\chapter{Functional inequalities on the sphere}\label{sec:inequalities} In this section, we prove the functional inequalities used in the rest of the paper. They concern the Langevin generator of a quadratic potential on $\Sph$ and its Gibbs measure, which is a Bingham distribution. We note that all statements of this section are deterministic. They hold for a fixed matrix, or for a fixed sequence of precisions, that satisfies explicit hypotheses. In the applications, these hypotheses are verified on events of the disorder, following the convention of \cref{sec:model-quantifiers}. In \cref{sec:ineq-setting}, we fix the conventions and collect standard tools. In \cref{sec:ineq-calculus}, we write the Bochner formula for the generator in ambient coordinates (\cref{sgl-ambient-bochner}). With a weight built from the equilibrium covariance, we then prove a weighted gradient estimate for the semigroup and a lower bound on the spectral gap (\cref{sgl-weighted-square,sgl-semigroup-gradient}). These results are used in \cref{sec:comparison,sec:process}. \Cref{sec:ineq-many} contains a lower bound on the spectral gap that is sharp when many precisions are comparable to the smallest one (\cref{ht-many-soft-sharp-gaps}), and the Poincar\'e inequality that follows from it (\cref{cg-many-soft-pi}). The first is used in \cref{sec:comparison}, and the second in \cref{sec:density,sec:cold-meso,sec:crit-profile}. In \cref{sec:ineq-lsi}, we extend functions from the sphere to Gaussian space and obtain logarithmic Sobolev inequalities with an additive error term (\cref{cg-canonical-tube-entropy}). We also bound the density of a weighted sum of squared Gaussian variables (\cref{cg-mean-level-density}). Together with a Poincar\'e inequality, these results give logarithmic Sobolev inequalities (\cref{cg-three-soft-lsi}). \Cref{sec:ineq-few} treats the case where only a few precisions are close to the smallest one (\cref{cg-folded-pi,cg-full-sphere-bridge}), which is the situation in the critical window. In \cref{sec:ineq-critical}, we combine these results. We bound the Poincar\'e and logarithmic Sobolev constants of the Gibbs measure by $CN^{2/3}$ in the critical window, and those of the Bingham comparison law at time $t\le KN^{2/3}$ by $C_K(1+t)$ (\cref{cg-critical-lsi}). This is used in \cref{sec:density,sec:crit-profile}. We use the definitions of \cref{sec:model}, the conditioned Gaussian laws $\mu_v$ of \cref{def:cond-finite}, the hypotheses \eqref{cg-weyl-hypotheses} on the critical half-precisions, and the heat estimates of \cref{cd-heat-profiles}. The analytic inputs are classical; we recall them in \cref{lem:ineq-tools}. They are the Bochner formula and the $\Gamma_2$ calculus of Bakry and \'Emery \cite{BakryEmery1985,BakryGentilLedoux2014}, the Gaussian logarithmic Sobolev inequality \cite{Gross1975}, Rothaus's lemma \cite{Rothaus1985}, the Brascamp--Lieb inequality \cite{BrascampLieb1976} in its version on convex domains \cite{KolesnikovMilman2017}, and the perturbation argument of Holley and Stroock \cite[Lemma~5.1, pp.~1183--1184]{HolleyStroock1987}. \section{Conventions and standard tools}\label{sec:ineq-setting} \paragraph{Dirichlet forms.} Throughout this section, $N\ge2$. For a function $f$ defined near $\Sph$, we write $\nabla_Sf$ for its gradient along $\Sph$ and set $\Gamma(f,g)=\langle\nabla_Sf,\nabla_Sg\rangle$ and $\Gamma(f)=\Gamma(f,f)=|\nabla_Sf|^2$. For a probability measure $\mu$ on $\Sph$ with a smooth positive density with respect to $\nu_N$, we write \[ \cE_\mu(f)=\int\Gamma(f)\,d\mu \] for the spherical Dirichlet form, defined on the closure $H^1(\Sph)$ of the smooth functions in the norm $(\mu(f^2)+\cE_\mu(f))^{1/2}$, and $C_{\mathrm P}(\mu)$ and $C_{\mathrm{LS}}(\mu)$ for the smallest constants such that the Poincar\'e and logarithmic Sobolev inequalities \eqref{eq:model-fi} hold. Recall from \cref{sec:model} that every such inequality in this paper refers to this spherical form, which is the Dirichlet form of the Langevin generator with noise coefficient $\sqrt2$. Ambient Euclidean and Gaussian forms appear only inside proofs, where they are named. The inequalities \eqref{eq:model-fi} for smooth $f$ extend to $H^1(\Sph)$: if $f_n\to f$ in $H^1(\Sph)$ and almost everywhere, then $\Var_\mu(f_n)\to\Var_\mu(f)$ and $\cE_\mu(f_n)\to\cE_\mu(f)$, and Fatou's lemma, applied to $f_n^2\log f_n^2+e^{-1}\ge0$, gives $\Ent_\mu(f^2)\le\liminf_n\Ent_\mu(f_n^2)$. We use this density argument without further comment. \paragraph{Quadratic potentials.} Let $A$ be a real symmetric $N\times N$ matrix with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$, and recall the generator $L_A$, the semigroup $P_t^A$ and the Gibbs measure $\pi_A$ from \eqref{eq:model-general}. In the notation of this section, the integration by parts formula \eqref{eq:model-dirichlet} reads \begin{equation}\label{sgl-euclidean-ibp} \int f\,(-L_Ag)\,d\pi_A=\int\Gamma(f,g)\,d\pi_A \end{equation} for smooth $f$ and $g$. By elliptic spectral theory on the compact connected weighted manifold $\Sph$ \cite[Section~2, p.~30]{Setti1998}, the closure of $-L_A$ is self-adjoint and nonnegative on $L^2(\pi_A)$, its spectrum is discrete, its eigenfunctions are smooth, and its kernel consists of the constant functions. In particular, the spectral gap $\lambda_{\mathrm{gap}}(L_A)$ of \cref{sec:model} is the smallest $\lambda>0$ for which there exists a nonconstant smooth $f$ with $L_Af=-\lambda f$, and $C_{\mathrm P}(\pi_A)=1/\lambda_{\mathrm{gap}}(L_A)$. We recall from \cref{def:model-multiplier} the Lagrange multiplier $z>\alpha_1$, the precisions $p_i=z-\alpha_i$, the variances $q_i=1/p_i$, which satisfy $\sum_iq_i=N$, the smallest and largest precisions $d=p_1$ and $s=p_N$, the ratio $\kappa=s/d$, and the half-precisions $r_i=p_i/2$. We also write $Q=(zI-A)^{-1}$. Unless stated otherwise, we work in an orthonormal eigenbasis of $A$, so that $A=\diag(\alpha_1,\dots,\alpha_N)$, $Q=\diag(q_1,\dots,q_N)$, and $x_1,\dots,x_N$ are the eigencoordinates. Recall from \cref{prop:cond-bingham} that $\pi_A$ is the conditioned Gaussian law $\mu_q$ of \cref{def:cond-finite}; on $\Sph$, it is proportional to $\exp(-\frac12\sum_ip_ix_i^2)\,\nu_N=\exp(-\sum_ir_ix_i^2)\,\nu_N$. By \eqref{eq:model-shift}, adding a multiple of the identity to $A$ changes neither $L_A$ nor $\pi_A$. It shifts $z$ by the same amount and leaves the precisions and variances unchanged. In particular, $L_A=L_{-P}$ and $\pi_A=\pi_{-P}$ for $P=\diag(p_1,\dots,p_N)$. Conversely, for every $q\in(0,\infty)^N$ with $\sum_iq_i=N$, the law $\mu_q$ equals $\pi_A$ for $A=-\diag(q_1^{-1},\dots,q_N^{-1})$, whose Lagrange multiplier is $z=0$ and whose variances are $q_1,\dots,q_N$. We use this identification without further comment: a statement about $\pi_A$ in terms of its precisions is a statement about every $\mu_q$. As in \cref{def:model-multiplier}, the variances $q_i$ are the parameters of the Gaussian vector before conditioning; they need not equal the second moments of $\pi_A$. \paragraph{Reflections.} For $1\le j\le N$, let $R_j$ be the reflection that changes the sign of $x_j$. When $A$ is diagonal, $R_j$ preserves $\nu_N$, $\pi_A$, $L_A$ and $\Gamma$ (see \cref{lem:model-symmetry}). Let $\mu$ be a probability measure on $\Sph$ with a smooth positive density that is invariant under $R_j$. We write $H_j^+=\{x\in\Sph:x_j\ge0\}$ and $\mu_j^+=\mu(\cdot\mid H_j^+)$, and we equip $\mu_j^+$ with the reflecting form \[ \cE_{\mu_j^+}(u)=\int_{H_j^+}\Gamma(u)\,d\mu_j^+,\qquad u\in H^1(H_j^+). \] No boundary condition is imposed on $u$. When $\mu$ is invariant under every $R_i$, we similarly equip the orthant $O=\{x\in\Sph:x_i\ge0\text{ for all }i\}$ with $\mu_O=\mu(\cdot\mid O)$ and the reflecting form. A function invariant under every $R_i$, that is, a function in the coordinate-even sector of \cref{def:model-sectors}, is determined by its restriction to $O$. We call $C_{\mathrm P}(\mu_O)$ and $C_{\mathrm{LS}}(\mu_O)$ the constants of the coordinate-even sector. When $j=1$ we write $\mu_+=\mu_1^+$. \begin{lemma}[Reflecting hemispheres and the coordinate-even sector]\label{lem:ineq-reflection} Let $\mu$ be invariant under $R_j$, and let $\alpha,\beta\ge0$. Suppose that \[ \Ent_\mu(f^2)\le\alpha\,\cE_\mu(f)+\beta\mu(f^2)\qquad\text{for all }f\in H^1(\Sph). \] Then the same inequality holds for $\mu_j^+$ and every $u\in H^1(H_j^+)$. Also, $C_{\mathrm P}(\mu_j^+)\le C_{\mathrm P}(\mu)$ and $C_{\mathrm{LS}}(\mu_j^+)\le C_{\mathrm{LS}}(\mu)$. If $\mu$ is invariant under every $R_i$, the same conclusions hold for $\mu_O$. \end{lemma} \begin{proof} Let $\phi_j(x)$ be $x$ with $x_j$ replaced by $|x_j|$. For $u\in H^1(H_j^+)$, the even extension $\tilde u=u\circ\phi_j$ belongs to $H^1(\Sph)$, and $\Gamma(\tilde u)=\Gamma(u)\circ\phi_j$ almost everywhere, because $R_j$ is an isometry of $\Sph$ whose fixed set is the smooth hypersurface $\{x_j=0\}$. We have $\int g\circ\phi_j\,d\mu=\int g\,d\mu_j^+$ for every measurable $g\ge0$ on $H_j^+$, because $\mu$ is $R_j$-invariant. Applying this to $g=u^2$, $u^2\log u^2$ and $\Gamma(u)$, we find that the variance, the entropy and the energy of $\tilde u$ under $\mu$ coincide with those of $u$ under $\mu_j^+$. This proves the claims for $\mu_j^+$. For $\mu_O$, use $\phi(x)=(|x_1|,\dots,|x_N|)$. \end{proof} \begin{lemma}[Standard tools]\label{lem:ineq-tools} The following hold. \begin{enumerate}[label=(\roman*)] \item\label{i:ineq-perturb} (Bounded perturbation.) Let $\mu$ and $\mu'=\rho\mu$ be probability measures on a finite-dimensional Euclidean space carrying an energy $\int\Gamma(u)\,d\mu$ with the same pointwise energy density $\Gamma(u)\ge0$ for both measures. If $\sup\rho/\inf\rho\le e^{\omega}$, then $C_{\mathrm P}(\mu')\le e^{\omega}C_{\mathrm P}(\mu)$. \item\label{i:ineq-rothaus} (Rothaus's lemma.) For every probability measure $\mu$ on $S_N$ with a smooth strictly positive density relative to $\nu_N$, every smooth real-valued $f$ on $S_N$, and every $a\in\R$, we have $\Ent_\mu(f^2)\le\Ent_\mu((f-a)^2)+2\mu((f-a)^2)$. If $\Ent_\mu(f^2)\le\alpha\cE_\mu(f)+\beta\mu(f^2)$ for all smooth $f$, with $\alpha,\beta\ge0$, then $C_{\mathrm{LS}}(\mu)\le\alpha+(\beta+2)C_{\mathrm P}(\mu)$. \item\label{i:ineq-ground} (Positive test functions.) For every smooth positive $w$ and every smooth $u$ on $\Sph$, \[ \int u^2\,\frac{-L_Aw}{w}\,d\pi_A\le\int\Gamma(u)\,d\pi_A . \] \item\label{i:ineq-gross} (Gaussian logarithmic Sobolev inequality.) If $\gamma$ is the standard Gaussian measure on $\R^m$, then $\Ent_\gamma(F^2)\le2\int|\nabla F|^2\,d\gamma$ for every smooth compactly supported $F$. \item\label{i:ineq-bl} (Brascamp--Lieb inequality on a convex domain.) Let $m\ge1$, let $\Omega\subset\R^m$ be a nonempty bounded convex open set with smooth boundary, let $\Omega'$ be an open set containing $\bar\Omega$, and let $U\in C^{1,1}(\Omega')$. Suppose that $\nabla^2U\succeq\Lambda$ almost everywhere on $\Omega'$, where $\Lambda$ is a continuous field of positive definite matrices on $\Omega'$. If $\mu$ is the probability measure on $\Omega$ proportional to $e^{-U}dy$, then \[ \Var_\mu(u)\le\int\langle\Lambda^{-1}\nabla u,\nabla u\rangle\,d\mu\qquad\text{for every }u\in C^1(\R^m). \] \item\label{i:ineq-sphere} (Spheres.) The uniform probability measure on the unit sphere $S^{k-1}\subset\R^k$, $k\ge2$, with its spherical gradient, has Poincar\'e constant $1/(k-1)$, and so does the uniform measure on the unit hemisphere $\{\omega\in S^{k-1}:\omega_1\ge0\}$ with its reflecting form. In particular, $C_{\mathrm P}(\nu_N)=N/(N-1)$. \end{enumerate} \end{lemma} \begin{proof} For \ref{i:ineq-perturb}, we have \[ \begin{aligned} \Var_{\mu'}(f)&\le\int(f-\mu f)^2\,d\mu'\le(\sup\rho)\Var_\mu(f)\\ &\le(\sup\rho)\,C_{\mathrm P}(\mu)\int\Gamma(f)\,d\mu\le\frac{\sup\rho}{\inf\rho}\,C_{\mathrm P}(\mu)\int\Gamma(f)\,d\mu'. \end{aligned} \] This is the Poincar\'e version of the perturbation argument of Holley and Stroock \cite[Lemma~5.1, pp.~1183--1184]{HolleyStroock1987}. The inequality in \ref{i:ineq-rothaus} is due to Rothaus \cite{Rothaus1985}; see also \cite[Lemma~4]{BartheRoberto2003}. Applying it with $a=\mu(f)$, and then the hypothesis to $f-\mu(f)$, we obtain \[ \Ent_\mu(f^2)\le\alpha\,\cE_\mu(f)+(\beta+2)\Var_\mu(f)\le\big(\alpha+(\beta+2)C_{\mathrm P}(\mu)\big)\cE_\mu(f). \] For a reflecting hemisphere or orthant obtained from a reflection-invariant smooth positive spherical law, the same Rothaus inequality follows by even extension and approximation in $H^1$, as in \cref{lem:ineq-reflection}. Consequently the tightening implication also holds for each of these reflecting forms, using its own Poincar\'e constant. For \ref{i:ineq-ground}, \eqref{sgl-euclidean-ibp} with $f=u^2/w$ and $g=w$ gives \[ \begin{aligned} \int u^2\,\frac{-L_Aw}{w}\,d\pi_A&=\int\Gamma\Big(\frac{u^2}w,w\Big)\,d\pi_A\\ &=\int\big(2u\,\Gamma(u,\log w)-u^2\,\Gamma(\log w)\big)\,d\pi_A\le\int\Gamma(u)\,d\pi_A, \end{aligned} \] where in the last step we used $|\nabla_Su-u\nabla_S\log w|^2\ge0$. This elementary inequality is the basis of the Lyapunov-function method; see \cite[Section~4.6.1, (4.6.4)--(4.6.5), p.~202]{BakryGentilLedoux2014}. For the marginal generators below, the diffusion matrices are symmetric and positive definite on the open chart. The same proof then applies once the integration by parts produces no boundary term; we check this in each use. Item \ref{i:ineq-gross} is Gross's inequality \cite[Corollary~4.2, (3.4), p.~1073]{Gross1975}, multiplied by $2$. For \ref{i:ineq-bl}, we first suppose that $U$ is smooth on $\bar\Omega$ and that $\nabla^2U$ is positive definite there. Then the claim with $\Lambda=\nabla^2U$ is \cite[Theorem~1.2(1)]{KolesnikovMilman2017}, applied with $1/N=0$ to the weighted manifold $\bar\Omega$ with the Euclidean metric and the measure $e^{-U}dy$. Its boundary is locally convex because $\Omega$ is convex, and its generalized Ricci tensor is $\nabla^2U$. For $\Omega=\R^m$ this is the inequality of Brascamp and Lieb \cite{BrascampLieb1976}. In general, let $U_\varepsilon=U*\phi_\varepsilon$ for a standard mollifier $\phi_\varepsilon$, defined on $\bar\Omega$ for small $\varepsilon$. The gradient $\nabla U$ is locally Lipschitz, and $\nabla^2U_\varepsilon=(\nabla^2U)*\phi_\varepsilon$. Fix $\theta\in(0,1)$. There exists $\varepsilon_0>0$ such that $\Lambda(y-w)\succeq(1-\theta)\Lambda(y)$ for $y\in\bar\Omega$ and $|w|\le\varepsilon_0$, because $\Lambda$ is continuous and positive definite on a compact neighborhood of $\bar\Omega$. Then $\nabla^2U_\varepsilon\succeq(1-\theta)\Lambda$ on $\bar\Omega$ for $\varepsilon\le\varepsilon_0$, and the smooth case gives the claim for $e^{-U_\varepsilon}$ with the factor $(1-\theta)^{-1}$. Finally, $U_\varepsilon\to U$ uniformly on $\bar\Omega$, and we may let $\varepsilon\to0$ and then $\theta\to0$. For \ref{i:ineq-sphere}, the eigenvalues of the Laplacian of $S^{k-1}$ are $\ell(\ell+k-2)$, $\ell\ge0$, and the first positive one, $k-1$, has the linear functions as eigenfunctions. For $k\ge3$, this value also attains the Lichnerowicz bound \cite[Section~4.8, (4.8.1), p.~213]{BakryGentilLedoux2014}. The reflecting form of the hemisphere corresponds to the functions on $S^{k-1}$ that are even in $\omega_1$, as in \cref{lem:ineq-reflection}, and $\omega_2$ is such an eigenfunction. Scaling by $\sqrt N$ gives $C_{\mathrm P}(\nu_N)=N/(N-1)$. \end{proof} \section{Ambient calculus for the quadratic generator}\label{sec:ineq-calculus}\label{sgl-euclidean-section} The results of this section use only Euclidean derivatives, matrix algebra and \eqref{sgl-euclidean-ibp}. We fix a real symmetric matrix $A$, which we do not assume to be diagonal. For $x\in\Sph$, let \[ \Pi_x=I-\frac{xx^\T}N,\qquad e_A(x)=\frac{x^\T Ax}N,\qquad b_A(x)=Ax-\Big(e_A(x)+\frac{N-1}N\Big)x. \] For a smooth function $u$ defined near $\Sph$, we write $Du$ and $D^2u$ for its Euclidean gradient and Hessian. Then \begin{equation}\label{sgl-euclidean-generator} L_Au=\operatorname{tr}(\Pi_xD^2u)+b_A(x)\cdot Du,\qquad\Gamma(u,v)=(Du)^\T \Pi_xDv \end{equation} on $\Sph$. The second formula holds because $\Pi_xDu$ is the tangential part of $Du$. For the first, we note that its right side depends only on the restriction of $u$ to $\Sph$. Indeed, suppose $u=0$ on $\Sph$. Then $\Pi_xDu=0$. For a unit vector $e$ orthogonal to $x$, the curve $\gamma_e(r)=(x+re)/(1+r^2/N)^{1/2}$ stays on $\Sph$ and satisfies $\gamma_e'(0)=e$ and $\gamma_e''(0)=-x/N$. Differentiating $u(\gamma_e(r))=0$ twice at $r=0$ gives $e^\T D^2u\,e=x\cdot Du/N$, and summing over an orthonormal basis of $x^\perp$, we find $\operatorname{tr}(\Pi_xD^2u)=(N-1)x\cdot Du/N$. Also, $Du$ is parallel to $x$, which gives $b_A(x)\cdot Du=(x\cdot Du)(x^\T Ax-Ne_A(x)-(N-1))/N=-(N-1)x\cdot Du/N$. The two terms cancel, and the right side vanishes. It suffices to check \eqref{sgl-euclidean-generator} for an extension that is constant along rays. For such $u$, the identities $x\cdot Du=0$ and $x^\T D^2u\,x=0$ imply that $\operatorname{tr}(\Pi_xD^2u)=\Delta u=\Delta_Su$, and $b_A(x)\cdot Du=(Ax)^\T \Pi_xDu$ is the drift term of $L_A$. For a smooth positive function $w$ on $\Sph$, we define \[ \Gamma_2(f)=\tfrac12\big(L_A\Gamma(f)-2\Gamma(f,L_Af)\big),\qquad \Gamma_{w,2}(f)=\tfrac12\big(L_A(w\Gamma(f))-2w\,\Gamma(f,L_Af)\big). \] \begin{lemma}[Bochner formula in ambient coordinates]\label{sgl-ambient-bochner} Let $f$ be smooth on $\Sph$, and extend it to $\{\sqrt N/2<|x|<2\sqrt N\}$ by $f(x)=f(\sqrt Nx/|x|)$. On $\Sph$, let \[ g=Df,\qquad E=D^2f,\qquad \nabla_S^2f=\Pi_xE\Pi_x. \] Then $x\cdot g=0$, $\Gamma(f)=|g|^2$, and \begin{align} Ex&=-g,\qquad E=\nabla_S^2f-\frac{gx^\T+xg^\T}N,\label{sgl-block-decomposition}\\ \|E\|_{\mathrm{HS}}^2&=\|\nabla_S^2f\|_{\mathrm{HS}}^2+\frac{2|g|^2}N,\qquad \operatorname{tr}(\Pi_xE^2)=\|\nabla_S^2f\|_{\mathrm{HS}}^2+\frac{|g|^2}N.