\chapter{Density bounds relative to time-dependent Bingham laws}\label{sec:density}

In this section, we bound the density of the time-$t$ law with respect to the Gibbs measure in every space $L^p$, at times when the time-$t$ law is still far from equilibrium in total variation.
The general result is \cref{thm:dens-main}.
We prove it for deterministic half-precisions $r_1,\dots,r_N$ that satisfy the hypotheses of \cref{def:dens-hyp} on a time interval $[0,S]$, and it states that, for every fixed $p>1$,
\[
\log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\pi_r}\Big)^p\,d\pi_r\le C_pN(1+t)^{-3/2},\qquad 0\le t\le S,
\]
where $\pi_r$ and $P^r_t$ are the Gibbs measure and the semigroup with these half-precisions.
In the critical window, the hypotheses hold on $[0,KN^{2/3}]$ for every fixed $K$ and all large $N$, and the right side is bounded at times of order $N^{2/3}$.
We record the consequences in \cref{sec:dens-critical}.
The density of the time-$t$ law has bounded $L^p(\pi_r)$ norm at every fixed positive multiple of $N^{2/3}$ (\cref{cl3-actual-source-density}).
Together with the stationary Poincar\'e inequality, this gives exponential decay of the total variation distance on the scale $N^{2/3}$ (\cref{cg-source-l2-decay}).

We compare $\nu_NP^r_t$ with the Bingham comparison law $\mu_s$ of \cref{sec:heat}, evaluated at the earlier time $s=\gamma t$ for a small constant $\gamma>0$.
The logarithm of the $p$-th moment of the density of $\nu_NP^r_t$ with respect to $\mu_{\gamma t}$ is a multiple of a R\'enyi divergence. We differentiate it in $t$ along the forward equation (\cref{lem:dens-derivative}).
The derivative contains a dissipation term and two error terms.
Using a logarithmic Sobolev inequality for $\mu_{\gamma t}$, we bound the Fisher information in the dissipation term below by a multiple of the divergence itself.
The error terms come from the difference between the potential of the dynamics and that of $\mu_{\gamma t}$, and from the motion of $\mu_{\gamma t}$; both are quadratic functions of the coordinates.
Their coefficients are small, because in each of them a term common to all coordinates cancels exactly (\cref{lem:dens-coefficients}).
We control their exponential moments under $\mu_{\gamma t}$ with the exact formula of \cref{gp-real-tilt} (\cref{lem:dens-tilt}) and close the resulting differential inequality by a scalar argument (\cref{cg-renyi-differential-inequality}).
Finally, we change the reference measure from $\mu_{\gamma t}$ to $\pi_r$, using the same exponential moments.
All statements concern the time-$t$ law of the diffusion started from $\nu_N$.
The comparison laws enter only as reference measures: $\mu_s$ need not coincide with the time-$s$ law, and no such identification is used.

We use the conventions of \cref{sec:model}, the conditioned Gaussian laws of \cref{sec:conditioning} (in particular \cref{lem:cond-radial,gp-real-tilt}), the comparison laws of \cref{sec:heat} and the estimate \cref{cd-heat-profiles}, and \cref{cg-mean-level-density,cg-critical-lsi} of \cref{sec:inequalities}.
In \cref{sec:cold-meso}, we use \cref{thm:dens-main} below the critical temperature to prove \cref{mc-renyi-thm}.
\Cref{cl3-actual-source-density,cg-source-l2-decay} are used for the critical profile in \cref{sec:crit-profile}.

R\'enyi divergences were introduced in \cite[Section~3, (3.3), p.~554]{Renyi1961}; we follow the presentation of van Erven and Harremo\"es \cite{vanErvenHarremoes2014}.
In \cite[Theorem~1.2]{CaoLuLu2019}, Cao, Lu, and Lu proved exponential decay of R\'enyi divergences along Fokker--Planck equations with uniformly convex potentials, $D^2V\succeq\kappa I$ for some $\kappa>0$, for smooth initial data that decay sufficiently fast.
Vempala and Wibisono computed the time derivative of a R\'enyi divergence along the Langevin dynamics, and proved its exponential decay when the stationary measure satisfies a logarithmic Sobolev inequality \cite[Lemmas~5 and~6, Theorem~2]{VempalaWibisono2019}.
In our argument, the reference measure need not be the stationary measure.
It moves, its logarithmic Sobolev constant grows linearly in time, and we must estimate the error terms that its motion produces.
We compute the exponential moments of these error terms exactly through the radial densities of \cref{sec:conditioning}, and this is where the scale $N(1+t)^{-3/2}$ comes from.

\section{Setting and notation}\label{sec:dens-setting}

\paragraph{Half-precisions and the dynamics.}
Throughout this section, $N\ge3$ and $r=(r_1,\dots,r_N)$ is a vector of positive numbers such that
\begin{equation}\label{eq:dens-trace}
\sum_{i=1}^N\frac1{2r_i}=N .
\end{equation}
Let $A_r=-2\diag(r_1,\dots,r_N)$.
Note that, by \eqref{eq:dens-trace}, the Lagrange multiplier of $A_r$ (\cref{def:model-multiplier}) is $z=0$, so that its precisions are $2r_1,\dots,2r_N$ and its half-precisions are $r_1,\dots,r_N$.
We write $F_r(x)=\sum_ir_ix_i^2$ and
\begin{equation}\label{eq:dens-dynamics}
L_r=L_{A_r}=\Delta_{S_N}-\nabla_{S_N}F_r\cdot\nabla_{S_N},\qquad
P^r_t=e^{tL_r},\qquad
\pi_r(dx)=\frac1{Z_r}e^{-F_r(x)}\,\nu_N(dx),
\end{equation}
where $Z_r=\int_{S_N}e^{-F_r}\,d\nu_N$.
Recall from \eqref{eq:model-precision-form} and \eqref{sp-canonical-definition} that the model with rescaled inverse temperature $c$ is the case where $r_i=(z_N(c)-c\lambda_i)/2$ are its half-precisions; then $L_r=L$, $P^r_t=P_t$, $\pi_r=\pi_{N,c}$, and $\nu_NP^r_t$ is the time-$t$ law.
Recall from \cref{prop:cond-bingham} that $\pi_r$ is the conditioned Gaussian law $\mu_q$ with variances $q_i=(2r_i)^{-1}$.
We write $\Gamma(f)=|\nabla_{S_N}f|^2$, and logarithmic Sobolev inequalities are understood as in \eqref{eq:model-fi}.
Recall from \eqref{eq:model-dirichlet} that $-\int fL_rg\,d\pi_r=\int\ip{\nabla_{S_N}f}{\nabla_{S_N}g}\,d\pi_r$ for smooth $f$ and $g$.
We also use this identity for $f\in C^1(S_N)$ and $g\in C^2(S_N)$: approximate them by smooth functions uniformly together with their derivatives through orders one and two, respectively, and pass to the limit on the compact sphere. The same approximation through order one extends every logarithmic Sobolev inequality below from its smooth core to $C^1$ tests, because both the entropy and the energy converge under this approximation.

\paragraph{The comparison law.}
Let $\ind=(1,\dots,1)^\T$, and let
\begin{equation}\label{eq:dens-heat}
H=-4\diag(r_1,\dots,r_N)+\frac2N\ind\ind^\T,\qquad
h(t)=e^{tH}\ind,
\end{equation}
and, for $t\ge0$,
\begin{equation}\label{eq:dens-normalized}
\bar h(t)=\frac1N\sum_{i=1}^Nh_i(t),\qquad
v_i(t)=\frac{h_i(t)}{\bar h(t)} .
\end{equation}
The matrix $H$ is the matrix of the covariance evolution of \cref{sec:heat} for $A_r$, whose precisions are $2r_i$; the vector $h(t)$ is its heat vector, and $v_1(t),\dots,v_N(t)$ are its normalized variances.
They are positive, $\sum_iv_i(t)=N$, and $v_i(0)=1$.
Recall from \cref{def:heat-covariance,prop:cond-bingham} that the Bingham comparison law of $A_r$ at time $t\ge0$ is the conditioned Gaussian law $\mu_t=\mu_{v(t)}$ of \cref{def:cond-finite}, that is,
\begin{equation}\label{eq:dens-comparison}
\mu_t(dx)=\frac{1}{\mathcal Z(t)}\exp\Big(-\sum_{i=1}^N\frac{x_i^2}{2v_i(t)}\Big)\nu_N(dx),
\end{equation}
with $\mathcal Z(t)$ the normalizing constant.
Note that $\mu_0=\nu_N$, since $v_i(0)=1$.
The numbers $v_i(t)$ need not equal the second moments of $\mu_t$ or of the time-$t$ law.
We define
\begin{equation}\label{eq:dens-zeta}
\ell(t)=\frac{\bar h'(t)}{\bar h(t)},\qquad
\zeta_i(t)=r_i-\frac1{2v_i(t)}+\frac{\ell(t)}4 .
\end{equation}
Using $\sum_ix_i^2=N$ on $S_N$, we find that $F_r(x)-\sum_ix_i^2/(2v_i(t))=\sum_i\zeta_i(t)x_i^2-N\ell(t)/4$ there, which yields
\begin{equation}\label{eq:dens-ratio}
\frac{d\mu_t}{d\pi_r}(x)=\frac{\exp\big(\sum_i\zeta_i(t)x_i^2\big)}{\int_{S_N}\exp\big(\sum_i\zeta_i(t)y_i^2\big)\,\pi_r(dy)} .
\end{equation}
The coefficients $\zeta_i(t)$ measure the difference between $\mu_t$ and $\pi_r$.
The common term $\ell(t)/4$ does not change \eqref{eq:dens-ratio}; it is included because it makes all $\zeta_i(t)$ small at once (\cref{lem:dens-coefficients}).

