^2/2$ for $0\le r\le\rho_0$, and the choices $r=|\nabla f|/M_2$ or $r=\rho_0$ yield $|\nabla f|^2\le2M_2f$ or $|\nabla f|\le2f/\rho_0$. The following quantities are then finite:
\begin{enumerate}[label=(\alph*)]
\item $\int w\Gamma(\log w)\,dG\le2\int w\,\Gamma(\log\eta)\,dG+2\int w\,\Gamma(\log f)\,dG$, where $w\Gamma(\log\eta)=k^2Z^{-1}f\zeta^{k-2}\Gamma(\zeta)$ and $w\Gamma(\log f)\le Z^{-1}\eta|\nabla f|^2/f$ are bounded;
\item $\int\bar w|\nabla h|^2\,dG_{\mathrm z}\le\int w|\nabla_z\log w|^2\,dG$, by the Cauchy--Schwarz inequality on each fiber; hence $\mathscr D\le2\int w\Gamma(\log w)\,dG+2\int\bar w|\nabla h|^2\,dG_{\mathrm z}$ is finite;
\item $\int f\zeta^{k-2}(|\log w|^2+|\log\bar w|^2)\,dG$ on $\{w>0\}$, by \Cref{cce-lem:weighted-logs} applied to $f\ind_{\mathcal K}$, $\eta$, and $\alpha=(k-2)/k\ge1/3$; and $\mathscr H\le\int w(|\log w|+|\log\bar w|)\,dG$.
\end{enumerate}
Let $0<e\le1$ and $g_e=\log(w+e)-\log(\bar w+e)$, a smooth function. Let $\mathscr H_e=\int\Phi_e(w)\,dG-\int\Phi_e(\bar w)\,dG_{\mathrm z}$ with $\Phi_e(x)=(x+e)\log(x+e)-x$, so that $\Phi_e'(x)=\log(x+e)$. Then $\mathscr H_e$ is continuously differentiable on $[\alpha,\beta]$ with $\mathscr H_e'=\int\partial_tw\,g_e\,dG$, and $\eta g_e\in C^\infty_c(D^+)$. The formal computation up to \eqref{eq:score-identity-pre}, with $g_e$ in place of $g$ inside the integrals and before any use of $\nabla w=w\nabla(g+h)$, involves only smooth functions and is valid. We obtain $\mathscr H_e(\beta)-\mathscr H_e(\alpha)=\int_\alpha^\beta R_e(t)\,dt$, where
\[
\begin{aligned}
R_e={}&-\int\Gamma(w,g_e)\,dG+\int w\,\frac yN\cdot\nabla g_e\,dG\\
&+\frac1Z\int fg_e\,L\eta\,dG+\frac2Z\int f\,\Gamma(\eta,g_e)\,dG-\frac{Z'}Z\int wg_e\,dG .
\end{aligned}
\]
Now let $e\to0$. We have $\nabla g_e=\nabla w/(w+e)-\nabla\bar w/(\bar w+e)$, and $\nabla w=0$ where $w=0$, $\nabla\bar w=0$ where $\bar w=0$. The terms of $R_e$ converge pointwise to the corresponding terms with $g$, and they are dominated as follows.
\begin{itemize}
\item By item (a), $\Gamma(w)/(w+e)\le\Gamma(w)/w$ is integrable.
\item By items (a) and (b), the functions $|\Gamma(w,\bar w)|/(\bar w+e)\le(\Gamma(w)/w)^{1/2}w^{1/2}|\nabla\bar w|/\bar w$ and $w|\nabla g_e|\le|\nabla w|+w|\nabla\bar w|/\bar w$ are integrable, and so is $|f\Gamma(\eta,g_e)|\le Zw\,\Gamma(\log\eta)^{1/2}(\Gamma(\log w)^{1/2}+|\nabla h|)$.
\item Since $|\log(x+e)|\le|\log x|+\log2$ for $x>0$ and $|L\eta|\le C\zeta^{k-2}$, the remaining terms are dominated using item (c).
\end{itemize} Here we used that $f=0$ wherever $\eta>0$ and $w=0$, and that $L\eta=0$ where $\zeta=0$ because $k\ge3$. By dominated convergence, $\mathscr H_e(t)\to\mathscr H(t)$ and $R_e(t)\to R(t)$, the right side of \eqref{eq:score-identity-pre}, with $|R_e(t)|$ bounded uniformly in $e$ and $t$. We conclude that $\mathscr H(\beta)-\mathscr H(\alpha)=\int_\alpha^\beta R(t)\,dt$. In particular, $\mathscr H$ is absolutely continuous and \eqref{eq:score-identity-pre} holds almost everywhere. The identity \eqref{eq:score-conditional-mean-zero} holds because $w\nabla_zg$ is integrable by items (a) and (b). For the second integration by parts, we apply \eqref{eq:score-chart-ibp} to the smooth function $f(g_e-\mathscr H)$ and to $\eta$, and pass to the limit in the same way; here $|g_e\,\Gamma(f,\eta)|\le C|g_e|\,f^{1/2}\zeta^{k-1}$ by $|\nabla f|^2\le C_{\mathcal K}f$, which is dominated by item (c) and the Cauchy--Schwarz inequality. This proves the proposition.
\end{proof}