\label{sgl-block-norms} \end{align} Moreover, \begin{equation}\label{sgl-euclidean-bochner} \Gamma_2(f)=\|\nabla_S^2f\|_{\mathrm{HS}}^2+g^\T\Big[\Big(e_A+1-\frac2N\Big)I-A\Big]g, \end{equation} and for every smooth positive $w$ on $\Sph$, with $a=\nabla_S\log w$, \begin{equation}\label{sgl-euclidean-weighted-bochner} \Gamma_{w,2}(f)=w\|\nabla_S^2f\|_{\mathrm{HS}}^2+w\,g^\T\Big[\Big(e_A+1-\frac2N+\frac{L_Aw}{2w}\Big)I-A\Big]g+2w\,\nabla_S^2f(a,g). \end{equation} \end{lemma} Here $\nabla_S^2f(a,g)=a^\T(\nabla_S^2f)g$. The matrix $\nabla_S^2f$ represents the Hessian of $f$ on $\Sph$, since the extension is constant along rays, and \eqref{sgl-euclidean-bochner} is Bochner's formula for $L_A$, written in ambient coordinates; compare \cite[equations (1.38)--(1.41)]{ChafaiFathi2025}, with potential $V=-\frac12x^\T Ax$. For $A=0$, the identity \eqref{sgl-euclidean-bochner} implies that $\Gamma_2(f)\ge(1-2/N)\Gamma(f)$, the curvature bound $\mathrm{Ric}=(N-2)/N$ of $\Sph$. The semigroup interpolation argument based on $\Gamma_2$ is the method of Bakry and \'Emery \cite{BakryEmery1985}; see \cite[Corollary~3.3.19, p.~163]{BakryGentilLedoux2014}. The weighted expression used here is computed directly in \eqref{sgl-euclidean-weighted-bochner} and specialized to the covariance weight in \cref{sgl-weighted-square}. \begin{proof} Since the extension is constant along rays, Euler's identity implies that $x\cdot Df=0$. Differentiating it once, we obtain $g+Ex=0$, and multiplying by $x^\T$ yields $x^\T Ex=0$. We decompose $E$ with the orthogonal projections $\Pi_x$ and $xx^\T/N$. Using $Ex=-g$ and $\Pi_xg=g$, we find $\Pi_xE(xx^\T/N)=-gx^\T/N$. Its transpose is $(xx^\T/N)E\Pi_x=-xg^\T/N$, and $(xx^\T/N)E(xx^\T/N)=0$. This proves \eqref{sgl-block-decomposition}. The three matrices $\nabla_S^2f$, $gx^\T/N$ and $xg^\T/N$ are orthogonal for the Hilbert--Schmidt inner product, since $(\nabla_S^2f)x=0$ and $x\cdot g=0$, and the last two have squared norm $|g|^2|x|^2/N^2=|g|^2/N$. This proves the first identity in \eqref{sgl-block-norms}. For the second, we compute $\Pi_xE=\nabla_S^2f-gx^\T/N$ and \[ \Pi_xE^2=\Big(\nabla_S^2f-\frac{gx^\T}N\Big)\Big(\nabla_S^2f-\frac{gx^\T+xg^\T}N\Big) =(\nabla_S^2f)^2-\frac{(\nabla_S^2f)g\,x^\T}N+\frac{gg^\T}N, \] whose trace is $\|\nabla_S^2f\|_{\mathrm{HS}}^2+|g|^2/N$. Let $G=|Df|^2$ on the annulus. The function $G$ is homogeneous of degree $-2$, because the gradient of a function that is homogeneous of degree zero is homogeneous of degree $-1$. By Euler's identity, we have $x\cdot DG=-2G$ and $x^\T D^2G\,x=6G$. Also $DG=2Eg$ and $D^2G=2E^2+2\sum_jg_jD^2(\partial_jf)$; taking the trace, we find $\operatorname{tr}(D^2G)=2\|E\|_{\mathrm{HS}}^2+2g\cdot D\Delta f$. By \eqref{sgl-euclidean-generator}, using $x^\T Eg=-|g|^2$, we find \[ L_AG=2\|E\|_{\mathrm{HS}}^2+2g\cdot D\Delta f-\frac{6G}N+2g^\T EAx+2\Big(e_A+\frac{N-1}N\Big)G. \] By \eqref{sgl-euclidean-generator}, the function $\Delta f+Ax\cdot Df$ is an extension of $L_Af$, since $x\cdot Df=0$ and $x^\T Ex=0$. Its gradient is $D\Delta f+Ag+EAx$, and taking the inner product with $g$, we find \[ 2\Gamma(f,L_Af)=2g\cdot D\Delta f+2g^\T Ag+2g^\T EAx. \] Subtracting and using the first identity in \eqref{sgl-block-norms}, we have \[ \Gamma_2(f)=\|E\|_{\mathrm{HS}}^2+\Big(e_A+1-\frac4N\Big)|g|^2-g^\T Ag, \] which is \eqref{sgl-euclidean-bochner}. Finally, by the product rule $L_A(wG)=wL_AG+GL_Aw+2\Gamma(w,G)$, we have \[ \Gamma_{w,2}(f)=w\Gamma_2(f)+\tfrac12G\,L_Aw+\Gamma(w,G). \] The tangential part of $DG=2Eg$ is $2\Pi_xEg=2(\nabla_S^2f)g$ by \eqref{sgl-block-decomposition}. Since $a=\nabla_S\log w$, we have $\Gamma(w,G)=2w\,\nabla_S^2f(a,g)$, which proves \eqref{sgl-euclidean-weighted-bochner}. \end{proof} \begin{corollary}[Gradient of the norm of the gradient]\label{sgl-gradient-norm} In the notation of \cref{sgl-ambient-bochner}, for every $\varepsilon>0$ the function $u_\varepsilon=(\Gamma(f)+\varepsilon^2)^{1/2}$ satisfies \[ \Gamma(u_\varepsilon)=\frac{|(\nabla_S^2f)g|^2}{|g|^2+\varepsilon^2}\le\|\nabla_S^2f\|_{\mathrm{HS}}^2. \] \end{corollary} \begin{proof} As shown at the end of the previous proof, the tangential gradient of $\Gamma(f)=G$ is $2(\nabla_S^2f)g$. Using the chain rule, we find $\nabla_Su_\varepsilon=(\nabla_S^2f)g/u_\varepsilon$, and the claim follows from $|(\nabla_S^2f)g|\le\|\nabla_S^2f\|_{\mathrm{HS}}|g|$. \end{proof} We now use the equilibrium covariance $Q=(zI-A)^{-1}$ as a weight. Let \[ \varphi(x)=\frac{x^\T Qx}N,\qquad x\in\Sph. \] Note that $1/s\le\varphi\le1/d$ on $\Sph$, because the eigenvalues of $Q$ lie in $[1/s,1/d]$. \begin{lemma}[The weight $\varphi$ and a weighted curvature bound]\label{sgl-weighted-square} Let $a=\nabla_S\log\varphi$ and $J=(\kappa+\kappa^{-1}-2)/N$. Then \begin{equation}\label{sgl-F-cancellation} L_A\varphi=2(z-1-e_A)\varphi,\qquad |a|^2\le J, \end{equation} and, in the notation of \cref{sgl-ambient-bochner}, \begin{equation}\label{sgl-canonical-weighted-identity} \Gamma_{\varphi,2}(f)=\varphi\Big\{\|\nabla_S^2f\|_{\mathrm{HS}}^2+2\nabla_S^2f(a,g)+g^\T\Big[(zI-A)-\frac2NI\Big]g\Big\}. \end{equation} In particular, with $r_0=d-(\kappa+\kappa^{-1})/N$, \begin{equation}\label{sgl-canonical-curvature} \Gamma_{\varphi,2}(f)\ge r_0\,\varphi\,\Gamma(f). \end{equation} \end{lemma} The identity \eqref{sgl-canonical-weighted-identity} and the bound \eqref{sgl-canonical-curvature} hold for every symmetric $A$, also when $r_0\le0$. \begin{proof} By \eqref{sgl-euclidean-generator} with $D\varphi=2Qx/N$ and $D^2\varphi=2Q/N$, and using $\operatorname{tr}Q=N$, we have \[ \begin{aligned} L_A\varphi&=\frac2N\Big(\operatorname{tr}Q-\frac{x^\T Qx}N\Big)+\frac2N\Big(x^\T AQx-\Big(e_A+\frac{N-1}N\Big)x^\T Qx\Big)\\ &=2-2\varphi+\frac2Nx^\T AQx-2e_A\varphi. \end{aligned} \] Recall that $AQ=zQ-I$. Then $x^\T AQx=Nz\varphi-N$, and the first identity in \eqref{sgl-F-cancellation} follows. For the second, we work in an eigenbasis of $A$ and write $\omega_i=x_i^2/N$, so that $\omega_i\ge0$ and $\sum_i\omega_i=1$. We have \[ |a|^2=\frac{4}{N^2\varphi^2}\Big(|Qx|^2-N\varphi^2\Big)=\frac4N\bigg(\frac{\sum_i\omega_iq_i^2}{(\sum_i\omega_iq_i)^2}-1\bigg), \] where we used that the tangential part of $D\varphi$ is $2Qx/N-2\varphi x/N$. Let $\ell=1/s$ and $\bar u=1/d$. Then $q_i\in[\ell,\bar u]$, and therefore $q_i^2\le(\ell+\bar u)q_i-\ell\bar u$. With $m=\sum_i\omega_iq_i\in[\ell,\bar u]$, the ratio above is at most $((\ell+\bar u)m-\ell\bar u)/m^2$. As a function of $m>0$, this is maximized at $m=2\ell\bar u/(\ell+\bar u)$, where it equals $(\ell+\bar u)^2/(4\ell\bar u)=(\kappa+2+\kappa^{-1})/4$. We conclude that $|a|^2\le(\kappa+\kappa^{-1}-2)/N=J$. This is the elementary form of the Kantorovich inequality \cite[pp.