\paragraph{R\'enyi divergences.}
For probability measures $\mu\ll\eta$ and $p>1$, we write
\begin{equation}\label{eq:dens-renyi}
\mathcal R_p(\mu\Vert\eta)=\log\int\Big(\frac{d\mu}{d\eta}\Big)^p\,d\eta\in[0,\infty].
\end{equation}
This is $p-1$ times the R\'enyi divergence of order $p$ of $\mu$ from $\eta$ \cite[Definition~2, (9)]{vanErvenHarremoes2014}.
It is nonnegative by Jensen's inequality.
By Jensen's inequality under $\mu$, we also have
\begin{equation}\label{eq:dens-kl-renyi}
\KL(\mu\Vert\eta)=\frac1{p-1}\int\log\Big(\frac{d\mu}{d\eta}\Big)^{p-1}d\mu
\le\frac1{p-1}\log\int\Big(\frac{d\mu}{d\eta}\Big)^{p-1}d\mu=\frac{\mathcal R_p(\mu\Vert\eta)}{p-1},
\end{equation}
which is the monotonicity of R\'enyi divergences in their order \cite[Theorems~3 and~5]{vanErvenHarremoes2014}.

\paragraph{Gaussian squares.}
For positive $w=(w_1,\dots,w_N)$, let $p_w$ be the continuous density of $T_w=\sum_iw_i(g_i^2-1)$ from \cref{sec:cond-notation} and \cref{lem:cond-three}, where $g_1,\dots,g_N$ are independent standard Gaussian variables.
Thus $p_w(N-\sum_jw_j)$ is the density of $\sum_iw_ig_i^2$ at $N$, and $p_w(0)$ is its density at its mean.
We use two facts.
The first is \cref{gp-real-tilt} with $t=1$ and with $2v_i\omega_i$ in place of the coefficients $w_i$ there, written in the coordinates $x_i=\sqrt{v_i}\,Z_i$: if $\sum_iv_i=N$ and $\omega_1,\dots,\omega_N$ are real numbers with $2v_i\omega_i<1$ for all $i$, then
\begin{equation}\label{eq:dens-tilt-identity}
\int_{S_N}e^{\sum_i\omega_ix_i^2}\,\mu_v(dx)
=\prod_{i=1}^N(1-2v_i\omega_i)^{-1/2}\,\frac{p_{w}(N-\sum_jw_j)}{p_v(0)},\qquad
w_i=\frac{v_i}{1-2v_i\omega_i}.
\end{equation}
The second fact is the following restatement of \cref{cg-mean-level-density}.

\begin{lemma}[Weighted sums of squares]\label{lem:dens-mean-level}
Let $\delta\in(0,1]$, let $w=(w_1,\dots,w_N)$ be positive, and suppose that there exist three distinct indices $i$ with $w_i\ge\delta\max_jw_j$.
There exists $C_\delta\ge1$, depending only on $\delta$, such that
\begin{equation}\label{eq:dens-mean-level}
p_w(0)\ge C_\delta^{-1}\Big(\sum_iw_i^2\Big)^{-1/2},\qquad
\sup_{y\in\R}p_w(y)\le C_\delta\Big(\sum_iw_i^2\Big)^{-1/2}.
\end{equation}
If moreover $\sum_iw_i=N$, then $\int_{S_N}x_i^2\,\mu_w(dx)\le C_\delta w_i$ for every $i$.
\end{lemma}

\begin{proof}
Since $T_w$ is the translate of $Q_w=\sum_iw_ig_i^2$ by $\sum_iw_i$, the lower bound in \eqref{eq:dens-mean-level} is the lower bound of \cref{cg-mean-level-density} for the density of $Q_w$ at its mean, with $b=w$.
For the upper bound, fix an index $k$ and write $Q_w=Q_{w,-k}+w_kg_k^2$, with independent summands.
If $h_{-k}$ is the density of $Q_{w,-k}$ and $f_k$ is the density of $w_kg_k^2$, then the density of $Q_w$ is $h_{-k}*f_k$, which is continuous because $h_{-k}$ is bounded and $f_k$ is integrable.
Hence $\sup p_w=\|h_{-k}*f_k\|_\infty\le\|h_{-k}\|_\infty\|f_k\|_1=\|h_{-k}\|_\infty$, and the bound of \cref{cg-mean-level-density} for the density with one coordinate removed gives the upper bound in \eqref{eq:dens-mean-level}.
For the last claim, \cref{cg-mean-level-density} bounds by $C_\delta w_i$ the conditional expectation of $w_ig_i^2$ given $\sum_jw_jg_j^2=N$, defined there through the joint density at the level $N$.
By the third part of \cref{lem:cond-radial}, applied with $F(y)=y_i^2$, the integral $\int x_i^2\,\mu_w(dx)$ is the same ratio of densities at the level $N$: in both cases the numerator is the value at that level of a continuous density of the measure $B\mapsto\E[w_ig_i^2;\sum_jw_jg_j^2\in B]$, and two continuous densities of the same measure agree.
This proves the last claim.
\end{proof}

\paragraph{Hypotheses.}
The general results of this section are stated under the following hypotheses.

\begin{definition}[{Hypotheses on the time interval $[0,S]$}]\label{def:dens-hyp}
Let $S>0$ and $C_*\ge1$.
We say that $r$ satisfies \textup{(H1)}--\textup{(H4)} on $[0,S]$ with constant $C_*$ if the following hold.
\begin{enumerate}
\item[\textup{(H1)}] $\max_ir_i\le C_*$ and $N(1+S)^{-3/2}\ge C_*^{-1}$.
\item[\textup{(H2)}] For all $t\in[0,S]$ and all $i$,
\[
\frac{1}{C_*\sqrt{1+t}}\le\bar h(t)\le\frac{C_*}{\sqrt{1+t}},\qquad
\frac{\sqrt{1+t}}{C_*(1+r_it)}\le h_i(t)\le\frac{C_*\sqrt{1+t}}{1+r_it}.
\]
\item[\textup{(H3)}] For all $t\in[0,S]$, $\sum_i(1+r_it)^{-2}\le C_*N(1+t)^{-3/2}$.
\item[\textup{(H4)}] There exist three distinct indices $i$ with $r_iS\le C_*$.
\end{enumerate}
We say that $r$ satisfies \textup{(H1)}--\textup{(H5)} on $[0,S]$ with constant $C_*$ if moreover the following holds.
\begin{enumerate}
\item[\textup{(H5)}] For every $t\in[0,S]$, the comparison law $\mu_t$ satisfies a logarithmic Sobolev inequality with constant $C_*(1+t)$:
\[
\Ent_{\mu_t}(f^2)\le C_*(1+t)\int_{S_N}\Gamma(f)\,d\mu_t\qquad\text{for all smooth }f\text{ on }S_N .
\]
\end{enumerate}
\end{definition}

The heat bounds \textup{(H2)} say that $h(t)$ behaves as in the critical window, where they are \cref{cd-heat-profiles}.
The counting bound \textup{(H3)} holds when the half-precisions grow like $(i/N)^{2/3}$, by \cref{lem:dens-count} below.
The hypothesis \textup{(H4)} ensures that three variances $v_i(t)$ are comparable to the largest one, so that \cref{lem:dens-mean-level} applies to $v(t)$.
We verify all five hypotheses in the critical window in \cref{lem:dens-critical-hyp}.

\section{The coefficients of the comparison law}\label{sec:dens-coefficients}

We begin with an elementary counting bound.

\begin{lemma}[Sums over the half-precisions]\label{lem:dens-count}
Let $c_0>0$ and $k_0\ge0$, and suppose that $\#\{i:r_i\le u\}\le k_0+N(u/c_0)^{3/2}$ for every $u>0$.
Then for every $j>3/2$ there exists $C=C(j,c_0)$ such that
\[
\sum_{i=1}^N\frac1{(1+r_is)^j}\le k_0+CN(1+s)^{-3/2}\qquad\text{for all }s\ge0 .
\]
The hypothesis holds with $k_0=0$ if $r_i\ge c_0(i/N)^{2/3}$ for all $i$, and with $k_0=1$ if $r_i\ge c_0((i-1)/N)^{2/3}$ for all $i\ge2$.
\end{lemma}

\begin{proof}
If $r_i\ge c_0(i/N)^{2/3}$ and $r_i\le u$, then $i\le N(u/c_0)^{3/2}$, which gives the hypothesis with $k_0=0$; the case $k_0=1$ is the same, with $i-1$ in place of $i$.
For $s\le1$, the sum is at most $N\le2^{3/2}N(1+s)^{-3/2}$.
Let $s\ge1$.
For each $i$, we have $(1+r_is)^{-j}=\int_{r_i}^\infty js(1+us)^{-j-1}\dd u$.
Summing over $i$ and exchanging the sum and the integral, we obtain
\[
\sum_{i=1}^N\frac1{(1+r_is)^j}=\int_0^\infty\frac{js\,\#\{i:r_i\le u\}}{(1+us)^{j+1}}\,\dd u
\le k_0+\frac{N}{c_0^{3/2}}\int_0^\infty\frac{js\,u^{3/2}}{(1+us)^{j+1}}\,\dd u .
\]
The substitution $y=us$ shows that the last integral is $s^{-3/2}j\int_0^\infty y^{3/2}(1+y)^{-j-1}\dd y$, and the integral in $y$ is finite because $j>3/2$.
Since $s^{-3/2}\le2^{3/2}(1+s)^{-3/2}$ for $s\ge1$, this completes the proof.
\end{proof}

The next lemma is the main deterministic input.
The comparability \eqref{cg-reference-comparability} is a restatement of \textup{(H2)}.
The estimates \eqref{cg-reference-e} and \eqref{cg-reference-derivative} use cancellations.
For $t\ge1$ and $r_it\ge1$, both $r_i$ and $(2v_i(t))^{-1}$ are comparable to $(1+r_it)/(1+t)$, whereas the difference $\zeta_i(t)$, which includes the common term $\ell(t)/4$, is bounded by this scale times $C(1+r_it)^{-2}$.