\subsection{Entropy dissipation uniformly in the top fraction}\label{sec:score-dissipation}\label{cce-sec:tail}

We now bound the right side of \eqref{mc-eq:te22} for a specific cutoff function. It is adapted to the approach to the critical temperature from below, where the top fraction $\theta$ tends to zero and all estimates must be uniform in $\theta$. The point is that every error term is expressed through the dimensionless quantity $m=\theta Na_2$ and through a bound $p_*$ on the mass of a region outside which the cutoff function equals one.

\paragraph{Setting.}
We assume that $0=a_1<a=a_2\le a_3\le\cdots\le a_N$, and let
\begin{equation}\label{eq:score-cold-setting}
\theta=1-\frac1N\sum_{i=2}^N\frac1{2a_i},\qquad m=\theta Na,\qquad b=a_{J+1},
\end{equation}
where $J\ge2$ is fixed with $a_J\le C_Ja$. For the scaled eigenvalue gaps of the model, $\theta$ is the top fraction $\theta_N$ of \eqref{eq:model-cold-parameters}. We assume $\theta>0$, $m\ge2$, and $N\theta^3\le C_0m^3$. In the chart, we write
\begin{equation}\label{mc-eq:te2}
|x_{\le J}|^2=\sum_{k=1}^Jx_k^2=N-|v|^2,\qquad u=\frac{x_1^2}{\theta N},\qquad\bar u=\frac{|x_{\le J}|^2}{\theta N},\qquad\xi=\frac{b\sum_{i>J}v_i^2/a_i}{|x_{\le J}|^2}.
\end{equation}
Set $\xi=0$ where $|x_{\le J}|=0$. The quantities $\bar u$ and $\xi$ are invariant under all rotations of $(x_1,\dots,x_J)$. Fix $0<s<L_0<L_1<L$ and functions $\chi_1,\chi_2,\chi_3\in C^\infty(\R;[0,1])$ such that $\chi_1=0$ on $(-\infty,s/2]$ and $\chi_1=1$ on $[s,\infty)$, $\chi_2=1$ on $(-\infty,L_0]$ and $\chi_2=0$ on $[L_1,\infty)$, and $\chi_3=1$ on $(-\infty,1/2]$ and $\chi_3=0$ on $[1,\infty)$. Fix an integer $k\ge16$ and $0<\delta\le1$, and let
\begin{equation}\label{mc-eq:te3}
\eta=\zeta^k,\qquad\zeta=\chi_1(u)\,\chi_2(\bar u)\,\chi_3(\xi/\delta).
\end{equation}
Since $\zeta=0$ on $\{u<s/2\}$, an open set containing $\{|x_{\le J}|=0\}$ and a neighborhood of $\partial U_N$, the function $\zeta$ is smooth. Let
\begin{equation}\label{mc-eq:te4}
\mathcal C=\{u<s\}\cup\{\bar u>L_0\}\cup\{\xi>\delta/2\}.
\end{equation}
Then $\eta=1$ outside $\mathcal C$, and all derivatives of $\zeta$ and $\eta$ vanish outside $\mathcal C$. Thus $\mathcal C$ is a region outside which the cutoff function equals one, and it contains the set $\{\eta=0\}$.
We regard $\eta$ and $\mathcal C$ also as a function and a set on $S_N$, through $x\mapsto y(x)$.

\paragraph{Endpoint measures with weighting and killing.}
Let $X$ solve \eqref{mc-eq:sr1} with $X_0\sim\nu_N$, and let $\rho_t=\nu_NP_t$ be its time-$t$ law. Let $\pi\propto e^{-F}\nu_N$. Fix a deterministic time $T\ge0$, an $\mathcal F_T$-measurable random variable $\Psi$ with values in $[0,1]$ such that the measure $A\mapsto\E[\Psi\ind_{\{X_T\in A\}}]$ is invariant under $R_1$, and the open set
\[
D=\{x\in S_N:|x_{\le J}|^2<L\theta N\}.
\]
Let $\sigma_t$, $t\ge T$, be the endpoint subprobability measures \eqref{eq:score-sigma}: the path is weighted by $\Psi$ and killed when it leaves $D$ after time $T$. Then $\sigma_t\le\rho_t$ as measures. With $\zeta$ as in \eqref{mc-eq:te3}, which has compact support in $D^+$, let $f_t$, $Z_t$, $w_t$, $\bar w_t$, $h_t$, $g_t$, $\mathscr H(t)$, $\mathscr D(t)$, and $S_t$ be as in \Cref{sec:score-entropy-identity}.