~142--143]{Kantorovich1948}. For the equivalent second-moment form and its earlier history, including the work of Frucht, see \cite[Section~1, (1.3)]{Moslehian2013}. Inserting $L_A\varphi/(2\varphi)=z-1-e_A$ into \eqref{sgl-euclidean-weighted-bochner}, we find that the function $e_A$ cancels, which proves \eqref{sgl-canonical-weighted-identity}. For \eqref{sgl-canonical-curvature}, let $T=(ag^\T+ga^\T)/2$. Then $\langle\nabla_S^2f,T\rangle_{\mathrm{HS}}=\nabla_S^2f(a,g)$ and $\|T\|_{\mathrm{HS}}^2=(|a|^2|g|^2+(a\cdot g)^2)/2\le|a|^2|g|^2$. Completing the square, we find \[ \|\nabla_S^2f\|_{\mathrm{HS}}^2+2\nabla_S^2f(a,g)=\|\nabla_S^2f+T\|_{\mathrm{HS}}^2-\|T\|_{\mathrm{HS}}^2\ge-|a|^2|g|^2\ge-J|g|^2. \] The eigenvalues of $zI-A$ are at least $d$. Inserting these two bounds into \eqref{sgl-canonical-weighted-identity}, we obtain $\Gamma_{\varphi,2}(f)\ge\varphi(d-2/N-J)|g|^2=r_0\varphi\Gamma(f)$. \end{proof} \begin{theorem}[Weighted gradient estimate and spectral gap]\label{sgl-semigroup-gradient}\label{ht-trace-contraction} Let $r_0=d-(\kappa+\kappa^{-1})/N$. For every $t\ge0$ and every smooth $f$ on $\Sph$, \begin{equation}\label{sgl-weighted-gradient} \varphi\,\Gamma(P_t^Af)\le e^{-2r_0t}P_t^A\big(\varphi\,\Gamma(f)\big),\qquad \Gamma(P_t^Af)\le\kappa\,e^{-2r_0t}P_t^A\Gamma(f). \end{equation} If $r_0>0$, then $\lambda_{\mathrm{gap}}(L_A)\ge r_0$. These conclusions remain true when $r_0$ is replaced by any smaller number, for instance by $r=d-(\kappa+1+\kappa^{-1})/N=r_0-1/N$. \end{theorem} \begin{proof} Fix $t>0$. The function $(u,x)\mapsto P_u^Af(x)$ is smooth on $[0,t]\times\Sph$, by parabolic regularity on the compact manifold $\Sph$. For $0\le u\le t$, let \[ \mathcal K(u)=e^{-2r_0u}P_u^A\big(\varphi\,\Gamma(P^A_{t-u}f)\big). \] Using $\partial_uP^A_{t-u}f=-L_AP^A_{t-u}f$ and the definition of $\Gamma_{\varphi,2}$, we find \[ \begin{aligned} \mathcal K'(u)&=e^{-2r_0u}P_u^A\Big(L_A\big(\varphi\Gamma(P^A_{t-u}f)\big)-2\varphi\Gamma\big(P^A_{t-u}f,L_AP^A_{t-u}f\big)-2r_0\varphi\Gamma(P^A_{t-u}f)\Big)\\ &=2e^{-2r_0u}P_u^A\Big(\Gamma_{\varphi,2}(P^A_{t-u}f)-r_0\varphi\Gamma(P^A_{t-u}f)\Big), \end{aligned} \] which is nonnegative by \eqref{sgl-canonical-curvature} and the positivity of $P_u^A$. Comparing $\mathcal K(0)$ and $\mathcal K(t)$ proves the first inequality in \eqref{sgl-weighted-gradient}. The second follows from $1/s\le\varphi\le1/d$. Let $f$ be a nonconstant smooth eigenfunction, $L_Af=-\lambda f$. By \eqref{sgl-euclidean-ibp}, we have $\int L_A(\varphi\Gamma(f))\,d\pi_A=0$. Using also $\Gamma(f,L_Af)=-\lambda\Gamma(f)$, we find \[ \int\Gamma_{\varphi,2}(f)\,d\pi_A=\lambda\int\varphi\,\Gamma(f)\,d\pi_A. \] The last integral is positive. Integrating \eqref{sgl-canonical-curvature}, we find $\lambda\ge r_0$. Since this holds for every nonconstant eigenfunction, we conclude that $\lambda_{\mathrm{gap}}(L_A)\ge r_0$. The last assertion is immediate, since $e^{-2r_0t}\le e^{-2rt}$ and $r\le r_0$. \end{proof} The rate $r_0$ is positive only when $d>(\kappa+\kappa^{-1})/N$. Near the critical temperature, $\kappa$ is of order $d^{-1}$, and this condition requires $Nd^2$ to be large. The next section proves a lower bound on $\lambda_{\mathrm{gap}}(L_A)$ without this restriction, and \cref{sec:ineq-few,sec:ineq-critical} treat the critical window, where $Nd^2$ tends to zero. \section{Many precisions comparable to the smallest one}\label{sec:ineq-many} In this section, $A$ is diagonal, with precisions $p_i$, variances $q_i$, smallest precision $d=p_1$ and weight $\varphi=x^\T Qx/N$, as in \cref{sec:ineq-setting,sec:ineq-calculus}. Let $L\ge1$, and let $K\subset\{1,\dots,N\}$ be a set of $k\ge3$ indices such that \begin{equation}\label{eq:ineq-many-hyp} p_i\le L\,d\qquad\text{for every }i\in K. \end{equation} We write $\ell=k-2$ and $S_K(x)=\sum_{i\in K}x_i^2$. The gradient of $\log\varphi$ can be large where the coordinates in $K$ are small. The following inequality, of Hardy type, controls its square on average by $1/S_K$, which is small on average when $k$ is large. \begin{lemma}[A Hardy-type inequality]\label{lem:ineq-hardy} Assume \eqref{eq:ineq-many-hyp}. For every smooth $u$ on $\Sph$, \begin{equation}\label{cg-many-soft-hardy} \pi_A\big(\Gamma(\log\varphi)\,u^2\big)\le\frac{8L^2d}{k-2}\,\pi_A(u^2)+\frac{32L(1+L)}{(k-2)^2}\,\pi_A\big(\Gamma(u)\big). \end{equation} \end{lemma} \begin{proof} Since $L_A=L_{-P}$ and $\pi_A=\pi_{-P}$, we may assume that $A=-P$, so that $z=0$ and $e_A=-\bar p$, where $\bar p(x)=N^{-1}\sum_ip_ix_i^2$. By \eqref{sgl-euclidean-generator}, for every $i$, \[ L_Ax_i^2=2-2x_i^2-2p_ix_i^2+2\bar p\,x_i^2. \] Multiplying by $q_i/N$ and summing over $i$, and using $\sum_iq_i=N$ and $|x|^2=N$, we find $L_A\varphi=2(\bar p-1)\varphi$, in agreement with \eqref{sgl-F-cancellation}. It follows that $L_A\log\varphi=2\bar p-2-\Gamma(\log\varphi)$. Summing over $i\in K$ instead, and using $\Gamma(S_K)=4S_K-4S_K^2/N$, we find \[ L_A\log S_K=\frac{L_AS_K}{S_K}-\frac{\Gamma(S_K)}{S_K^2}=\frac{2(k-2)}{S_K}-2-2\bar p_K+2\bar p+\frac4N, \qquad \bar p_K=\frac{\sum_{i\in K}p_ix_i^2}{S_K}. \] Let $\Upsilon=\log(S_K/\varphi)$. Subtracting the two identities, we obtain \begin{equation}\label{eq:ineq-hardy-drift} L_A\Upsilon=\frac{2(k-2)}{S_K}-2\bar p_K+\frac4N+\Gamma(\log\varphi). \end{equation} By the computation of $|a|^2$ in the proof of \cref{sgl-weighted-square}, \[ \Gamma(\log\varphi)=\frac{4\sum_iq_i^2x_i^2}{(\sum_iq_ix_i^2)^2}-\frac4N\le\frac{4q_1}{\sum_{i\in K}q_ix_i^2}\le\frac{4L}{S_K}, \] where we used $q_i\le q_1=1/d$ for all $i$ and $q_i\ge1/(Ld)$ for $i\in K$. Together with $\Gamma(\log S_K)\le4/S_K$, this gives $\Gamma(\Upsilon)\le2\Gamma(\log S_K)+2\Gamma(\log\varphi)\le8(1+L)/S_K$. We claim that \begin{equation}\label{eq:ineq-hardy-ibp} \int u^2L_A\Upsilon\,d\pi_A=-2\int u\,\Gamma(u,\Upsilon)\,d\pi_A. \end{equation} If $k=N$, then $S_K=N$, the function $\Upsilon$ is smooth, and \eqref{eq:ineq-hardy-ibp} is a case of \eqref{sgl-euclidean-ibp}. If $k0\}$, and $|L_A\Upsilon|$ and $\Gamma(\Upsilon)$ are bounded by a constant multiple of $1/S_K$. Under $\nu_N$, the variable $S_K/N$ has the Beta$(k/2,(N-k)/2)$ distribution, whose density is proportional to $s^{k/2-1}(1-s)^{(N-k)/2-1}$, and $1/S_K$ is integrable because $k\ge3$; the same holds under $\pi_A$, whose density with respect to $\nu_N$ is bounded. For $0<\varepsilon\varepsilon\}$. Its boundary $\{S_K=\varepsilon\}$ is a product of spheres of dimensions $k-1$ and $N-k-1$ with radii $\sqrt\varepsilon$ and $\sqrt{N-\varepsilon}$, and its $(N-2)$-dimensional volume is at most $C_{N,k}\varepsilon^{(k-1)/2}$. The normal derivative of $\Upsilon$ there is at most $\Gamma(\Upsilon)^{1/2}\le(8(1+L)/\varepsilon)^{1/2}$. The boundary term is $O(\varepsilon^{(k-2)/2})$, which tends to zero, and the two integrals over $\{S_K>\varepsilon\}$ converge to those in \eqref{eq:ineq-hardy-ibp} by dominated convergence. This proves \eqref{eq:ineq-hardy-ibp}. Let $n=\pi_A(u^2)$, $U=\pi_A(u^2/S_K)$ and $E=\pi_A(\Gamma(u))$. In \eqref{eq:ineq-hardy-drift}, we drop the nonnegative terms $4/N+\Gamma(\log\varphi)$ and use $\bar p_K\le Ld$. Then \eqref{eq:ineq-hardy-ibp} and the Cauchy--Schwarz inequality give \[ \begin{aligned} 2(k-2)U-2Ld\,n&\le2\big(\pi_A(u^2\Gamma(\Upsilon))\big)^{1/2}E^{1/2}\\ &\le2\big(8(1+L)UE\big)^{1/2}\le(k-2)U+\frac{8(1+L)}{k-2}E. \end{aligned} \] Absorbing the term $(k-2)U$, we find $U\le2Ld\,n/(k-2)+8(1+L)E/(k-2)^2$. Multiplying by $4L$ and using $\Gamma(\log\varphi)\le4L/S_K$, we obtain \eqref{cg-many-soft-hardy}. \end{proof} \begin{theorem}[Spectral gap when many precisions are comparable to the smallest one]\label{ht-many-soft-sharp-gaps} Assume \eqref{eq:ineq-many-hyp}, and set \[ A_{\mathrm H}=\frac{8L^2d}{\ell},\qquad B_{\mathrm H}=\frac{32L(1+L)}{\ell^2},\qquad \rho_\varepsilon=\frac12+\frac1{4\varepsilon}\quad(\varepsilon>0). \] \begin{enumerate}[label=(\roman*)] \item For every smooth $f$ on $\Sph$, \begin{equation}\label{ht-many-soft-gradient-hardy} \pi_A\big(\Gamma(\log\varphi)\Gamma(f)\big)\le A_{\mathrm H}\,\pi_A\big(\Gamma(f)\big)+B_{\mathrm H}\,\pi_A\big(\|\nabla_S^2f\|_{\mathrm{HS}}^2\big). \end{equation} \item Let $f$ be a nonconstant smooth function with $L_Af=-\lambda f$, and write $E_1=\pi_A(\Gamma(f))$ and $E_2=\pi_A(\|\nabla_S^2f\|_{\mathrm{HS}}^2)$. Then for every $\varepsilon>0$, \begin{equation}\label{ht-many-soft-bochner-lower} \lambda E_1\ge(1-\varepsilon-\rho_\varepsilon B_{\mathrm H})\,E_2+\Big(d-\frac2N-\rho_\varepsilon A_{\mathrm H}\Big)E_1. \end{equation} \item If $\ell\ge64L^2$, then \[ \lambda_{\mathrm{gap}}(L_A)\ge d-\frac2N-\frac{8L^2d}{\ell}. \] \end{enumerate} \end{theorem} \begin{proof} For (i), we apply \cref{lem:ineq-hardy} in the form \begin{equation}\label{ht-many-soft-hardy-coefficients} \pi_A\big(\Gamma(\log\varphi)u^2\big)\le A_{\mathrm H}\,\pi_A(u^2)+B_{\mathrm H}\,\pi_A\big(\Gamma(u)\big) \end{equation} to the smooth function $u_\varepsilon=(\Gamma(f)+\varepsilon^2)^{1/2}$. Using $\Gamma(f)\le u_\varepsilon^2$ and the bound $\Gamma(u_\varepsilon)\le\|\nabla_S^2f\|_{\mathrm{HS}}^2$ of \cref{sgl-gradient-norm}, we find \[ \pi_A\big(\Gamma(\log\varphi)\Gamma(f)\big)\le\pi_A\big(\Gamma(\log\varphi)u_\varepsilon^2\big)\le A_{\mathrm H}\big(E_1+\varepsilon^2\big)+B_{\mathrm H}E_2, \] and we let $\varepsilon\to0$. For (ii), we again take $A=-P$, so that $e_A=-\bar p$ and \eqref{sgl-euclidean-bochner} reads \[ \Gamma_2(f)=\|\nabla_S^2f\|_{\mathrm{HS}}^2+\sum_ip_ig_i^2+\Big(1-\frac2N-\bar p\Big)|g|^2, \] with $g=Df$ as in \cref{sgl-ambient-bochner}. Recall from the proof of \cref{lem:ineq-hardy} that $1-\bar p=-\frac12L_A\log\varphi-\frac12\Gamma(\log\varphi)$. By \eqref{sgl-euclidean-ibp}, $\pi_A(\Gamma_2(f))=\pi_A((L_Af)^2)=\lambda E_1$. Also by \eqref{sgl-euclidean-ibp}, and since the tangential gradient