\begin{lemma}[Coefficients of the comparison law]\label{lem:dens-coefficients}
Let $S>0$ and $C_*\ge1$, and suppose that $\max_ir_i\le C_*$ and that $r$ satisfies \textup{(H2)} on $[0,S]$ with constant $C_*$.
There exists $C=C(C_*)$ such that, for all $t\in[0,S]$ and all $i$,
\begin{align}
C^{-1}\frac{1+t}{1+r_it}\le v_i(t)&\le C\frac{1+t}{1+r_it},\label{cg-reference-comparability}\\
|\zeta_i(t)|&\le\frac{C}{(1+t)(1+r_it)},\label{cg-reference-e}\\
|\zeta_i'(t)|&\le\frac{C}{(1+t)^2(1+r_it)}.\label{cg-reference-derivative}
\end{align}
If $r$ also satisfies \textup{(H3)} on $[0,S]$ with constant $C_*$, then, for all $t\in[0,S]$,
\begin{equation}\label{eq:dens-coefficient-sums}
\sum_iv_i(t)\zeta_i(t)^2\le CN(1+t)^{-5/2},\qquad
\sum_iv_i(t)|\zeta_i'(t)|\le CN(1+t)^{-5/2}.
\end{equation}
\end{lemma}

\begin{proof}
The proof has five steps.
We first record bounds on the derivatives of $\bar h$.
Then we derive an exact equation for $v_i$, which settles the bounded times and the indices with $r_it\le1$.
For $t\ge1$ and $r_it\ge1$, the bound \eqref{cg-reference-e} comes from an exact cancellation in the Duhamel formula for $h_i$, and \eqref{cg-reference-derivative} from a second-order expansion of $h_i$ combined with the exact equation for $v_i$.
Throughout the proof, $C$ denotes a constant depending only on $C_*$, whose value may change from line to line.
The comparability \eqref{cg-reference-comparability} follows from \textup{(H2)}, with $C=C_*^2$, because $v_i=h_i/\bar h$.

\paragraph{Step 1: derivatives of $\bar h$.}
For $y\in\R^N$, we have $(\sum_iy_i)^2\le\sum_i(2r_i)^{-1}\sum_i2r_iy_i^2=2N\sum_ir_iy_i^2$, by the Cauchy--Schwarz inequality and \eqref{eq:dens-trace}.
We obtain
\[
-4C_*|y|^2\le-4\sum_ir_iy_i^2\le y^\T Hy=-4\sum_ir_iy_i^2+\frac2N\Big(\sum_iy_i\Big)^2\le0 .
\]
Using the spectral theorem, we write $\bar h(t)=N^{-1}\ip{\ind}{e^{tH}\ind}=\int e^{-\lambda t}\,\mathsf m(d\lambda)$, where $\mathsf m$ is a positive measure supported in $[0,4C_*]$; namely, $\mathsf m$ puts mass $N^{-1}\ip{\ind}{u}^2$ at $\lambda$ for each unit eigenvector $u$ of $-H$ with eigenvalue $\lambda$.
Differentiating under the integral, we find $(-1)^j\bar h^{(j)}(t)=\int\lambda^je^{-\lambda t}\,\mathsf m(d\lambda)\ge0$; in particular, $\bar h$ is nonincreasing.
For $t\ge1$, the inequality $x^je^{-x/2}\le(2j)^j$ gives $\lambda^je^{-\lambda t}\le(2j)^jt^{-j}e^{-\lambda t/2}$, so that $|\bar h^{(j)}(t)|\le(2j)^jt^{-j}\bar h(t/2)$.
For $t\le1$, we have $|\bar h^{(j)}(t)|\le(4C_*)^j\bar h(t)$.
Also, \textup{(H2)} implies that $\bar h(u)\le C\bar h(t)$ whenever $t/4\le u\le t\le S$.
Combining these bounds, we find, for $j=1,2,3$,
\begin{equation}\label{eq:dens-V-derivatives}
|\bar h^{(j)}(u)|\le C(1+t)^{-j}\,\bar h(t),\qquad t/2\le u\le t\le S .
\end{equation}
In particular, $|\ell(t)|\le C(1+t)^{-1}$, and $\ell'=\bar h''/\bar h-\ell^2$ satisfies $|\ell'(t)|\le C(1+t)^{-2}$.

\paragraph{Step 2: an exact equation.}
Differentiating \eqref{eq:dens-heat}, we see that the coordinates of $h$ solve
\begin{equation}\label{eq:dens-heat-ode}
h_i'(t)=-4r_ih_i(t)+2\bar h(t),\qquad h_i(0)=1,
\end{equation}
and variation of constants gives
\begin{equation}\label{eq:dens-duhamel}
h_i(t)=e^{-4r_it}+2\int_0^te^{-4r_i(t-u)}\bar h(u)\dd u .
\end{equation}
We divide \eqref{eq:dens-heat-ode} by $h_i$, subtract $\ell$, and use $2\bar h/h_i=2/v_i$.
This gives $(\log v_i)'=h_i'/h_i-\ell=-4r_i+2/v_i-\ell=-4\zeta_i$, that is,
\begin{equation}\label{cg-q-riccati}
v_i'(t)=-4\zeta_i(t)v_i(t),\qquad 4\zeta_i(t)=\ell(t)-\frac{h_i'(t)}{h_i(t)} .
\end{equation}
Differentiating \eqref{eq:dens-zeta} and using \eqref{cg-q-riccati}, we find
\begin{equation}\label{eq:dens-zeta-prime}
\zeta_i'=\frac{v_i'}{2v_i^2}+\frac{\ell'}4=-\frac{2\zeta_i}{v_i}+\frac{\ell'}4 .
\end{equation}

\paragraph{Step 3: bounded times and small $r_it$.}
Let $t\le1$.
Then $v_i(t)\ge C^{-1}$ by \eqref{cg-reference-comparability}, so that $|\zeta_i(t)|\le r_i+(2v_i)^{-1}+|\ell|/4\le C$ and, by \eqref{eq:dens-zeta-prime}, $|\zeta_i'(t)|\le C$.
These are the claimed bounds, because $(1+t)(1+r_it)\le2(1+C_*)$.
Now let $t\ge1$ and $r_it\le1$.
Then $v_i(t)\ge C^{-1}t$, so that $(2v_i)^{-1}\le C/t$, and also $r_i\le1/t$ and $|\ell|\le C/t$.
It follows that $|\zeta_i|\le C/t$, and \eqref{eq:dens-zeta-prime} implies that $|\zeta_i'|\le2|\zeta_i|/v_i+|\ell'|/4\le C/t^2$.
As $1+r_it\le2$ and $1+t\le2t$, these are again the claimed bounds.

In the remaining two steps, we fix $i$ and $t\in[1,S]$ with $\xi=r_it\ge1$.
Recall from \textup{(H2)} that $h_i(t)\ge C^{-1}\sqrt{1+t}/(1+\xi)\ge C^{-1}\sqrt t/\xi$, and that $\bar h(u)\le Ct^{-1/2}$ for $u\in[t/2,t]$.

\paragraph{Step 4: the first cancellation.}
We split \eqref{eq:dens-duhamel} at $u=t/2$ and write $h_i=I_1+I_2$, where
\[
I_1(t)=e^{-4r_it}+2\int_0^{t/2}e^{-4r_i(t-u)}\bar h(u)\dd u,\qquad
I_2(t)=2\int_0^{t/2}e^{-4r_i\tau}\bar h(t-\tau)\dd\tau .
\]
Each integral has a moving endpoint, and each endpoint contributes $e^{-2\xi}\bar h(t/2)$ to the derivative.
Using $\bar h'(t-\tau)=\ell(t-\tau)\bar h(t-\tau)$ and subtracting $\ell(t)h_i(t)=\ell(t)I_1(t)+\ell(t)I_2(t)$, we obtain the exact identity
\begin{multline}\label{eq:dens-first-cancellation}
h_i'(t)-\ell(t)h_i(t)=-\big(4r_i+\ell(t)\big)I_1(t)+2e^{-2\xi}\bar h(t/2)\\
+2\int_0^{t/2}e^{-4r_i\tau}\bar h(t-\tau)\big(\ell(t-\tau)-\ell(t)\big)\dd\tau .
\end{multline}
The term $\ell(t)h_i(t)$, common to all coordinates, has been removed exactly; only the variation of $\ell$ over $[t/2,t]$ remains.
We bound the three terms on the right side.
For $u\le t/2$, we have $e^{-4r_i(t-u)}\le e^{-2\xi}$, and $\int_0^{t/2}\bar h(u)\dd u\le2C_*\sqrt{1+t}$ by \textup{(H2)}.
This gives $I_1(t)\le e^{-4\xi}+4C_*e^{-2\xi}\sqrt{1+t}\le Ce^{-2\xi}\sqrt t$.
Using $4r_i=4\xi/t$ and $|\ell(t)|\le C/t\le C\xi/t$, we see that the first term on the right side of \eqref{eq:dens-first-cancellation} is at most $C\xi e^{-2\xi}t^{-1/2}$ in absolute value.
The second term is at most $Ce^{-2\xi}t^{-1/2}$, since $\bar h(t/2)\le Ct^{-1/2}$.
For the third term, the bound on $\ell'$ in Step~1 gives $|\ell(t-\tau)-\ell(t)|\le C\tau t^{-2}$ for $0\le\tau\le t/2$, and $\bar h(t-\tau)\le Ct^{-1/2}$.
Using $\int_0^\infty\tau e^{-4r_i\tau}\dd\tau=(4r_i)^{-2}$, we find that the third term is at most $Ct^{-5/2}r_i^{-2}=Ct^{-1/2}\xi^{-2}$.
By \eqref{cg-q-riccati}, we have $4|\zeta_i(t)|=|h_i'(t)-\ell(t)h_i(t)|/h_i(t)$.
Dividing the three bounds by $h_i(t)\ge C^{-1}\sqrt t/\xi$, we conclude that
\[
|\zeta_i(t)|\le\frac Ct\Big(\xi^2e^{-2\xi}+\xi e^{-2\xi}+\xi^{-1}\Big)\le\frac{C}{t\xi},
\]
where we used that $\xi^3e^{-2\xi}$ is bounded.
Since $(1+t)(1+\xi)\le4t\xi$, \eqref{cg-reference-e} follows.