\paragraph{Hypotheses.}
Let $t_0\ge T+1/a$ and $t_1\ge t_0$. We assume that there exist constants $p>1$, $C_p$, $C_\pi$, $z_0>0$, $C_4$, and a number $p_*\ge0$ such that for every $t\in[t_0,t_1]$:
\begin{equation}\label{mc-eq:te1}
\log\int\Big(\frac{d\rho_t}{d\pi}\Big)^pd\pi\le C_pN\theta^3 ;
\end{equation}
\begin{equation}\label{eq:score-graph-hyp}
\frac{d\pi^+}{dG}\le C_\pi\quad\text{on }\{y\in U_N:x_1(y)^2\ge s\theta N/4\};
\end{equation}
\begin{equation}\label{mc-eq:te6}
Z_t\ge z_0,\qquad\sigma_t(\mathcal C)\le p_* ;
\end{equation}
\begin{equation}\label{mc-eq:te7}
\Big(\int x_j^4\,d\sigma_t\Big)^{1/2}\le C_4\big\{\theta Ne^{-4a_j(t-T)}+a_j^{-1}\big\}\qquad(2\le j\le J).
\end{equation}
Here \eqref{eq:score-graph-hyp} does not depend on $t$. The roles of the parameters are summarized in \Cref{tab:score-parameters}.

\begin{table}[ht]
\centering
\caption{Parameters in \Cref{cce-thm:consumer}. The constants in the conclusion depend only on the fixed parameters in the first three rows.}
\label{tab:score-parameters}
\begin{tabular}{@{}p{0.36\textwidth}p{0.56\textwidth}@{}}
\toprule
Parameter & Role\\
\midrule
$J$, $C_J$, $C_0$ & number of retained coordinates; ratio bound $a_J\le C_Ja$; bound $N\theta^3\le C_0m^3$\\
$s,L_0,L_1,L$, $\chi_1,\chi_2,\chi_3$, $k$, $\varepsilon$ & cutoff function \eqref{mc-eq:te3}, killing set $D$, and the fraction $\varepsilon$ of dissipation spent on errors\\
$p,C_p,C_\pi,z_0,C_4$ & constants in the hypotheses \eqref{mc-eq:te1}--\eqref{mc-eq:te7}\\
\midrule
$N$, $\theta$, $a_2,\dots,a_N$ & free, subject to the setting; $m=\theta Na_2\ge2$\\
$\delta\in(0,1]$, $p_*\ge0$ & width of the cone condition $\xi\le\delta$; bound on $\sigma_t(\mathcal C)$\\
$T\le t_0-1/a$, $t_1\ge t_0$ & time of the weight $\Psi$ and of the start of killing; time interval\\
\bottomrule
\end{tabular}
\end{table}

\begin{theorem}[Entropy dissipation for the remaining coordinates]\label{cce-thm:consumer}
In this setting, the function $\mathscr H$ is absolutely continuous on $[t_0,t_1]$. For every $\varepsilon\in(0,1/4)$, for almost every $t\in[t_0,t_1]$,
\begin{equation}\label{mc-eq:te8}
\mathscr H'\le-(1-2\varepsilon)\mathscr D+\frac{Ca}m\big\{1+m^2e^{-8a(t-T)}\big\}+Cam^5\delta^{-2}p_*^{1/2},
\end{equation}
and for every $t\in[t_0,t_1]$,
\begin{equation}\label{mc-eq:te9}
\mathscr H\le\frac{1+2L_1\delta/s}{4b}\,\mathscr D,\qquad0\le\mathscr H\le Cm^3 .
\end{equation}
The constants $C$ depend only on the fixed parameters of \Cref{tab:score-parameters}; in particular, they do not depend on $N$, $\theta$, $\delta$, $p_*$, $T$, $t_0$, or $t_1$.
\end{theorem}

\begin{remark}\label{rem:score-consumer}
\Cref{cce-thm:consumer} concerns the endpoint subprobability measures $\sigma_t$. The mass removed by the weight $\Psi$ and by killing does not enter \eqref{mc-eq:te8}; it is accounted for separately, in total variation, with \Cref{lem:score-domination} and \Cref{ccf-lem:finite-product-closure}. The hypothesis \eqref{mc-eq:te7} enters only through the bound \eqref{mc-eq:te15} on $\|S\|_{L^2(w)}$. Its proof below shows that an additional term $\theta Np_*^{1/2}$ on the right side of \eqref{mc-eq:te7} changes \eqref{mc-eq:te8} only by a constant factor, provided $mp_*^{1/2}\le C_*$ for a fixed constant $C_*$; the constant $C$ in \eqref{mc-eq:te8} may then also depend on $C_*$.
\end{remark}