of $\Gamma(f)$ is $2(\nabla_S^2f)g$, \[ -\frac12\int(L_A\log\varphi)\,\Gamma(f)\,d\pi_A=\frac12\int\Gamma\big(\log\varphi,\Gamma(f)\big)\,d\pi_A=\int\nabla_S^2f(a,g)\,d\pi_A, \] where $a=\nabla_S\log\varphi$. Integrating the expression for $\Gamma_2(f)$, we obtain \[ \lambda E_1=E_2+\pi_A\Big(\sum_ip_ig_i^2\Big)-\frac2NE_1+\pi_A\big(\nabla_S^2f(a,g)\big)-\frac12\pi_A\big(\Gamma(\log\varphi)\Gamma(f)\big). \] We have $\sum_ip_ig_i^2\ge d|g|^2$, and $|\nabla_S^2f(a,g)|\le\varepsilon\|\nabla_S^2f\|_{\mathrm{HS}}^2+|a|^2|g|^2/(4\varepsilon)$ with $|a|^2=\Gamma(\log\varphi)$. Inserting these bounds, we have \[ \lambda E_1\ge(1-\varepsilon)E_2+\Big(d-\frac2N\Big)E_1-\rho_\varepsilon\,\pi_A\big(\Gamma(\log\varphi)\Gamma(f)\big), \] and \eqref{ht-many-soft-bochner-lower} follows from \eqref{ht-many-soft-gradient-hardy}. For (iii), we take $\varepsilon=1/2$, so that $\rho_\varepsilon=1$. If $\ell\ge64L^2$, then $B_{\mathrm H}\le64L^2/\ell^2\le1/(64L^2)<1/2$, and the coefficient of $E_2$ in \eqref{ht-many-soft-bochner-lower} is nonnegative. Dropping this term and dividing by $E_1>0$, we find $\lambda\ge d-2/N-A_{\mathrm H}$ for every nonconstant eigenfunction, which proves (iii). \end{proof} \begin{corollary}[Poincar\'e inequality when many precisions are comparable to the smallest one]\label{cg-many-soft-pi} Assume \eqref{eq:ineq-many-hyp} with $\ell\ge64L^2$, and suppose that $Nd\ge8$. Then $C_{\mathrm P}(\pi_A)\le2/d$. The same bound holds for every coordinate hemisphere with its reflecting form and for the coordinate-even sector. Equivalently, in terms of the half-precisions $r_i=p_i/2$ and $r_{\min}=d/2$: if $r_i\le Lr_{\min}$ for $i\in K$, $k-2\ge64L^2$, and $Nr_{\min}\ge4$, then $C_{\mathrm P}(\pi_A)\le1/r_{\min}$. \end{corollary} \begin{proof} Under these hypotheses, $8L^2d/\ell\le d/8$ and $2/N\le d/4$. \Cref{ht-many-soft-sharp-gaps} implies that $\lambda_{\mathrm{gap}}(L_A)\ge5d/8\ge d/2$, that is, $C_{\mathrm P}(\pi_A)\le2/d$. The other cases follow from \cref{lem:ineq-reflection}. \end{proof} Several later arguments apply these inequalities to conditional laws on spheres of other radii. We record the scaling. \begin{remark}[Spheres of other radii]\label{rem:ineq-scaling} Let $n\ge2$, $S>0$ and $v\in(0,\infty)^n$ with $\sum_iv_i=S$, and let $\mu_v$ be the probability measure on $\{y\in\R^n:|y|^2=S\}$ proportional to $\exp(-\sum_iy_i^2/(2v_i))$ times the surface measure, as in \eqref{gp-pin-definition}, with the spherical Dirichlet form of that sphere. The map $x\mapsto(S/n)^{1/2}x$ carries $\mu_{nv/S}$ on $S_n$ to $\mu_v$. If $f$ is a function on $\{|y|^2=S\}$ and $f'(x)=f((S/n)^{1/2}x)$, then $\Gamma(f')=(S/n)\Gamma(f)$ at corresponding points, while variances and entropies are unchanged. It follows that $C_{\mathrm P}(\mu_v)=(S/n)C_{\mathrm P}(\mu_{nv/S})$, and similarly for $C_{\mathrm{LS}}$. With $r_i=1/(2v_i)$ and $r_{\min}=\min_ir_i$, \cref{cg-many-soft-pi} reads as follows: if $r_i\le Lr_{\min}$ for $k$ indices $i$, with $k-2\ge64L^2$, and $Sr_{\min}\ge4$, then $C_{\mathrm P}(\mu_v)\le1/r_{\min}$. Likewise, the inequalities of \cref{cg-three-soft-lsi} below hold for $\mu_v$ in the form $C_{\mathrm{LS}}(\mu_v)\le C_\delta\max_iv_i+C_\delta C_{\mathrm P}(\mu_v)$, since this form is invariant under the scaling. \end{remark} \section{Logarithmic Sobolev inequalities from a Gaussian extension}\label{sec:ineq-lsi}\label{cg-tube-section} In this section, $q\in(0,\infty)^N$ satisfies $\sum_iq_i=N$, and $\mu=\mu_q$; recall from \cref{sec:ineq-setting} that $\mu_q=\pi_A$ for $A=-\diag(q_1^{-1},\dots,q_N^{-1})$, whose smallest precision is $1/\max_iq_i$. We write $|q|^2=\sum_iq_i^2$. Let $G_1,\dots,G_N$ be independent standard Gaussian variables, and let $h_q$ be the density of $\sum_iq_iG_i^2$. The number $h_q(N)$ is the value of this density at its mean $\E\sum_iq_iG_i^2=N$. Recall that $\mu_q$ is the conditional law of a Gaussian vector given that its squared norm equals $N$. In the next proof, we extend a function $f$ on $\Sph$ to Gaussian space, in a thin layer outside the sphere, so that the Gaussian norm, energy and entropy of the extension are controlled by those of $f$. We then apply the Gaussian logarithmic Sobolev inequality to the extension. The extension must compensate for the Jacobian of the coarea formula, which varies along the sphere. We obtain a logarithmic Sobolev inequality with an additive error term, which we then remove with a Poincar\'e inequality. \begin{theorem}[A logarithmic Sobolev inequality with an error term]\label{cg-canonical-tube-entropy} After relabeling the coordinates, let $q_1\ge q_2\ge\dots\ge q_N$, and write $\osc q=q_1-q_N$. Fix $\eta\in(0,1]$ and suppose that \begin{equation}\label{eq:ineq-tube-hyp} |q|^2\ge(1+\eta)(\osc q)^2. \end{equation} There exists $C_\eta<\infty$, depending only on $\eta$, such that for every $f\in H^1(\Sph)$, \begin{align} \Ent_\mu(f^2)&\le C_\eta\,q_1\,\cE_\mu(f)+\big[C_\eta+\log\big(h_q(N)^2|q|^2\big)\big]\mu(f^2),\label{cg-canonical-tube-defect}\\ \Ent_\mu(f^2)&\le C_\eta\,q_1\,\cE_\mu(f)+\Big[C_\eta+\log\frac{|q|^2}{q_1q_2}\Big]\mu(f^2).\label{cg-tube-physical-defective-entropy} \end{align} Moreover, \begin{equation}\label{cg-tube-physical-lsi-from-actual-pi} C_{\mathrm{LS}}(\mu)\le C_\eta\,q_1+\Big[C_\eta+2+\log\frac{|q|^2}{q_1q_2}\Big]C_{\mathrm P}(\mu). \end{equation} The same three inequalities hold for every coordinate hemisphere and for the coordinate-even sector, with $\Ent$, $\cE$ and $C_{\mathrm P}$ of that form. In particular, if $q_2\ge\delta q_1$ and $|q|^2\le K'q_1^2$ for some $\delta\in(0,1]$ and $K'<\infty$, then $C_{\mathrm{LS}}(\mu)\le C_\delta q_1+C_{\delta,K'}C_{\mathrm P}(\mu)$. \end{theorem} In terms of the half-precisions $r_i=1/(2q_i)$, the first term is $C_\eta q_1=C_\eta/(2\min_ir_i)$. \begin{proof} The proof has seven steps. Step~1 transfers the problem to an ellipsoid in Gaussian coordinates. Step~2 proves a weighted estimate on the ellipsoid. Steps~3--5 construct the extension along a normal flow and bound its norm and energy. Step~6 applies the Gaussian logarithmic Sobolev inequality and absorbs a logarithmic weight. Step~7 bounds $h_q(N)$ and concludes. We may take $f$ smooth, by the density argument of \cref{sec:ineq-setting}. By \cref{lem:ineq-reflection}, it suffices to prove \eqref{cg-canonical-tube-defect} and \eqref{cg-tube-physical-defective-entropy} on the full sphere: they then hold for every hemisphere and for the orthant, and \eqref{cg-tube-physical-lsi-from-actual-pi} for each of these forms follows from \cref{lem:ineq-tools}\ref{i:ineq-rothaus}, since the bracket in \eqref{cg-tube-physical-defective-entropy} is nonnegative (see Step~7). \paragraph{Step 1: an ellipsoid in Gaussian coordinates.} Let $Q=\diag(q_1,\dots,q_N)$, $\mathcal Q(y)=y^\T Qy$, and $\varpi(y)=y^\T Q^2y=|\nabla\mathcal Q(y)|^2/4$. For $\ell>0$, the level set $E_\ell=\{\mathcal Q=\ell\}$ is a smooth ellipsoid with unit normal $n=Qy/\varpi^{1/2}$. Let $\gamma$ be the standard Gaussian measure on $\R^N$ and $\sigma_\ell$ the surface measure of $E_\ell$. By the coarea formula \cite[Chapter~2, Section~6, (6.3), p.~66]{Simon2014}, for bounded measurable $g$, \[ \int g\,d\gamma=\int_0^\infty\int_{E_\ell}g(y)\,\frac{e^{-|y|^2/2}}{(2\pi)^{N/2}\,2\varpi(y)^{1/2}}\,\sigma_\ell(dy)\,d\ell . \] In particular, the inner integral with $g=1$ is $h_q(\ell)$, and the probability measure $\mu^{(\ell)}$ on $E_\ell$ with density proportional to $e^{-|y|^2/2}\varpi^{-1/2}$ with respect to $\sigma_\ell$ is the conditional law of a standard Gaussian vector $G$ given $\mathcal Q(G)=\ell$. The vector $X=Q^{1/2}G$ has independent $\cN(0,q_i)$ coordinates and $|X|^2=\mathcal Q(G)$. By the coarea formula in the coordinates $x$, where $|\nabla|x|^2|$ is constant on each sphere, the conditional law of $X$ given $|X|^2=\ell$ is proportional to $e^{-\sum_ix_i^2/(2q_i)}$ times the surface measure; for $\ell=N$ this is $\mu_q$. Both families of conditional laws depend continuously on $\ell$, and they describe the same random vector given the same variable. We conclude that $\mu^E=\mu^{(N)}$ is the image of $\mu_q$ under $x\mapsto Q^{-1/2}x$. We write $E=E_N$. Let $\Gamma_E(\hat f)=|(I-nn^\T)\nabla\hat f|^2$ be the Euclidean tangential energy on $E$. For $f$ smooth near $\Sph$, let $\hat f(y)=f(Q^{1/2}y)$. Using $\partial_{y_i}\hat f=q_i^{1/2}\partial_if$ and $\nabla\mathcal Q=2Qy$, we find, at $x=Q^{1/2}y$, \[ \Gamma_E(\hat f)=\min_{b\in\R}\big|\nabla\hat f-2bQy\big|^2=\min_{b\in\R}\sum_iq_i\big(\partial_if-2bx_i\big)^2\le q_1\min_{b\in\R}\sum_i\big(\partial_if-2bx_i\big)^2=q_1\Gamma(f). \] Here $\partial_if$ are the derivatives of any smooth extension of $f$, and both minima are tangential energies, which do not depend on the extension. The image map preserves $\Ent$ and $\mu(f^2)$, and it suffices to show that for every smooth $f$ on $E$, \begin{equation}\label{eq:ineq-tube-goal} \Ent_{\mu^E}(f^2)\le C_\eta\int\Gamma_E(f)\,d\mu^E+\big[C_\eta+\log\big(h_q(N)^2|q|^2\big)\big]\mu^E(f^2). \end{equation} \paragraph{Step 2: a weighted estimate on $E$.