\paragraph{Step 5: the second cancellation.}
To bound $\zeta_i'$, we cannot differentiate the error terms of Step~4.
Instead, we expand $h_i$ to second order and use an exact equation.
For $u\in[t/2,t]$, let
\[
\varphi(u)=\frac1{2r_i}\Big(\bar h(u)-\frac{\bar h'(u)}{4r_i}+\frac{\bar h''(u)}{16r_i^2}\Big),\qquad
\eta(u)=h_i(u)-\varphi(u).
\]
Using \eqref{eq:dens-heat-ode}, we find that the terms with $\bar h$, $\bar h'$ and $\bar h''$ cancel in $\eta'+4r_i\eta$, so that $\eta'(u)+4r_i\eta(u)=-\bar h'''(u)/(32r_i^3)$.
Variation of constants on $[t/2,t]$ gives
\[
\eta(t)=e^{-2\xi}\eta(t/2)-\frac1{32r_i^3}\int_{t/2}^te^{-4r_i(t-u)}\bar h'''(u)\dd u .
\]
We have $h_i(t/2)\le C\sqrt t/\xi$ and $\bar h(t)/r_i\ge C^{-1}t^{-1/2}r_i^{-1}=C^{-1}\sqrt t/\xi$, by \textup{(H2)}.
By \eqref{eq:dens-V-derivatives} and $4r_it\ge4$, we have $|\varphi(t/2)|\le C\bar h(t)/r_i$.
Combining the last two bounds, we have $|\eta(t/2)|\le C\bar h(t)/r_i$.
Also $|\bar h'''(u)|\le Ct^{-3}\bar h(t)$ for $u\in[t/2,t]$, by \eqref{eq:dens-V-derivatives}.
Inserting these bounds into the formula for $\eta(t)$ and using $e^{-2\xi}\le C\xi^{-3}$, we find
\[
|\eta(t)|\le\frac{C\bar h(t)}{r_i}e^{-2\xi}+\frac{C\bar h(t)}{r_i^4t^3}\le\frac{C\bar h(t)}{r_i^4t^3} .
\]
Let $\Theta=2r_iv_i=2r_ih_i/\bar h$ and $d=\bar h''/\bar h$.
The definitions of $\varphi$ and $\eta$ give
\begin{equation}\label{eq:dens-theta-expansion}
\Theta=1-\frac{\ell}{4r_i}+\frac{d}{16r_i^2}+\epsilon,\qquad
\epsilon=\frac{2r_i\eta}{\bar h},\qquad|\epsilon|\le\frac{C}{\xi^3},
\end{equation}
all evaluated at time $t$.
Since $|\ell|\le C/t$ and $|d|\le C/t^2$, we have $|\Theta-1|\le C/\xi$; also $C^{-1}\le\Theta\le C$, by \eqref{cg-reference-comparability}.
On the other hand, by \eqref{cg-q-riccati}, $\Theta$ satisfies the exact equation $\Theta'=-4\zeta_i\Theta=4r_i-(4r_i+\ell)\Theta$.
Inserting \eqref{eq:dens-theta-expansion} into its right side, without differentiating $\epsilon$, and using $\ell'=d-\ell^2$, we find
\[
\Theta'=-\frac{\ell'}{4r_i}-\frac{\ell d}{16r_i^2}-(4r_i+\ell)\epsilon .
\]
The last two terms are at most $C/(r_i^2t^3)$ in absolute value, since $|\ell d|\le Ct^{-3}$ and $(4r_i+|\ell|)|\epsilon|\le Cr_i\xi^{-3}$.
Differentiating $(2v_i)^{-1}=r_i/\Theta$, we have $((2v_i)^{-1})'=-r_i\Theta'/\Theta^2$, and therefore
\[
\zeta_i'=\frac{\ell'}4-\Big(\frac1{2v_i}\Big)'=\frac{\ell'}4\Big(1-\frac1{\Theta^2}\Big)+\epsilon_2,\qquad|\epsilon_2|\le\frac{C}{r_it^3},
\]
where $\epsilon_2=-r_i\Theta^{-2}\big(\ell d/(16r_i^2)+(4r_i+\ell)\epsilon\big)$.
Using $|1-\Theta^{-2}|\le C|\Theta-1|\le C/\xi$ and $|\ell'|\le Ct^{-2}$, we conclude that $|\zeta_i'(t)|\le C/(t^2\xi)$.
Since $(1+t)^2(1+\xi)\le8t^2\xi$, \eqref{cg-reference-derivative} follows.

Finally, assume \textup{(H3)}.
By \eqref{cg-reference-comparability}, \eqref{cg-reference-e}, and \eqref{cg-reference-derivative}, we have
\[
\sum_iv_i\zeta_i^2\le\frac C{1+t}\sum_i\frac1{(1+r_it)^3},\qquad
\sum_iv_i|\zeta_i'|\le\frac C{1+t}\sum_i\frac1{(1+r_it)^2},
\]
and \textup{(H3)} bounds both sums by $CN(1+t)^{-3/2}$.
This proves \eqref{eq:dens-coefficient-sums}.
\end{proof}

The identities in Steps~2, 4 and~5 are exact consequences of \eqref{eq:dens-heat-ode}; they can be checked by expanding both sides in terms of $h_i$, $\bar h$ and the derivatives of $\bar h$.

\section{Exponential moments of quadratic functions}\label{sec:dens-tilts}

The next lemma controls exponential moments under the comparison law of quadratic functions whose coefficients have the size of the bounds in \cref{lem:dens-coefficients}.
The constant $\Lambda$ need not be small, and the coefficients may have either sign.

\begin{lemma}[Exponential moments under the comparison law]\label{lem:dens-tilt}
Let $S>0$ and $C_*\ge1$, and suppose that $r$ satisfies \textup{(H1)}--\textup{(H4)} on $[0,S]$ with constant $C_*$.
Let $\Lambda\ge0$ and $s\in[0,S]$, and let $\omega_1,\dots,\omega_N$ be real numbers with
\begin{equation}\label{eq:dens-tilt-envelope}
|\omega_i|\le\frac{\Lambda}{(1+s)(1+r_is)}\qquad\text{for all }i .
\end{equation}
There exists $C=C(\Lambda,C_*)$ such that
\begin{equation}\label{eq:dens-tilt-bound}
\log\int_{S_N}\exp\Big(\sum_i\omega_ix_i^2\Big)\,\mu_s(dx)\le CN(1+s)^{-3/2} .
\end{equation}
\end{lemma}

\begin{proof}
Throughout the proof, $C$ denotes a constant depending only on $\Lambda$ and $C_*$.
We write $v_i=v_i(s)$ and $T=1+s$.
Recall from \eqref{cg-reference-comparability} that $C^{-1}T/(1+r_is)\le v_i\le CT/(1+r_is)$.
Thus $v_i\le CT$ for all $i$, and $v_i\ge C^{-1}T$ for the three indices of \textup{(H4)}, since $r_is\le r_iS\le C_*$ for them.
Combining the coefficient bound \eqref{eq:dens-tilt-envelope} with \textup{(H3)}, we obtain
\begin{equation}\label{cl3-coarea-count-inputs}
\sum_iv_i|\omega_i|\le C\sum_i\frac1{(1+r_is)^2}\le CNT^{-3/2},\qquad
\frac1{T^2}\sum_iv_i^2\le C\sum_i\frac1{(1+r_is)^2}\le CNT^{-3/2} .
\end{equation}

We first shift all coefficients by the same amount.
On $S_N$, we have $\sum_i\omega_ix_i^2=\Lambda N/T+\sum_i(\omega_i-\Lambda/T)x_i^2$.
Let $u_i=2v_i(\Lambda/T-\omega_i)$.
The coefficient bound \eqref{eq:dens-tilt-envelope} implies that $0\le u_i\le4\Lambda v_i/T\le C$.
Applying \eqref{eq:dens-tilt-identity} with the coefficients $\omega_i-\Lambda/T$, for which $1-2v_i(\omega_i-\Lambda/T)=1+u_i>0$, we have
\begin{equation}\label{cl3-coarea-normalizer-identity}
\log\int e^{\sum_i\omega_ix_i^2}\,d\mu_s=\frac{\Lambda N}T-\frac12\sum_i\log(1+u_i)+\log\frac{p_w(N-\sum_jw_j)}{p_v(0)},\qquad
w_i=\frac{v_i}{1+u_i} .
\end{equation}
The shift ensures that all the variances $w_i$ are positive, whatever the signs of the $\omega_i$.

We bound the last term in \eqref{cl3-coarea-normalizer-identity}.
Since $0\le u_i\le C$, we have $(1+C)^{-1}v_i\le w_i\le v_i$.
Then $w_i\ge C^{-1}\max_jw_j$ for the three indices of \textup{(H4)}, and $\sum_iw_i^2\ge(1+C)^{-2}\sum_iv_i^2$; the same three indices satisfy $v_i\ge C^{-1}\max_jv_j$.
Then \cref{lem:dens-mean-level}, applied to $v$ for the lower bound at its mean $N$ and to $w$ for the upper bound on its density, implies that
\begin{equation}\label{cl3-coarea-density-ratio}
\frac{p_w(N-\sum_jw_j)}{p_v(0)}\le\frac{C(\sum_iw_i^2)^{-1/2}}{C^{-1}(\sum_iv_i^2)^{-1/2}}\le C .
\end{equation}
No lower bound for the density of $\sum_iw_ig_i^2$ away from its mean is needed.