The proof bounds each term of \eqref{mc-eq:te22}. Step~1 bounds weighted logarithms of $f_t$ by $m^3$, using \eqref{mc-eq:te1} and \eqref{eq:score-graph-hyp}. Step~2 bounds the scores of the normalized cutoff-weighted law $\eta\sigma_t/Z_t$ along the rotations $K_{1j}$, using \Cref{csr-cor:cold-scale}. Step~3 converts them into bounds for $S=z\cdot\nabla h$, using \Cref{lem:score-marginal-score} and \eqref{mc-eq:te7}. Step~4 computes the derivatives of the cutoff function. Steps~5 and~6 bound the term with $1+S$ and the terms supported in $\mathcal C$, and Step~7 concludes with \Cref{prop:score-conditional-lsi}.

\begin{proof}
Throughout the proof, $C$ denotes a constant depending only on the fixed parameters of \Cref{tab:score-parameters}. We may assume $p_*\le1$, since $\sigma_t(\mathcal C)\le\sigma_t(S_N)\le1$. Note that $\theta<1$, that $1/(\theta N)=a/m$, and that on $\supp\zeta$ we have
\begin{equation}\label{eq:score-support}
x_1^2\ge s\theta N/2,\qquad|x_{\le J}|^2\le L_1\theta N,\qquad|z|^2\le L_1\theta N,\qquad\xi\le\delta .
\end{equation}

\paragraph{Step 0: the entropy identity.}
The set $D$ is open and invariant under $R_1$, and $\zeta$ has compact support in $D^+$ by \eqref{eq:score-support}, because $L_1<L$. By \eqref{mc-eq:te6} and continuity, $Z_t>0$ on an open interval $\mathcal T\supset[t_0,t_1]$, and $t_0>T$. \Cref{prop:score-entropy-identity} shows that $\mathscr H$ is absolutely continuous on $[t_0,t_1]$ and that \eqref{mc-eq:te22} holds for almost every $t\in[t_0,t_1]$. We fix $t\in[t_0,t_1]$. Steps~1--4 below hold for every such $t$. In Steps~5--7, we also assume that \eqref{mc-eq:te22} holds at $t$.

\paragraph{Step 1: weighted logarithms.}
Let $\bar f=f\ind_{\{x_1^2\ge s\theta N/4\}}$, which agrees with $f$ on $\supp\eta$. Let $r_t=d\sigma_t/d\pi\le d\rho_t/d\pi$. Since $\sigma_t$ and $\pi$ are invariant under $R_1$, the density of $\sigma_t^+$ with respect to $\pi^+$ is $r_t\circ x^+$, and $\int(r_t\circ x^+)^p\,d\pi^+=\int r_t^p\,d\pi$. By \eqref{eq:score-graph-hyp} and \eqref{mc-eq:te1},
\begin{equation}\label{mc-eq:te11}
\int\bar f^p\,dG=\int_{\{x_1^2\ge s\theta N/4\}}(r_t\circ x^+)^p\Big(\frac{d\pi^+}{dG}\Big)^{p-1}d\pi^+\le C_\pi^{p-1}\int\Big(\frac{d\rho_t}{d\pi}\Big)^pd\pi\le e^{Cm^3},
\end{equation}
where we used $N\theta^3\le C_0m^3$ and $m\ge2$. Also $\int\bar f\,dG\le\sigma_t(S_N)\le1$ and $\int\eta\bar f\,dG=Z_t\ge z_0$. We may therefore use \Cref{cce-lem:weighted-logs} with $A=Cm^3$, and we obtain, for every $r\in[1,\infty)$ and $\alpha_0\in(0,1]$,
\begin{equation}\label{eq:score-weighted-logs-applied}
\sup_{\alpha\in[\alpha_0,1]}\int f\eta^\alpha|g|^r\,dG\le Cm^{3r},\qquad0\le\mathscr H\le Cm^3 ,
\end{equation}
where $C$ also depends on $r$ and $\alpha_0$. This proves the second inequality in \eqref{mc-eq:te9}. We write $A_*=Cm^3$ for the constant in the second bound.