} For $y\in E$, let $w_i(y)=q_i^2y_i^2/\varpi(y)$. These weights form a probability vector, and we write $\langle q\rangle_w=\sum_iw_iq_i$ and $\Var_w(q)=\sum_iw_iq_i^2-\langle q\rangle_w^2$. Then $\langle q\rangle_w\in[q_N,q_1]$ and $\Var_w(q)\le(\osc q)^2/4$. The generator $L_E$ of the form $\int\Gamma_E\,d\mu^E$ is $\Delta_E+\nabla_E\log\rho\cdot\nabla_E$, where $\rho\propto e^{-|y|^2/2}\varpi^{-1/2}$ and $\Delta_E$ is the Laplace--Beltrami operator of $E$. For a smooth function $f$ defined near $E$, we have $\Delta_Ef=\operatorname{tr}((I-nn^\T)D^2f)-(\operatorname{div}n)\,n\cdot Df$, where $\operatorname{div}n=(\operatorname{tr}Q-\langle q\rangle_w)/\varpi^{1/2}$ is the mean curvature of $E$. The tangential part of $\nabla\log\rho=-y-Q^2y/\varpi$ is $-y+\mathcal Q(y)Qy/\varpi-Q^2y/\varpi+\langle q\rangle_wQy/\varpi$, since $n\cdot y=\mathcal Q(y)/\varpi^{1/2}$ and $n\cdot Q^2y=\langle q\rangle_w\varpi^{1/2}$. Adding the term $-(\operatorname{div}n)\,n\cdot Df$ and using $\operatorname{tr}Q=N=\mathcal Q(y)$ on $E$, we obtain \[ L_Ef=\operatorname{tr}\big((I-nn^\T)D^2f\big)+\Big(-y-\frac{Q^2y}{\varpi}+\frac{2\langle q\rangle_wQy}{\varpi}\Big)\cdot Df . \] For $\varpi$, with $D\varpi=2Q^2y$ and $D^2\varpi=2Q^2$, this formula gives \begin{equation}\label{cg-tube-conditional-p-identities} \begin{gathered} L_E\varpi=2|q|^2-2\varpi-4\Var_w(q),\qquad\Gamma_E(\varpi)=4\varpi\Var_w(q),\\ L_E\log\varpi=\frac{2|q|^2}{\varpi}-2-\frac{8\Var_w(q)}{\varpi}. \end{gathered} \end{equation} The formula for $\Gamma_E(\varpi)$ uses only the normal $n$ and holds on every level set $E_\ell$. Let $\mathcal W=\varpi^{-\eta}$. Then \[ \frac{-L_E\mathcal W}{\mathcal W}=\eta L_E\log\varpi-\eta^2\Gamma_E(\log\varpi)=\frac{2\eta|q|^2-(8\eta+4\eta^2)\Var_w(q)}{\varpi}-2\eta\ge\frac{\Lambda_\eta}{\varpi}-2\eta, \] with $\Lambda_\eta=\eta[2|q|^2-(2+\eta)(\osc q)^2]$. We claim $\Lambda_\eta\ge\eta^2q_1^2/4$. If $\osc q\ge q_1/2$, then \eqref{eq:ineq-tube-hyp} implies that $2|q|^2-(2+\eta)(\osc q)^2\ge\eta(\osc q)^2$, and $\Lambda_\eta\ge\eta^2(\osc q)^2\ge\eta^2q_1^2/4$. If $\osc q0$, and $V$ is smooth with linear growth. The solution exists for all $\ell\ge N$, and the flow, with its inverse obtained by running it backward, maps $E$ diffeomorphically onto $E_\ell$. We write $u=\ell-N$, $\varpi_\ell(\omega)=\varpi(Y_\ell(\omega))$ and $\varpi_0=\varpi_N$. Using $d\varpi_\ell/d\ell=D\varpi\cdot V=\langle q\rangle_w(Y_\ell)$, we find \begin{equation}\label{cg-tube-normal-p-growth} \frac{d\varpi_\ell}{d\ell}\in[q_N,q_1],\qquad\varpi_0\le\varpi_\ell\le\varpi_0+q_1u. \end{equation} For tangent vectors $\xi,\xi'$ to $E$ at $\omega$, let $\xi_\ell=DY_\ell(\omega)\xi$ and $\xi'_\ell=DY_\ell(\omega)\xi'$, which are tangent to $E_\ell$. Then \[ \frac d{d\ell}\langle\xi_\ell,\xi'_\ell\rangle=\langle DV\xi_\ell,\xi'_\ell\rangle+\langle\xi_\ell,DV\xi'_\ell\rangle=\frac{\xi_\ell^\T Q\xi'_\ell}{\varpi_\ell}, \] because $DV\xi=Q\xi/(2\varpi)-Qy\,(D\varpi\cdot\xi)/(2\varpi^2)$ and $Qy\perp\xi'_\ell$. In particular $|\xi_\ell|$ is nondecreasing in $\ell$. We say that a function $\tilde g$ on $E_\ell$ is transported from $E_{\ell'}$, $N\le\ell'\le\ell$, if $\tilde g(Y_\ell(\omega))=g(Y_{\ell'}(\omega))$ for a function $g$ on $E_{\ell'}$. Writing $\Gamma_\ell$ for the tangential energy on $E_\ell$, we see from the monotonicity of $|\xi_\ell|$ that \begin{equation}\label{cg-tube-tangent-energy-monotonicity} \Gamma_\ell(\tilde g)\big(Y_\ell(\omega)\big)\le\Gamma_{\ell'}(g)\big(Y_{\ell'}(\omega)\big). \end{equation} Applying \eqref{cg-tube-conditional-p-identities} on each level, we find that $\Gamma_\ell(\log\varpi)\le(\osc q)^2/\varpi$ on $E_\ell$. Combined with \eqref{cg-tube-tangent-energy-monotonicity} and \eqref{cg-tube-normal-p-growth}, this shows that for every $N\le\ell'\le\ell$ the function $\log\varpi_{\ell'}$, viewed on $E_\ell$, satisfies $\Gamma_\ell(\log\varpi_{\ell'})\le(\osc q)^2/\varpi_0$. We now compute the Gaussian density in the coordinates $(\omega,\ell)\in E\times[N,\infty)$. We write $\gamma(dy)=j_\ell(\omega)\,\sigma_N(d\omega)\,d\ell$ on $\{\mathcal Q\ge N\}$. At $\ell=N$, the factor $j_N(\omega)=(2\pi)^{-N/2}e^{-|\omega|^2/2}/(2\varpi_0^{1/2})$ is the coarea density of Step~1, and $j_N(\omega)\,\sigma_N(d\omega)=h_q(N)\,\mu^E(d\omega)$. By the Liouville formula, $\partial_\ell\log j_\ell=(\operatorname{div}V-y\cdot V)(Y_\ell)$, where $y\cdot V$ comes from the Gaussian factor. We have $\operatorname{div}V=\operatorname{tr}Q/(2\varpi)-\langle q\rangle_w/\varpi$ and $y\cdot V=\mathcal Q/(2\varpi)$. On $E_\ell$, where $\mathcal Q=N+u$ and $\operatorname{tr}Q=N$, this gives $\partial_\ell\log j_\ell=-u/(2\varpi_\ell)-\langle q\rangle_w/\varpi_\ell$. Integrating, and using $\partial_\ell\log\varpi_\ell=\langle q\rangle_w/\varpi_\ell$, we arrive at \begin{equation}\label{cg-tube-exact-coarea-jacobian} \frac{j_\ell}{j_N}=\frac{\varpi_0}{\varpi_\ell}\,e^{-I_\ell},\qquad I_\ell=\int_0^u\frac{v}{2\varpi_{N+v}}\,dv . \end{equation} For $0\le u\le\varpi_0^{1/2}$, we have $0\le I_\ell\le u^2/(4\varpi_0)\le1/4$. \paragraph{Step 4: the extension and its norm.} Fix once and for all a nonzero $\chi\in C_c^\infty((1/4,3/4))$, and let $\chi_0=\int\chi^2$, $\chi_1=\int(\chi')^2$ and $\chi_2=\int\chi^2\log\chi^2$. For smooth $f$ on $E$, we define $\mathcal F=0$ on $\{\mathcal QN\}$, the gradient splits into its tangential part on $E_\ell$ and its normal part. As $V=n/(2\varpi^{1/2})$, the derivative $\partial_\ell$ along the flow at fixed $\omega$ satisfies $n\cdot\nabla\mathcal G=2\varpi_\ell^{1/2}\partial_\ell\mathcal G$, and so \[ |\nabla\mathcal G|^2=\Gamma_\ell(\mathcal G)+4\varpi_\ell\,(\partial_\ell\mathcal G)^2 . \] Let $\tilde f$ be $f$ transported along the flow, so that $\partial_\ell\tilde f=0$ and $\Gamma_\ell(\tilde f)\le\Gamma_E(f)$ by \eqref{cg-tube-tangent-energy-monotonicity}. Let $H_0=q_1^2/\varpi_0$, a function on $E$, and recall $\osc q\le q_1$. We bound the derivatives of the two other factors in \eqref{cg-tube-explicit-gaussian-extension} on the support of $\mathcal F$. First, $\log\mathfrak a_\ell=\frac12\log\varpi_\ell-\frac34\log\varpi_0+\frac12I_\ell$. Tangential differentiation fixes $u$. By the bound at the end of Step~3 and Minkowski's inequality, \[ \Gamma_\ell(I_\ell)^{1/2}\le\int_0^u\frac{v}{2\varpi_{N+v}}\,\Gamma_\ell(\log\varpi_{N+v})^{1/2}\,dv\le\frac{\osc q}{\varpi_0^{1/2}}\int_0^u\frac{v}{2\varpi_0}\,dv\le\frac{\osc q}{4\varpi_0^{1/2}}, \] since $u\le\varpi_0^{1/2}$. Using the inequality $(\theta_1+\theta_2+\theta_3)^2\le3(\theta_1^2+\theta_2^2+\theta_3^2)$ and the bounds of Step~3, we obtain \[ \Gamma_\ell(\log\mathfrak a_\ell)\le3\Big(\frac14+\frac9{16}+\frac14\cdot\frac1{16}\Big)\frac{(\osc q)^2}{\varpi_0}=\frac{159}{64}\,\frac{(\osc q)^2}{\varpi_0}\le\frac{159}{64}H_0. \] In the normal direction, $\partial_\ell\log\mathfrak a_\ell=\langle q\rangle_w/(2\varpi_\ell)+u/(4\varpi_\ell)$. Since $\langle q\rangle_w\le q_1$, $\varpi_\ell\ge\varpi_0$ and $u\le\varpi_0^{1/2}$, we have \[ 4\varpi_\ell(\partial_\ell\log\mathfrak a_\ell)^2\le\frac{2\langle q\rangle_w^2}{\varpi_\ell}+\frac{u^2}{2\varpi_\ell}\le2H_0+\frac12 . \] Together, $|\nabla\log\mathfrak a_\ell|^2\le5H_0+1$. Second, let $t=u/\varpi_0^{1/2}\le1$. Then $\partial_\ell t=\varpi_0^{-1/2}$, and the tangential gradient of $t$ is $-(t/2)$ times that of $\log\varpi_0$, and $\Gamma_\ell(t)\le(\osc q)^2/(4\varpi_0)\le H_0/4$. Using \eqref{cg-tube-normal-p-growth} and $4H_0^{1/2}\le2+2H_0$, we find \[ |\nabla\chi(t)|^2=\chi'(t)^2\Big(\Gamma_\ell(t)+\frac{4\varpi_\ell}{\varpi_0}\Big)\le\chi'(t)^2\Big(\frac{H_0}4+4+\frac{4q_1u}{\varpi_0}\Big)\le\chi'(t)^2\Big(6+\frac94H_0\Big). \] By the product rule and the same elementary inequality, \[ |\nabla\mathcal F|^2\le3\mathfrak a_\ell^2\Big(\tilde f^2\chi(t)^2(5H_0+1)+\chi(t)^2\,\Gamma_E(f)+\tilde f^2\chi'(t)^2\Big(6+\frac94H_0\Big)\Big). \] We integrate against $j_\ell(\omega)\,\sigma_N(d\omega)\,d\ell$, use $\mathfrak a_\ell^2j_\ell=j_N\varpi_0^{-1/2}$ and $d\ell=\varpi_0^{1/2}dt$, and obtain \begin{equation}\label{cg-tube-extension-full-energy} \int|\nabla\mathcal F|^2\,d\gamma\le3h_q(N)\Big\{\chi_0\int\Gamma_E(f)\,d\mu^E+(\chi_0+6\chi_1)\,\mu^E(f^2)+\Big(5\chi_0+\frac94\chi_1\Big)\mu^E(f^2H_0)\Big\}. \end{equation} By \eqref{cg-tube-weighted-normal-occupation}, the right side is at most $\chi_0h_q(N)\big(A_\eta\int\Gamma_E(f)\,d\mu^E+B_\eta\,\mu^E(f^2)\big)$, where \[ A_\eta=3+\frac{12}{\eta^2}\Big(5+\frac{9\chi_1}{4\chi_0}\Big),\qquad B_\eta=3\Big(1+\frac{6\chi_1}{\chi_0}\Big)+\frac{24}\eta\Big(5+\frac{9\chi_1}{4\chi_0}\Big). \] \paragraph{Step 6: the Gaussian inequality and the absorption of a weight.