We now bound the first two terms in \eqref{cl3-coarea-normalizer-identity}.
Using $\log(1+u)\ge u-u^2/2$ for $u\ge0$, and $\sum_iu_i=2\Lambda N/T-2\sum_iv_i\omega_i$ (which follows from $\sum_iv_i=N$), we find
\[
\frac{\Lambda N}T-\frac12\sum_i\log(1+u_i)\le\frac{\Lambda N}T-\frac12\sum_iu_i+\frac14\sum_iu_i^2
=\sum_iv_i\omega_i+\frac14\sum_iu_i^2 .
\]
The terms $\Lambda N/T$ cancel exactly.
Using \eqref{cl3-coarea-count-inputs} and $u_i^2\le16\Lambda^2v_i^2/T^2$, we see that the right side is at most $CNT^{-3/2}$.
Finally, the logarithm of \eqref{cl3-coarea-density-ratio} is at most $C\le CC_*N(1+s)^{-3/2}$, because $N(1+s)^{-3/2}\ge N(1+S)^{-3/2}\ge C_*^{-1}$ by \textup{(H1)}.
This proves \eqref{eq:dens-tilt-bound}.
\end{proof}

\section{A differential inequality for R\'enyi divergences}\label{sec:dens-renyi}

The following scalar lemma closes the differential inequality for the R\'enyi divergence.
It is stated for general probability measures $\mu_t$ on $S_N$; in our application, $\mu_t$ will be the comparison law at time $\gamma t$.

\begin{lemma}[A differential inequality for R\'enyi divergences]\label{cg-renyi-differential-inequality}
Fix $p>1$, $C_1>0$, $C_2,C_3\ge0$, an integer $N\ge3$, and $\gamma$ with $0<\gamma\le\min\{1,(4pC_1)^{-1}\}$.
Let $t_1>0$.
For $t\in[0,t_1]$, let $\mu_t$ be a probability measure on $S_N$, let $f_t$ be a strictly positive function of class $C^2$ on $S_N$ with $\int f_t\,d\mu_t=1$, and let $U_t\ge0$ be measurable.
Let
\[
\mathcal R(t)=\log\int f_t^p\,d\mu_t,\qquad
\psi_t=f_t^pe^{-\mathcal R(t)},\qquad
J(t)=\int\psi_t\,\Gamma(\log\psi_t)\,d\mu_t .
\]
Assume that $J(t)$ and $\Ent_{\mu_t}(\psi_t)$ are finite, that $\mathcal R$ is continuously differentiable on $[0,t_1]$ with $\mathcal R(0)=0$, and that for every $t\in[0,t_1]$,
\begin{gather}
\Ent_{\mu_t}(\psi_t)\le\frac{C_1(1+\gamma t)}4J(t),\label{eq:dens-cert-lsi}\\
\log\int\exp\big(pC_1(1+\gamma t)U_t\big)\,d\mu_t\le C_2N(1+\gamma t)^{-3/2},\label{eq:dens-cert-mgf}\\
\mathcal R'(t)\le-\frac{p-1}{2p}J(t)+(p-1)\int U_t\,\psi_t\,d\mu_t+(p-1)C_3N(1+\gamma t)^{-5/2}.\label{eq:dens-cert-derivative}
\end{gather}
Let $a=(pC_1\gamma)^{-1}$ and $Q=(p-1)[C_2/(pC_1)+C_3]$.
Then $a\ge4$, and for every $t\in[0,t_1]$,
\[
\mathcal R(t)\le\frac{Q}{\gamma(a-3/2)}N(1+\gamma t)^{-3/2}\le2(p-1)(C_2+pC_1C_3)N(1+\gamma t)^{-3/2} .
\]
\end{lemma}

\begin{proof}
Fix $t$, and write $\mu=\mu_t$, $\psi=\psi_t$, and $\lambda=pC_1(1+\gamma t)$.
We first bound the forcing term.
By Jensen's inequality under the probability measure $\psi\mu$, applied to $\min(U_t,M)$ and followed by the limit $M\to\infty$, we have
\[
\log\int e^{\lambda U_t}\,d\mu\ge\log\int e^{\lambda U_t-\log\psi}\,\psi\,d\mu\ge\lambda\int U_t\,\psi\,d\mu-\Ent_\mu(\psi) .
\]
This is the classical duality between relative entropy and exponential moments.
Combined with \eqref{eq:dens-cert-lsi} and \eqref{eq:dens-cert-mgf}, this gives
\[
\int U_t\,\psi\,d\mu\le\frac{\Ent_\mu(\psi)}{\lambda}+\frac{C_2N(1+\gamma t)^{-3/2}}{\lambda}
\le\frac{J(t)}{4p}+\frac{C_2}{pC_1}N(1+\gamma t)^{-5/2} .
\]
Inserting this into \eqref{eq:dens-cert-derivative}, we obtain
\begin{equation}\label{cl3-renyi-closed-information}
\mathcal R'(t)\le-\frac{p-1}{4p}J(t)+QN(1+\gamma t)^{-5/2} .
\end{equation}

Next, we bound $J(t)$ below by $\mathcal R(t)$.
Since $f_t=e^{\mathcal R(t)/p}\psi^{1/p}$ and $\int f_t\,d\mu=1$, we have
\[
e^{-\mathcal R(t)/p}=\int\psi^{1/p}\,d\mu=\int\exp\Big(-\frac{p-1}p\log\psi\Big)\,\psi\,d\mu
\ge\exp\Big(-\frac{p-1}p\Ent_\mu(\psi)\Big),
\]
where the inequality is Jensen's inequality under $\psi\mu$.
Taking logarithms, we find $\Ent_\mu(\psi)\ge\mathcal R(t)/(p-1)$.
Together with \eqref{eq:dens-cert-lsi}, we find that $J(t)\ge4\mathcal R(t)/((p-1)C_1(1+\gamma t))$, and \eqref{cl3-renyi-closed-information} becomes
\begin{equation}\label{cl3-renyi-closed-scalar}
\mathcal R'(t)\le-\frac{\mathcal R(t)}{pC_1(1+\gamma t)}+QN(1+\gamma t)^{-5/2}=-\frac{a\gamma\,\mathcal R(t)}{1+\gamma t}+QN(1+\gamma t)^{-5/2} .
\end{equation}

Finally, we integrate.
Since $\gamma\le(4pC_1)^{-1}$, we have $a\ge4$.
Then \eqref{cl3-renyi-closed-scalar} implies that the derivative of $(1+\gamma t)^a\mathcal R(t)$ is at most $QN(1+\gamma t)^{a-5/2}$.
Integrating from $0$ and using $\mathcal R(0)=0$, we have
\[
(1+\gamma t)^a\mathcal R(t)\le QN\int_0^t(1+\gamma u)^{a-5/2}\dd u=\frac{QN}{\gamma(a-3/2)}\Big((1+\gamma t)^{a-3/2}-1\Big),
\]
which gives the first bound.
For the second, note that $\gamma a=(pC_1)^{-1}$ and $a-3/2\ge5a/8$, so that $Q/(\gamma(a-3/2))\le8pC_1Q/5\le2(p-1)(C_2+pC_1C_3)$.
\end{proof}

The next lemma computes the derivative of the R\'enyi divergence of the time-$t$ law from a time-dependent Bingham law, and estimates it in the form \eqref{eq:dens-cert-derivative}.
The two real numbers $a$ and $b$ are free; they express that adding a common multiple of $\sum_ix_i^2$ to the potential of the reference measure, or to its time derivative, changes nothing on $S_N$.

\begin{lemma}[R\'enyi divergence from a time-dependent Bingham law]\label{lem:dens-derivative}
Let $r_1,\dots,r_N>0$, let $\gamma>0$ and $t_1>0$, and let $k:[0,\gamma t_1]\to(0,\infty)^N$ be continuously differentiable with $k_i(0)=1/2$ for all $i$.
For $s\in[0,\gamma t_1]$, let $\mu^k_s$ be the probability measure on $S_N$ proportional to $\exp(-\sum_ik_i(s)x_i^2)\,\nu_N(dx)$.
Fix $p>1$, and for $t\in[0,t_1]$ let
\begin{gather*}
f_t=\frac{d(\nu_NP^r_t)}{d\mu^k_{\gamma t}},\qquad
\mathcal R(t)=\log\int f_t^p\,d\mu^k_{\gamma t},\\
\psi_t=f_t^pe^{-\mathcal R(t)},\qquad
J(t)=\int\psi_t\,\Gamma(\log\psi_t)\,d\mu^k_{\gamma t} .
\end{gather*}
Then, for each $t$, $f_t$ is a positive function of class $C^2$ on $S_N$; moreover, $\mathcal R$ is continuously differentiable on $[0,t_1]$, and $\mathcal R(0)=0$.
Moreover, for every $t\in[0,t_1]$ and all real numbers $a$ and $b$,
\begin{equation}\label{cl3-renyi-differential-prefix}
\mathcal R'(t)\le-\frac{p-1}{2p}J(t)+(p-1)\int U_t\,\psi_t\,d\mu^k_{\gamma t}
+(p-1)\gamma\int\sum_i|k_i'(\gamma t)-b|\,x_i^2\,d\mu^k_{\gamma t},
\end{equation}
where
\[
U_t(x)=\sum_i\Big(2p\big(r_i-k_i(\gamma t)+a\big)^2+\gamma\big|k_i'(\gamma t)-b\big|\Big)x_i^2 .
\]
\end{lemma}