\paragraph{Step 2: scores after weighting by the cutoff.}
Let $2\le j\le J$. The normalized measure $\sigma_t/\sigma_t(S_N)$ is the endpoint law weighted by $W=\Psi\ind_{\{\tau>t\}}$, where $\tau$ is the exit time from $D$ after $T$. This is a path weight for the final interval $[t-1/a,t]$ and the set $D$ in the sense of \eqref{mc-eq:sr3}: $\Psi$ and the event that the path stays in $D$ during $[T,t-1/a]$ are $\mathcal F_{t-1/a}$-measurable, and the event that the path stays in $D$ during $[t-1/a,t]$ is invariant under the rotations $\Theta^{(1j)}_\varepsilon$, because $D$ is invariant under all rotations of $(x_1,\dots,x_J)$. Its mass is $\sigma_t(S_N)\ge Z_t\ge z_0$, and \Cref{csr-cor:cold-scale} implies that
\begin{equation}\label{mc-eq:te12}
\E_{\sigma_t/\sigma_t(S_N)}\Big|K_{1j}\log\frac{d\sigma_t}{d\nu_N}\Big|^{4}\le Cm^{2};
\end{equation}
here the normalization by $\sigma_t(S_N)$ does not change the score. The quantities $\bar u$ and $\xi$ are invariant under the rotations generated by $K_{1j}$, and $K_{1j}u=-2x_1x_j/(\theta N)$. On $\supp\zeta$, we have $|K_{1j}u|\le(x_1^2+x_j^2)/(\theta N)\le\bar u\le L_1$, and therefore
\begin{equation}\label{mc-eq:te13}
|K_{1j}\eta|=k\zeta^{k-1}|\chi_1'(u)|\,|K_{1j}u|\,\chi_2\chi_3\le C\eta^{1-1/k}\le C\eta^{3/4},
\end{equation}
since $k\ge16$. Let $\widehat\sigma_t=\eta\sigma_t/Z_t$ be the normalized cutoff-weighted law, with normalizing mass $Z_t=\int\eta\,d\sigma_t$; it is a probability measure with $\widehat\sigma_t^+=w_tG$. \Cref{csh-lem:cut-score} with $p=2$ implies that its weak $K_{1j}$-score $\vartheta_j$ satisfies $\E_{\widehat\sigma_t}|\vartheta_j|^{4}\le Cm^{2}$.

\paragraph{Step 3: the marginal score.}
The measure $\widehat\sigma_t$ is invariant under $R_1$ and satisfies $\widehat\sigma_t(\{x_1^2<s\theta N/2\})=0$. By \Cref{lem:score-marginal-score}, we have \eqref{mc-eq:te14} with $w=w_t$, and $S=\sum_{j=2}^Jz_j\partial_jh$ is the sum of the three terms
\[
T_{1,j}=\E_w\Big[\frac{z_j}{x_1}\vartheta_j\circ x^+\,\Big|\,z\Big],\qquad T_{2,j}=\E_w\Big[\frac{z_j^2}{x_1^2}\,\Big|\,z\Big],\qquad T_{3,j}=2a_jz_j^2 .
\]
For the first term, by Jensen's inequality, \eqref{eq:score-support}, and the Cauchy--Schwarz inequality,
\[
\E_wT_{1,j}^2\le\frac2{s\theta N}\E_w\big[z_j^2\vartheta_j^2\big]\le\frac2{s\theta N}\big(\E_wz_j^4\big)^{1/2}\big(\E_w|\vartheta_j\circ x^+|^4\big)^{1/2}.
\]
We have $\E_wz_j^4=Z_t^{-1}\int\eta x_j^4\,d\sigma_t\le z_0^{-1}\int x_j^4\,d\sigma_t$ by \eqref{mc-eq:te7}, so $(\E_wz_j^4)^{1/2}\le C\{\theta Ne^{-4a_j(t-T)}+a_j^{-1}\}$. With $\E_w|\vartheta_j\circ x^+|^4=\E_{\widehat\sigma_t}\vartheta_j^4\le Cm^2$, by the remark after \Cref{lem:score-chart}, and $a_j\ge a$, we find $\E_wT_{1,j}^2\le C\{me^{-4a(t-T)}+m/(\theta Na_j)\}\le C\{1+me^{-4a(t-T)}\}$. The second term satisfies $0\le T_{2,j}\le|z|^2/x_1^2\le2L_1/s$ by \eqref{eq:score-support}. For the third term, $\E_wT_{3,j}^2=4a_j^2\E_wz_j^4\le C\{\theta Na_je^{-4a(t-T)}+1\}^2$, and $\theta Na_j\le C_Jm$. Summing over $j\le J$, we have
\begin{equation}\label{mc-eq:te15}
\|S\|_{L^2(w)}\le C\big\{1+me^{-4a(t-T)}\big\},\qquad\|S\|_{L^4(w)}\le Cm .
\end{equation}
For the second bound, we have $|T_{1,j}|^4\le\E_w[(z_j/x_1)^4\vartheta_j^4\mid z]$ with $(z_j/x_1)^4\le(2L_1/s)^2$, which gives $\E_wT_{1,j}^4\le Cm^2$. Further, $T_{2,j}$ is bounded, and $0\le T_{3,j}\le2a_jL_1\theta N\le Cm$ by \eqref{eq:score-support}. This bound uses no moment of $x_j$.