} Let $n_f=\mu^E(f^2)$. Computing $\int\mathcal F^2\log\mathcal F^2\,d\gamma$ as in \eqref{cg-tube-extension-exact-norm} and using $\int\mathcal F^2\,d\gamma=\chi_0h_q(N)n_f$, we obtain \[ \frac{\Ent_\gamma(\mathcal F^2)}{\chi_0h_q(N)}=\Ent_{\mu^E}(f^2)+\frac1{\chi_0}\int f^2\Big(\int_0^\infty\chi(t)^2\log\big(\mathfrak a_\ell^2\chi(t)^2\big)\,dt\Big)\,d\mu^E-n_f\log\big(\chi_0h_q(N)\big). \] We have $\log\mathfrak a_\ell^2=\log(\varpi_\ell/\varpi_0)+I_\ell-\frac12\log\varpi_0\ge-\frac12\log\varpi_0$, and the inner integral is at least $\chi_2-\frac{\chi_0}2\log\varpi_0$. By \cref{lem:ineq-tools}\ref{i:ineq-gross} and Step~5, \begin{equation}\label{cg-tube-entropy-before-absorption} \Ent_{\mu^E}(f^2)\le2A_\eta\int\Gamma_E(f)\,d\mu^E+C'_\eta\,n_f+\frac12\int f^2\log\varpi\,d\mu^E+n_f\log h_q(N), \end{equation} with $C'_\eta=2B_\eta+\log\chi_0-\chi_2/\chi_0$. The weight $\log\varpi$ must be paid with the factor $f^2$. By the variational formula for the entropy, $\int f^2g\,d\mu^E\le\Ent_{\mu^E}(f^2)+n_f\log\int e^g\,d\mu^E$ for every bounded measurable real function $g$; with $g=\log(\varpi/q_1^2)$, which is bounded since $E$ is compact and $\varpi>0$, and the bound $\mu^E(\varpi)\le|q|^2$ from Step~2, \[ \int f^2\log\frac{\varpi}{q_1^2}\,d\mu^E\le\Ent_{\mu^E}(f^2)+n_f\log\frac{|q|^2}{q_1^2}. \] Writing $\log\varpi=\log(\varpi/q_1^2)+2\log q_1$, we find $\frac12\int f^2\log\varpi\,d\mu^E\le\frac12\Ent_{\mu^E}(f^2)+\frac12n_f\log|q|^2$. Inserting this into \eqref{cg-tube-entropy-before-absorption} and absorbing half of the entropy, we obtain \eqref{eq:ineq-tube-goal} with $C_\eta=\max\{4A_\eta,2C'_\eta,0\}$, which depends only on $\eta$ since $\chi$ is fixed. Together with Step~1, this proves \eqref{cg-canonical-tube-defect}. \paragraph{Step 7: the density at the mean.} The variable $q_iG_i^2$ has density $e^{-y/(2q_i)}/(2\pi q_iy)^{1/2}$ on $(0,\infty)$. The density of $q_1G_1^2+q_2G_2^2$ at $y>0$ is the convolution of these densities, which is at most \[ \frac1{2\pi(q_1q_2)^{1/2}}\int_0^y\frac{dv}{(v(y-v))^{1/2}}=\frac1{2(q_1q_2)^{1/2}} . \] Convolution with the law of $\sum_{i\ge3}q_iG_i^2$ preserves this bound. Thus $h_q(N)^2\le1/(4q_1q_2)$, and \eqref{cg-canonical-tube-defect} implies \begin{equation}\label{cg-tube-induced-defective-entropy} \Ent_\mu(f^2)\le C_\eta\,q_1\,\cE_\mu(f)+\Big[C_\eta+\log\frac{|q|^2}{4q_1q_2}\Big]\mu(f^2), \end{equation} which gives \eqref{cg-tube-physical-defective-entropy}. The bracket in \eqref{cg-tube-physical-defective-entropy} is nonnegative, because $|q|^2\ge q_1^2+q_2^2\ge2q_1q_2$, and \cref{lem:ineq-tools}\ref{i:ineq-rothaus} implies \eqref{cg-tube-physical-lsi-from-actual-pi}. Finally, if $q_2\ge\delta q_1$ and $|q|^2\le K'q_1^2$, then $|q|^2\ge(1+\delta^2)q_1^2\ge(1+\delta^2)(\osc q)^2$, which is \eqref{eq:ineq-tube-hyp} with $\eta=\delta^2$, and $\log(|q|^2/(q_1q_2))\le\log(K'/\delta)$. This completes the proof. \end{proof} The second inequality is useful only when $|q|^2/q_1^2$ is bounded. When at least three variances are comparable to the largest one, the density $h_q(N)$ in \eqref{cg-canonical-tube-defect} is bounded by a constant multiple of $|q|^{-1}$, whatever the size of $|q|^2/q_1^2$. This is the content of the next lemma, which is also used in \cref{sec:density,sec:cold-meso,sec:crit-profile}. \begin{lemma}[Densities of weighted sums of squared Gaussians]\label{cg-mean-level-density} Let $b_1,\dots,b_n>0$, $\beta=\max_ib_i$, $M_1=\sum_ib_i$ and $|b|^2=\sum_ib_i^2$. Let $G_1,\dots,G_n$ be independent standard Gaussian variables, let $h_b$ be the density of $Q_b=\sum_ib_iG_i^2$, let $f_i$ be the density of $b_iG_i^2$, and let $h_{b,-i}$ be the density of $Q_b-b_iG_i^2$. Fix $\delta\in(0,1]$ and suppose that there exist three distinct indices $i$ with $b_i\ge\delta\beta$. Then there exist $c_\delta,C_\delta>0$, depending only on $\delta$, such that \[ h_b(M_1)\ge\frac{c_\delta}{|b|},\qquad\sup_yh_b(y)\le\frac{C_\delta}{|b|},\qquad\sup_yh_{b,-i}(y)\le\frac{C_\delta}{|b|}\quad\text{for every }i. \] Moreover, for every $i$, the conditional expectation at the exact level, \[ \E\big[b_iG_i^2\,\big|\,Q_b=M_1\big]=\frac1{h_b(M_1)}\int_0^{M_1}y\,f_i(y)\,h_{b,-i}(M_1-y)\,dy, \] satisfies $\E[b_iG_i^2\mid Q_b=M_1]\le C_\delta b_i$. \end{lemma} \begin{proof} Let $R=|b|^2/\beta^2\ge1$. We distinguish bounded and large values of $R$, with a threshold $R_0$ fixed below that depends on nothing. \paragraph{Large $R$.} Let $\sigma^2=2|b|^2$, which is the variance of $Q_b$, let $Z=(Q_b-M_1)/\sigma$, and let $\widehat h(t)=\E e^{\mathrm itZ}$. With $u_j=b_j/\beta\le1$, we have $\sum_ju_j^2=R$ and \[ \widehat h(t)=\prod_je^{-\mathrm itb_j/\sigma}\big(1-2\mathrm itb_j/\sigma\big)^{-1/2},\qquad |\widehat h(t)|=\prod_j\Big(1+\frac{2t^2u_j^2}R\Big)^{-1/4}\le\Big(1+\frac{2t^2}R\Big)^{-R/4}, \] where we used $\log(1+xs)\ge s\log(1+x)$ for $x\ge0$ and $s=u_j^2\in[0,1]$, which holds because the left side is concave in $s$. For $|t|\le R^{1/2}$, we have $|\widehat h(t)|\le e^{-t^2/6}$ by the bound $\log(1+x)\ge x/3$ on $[0,2]$. For $|t|\ge R^{1/2}$, the bound $1+2y^2\ge3y$ for $y\ge1$ gives $\int_{|t|\ge R^{1/2}}|\widehat h(t)|\,dt\le2R^{1/2}3^{-R/4}/(R/4-1)$ when $R>4$. Next, with $w_j=2\mathrm itb_j/\sigma$, we have $|w_j|\le\sqrt2|t|R^{-1/2}$ and $-\frac12\log(1-w)=\frac12w+\frac14w^2+\frac12\epsilon(w)$ with $|\epsilon(w)|\le\frac23|w|^3$ for $|w|\le\frac12$. The linear terms cancel the factors $e^{-\mathrm itb_j/\sigma}$, the quadratic terms sum to $-t^2|b|^2/\sigma^2=-t^2/2$, and since $\sum_jb_j^3\le\beta|b|^2$, the remainders sum to at most $|t|^3R^{-1/2}$ in absolute value. It follows that $\widehat h(t)\to e^{-t^2/2}$ uniformly on compact sets of $t$ as $R\to\infty$, uniformly in $b$. Let $\epsilon_0=(2\pi)^{-1/2}/4$. We first choose $T_0$ such that $\pi^{-1}\int_{T_0}^\infty(e^{-t^2/6}+e^{-t^2/2})\,dt\le\epsilon_0/3$, and then $R_F\ge\max\{8,8T_0^2\}$ such that, for $R\ge R_F$, the integral of $\pi^{-1}|\widehat h(t)-e^{-t^2/2}|$ over $[0,T_0]$ and the quantity $\pi^{-1}R^{1/2}3^{-R/4}/(R/4-1)$ are each at most $\epsilon_0/3$. For $R\ge R_F$, Fourier inversion shows that $Z$ has a continuous density $g_Z$ with $|g_Z(x)-(2\pi)^{-1/2}e^{-x^2/2}|\le\epsilon_0$ for all $x$. In particular, $g_Z(0)\ge\frac34(2\pi)^{-1/2}$ and $\sup g_Z\le\frac54(2\pi)^{-1/2}$. The identity $h_b(y)=\sigma^{-1}g_Z((y-M_1)/\sigma)$ then yields the first two bounds with absolute constants when $R\ge R_F$. Let $R_0=R_F+1$ and suppose $R>R_0$. Deleting coordinate $i$ leaves weights $b^{(-i)}$ with $|b^{(-i)}|^2\ge|b|^2-\beta^2\ge|b|^2/2$ and maximum at most $\beta$, and the ratio of $|b^{(-i)}|^2$ to the square of this maximum is at least $R-1>R_F$. Applying the previous paragraph to these weights, we find that $\sup h_{b,-i}\le C|b^{(-i)}|^{-1}\le\sqrt2C|b|^{-1}$. \paragraph{Bounded $R$.} Suppose $R\le R_0$. Let $i_1$ be an index with $b_{i_1}=\beta$, let $i_2\ne i_1$ be one of the distinguished indices, so that $b_{i_2}\ge\delta\beta$, and let $U$ be the sum of the remaining terms $b_jG_j^2$. Then $\Var U\le2|b|^2\le2R_0\beta^2$. Let $s_0=(b_{i_1}+b_{i_2})/2\ge\beta/2$. By Cantelli's inequality, \[ \PP\big(U-\E U\le s_0\big)\ge\frac{s_0^2}{\Var U+s_0^2}\ge\frac1{8R_0+1}=:p_0, \] and by Chebyshev's inequality, $\PP(U-\E U<-L_0\beta)\le2R_0/L_0^2\le p_0/2$ if $L_0^2=4R_0/p_0$. On the event $\{-L_0\beta\le U-\E U\le s_0\}$, which has probability at least $p_0/2$, the variable $M_1-U=b_{i_1}+b_{i_2}-(U-\E U)$ lies in $[s_0,(2+L_0)\beta]$. The density of $b_{i_1}G_{i_1}^2+b_{i_2}G_{i_2}^2$ at $y$ is \[ \frac1{2\pi(b_{i_1}b_{i_2})^{1/2}}\int_0^y\frac{e^{-v/(2b_{i_1})-(y-v)/(2b_{i_2})}}{(v(y-v))^{1/2}}\,dv\ge\frac{e^{-y/(2b_{i_2})}}{2(b_{i_1}b_{i_2})^{1/2}}\ge\frac{e^{-(2+L_0)/(2\delta)}}{2\beta} \] for $y\le(2+L_0)\beta$, where we used $b_{i_2}\le b_{i_1}$ and $\int_0^y(v(y-v))^{-1/2}dv=\pi$. Averaging over $U$, we obtain $h_b(M_1)\ge(p_0/4)e^{-(2+L_0)/(2\delta)}\beta^{-1}$, and $\beta\le|b|\le R_0^{1/2}\beta$. For the upper bounds, after deleting any coordinate $i$, at least two of the three distinguished indices remain, say $j$ and $j'$. By the computation in Step~7 of the proof of \cref{cg-canonical-tube-entropy}, the density of $b_jG_j^2+b_{j'}G_{j'}^2$ is at most $(2(b_jb_{j'})^{1/2})^{-1}\le(2\delta\beta)^{-1}$, and convolution with the other terms preserves this bound. Using $|b|\le R_0^{1/2}\beta$, we find $\sup h_{b,-i}\le R_0^{1/2}/(2\delta|b|)$. \paragraph{Conclusion.