\begin{proof}
We begin with the regularity.
Let $g_0=d\nu_N/d\pi_r=Z_re^{F_r}$, a smooth positive function on $S_N$, and let $g_t=P^r_tg_0$.
By the symmetry \eqref{eq:model-dirichlet} of $L_r$ in $L^2(\pi_r)$, we have $(\nu_NP^r_t)(\phi)=\pi_r(g_0P^r_t\phi)=\pi_r(\phi\,g_t)$ for bounded measurable $\phi$, so that $g_t=d(\nu_NP^r_t)/d\pi_r$.
Since $L_r$ is an elliptic operator with smooth coefficients on the compact manifold $S_N$ and $g_0$ is smooth, the function $g_t$ is of class $C^2$ in space and $C^1$ in time on $[0,\infty)\times S_N$, with the time derivative at zero understood from the right, and solves $\partial_tg_t=L_rg_t$; this is standard parabolic regularity, and it also follows by expanding $g_0$ in eigenfunctions of the self-adjoint operator $L_r$ on $L^2(\pi_r)$, whose coefficients decay faster than any power of the eigenvalues, together with elliptic regularity.
Also $g_t\ge\min g_0>0$, since $P^r_t$ is a Markov semigroup.
Let $W_s(x)=\sum_ik_i(s)x_i^2$.
The density of $\mu^k_s$ with respect to $\pi_r$ is $\theta_s=e^{F_r-W_s}/\pi_r(e^{F_r-W_s})$, which is continuously differentiable in $s$, smooth in $x$, and positive.
Then $f_t=g_t/\theta_{\gamma t}$ is continuously differentiable in $t$, twice continuously differentiable in $x$, and positive, and
\[
\mathcal R(t)=\log\int g_t^p\,\theta_{\gamma t}^{1-p}\,d\pi_r .
\]
The integrand and its first time derivative are continuous on $[0,t_1]\times S_N$, so compactness permits differentiation under the integral, and $\mathcal R$ is continuously differentiable.
At $t=0$, we have $\nu_NP^r_0=\nu_N=\mu^k_0$, since $W_0=N/2$ on $S_N$; hence $f_0=1$ and $\mathcal R(0)=0$.

We now compute the derivative.
Fix $t$, and write $\mu=\mu^k_{\gamma t}$, $\theta=\theta_{\gamma t}$, $W=W_{\gamma t}$, $f=f_t$, $\psi=\psi_t$, and $\nabla=\nabla_{S_N}$.
The derivative of $\int g_t^p\theta_{\gamma t}^{1-p}d\pi_r$ is the sum of
\[
p\int f^{p-1}L_rg_t\,d\pi_r\qquad\text{and}\qquad(1-p)\int f^p\,\partial_t\log\theta_{\gamma t}\,d\mu .
\]
For the first term, we use \eqref{eq:model-dirichlet}, $g_t=f\theta$, and $\nabla\theta=\theta\nabla(F_r-W)$:
\[
p\int f^{p-1}L_rg_t\,d\pi_r=-p(p-1)\int f^{p-2}\ip{\nabla f}{\nabla f+f\nabla(F_r-W)}\,d\mu .
\]
For the second term, $\log\theta_s=F_r-W_s-\log\pi_r(e^{F_r-W_s})$ gives $\partial_s\log\theta_s=-\partial_sW_s+\int\partial_sW_s\,d\mu^k_s$, and $\partial_t\log\theta_{\gamma t}=\gamma(\partial_s\log\theta_s)|_{s=\gamma t}$.
Let $W'(x)=\sum_ik_i'(\gamma t)x_i^2$.
Dividing by $e^{\mathcal R(t)}=\int f^pd\mu$, and using $f^{p-2}|\nabla f|^2=f^p|\nabla\log f|^2$ and $\nabla\log\psi=p\nabla\log f$, we obtain the exact identity
\begin{multline}\label{eq:dens-derivative-identity}
\mathcal R'(t)=-\frac{p-1}pJ(t)-(p-1)\int\psi\,\ip{\nabla\log\psi}{\nabla(F_r-W)}\,d\mu\\
+(p-1)\gamma\Big(\int\psi\,W'\,d\mu-\int W'\,d\mu\Big) .
\end{multline}
Using Young's inequality in the form $|\ip{\nabla\log\psi}{\nabla(F_r-W)}|\le(2p)^{-1}|\nabla\log\psi|^2+(p/2)|\nabla(F_r-W)|^2$, we find that the second term on the right side of \eqref{eq:dens-derivative-identity} is at most
\[
\frac{p-1}{2p}J(t)+\frac{p(p-1)}2\int\psi\,\Gamma(F_r-W)\,d\mu .
\]
On $S_N$, we have $F_r-W=\sum_i(r_i-k_i(\gamma t)+a)x_i^2-aN$, and the constant $aN$ has zero gradient.
The tangential gradient of a function on $\R^N$ is the projection of its Euclidean gradient, so that $\Gamma(F_r-W)\le4\sum_i(r_i-k_i(\gamma t)+a)^2x_i^2$.
Similarly, $W'=\sum_i(k_i'(\gamma t)-b)x_i^2+bN$ on $S_N$, and the constant $bN$ cancels in the difference of the two integrals in \eqref{eq:dens-derivative-identity}.
We conclude that
\[
\int\psi\,W'\,d\mu-\int W'\,d\mu\le\int\psi\sum_i|k_i'(\gamma t)-b|\,x_i^2\,d\mu+\int\sum_i|k_i'(\gamma t)-b|\,x_i^2\,d\mu .
\]
Inserting the last three bounds into \eqref{eq:dens-derivative-identity}, we obtain \eqref{cl3-renyi-differential-prefix}.
\end{proof}

\section{Density bounds on the time interval \texorpdfstring{$[0,S]$}{[0,S]}}\label{sec:dens-main}

We now combine the previous lemmas.

\begin{theorem}[{Density bounds on the time interval $[0,S]$}]\label{thm:dens-main}
Let $S>0$ and $C_*\ge1$, and suppose that $r$ satisfies \textup{(H1)}--\textup{(H5)} on $[0,S]$ with constant $C_*$.
Fix $p>1$, and let $\gamma=\min\{1,(4pC_*)^{-1}\}$.
There exists $C=C(p,C_*)$ such that, for every $t\in[0,S]$,
\begin{gather}
\log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\mu_{\gamma t}}\Big)^p\,d\mu_{\gamma t}\le CN(1+t)^{-3/2},\label{eq:dens-main-reference}\\
\log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\pi_r}\Big)^p\,d\pi_r\le CN(1+t)^{-3/2}.\label{eq:dens-main-gibbs}
\end{gather}
Further, the function $t\mapsto\int(d(\nu_NP^r_t)/d\pi_r)^p\,d\pi_r$ is nonincreasing on $[0,\infty)$, and for every $t\ge0$,
\begin{equation}\label{eq:dens-main-entropy}
\KL(\nu_NP^r_t\Vert\pi_r)\le\frac1{p-1}\log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\pi_r}\Big)^p\,d\pi_r .
\end{equation}
\end{theorem}

\begin{proof}
We first prove \eqref{eq:dens-main-reference}.
We use \cref{lem:dens-derivative} for the comparison laws and verify the three hypotheses of \cref{cg-renyi-differential-inequality}: the differential inequality \eqref{eq:dens-cert-derivative} with a reference term of order $N(1+\gamma t)^{-5/2}$, the exponential moment bound \eqref{eq:dens-cert-mgf}, and the logarithmic Sobolev inequality \eqref{eq:dens-cert-lsi}.
Then we change the reference measure from $\mu_{\gamma t}$ to $\pi_r$, and finally we treat later times.
The parameters are chosen in the following order: $p$ and $C_*$ are given, then $\gamma=\min\{1,(4pC_*)^{-1}\}$, then the constants of \cref{lem:dens-coefficients,lem:dens-tilt,lem:dens-mean-level}, which depend only on $C_*$ and on a constant $\Lambda=\Lambda(p,C_*)$ specified below.
In this proof, $C$ denotes a constant depending only on $p$ and $C_*$, and none of the constants depends on $N$, $S$ or $t$.
The proof is deterministic.

\paragraph{Step 1: the derivative.}
For $s\in[0,S]$, let $k_i(s)=(2v_i(s))^{-1}$ be the half-precisions of the comparison law $\mu_s$.
The map $s\mapsto k(s)$ is smooth, with values in $(0,\infty)^N$, and $k_i(0)=1/2$.
The measure $\mu^k_s$ of \cref{lem:dens-derivative} is then $\mu_s$.
Note that $\gamma t\le S$ for $t\le S$, since $\gamma\le1$.
We apply \cref{lem:dens-derivative} with $t_1=S$, $a=\ell(\gamma t)/4$, and $b=\ell'(\gamma t)/4$.
Then $r_i-k_i(\gamma t)+a=\zeta_i(\gamma t)$ and $k_i'(\gamma t)-b=-\zeta_i'(\gamma t)$, by \eqref{eq:dens-zeta}.
With $f_t$, $\mathcal R(t)$, $\psi_t$, and $J(t)$ as in \cref{lem:dens-derivative}, the inequality \eqref{cl3-renyi-differential-prefix} reads
\[
\mathcal R'(t)\le-\frac{p-1}{2p}J(t)+(p-1)\int U_t\,\psi_t\,d\mu_{\gamma t}+(p-1)\gamma\sum_i|\zeta_i'(\gamma t)|\int x_i^2\,d\mu_{\gamma t},
\]
where $U_t(x)=\sum_i\big(2p\zeta_i(\gamma t)^2+\gamma|\zeta_i'(\gamma t)|\big)x_i^2$.