\paragraph{Step 4: derivatives of the cutoff function.}
We write $\mathfrak d=y/N$. The following computations involving $\xi$ are on $\{|x_{\le J}|>0\}$, which contains $\supp\zeta$ by \eqref{eq:score-support}. A direct computation from \eqref{mc-eq:te2} shows that
\begin{equation}\label{mc-eq:te16}
\begin{gathered}
\Gamma(u)=\frac{4u(1-\theta u)}{\theta N},\qquad\Gamma(\bar u)=\frac{4\bar u(1-\theta\bar u)}{\theta N},\\
\Gamma(\xi)=\frac{4\big(v^\T D_b^2v+\xi^2|x_{\le J}|^2\big)}{|x_{\le J}|^4}\le\frac{4\xi(1+\xi)}{|x_{\le J}|^2},
\end{gathered}
\end{equation}
where $D_b=\diag(b/a_i)_{i>J}$. For the last one, we use $\nabla_v\xi=2(D_bv+\xi v)/|x_{\le J}|^2$ and $v\cdot\nabla_v\xi=2N\xi/|x_{\le J}|^2$. Subtracting the rank-one part of $\mathfrak g$ leaves the displayed expression, and the inequality follows from $D_b^2\le D_b$, which holds because $b\le a_i$. Also
\begin{equation}\label{mc-eq:te17}
|\mathfrak d\cdot\nabla u|\le\frac2{\theta N},\qquad|\mathfrak d\cdot\nabla\bar u|\le\frac2{\theta N},\qquad\mathfrak d\cdot\nabla\xi=\frac{2\xi}{|x_{\le J}|^2},
\end{equation}
and, for every function $h$ of $z$ alone, with $S=z\cdot\nabla h$,
\begin{equation}\label{mc-eq:te18}
\Gamma(h,u)=-\frac{2uS}N,\qquad\Gamma(h,\bar u)=\frac{2|v|^2S}{\theta N^2},\qquad\Gamma(h,\xi)=-\frac{2\xi S}{|x_{\le J}|^2}.
\end{equation}
On $\supp\zeta$, the derivatives of $\chi_1(u)$, $\chi_2(\bar u)$, and $\chi_3(\xi/\delta)$ are nonzero only where $u\le s$, where $\bar u\le L_1$, and where $\delta/2\le\xi\le\delta$, respectively. In the last case, $\Gamma(\xi)/\delta^2\le8/(\delta|x_{\le J}|^2)\le16/(s\theta N\delta)$ and $|\mathfrak d\cdot\nabla\xi|/\delta\le4/(s\theta N)$ by \eqref{eq:score-support}. All these derivatives vanish outside $\mathcal C$. Since $\eta=\zeta^k$, $\Gamma(\eta)/\eta=k^2\zeta^{k-2}\Gamma(\zeta)$, and $1/(\theta N)=a/m$, we conclude that
\begin{equation}\label{mc-eq:te19}
\frac{\Gamma(\eta)}\eta\le\frac{Ca}{m\delta^2}\eta^{1-2/k}\ind_{\mathcal C},\qquad|\mathfrak d\cdot\nabla\eta|\le\frac{Ca}m\eta^{1-1/k}\ind_{\mathcal C},\qquad|\Gamma(h,\eta)|\le\frac{Ca}m|S|\,\eta^{1-1/k}\ind_{\mathcal C}.
\end{equation}
In the first bound, $\delta^{-1}$ would suffice, but the weaker bound with $\delta^{-2}$ is enough for our purposes.

\paragraph{Step 5: the term with $1+S$.}
By the first part of \Cref{lem:score-schur} and \eqref{eq:score-support}, we have $|v\cdot\nabla_vg|\le(v^\T A_v^{-1}v)^{1/2}\Gamma(g)^{1/2}$ and $v^\T A_v^{-1}v\le N^2/x_1^2\le2N/(s\theta)$. Young's inequality and \eqref{mc-eq:te15} then yield
\begin{equation}\label{mc-eq:te26}
\Big|\frac1N\E_w\big[(1+S)\,v\cdot\nabla_vg\big]\Big|\le\varepsilon\mathscr D+\frac{1}{2\varepsilon s\theta N}\E_w(1+S)^2\le\varepsilon\mathscr D+\frac{Ca}m\big\{1+m^2e^{-8a(t-T)}\big\}.
\end{equation}

\paragraph{Step 6: the terms supported in $\mathcal C$.}
Each of the remaining four terms of \eqref{mc-eq:te22} has the form $\E_w[\cdots]$ or $Z^{-1}\int f\cdots dG$ with an integrand that vanishes outside $\mathcal C$, by \eqref{mc-eq:te19}. We first record two consequences of \eqref{mc-eq:te6} and \eqref{eq:score-weighted-logs-applied}. If $\beta_1\ge0$, then $\int f\eta^{\beta_1}\ind_{\mathcal C}\,dG\le\sigma_t(\mathcal C)\le p_*$. Splitting $\eta^{1-2/k}=\eta^{1/4}\cdot\eta^{3/4-2/k}$ and using the Cauchy--Schwarz inequality, \eqref{eq:score-weighted-logs-applied} with $r=4$ and $\alpha_0=1/2$, and $3/2-4/k\ge0$, we find
\begin{equation}\label{mc-eq:te23}
\int f\eta^{1-2/k}\big(1+g^2\big)\ind_{\mathcal C}\,dG\le p_*+\Big(\int f\eta^{1/2}g^4\,dG\Big)^{1/2}\Big(\int f\eta^{3/2-4/k}\ind_{\mathcal C}\,dG\Big)^{1/2}\le CA_*^2p_*^{1/2},
\end{equation}
and in the same way $\int f\eta^{1-2/k}|g|\ind_{\mathcal C}\,dG\le CA_*p_*^{1/2}$ and $\int f\eta^{1-1/k}|g|\ind_{\mathcal C}\,dG\le CA_*p_*^{1/2}$. Recall also that $\mathscr H\le A_*$.

For the first term, we have $|\E_w[(1-g+\mathscr H)\Gamma(g,\log\eta)]|\le\varepsilon\mathscr D+(4\varepsilon)^{-1}\E_w[(1-g+\mathscr H)^2\Gamma(\log\eta)]$, by the Cauchy--Schwarz and Young inequalities. Since $w\Gamma(\log\eta)=Z^{-1}f\Gamma(\eta)/\eta$, we deduce from \eqref{mc-eq:te19} and \eqref{mc-eq:te23} that
\begin{equation}\label{mc-eq:te24}
\big|\E_w\big[(1-g+\mathscr H)\Gamma(g,\log\eta)\big]\big|\le\varepsilon\mathscr D+\frac{Ca}{m\delta^2}A_*^2p_*^{1/2}.
\end{equation}
For the second term, $w\Gamma(h,\log\eta)=Z^{-1}f\Gamma(h,\eta)$, and \eqref{mc-eq:te19} reduces it to $(Ca/m)\int f\eta^{1-1/k}|S||g-\mathscr H|\ind_{\mathcal C}\,dG$. By Hölder's inequality with exponents $4$, $4$, and $2$, with the weights $\eta^{1/4}$, $\eta^{1/8}$, and $\eta^{5/8-1/k}$, whose product is $\eta^{1-1/k}$,
\[
\begin{aligned}
\int f\eta^{1-\frac1k}|S||g-\mathscr H|\ind_{\mathcal C}\,dG&\le\Big(\int f\eta|S|^4\Big)^{\frac14}\Big(\int f\eta^{\frac12}|g-\mathscr H|^4\Big)^{\frac14}\Big(\int f\eta^{\frac54-\frac2k}\ind_{\mathcal C}\Big)^{\frac12}\\
&\le CmA_*p_*^{1/2},
\end{aligned}
\]
where we used $\int f\eta|S|^4\,dG=Z\|S\|_{L^4(w)}^4\le Cm^4$ by \eqref{mc-eq:te15}. After multiplication by $Ca/m$, the second term is at most $CaA_*p_*^{1/2}$. Similarly, \eqref{mc-eq:te19} and the bounds after \eqref{mc-eq:te23} give the bound $(Ca/(m\delta^2))A_*p_*^{1/2}$ for the third term and the bound $(Ca/m)A_*p_*^{1/2}$ for the last term. Since $A_*=Cm^3$, $m\ge2$, and $\delta\le1$, all four terms are at most $\varepsilon\mathscr D+Cam^5\delta^{-2}p_*^{1/2}$ in total.