\paragraph{Step 2: the reference term.}
Let $s=\gamma t$.
Using \eqref{cg-reference-comparability} and \textup{(H4)}, we find three distinct indices $i$ with $v_i(s)\ge\delta\max_jv_j(s)$, where $\delta\in(0,1]$ depends only on $C_*$.
Since $\sum_iv_i(s)=N$, we have $\int x_i^2\,d\mu_s\le Cv_i(s)$ for every $i$ by \cref{lem:dens-mean-level}.
By \eqref{eq:dens-coefficient-sums} and $\gamma\le1$, we have
\begin{equation}\label{cl3-renyi-reference-forcing}
\gamma\sum_i|\zeta_i'(s)|\int x_i^2\,d\mu_s\le C\sum_iv_i(s)|\zeta_i'(s)|\le C_3N(1+\gamma t)^{-5/2},
\end{equation}
for a constant $C_3=C_3(C_*)$.
Thus \eqref{eq:dens-cert-derivative} holds with this $C_3$.

\paragraph{Step 3: exponential moments of the forcing.}
Next, \eqref{cg-reference-e}, \eqref{cg-reference-derivative}, and $\gamma\le1$ imply that the coefficients of the quadratic function $pC_*(1+s)U_t$ are at most
\[
pC_*(1+s)\Big(\frac{2pC_\zeta^2}{(1+s)^2(1+r_is)^2}+\frac{C_\zeta}{(1+s)^2(1+r_is)}\Big)\le\frac{\Lambda}{(1+s)(1+r_is)},
\]
where $C_\zeta=C_\zeta(C_*)$ is the constant of \cref{lem:dens-coefficients} and $\Lambda=pC_*C_\zeta(2pC_\zeta+1)$.
By \cref{lem:dens-tilt}, there exists $C_2=C_2(p,C_*)$ such that
\[
\log\int\exp\big(pC_*(1+\gamma t)U_t\big)\,d\mu_{\gamma t}\le C_2N(1+\gamma t)^{-3/2} .
\]
This is \eqref{eq:dens-cert-mgf} with $C_1=C_*$.

\paragraph{Step 4: the logarithmic Sobolev inequality.}
The function $\psi_t$ is positive and of class $C^2$, and $\Gamma(\sqrt{\psi_t})=\psi_t\Gamma(\log\psi_t)/4$.
Applying the $C^1$ extension of \textup{(H5)} at time $\gamma t$ with $f=\sqrt{\psi_t}$, we have $\Ent_{\mu_{\gamma t}}(\psi_t)\le C_*(1+\gamma t)J(t)/4$, which is \eqref{eq:dens-cert-lsi} with $C_1=C_*$.
All the integrals involved are finite, because their integrands are continuous on the compact manifold $S_N$.

\paragraph{Step 5: conclusion for the comparison law.}
We conclude from Steps~1--4 that \cref{cg-renyi-differential-inequality} applies with $\mu_t$ there equal to $\mu_{\gamma t}$, $C_1=C_*$, and the constants $C_2,C_3$ above.
Recall that $\mathcal R(0)=0$ by \cref{lem:dens-derivative}.
We conclude that $\mathcal R(t)\le CN(1+\gamma t)^{-3/2}$.
Since $\gamma\le1$, we have $1+\gamma t\ge\gamma(1+t)$, and \eqref{eq:dens-main-reference} follows with a constant $C\gamma^{-3/2}$.

\paragraph{Step 6: change of reference measure.}
Let $\gamma_2=\min\{1,(8pC_*)^{-1}\}$, which is the value of $\gamma$ for the exponent $2p$, and let $s=\gamma_2t$.
We have $d\mu_s/d\pi_r=e^Q/\pi_r(e^Q)$ with $Q(x)=\sum_i\zeta_i(s)x_i^2$, by \eqref{eq:dens-ratio}.
Also $d\pi_r/d\mu_s=e^{-Q}/\mu_s(e^{-Q})$, which gives $\pi_r(e^Q)=1/\mu_s(e^{-Q})$, and hence
\begin{equation}\label{cl3-renyi-transfer-normalizer}
\theta:=\frac{d\mu_s}{d\pi_r}=\Big(\int e^{-Q}\,d\mu_s\Big)e^{Q} .
\end{equation}
Let $f=d(\nu_NP^r_t)/d\mu_s$, so that $d(\nu_NP^r_t)/d\pi_r=f\theta$.
By the Cauchy--Schwarz inequality under $\mu_s$,
\[
\int(f\theta)^p\,d\pi_r=\int f^p\theta^{p-1}\,d\mu_s
\le\Big(\int f^{2p}\,d\mu_s\Big)^{1/2}\Big(\int\theta^{2(p-1)}\,d\mu_s\Big)^{1/2} .
\]
Using \eqref{eq:dens-main-reference} with the exponent $2p$, we see that the logarithm of the first integral on the right side is at most $CN(1+t)^{-3/2}$.
For the second integral, we have
\[
\int\theta^{2(p-1)}\,d\mu_s=\Big(\int e^{-Q}\,d\mu_s\Big)^{2(p-1)}\int e^{2(p-1)Q}\,d\mu_s .
\]
By \eqref{cg-reference-e}, the coefficients of $-Q$ and $2(p-1)Q$ satisfy \eqref{eq:dens-tilt-envelope} with $\Lambda=\max\{1,2(p-1)\}C_\zeta$.
Then \cref{lem:dens-tilt} implies that the logarithm of the second integral is at most $CN(1+\gamma_2t)^{-3/2}\le C\gamma_2^{-3/2}N(1+t)^{-3/2}$.
This proves \eqref{eq:dens-main-gibbs}.

\paragraph{Step 7: later times and relative entropy.}
Let $g_t=d(\nu_NP^r_t)/d\pi_r$.
As in the proof of \cref{lem:dens-derivative}, the reversibility of $\pi_r$ gives $g_{t+u}=P^r_ug_t$ for $t,u\ge0$.
Since $P^r_u$ is a Markov operator, we have $(P^r_ug_t)^p\le P^r_u(g_t^p)$ by Jensen's inequality, and the invariance of $\pi_r$ gives
\[
\int g_{t+u}^p\,d\pi_r\le\int P^r_u(g_t^p)\,d\pi_r=\int g_t^p\,d\pi_r .
\]
Thus $t\mapsto\int g_t^p\,d\pi_r$ is nonincreasing.
Finally, \eqref{eq:dens-main-entropy} is \eqref{eq:dens-kl-renyi}.
\end{proof}

\begin{remark}[The comparison law at the earlier time $\gamma t$]\label{rem:dens-gamma}
The comparison law is run at the slower speed $\gamma$ for the following reason.
In \eqref{cl3-renyi-closed-scalar}, the dissipation rate $(pC_*(1+\gamma t))^{-1}$ comes from the logarithmic Sobolev constant of $\mu_{\gamma t}$, while the forcing decays like $(1+\gamma t)^{-5/2}$.
In the variable $\tau=\gamma t$, the divergence therefore relaxes at rate $a/(1+\tau)$ with $a=(pC_*\gamma)^{-1}$, and integrating the forcing produces the decay $(1+\tau)^{-3/2}$ only when $a>3/2$.
Taking $\gamma$ small makes $a$ large, at the cost of comparing the time-$t$ law with the comparison law at the earlier time $\gamma t$.
\end{remark}

In \cref{sec:cold-meso}, \cref{thm:dens-main} is applied below the critical temperature, on a time interval of length of order $\theta_N^{-2}$. There the counting bound \textup{(H3)} follows from \cref{lem:dens-count} with $k_0=1$, the extra term being absorbed by the second part of \textup{(H1)}.

\section{The critical window}\label{sec:dens-critical}\label{cl3-density-section}

We now specialize \cref{thm:dens-main} to the critical window.
Recall from \cref{def:spec-critical-conditions} that a nondecreasing sequence $r_1\le\dots\le r_N$ satisfies the critical spectral conditions with constants $(c_0,C_0,\delta)$ if \eqref{cg-weyl-hypotheses} holds: $\sum_i(2r_i)^{-1}=N$, $c_0(i/N)^{2/3}\le r_i\le C_0(i/N)^{2/3}$ for all $i$, and $r_2-r_1\ge\delta N^{-2/3}$.
Some of the results below use only the first two of these conditions, without the ordering and the lower bound on $r_2-r_1$; we then say that $r$ satisfies the first two conditions in \eqref{cg-weyl-hypotheses} with constants $(c_0,C_0)$.
The ordering and the lower bound on $r_2-r_1$ enter only through \cref{cg-critical-lsi}.
For the model with $c_N=1+\sigma N^{-1/3}$, by \cref{cg-canonical-weyl}, the half-precisions satisfy the critical spectral conditions at every fixed confidence (in the sense of \cref{sec:model-quantifiers}), simultaneously for all $\sigma$ in a fixed compact interval, with constants depending only on the confidence and on the interval.
All results below are deterministic.
In their statements, a constant or a threshold $N_0$ for the dimension depends only on $(c_0,C_0,\delta)$ and on the parameters written as its subscripts or arguments; it does not depend on $N$, on the time, or on $\sigma$.
The results therefore apply on the events of \cref{cg-canonical-weyl}, uniformly in $\sigma$ in a compact interval.
\Cref{tab:dens-parameters} lists the parameters in the order in which they are fixed.