\paragraph{Step 7: conclusion.}
Inserting \eqref{mc-eq:te26} and the bounds of Step~6 into \eqref{mc-eq:te22}, we obtain \eqref{mc-eq:te8}. For the first inequality of \eqref{mc-eq:te9}, note that on $\supp\zeta$ we have $\xi\le\delta$ and, by \eqref{eq:score-support}, $|x_{\le J}|^2\le L_1\theta N\le(2L_1/s)x_1^2$. Hence
\[
b\sum_{i>J}\frac{v_i^2}{a_i}=\xi\,|x_{\le J}|^2\le\frac{2L_1\delta}s\,x_1^2\qquad\text{on }\{w>0\},
\]
and \Cref{prop:score-conditional-lsi} with $d=2L_1\delta/s$ implies that $\mathscr H\le(1+2L_1\delta/s)\mathscr D/(4b)$. This completes the proof.
\end{proof}

\section{Combining the coordinates}\label{sec:score-product}

The last lemma combines a conditional relative entropy bound for a normalized weighted law with a total variation bound for the first coordinates of the unweighted law. In the applications, it is applied after the chart of \Cref{sec:score-coordinates}, with \eqref{eq:score-fold-tv}; the stationary law $\pi$ is then compared with $G$ separately.

\begin{lemma}[Combining a conditional relative entropy bound with the first coordinates]\label{ccf-lem:finite-product-closure}
Let $\rho$ be a probability measure on a product space $E_{\mathrm z}\times E_{\mathrm v}$, and let $\sigma\le\rho$ be a measure with mass $Z>0$ and $\widehat\rho=\sigma/Z$. Let $G=G_{\mathrm z}\otimes G_{\mathrm v}$ be a product probability measure, and let $\mu$ be a probability measure on $E_{\mathrm z}$. Write $\rho_{\mathrm z}$ and $\widehat\rho_{\mathrm z}$ for the marginals on $E_{\mathrm z}$. If $\widehat\rho\ll G$, let $H_{\mathrm v}=\KL(\widehat\rho\Vert\widehat\rho_{\mathrm z}\otimes G_{\mathrm v})$. Then
\begin{equation}\label{ccf-eq:finite-product-closure}
\|\rho-\mu\otimes G_{\mathrm v}\|_{\TV}\le2(1-Z)+\sqrt{H_{\mathrm v}/2}+\|\rho_{\mathrm z}-\mu\|_{\TV}.
\end{equation}
For every probability measure $\pi$ on $E_{\mathrm z}\times E_{\mathrm v}$,
\begin{equation}\label{ccf-eq:finite-profile-closure}
\Big|\|\rho-\pi\|_{\TV}-\|\mu-G_{\mathrm z}\|_{\TV}\Big|\le2(1-Z)+\sqrt{H_{\mathrm v}/2}+\|\rho_{\mathrm z}-\mu\|_{\TV}+\|\pi-G\|_{\TV}.
\end{equation}
\end{lemma}

\begin{proof}
By \Cref{lem:score-domination}, $\|\rho-\widehat\rho\|_{\TV}\le1-Z$, and the same bound holds for the marginals $\rho_{\mathrm z}$ and $\widehat\rho_{\mathrm z}$ by the second part of \Cref{lem:model-tv}. Pinsker's inequality (\Cref{lem:model-tv}) gives $\|\widehat\rho-\widehat\rho_{\mathrm z}\otimes G_{\mathrm v}\|_{\TV}\le\sqrt{H_{\mathrm v}/2}$. For probability measures $\alpha,\alpha'$ on $E_{\mathrm z}$, we have $\|\alpha\otimes G_{\mathrm v}-\alpha'\otimes G_{\mathrm v}\|_{\TV}=\|\alpha-\alpha'\|_{\TV}$: the inequality $\ge$ follows by projection, and $\le$ from $(\alpha-\alpha')\otimes G_{\mathrm v}(B)=\int(\alpha-\alpha')(B^v)\,G_{\mathrm v}(dv)$, where $B^v$ is the section of $B$ at $v$. Using the triangle inequality along $\rho$, $\widehat\rho$, $\widehat\rho_{\mathrm z}\otimes G_{\mathrm v}$, $\rho_{\mathrm z}\otimes G_{\mathrm v}$, $\mu\otimes G_{\mathrm v}$, we obtain
\[
\|\rho-\mu\otimes G_{\mathrm v}\|_{\TV}\le(1-Z)+\sqrt{H_{\mathrm v}/2}+(1-Z)+\|\rho_{\mathrm z}-\mu\|_{\TV},
\]
which is \eqref{ccf-eq:finite-product-closure}. For \eqref{ccf-eq:finite-profile-closure}, the reverse triangle inequality gives $|\|\rho-\pi\|_{\TV}-\|\mu\otimes G_{\mathrm v}-G\|_{\TV}|\le\|\rho-\mu\otimes G_{\mathrm v}\|_{\TV}+\|\pi-G\|_{\TV}$, and $\|\mu\otimes G_{\mathrm v}-G_{\mathrm z}\otimes G_{\mathrm v}\|_{\TV}=\|\mu-G_{\mathrm z}\|_{\TV}$.
\end{proof}

If $H_{\mathrm v}=\infty$, there is nothing to prove. The constant $2$ in front of $1-Z$ is not optimal, but suffices for the applications.