\begin{table}[ht]
\centering
\caption{Parameters in the critical window, in the order in which they are fixed.}\label{tab:dens-parameters}
\begin{tabular}{@{}l>{\raggedright\arraybackslash}p{4.9cm}>{\raggedright\arraybackslash}p{5.6cm}@{}}
\toprule
Parameter & Role & Depends on\\
\midrule
$c_0,C_0,\delta$ & constants in \eqref{cg-weyl-hypotheses} & for the GOE: the confidence and the range of $\sigma$ (\cref{cg-canonical-weyl})\\
$K$ & time range $t\le KN^{2/3}$ & fixed\\
$p$ & order of the moment & fixed\\
$C_{*,K}$ & constant in \textup{(H1)}--\textup{(H5)} & $c_0,C_0,\delta,K$\\
$\gamma_{p,K}$ & speed of the comparison law & $p,C_{*,K}$\\
$C_{p,K}$ & constant in the conclusions & $p,K,c_0,C_0,\delta$\\
$N_0(K)$ & threshold for the dimension & $c_0,C_0,\delta,K$; the bounds hold uniformly in $N\ge N_0(K)$\\
\bottomrule
\end{tabular}
\end{table}

\begin{lemma}[The hypotheses in the critical window]\label{lem:dens-critical-hyp}
Let $K>0$ and $S=KN^{2/3}$.
\begin{enumerate}
\item If $r$ satisfies the first two conditions in \eqref{cg-weyl-hypotheses} with constants $(c_0,C_0)$, then $r$ satisfies \textup{(H1)}--\textup{(H4)} on $[0,S]$ with a constant $C_*$ depending only on $c_0$, $C_0$, and $K$.
\item If $r$ satisfies the critical spectral conditions with constants $(c_0,C_0,\delta)$, then there exists $N_0(K)$ such that for $N\ge N_0(K)$, $r$ satisfies \textup{(H1)}--\textup{(H5)} on $[0,S]$ with a constant $C_{*,K}$.
\end{enumerate}
\end{lemma}

\begin{proof}
For \textup{(H1)}, we have $r_i\le C_0$, and $N(1+S)^{-3/2}\ge N((1+K)N^{2/3})^{-3/2}=(1+K)^{-3/2}$.
The heat bounds \textup{(H2)} are \cref{cd-heat-profiles}, whose constants depend only on $c_0$, $C_0$, and $K$.
The counting bound \textup{(H3)} is \cref{lem:dens-count} with $k_0=0$ and $j=2$.
For \textup{(H4)}, the indices $i=1,2,3$ satisfy $r_iS\le C_0(3/N)^{2/3}KN^{2/3}=3^{2/3}C_0K$.
This proves the first part.
For the second part, \textup{(H5)} is the bound $C_{\mathrm{LS}}(\mu_t)\le C_K(1+t)$ for $0\le t\le KN^{2/3}$ in the second part of \cref{cg-critical-lsi}, which holds for $N\ge N_0(K)$; both $N_0(K)$ and $C_K$ depend only on $c_0$, $C_0$, $\delta$, and $K$.
\end{proof}

The next corollary restates \cref{lem:dens-coefficients,lem:dens-tilt} in the critical window.

\begin{corollary}[The comparison law in the critical window]\label{cg-reference-cancellations}\label{cl3-shifted-coarea-mgf}
Suppose that $r$ satisfies the first two conditions in \eqref{cg-weyl-hypotheses} with constants $(c_0,C_0)$, and fix $K>0$ and $\Lambda\ge0$.
Then \eqref{cg-reference-comparability}, \eqref{cg-reference-e}, \eqref{cg-reference-derivative}, and \eqref{eq:dens-coefficient-sums} hold for $0\le t\le KN^{2/3}$, with a constant depending only on $c_0$, $C_0$, and $K$.
Further, for $0\le s\le KN^{2/3}$ and all real $\omega_1,\dots,\omega_N$ with $|\omega_i|\le\Lambda(1+s)^{-1}(1+r_is)^{-1}$,
\begin{equation}\label{cl3-shifted-coarea-mgf-bound}
\log\int_{S_N}\exp\Big(\sum_i\omega_ix_i^2\Big)\,\mu_s(dx)\le C_{\Lambda,K}N(1+s)^{-3/2},
\end{equation}
where $C_{\Lambda,K}$ depends only on $\Lambda$, $K$, $c_0$, and $C_0$.
\end{corollary}

\begin{proof}
By the first part of \cref{lem:dens-critical-hyp}, $r$ satisfies \textup{(H1)}--\textup{(H4)} on $[0,KN^{2/3}]$, with a constant depending only on $c_0$, $C_0$, and $K$.
The corollary therefore follows from \cref{lem:dens-coefficients,lem:dens-tilt}.
\end{proof}

\begin{proposition}[R\'enyi bounds relative to the comparison law]\label{cl3-actual-source-renyi}
Suppose that $r$ satisfies the critical spectral conditions with constants $(c_0,C_0,\delta)$, and fix $p>1$ and $0<K<\infty$.
There exist $N_0(K)$, $\gamma_{p,K}\in(0,1]$, and $C_{p,K}$ such that, with $\gamma=\gamma_{p,K}$, for $N\ge N_0(K)$ and $0\le t\le KN^{2/3}$,
\begin{equation}\label{cl3-reference-renyi-bound}
\log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\mu_{\gamma t}}\Big)^p\,d\mu_{\gamma t}\le C_{p,K}N(1+t)^{-3/2} .
\end{equation}
\end{proposition}

\begin{proof}
By the second part of \cref{lem:dens-critical-hyp}, for $N\ge N_0(K)$, $r$ satisfies \textup{(H1)}--\textup{(H5)} on $[0,KN^{2/3}]$ with the constant $C_{*,K}$.
The bound \eqref{cl3-reference-renyi-bound} is then \eqref{eq:dens-main-reference}, with $\gamma_{p,K}=\min\{1,(4pC_{*,K})^{-1}\}$.
\end{proof}

\begin{theorem}[Density moments in the critical window]\label{cl3-actual-source-density}
Suppose that $r$ satisfies the critical spectral conditions with constants $(c_0,C_0,\delta)$, and fix $p>1$ and $0<K<\infty$.
There exist $N_0(K)$ and $C_{p,K}$ such that for $N\ge N_0(K)$,
\begin{equation}\label{cl3-stationary-renyi-profile}
\log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\pi_r}\Big)^p\,d\pi_r\le C_{p,K}N(1+t)^{-3/2},\qquad0\le t\le KN^{2/3} .
\end{equation}
In particular, for every fixed $s_0>0$, there exists $C_{p,s_0}$ such that for $N\ge N_0(s_0)$,
\[
\sup_{t\ge s_0N^{2/3}}\Big\|\frac{d(\nu_NP^r_t)}{d\pi_r}\Big\|_{L^p(\pi_r)}\le C_{p,s_0}.
\]
\end{theorem}

\begin{proof}
The bound \eqref{cl3-stationary-renyi-profile} follows from the second part of \cref{lem:dens-critical-hyp} and \eqref{eq:dens-main-gibbs}.
For the second claim, we apply \eqref{cl3-stationary-renyi-profile} with $K=s_0$ at $t=s_0N^{2/3}$.
Since $N(1+s_0N^{2/3})^{-3/2}\le s_0^{-3/2}$, the logarithm of the $p$-th moment at this time is at most $C_{p,K}s_0^{-3/2}$ with $K=s_0$, and we may take $C_{p,s_0}=\exp(C_{p,K}s_0^{-3/2}/p)$.
The $p$-th moment is nonincreasing in $t$ by \cref{thm:dens-main}, and this gives the bound for all $t\ge s_0N^{2/3}$.
\end{proof}

The applications in \cref{sec:crit-profile} use \cref{cl3-actual-source-density} with $p=2$ and $p=4$; by the triangle inequality, it also bounds $\|d(\nu_NP^r_t)/d\pi_r-1\|_{L^p(\pi_r)}$ by $C_{p,s_0}+1$ for $t\ge s_0N^{2/3}$.

\begin{proposition}[Exponential relaxation in the critical window]\label{cg-source-l2-decay}
Fix $0<c_0\le C_0$ and $\delta>0$.
For every $a>0$, there exist $C_a,\kappa>0$ and $N_0$ such that, for every $N\ge N_0$ and every $r$ satisfying the critical spectral conditions with constants $(c_0,C_0,\delta)$,
\begin{equation}\label{cg-l2-upper}
\big\|\nu_NP^r_{sN^{2/3}}-\pi_r\big\|_{\TV}
\le C_a e^{-\kappa(s-a)},\qquad s\ge a.
\end{equation}
The constants and the threshold depend only on $c_0,C_0,\delta$, and $a$.
\end{proposition}

\begin{proof}
Write $b=N^{2/3}$ and $g_t=d(\nu_NP^r_t)/d\pi_r$.
By \cref{cl3-actual-source-density} with $p=4$, there exist $M_a>0$ and a dimension threshold such that $\|g_{ab/2}\|_{L^4(\pi_r)}\le M_a$.
Since $\pi_r$ is a probability measure, this also bounds $\|g_{ab/2}\|_{L^2(\pi_r)}$.
By \cref{cg-critical-lsi}, after increasing the threshold, $\pi_r$ has Poincar\'e constant at most $K_{\mathrm P}b$, where $K_{\mathrm P}>0$ depends only on $c_0,C_0,\delta$.

The density satisfies $g_{t+u}=P^r_ug_t$ and $\int g_t\,d\pi_r=1$.
At positive times it has the regularity used in the proof of \cref{lem:dens-derivative}.
Thus \eqref{eq:model-dirichlet} and the Poincar\'e inequality give
\[
\frac{d}{dt}\|g_t-1\|_{L^2(\pi_r)}^2
=-2\int\Gamma(g_t)\,d\pi_r
\le-\frac{2}{K_{\mathrm P}b}\|g_t-1\|_{L^2(\pi_r)}^2.
\]
Integrating from $ab/2$ to $sb$, where $s\ge a$, yields
\[
\|g_{sb}-1\|_{L^2(\pi_r)}
\le e^{-(s-a/2)/K_{\mathrm P}}\|g_{ab/2}-1\|_{L^2(\pi_r)}
\le (M_a+1)e^{-(s-a/2)/K_{\mathrm P}}.
\]
Finally, \cref{lem:model-tv} and the Cauchy--Schwarz inequality imply
\[
\big\|\nu_NP^r_{sb}-\pi_r\big\|_{\TV}
=\frac12\|g_{sb}-1\|_{L^1(\pi_r)}
\le\frac{M_a+1}{2}e^{-(s-a)/K_{\mathrm P}}.
\]
This proves \eqref{cg-l2-upper} with $C_a=(M_a+1)/2$ and $\kappa=K_{\mathrm P}^{-1}$.
\end{proof}