} In both cases, $\sup h_b\le\sup h_{b,-i}$, since $h_b=h_{b,-i}*f_i$. Finally, \[ \E\big[b_iG_i^2\,\big|\,Q_b=M_1\big]\le\frac{\sup h_{b,-i}}{h_b(M_1)}\int_0^\infty y\,f_i(y)\,dy=\frac{\sup h_{b,-i}}{h_b(M_1)}\,b_i\le C_\delta b_i . \] \end{proof} \begin{corollary}[Logarithmic Sobolev inequality when three variances are comparable to the largest one]\label{cg-three-soft-lsi} Let $q\in(0,\infty)^N$ with $\sum_iq_i=N$, write $q_{\max}=\max_iq_i$, and let $\delta\in(0,1]$. Suppose that there exist three distinct indices $i$ with $q_i\ge\delta q_{\max}$. There exists $C_\delta$, depending only on $\delta$, such that for the full sphere, every coordinate hemisphere with its reflecting form, and the coordinate-even sector, \[ \Ent_{\mu_q}(f^2)\le C_\delta\,q_{\max}\,\cE_{\mu_q}(f)+C_\delta\,\mu_q(f^2),\qquad C_{\mathrm{LS}}(\mu_q)\le C_\delta\,q_{\max}+C_\delta\,C_{\mathrm P}(\mu_q), \] where $\cE$, $\Ent$ and $C_{\mathrm P}$ refer to the corresponding form. \end{corollary} In terms of half-precisions, $q_{\max}=1/(2r_{\min})$, and the hypothesis says that three of the $r_i$ are at most $r_{\min}/\delta$. \begin{proof} At least two of the three indices differ from an index where the maximum is attained, and their squares contribute at least $2\delta^2q_{\max}^2$ to $|q|^2$. Using $\osc q\le q_{\max}$, we see that $|q|^2\ge(1+2\delta^2)q_{\max}^2\ge(1+\eta)(\osc q)^2$ with $\eta=\min\{1,2\delta^2\}$. By \cref{cg-mean-level-density} with $b=q$ and $M_1=N$, we have $h_q(N)\le C_\delta|q|^{-1}$, that is, $\log(h_q(N)^2|q|^2)\le2\log C_\delta$. Inserting this into \eqref{cg-canonical-tube-defect} proves the first inequality, for all three forms. The second follows from \cref{lem:ineq-tools}\ref{i:ineq-rothaus}, applied to each form with its own Poincar\'e constant. \end{proof} \section{A few precisions close to the smallest one}\label{sec:ineq-few}\label{cg-unequal-section} In the critical window, the smallest precision is of order $N^{-2/3}$, the gap between the two smallest precisions is of the same order, and the conditions \eqref{cg-weyl-hypotheses} place only a bounded number of precisions within a fixed factor of the smallest one. These conditions do not guarantee the hypotheses of \cref{cg-many-soft-pi}. In this section we prove a Poincar\'e inequality on the hemisphere $\{x_1\ge0\}$ with its reflecting form (\cref{cg-folded-pi}), and then pass to the full sphere (\cref{cg-full-sphere-bridge}). \paragraph{Setting.} Throughout this section, $k\ge5$ and $N>k$ are integers, and $a_1,\dots,a_N\ge0$ are quadratic potential coefficients such that, for some $0<\bar a\le a$, \begin{equation}\label{cg-folded-H1} a_1=0,\qquad 0k). \end{equation} Let $F(x)=\sum_ia_ix_i^2$, let $\mu$ be the probability measure proportional to $e^{-F}\nu_N$, and let $\mu_+=\mu(\cdot\mid x_1\ge0)$. Note that $\mu=\pi_A$ with $A=-2\diag(a_1,\dots,a_N)$: the coefficients $a_i$ are half-precisions shifted so that the smallest one is zero. We write $x=(\sqrt S\,\omega,y)$, where $y=(x_{k+1},\dots,x_N)\in B_N=\{y\in\R^{N-k}:|y|^2k}a_jy_j^2,\qquad\mathcal A(\omega)=\sum_{i\le k}a_i\omega_i^2, \end{equation} and, since the inverse of $I+yy^\T/S$ is $M_y=I-yy^\T/N$, \begin{equation}\label{eq:ineq-energy-split} \Gamma(f)=\Gamma_y(f)+S^{-1}|\nabla_\omega f|^2,\qquad\Gamma_y(f)=(\nabla_yf)^\T M_y\nabla_yf. \end{equation} Here $\nabla_\omega f$ is the gradient on $S^{k-1}$ of $\omega\mapsto f(\sqrt S\omega,y)$, and $\nabla_yf$ is the gradient of $y\mapsto f(\sqrt{N-|y|^2}\,\omega,y)$ at fixed $\omega$. The marginal law of $y$ under $\mu$, and also under $\mu_+$ since $\mathcal A$ is even in $\omega_1$, is \[ \mu_{\mathrm{rem}}(dy)\propto e^{-V(y)}S^{k/2-1}Z(S)\,dy,\qquad Z(S)=\int e^{-S\mathcal A}\,d\sigma^k=\int e^{-S\mathcal A}\,d\sigma^k_+ . \] Given $y$, the conditional law of $\omega$ is $\vartheta_S\propto e^{-S\mathcal A}\sigma^k$ under $\mu$, and $\vartheta_S^+=\vartheta_S(\cdot\mid\omega_1\ge0)$ under $\mu_+$. We write $\langle\mathcal A\rangle_S$ and $\Var_{\vartheta_S}(\mathcal A)$ for the mean and variance of $\mathcal A$ under $\vartheta_S$, which are the same under $\vartheta_S^+$. With $\psi=\log Z$, differentiation under the integral gives $\psi'(S)=-\langle\mathcal A\rangle_S$ and $\psi''(S)=\Var_{\vartheta_S}(\mathcal A)\ge0$. For functions $u$ of $y$ alone, \eqref{eq:ineq-energy-split} shows that the Dirichlet form of $\mu$ restricts to $\int\Gamma_y(u)\,d\mu_{\mathrm{rem}}$, whose generator on $B_N$ is $L_{\mathrm{rem}}u=\rho^{-1}\operatorname{div}(\rho M_y\nabla u)$, where $\rho$ is the density of $\mu_{\mathrm{rem}}$. \begin{lemma}[Moments of the angular law]\label{lem:ineq-angular} There exist absolute constants $C_1,C_2,C_A,C_V$ such that the following hold. \begin{enumerate}[label=(\roman*)] \item For $\alpha\ge2$ and $\tau\ge0$, let $\nu_{\alpha,\tau}$ be the probability measure on $(0,1)$ proportional to $v^{\alpha-1}(1-v)^{-1/2}e^{-\tau v}\,dv$. Then \begin{equation}\label{cg-angular-1} \E_{\nu_{\alpha,\tau}}[\tau v]\le\alpha+C_1\sqrt\alpha,\qquad\E_{\nu_{\alpha,\tau}}\big[(\tau v)^2\big]\le C_2\alpha^2. \end{equation} \item Let $a_1=0$ and $a_2,\dots,a_k\ge0$, with $k\ge5$, and let $\vartheta_S\propto e^{-S\mathcal A}\sigma^k$. Then for every $S>0$, \begin{equation}\label{cg-angular-2} S\langle\mathcal A\rangle_S\le\frac{k-1}2+C_1\sqrt{\frac{k-1}2}\le C_Ak,\qquad S^2\Var_{\vartheta_S}(\mathcal A)\le S^2\E_{\vartheta_S}[\mathcal A^2]\le C_Vk^2 . \end{equation} The same bounds hold under $\vartheta_S^+$. \end{enumerate} \end{lemma} \begin{proof} Let $p_\alpha(w)=w^{\alpha-1}e^{-w}/\Gamma(\alpha)$ be the Gamma$(\alpha,1)$ density. We first show that there exist absolute constants $c,C>0$ such that, for all $\alpha\ge2$, \begin{equation}\label{cg-angular-3} \begin{aligned} p_\alpha(w)&\ge\frac c{\sqrt\alpha}&&\text{for }\alpha-\tfrac12\sqrt\alpha\le w\le\alpha,\\ p_\alpha(w)&\le\frac C{\sqrt\alpha}e^{-(w-\alpha)^2/(16\alpha)}&&\text{for }\alpha\le w\le4\alpha,\\ p_\alpha(w)&\le\frac C{\sqrt\alpha}e^{-w/10}&&\text{for }w\ge2\alpha. \end{aligned} \end{equation} Stirling's formula gives $p_\alpha(\alpha)\asymp\alpha^{-1/2}$. For $w=\alpha(1+x)$, we have $\log(p_\alpha(w)/p_\alpha(\alpha))=(\alpha-1)\log(1+x)-\alpha x$. Using $\log(1+x)\le x-x^2/(2(1+x))$ for $x\ge0$ and $\alpha-1\ge\alpha/2$, we see that this is at most $-\alpha x^2/16$ for $0\le x\le3$, which proves the second bound. For $w\ge2\alpha$, it is at most $\alpha(\log(1+x)-x)\le-\alpha(1+x)/10=-w/10$, because $\log y-y+1\le-y/10$ for $y\ge2$; this proves the third bound. For $-1/(2\sqrt\alpha)\le x\le0$, the inequality $\log(1+x)\ge x-x^2/(2(1+x))$ shows that the logarithm is at least $-(\alpha-1)x^2\ge-1/4$, which gives the first bound. For (i), the claims are immediate if $\tau\le\alpha$, since $\tau v\le\tau$. Let $\tau>\alpha$. The variable $w=\tau v$ has density proportional to $p_\alpha(w)(\tau-w)^{-1/2}$ on $(0,\tau)$, with normalizing integral $H(\tau)$. Integrating the first bound in \eqref{cg-angular-3} over its interval, we find \begin{equation}\label{cg-angular-4} H(\tau)\ge\frac c2\,(\tau-\alpha+\sqrt\alpha)^{-1/2}. \end{equation} Then $\E w\le\alpha+K(\tau)/H(\tau)$ with $K(\tau)=\int_\alpha^\tau(w-\alpha)(\tau-w)^{-1/2}p_\alpha(w)\,dw$. Suppose first $\alpha<\tau\le4\alpha$, and let $T=(\tau-\alpha)/\sqrt\alpha$. Substituting $w=\alpha+\sqrt\alpha\,x$ and using the second bound in \eqref{cg-angular-3}, we find \[ K(\tau)\le C\alpha^{1/4}\int_0^Tx(T-x)^{-1/2}e^{-x^2/16}\,dx\le\frac{C\alpha^{1/4}}{\sqrt{T+1}} . \] For the last inequality: if $T\le2$, the integral is at most $\int_0^Tx(T-x)^{-1/2}dx=\frac43T^{3/2}$; if $T>2$, the part over $[0,T/2]$ is at most $(T/2)^{-1/2}\int_0^\infty xe^{-x^2/16}dx=8(2/T)^{1/2}$, and the part over $[T/2,T]$ is at most $Te^{-T^2/64}(2T)^{1/2}$. Then \eqref{cg-angular-4} implies that $K/H\le C\sqrt\alpha$, because $\tau-\alpha+\sqrt\alpha=\sqrt\alpha(T+1)$. Suppose next $\tau\ge4\alpha$, and split $K$ at $\tau/2\ge2\alpha$. On $[\alpha,\tau/2]$, we have $(\tau-w)^{-1/2}\le(2/\tau)^{1/2}$ and $\int(w-\alpha)_+p_\alpha(w)\,dw\le\sqrt\alpha$, the standard deviation of $p_\alpha$. On $[\tau/2,\tau]$, the third bound in \eqref{cg-angular-3} gives at most $\tau\cdot C\alpha^{-1/2}e^{-\tau/20}(2\tau)^{1/2}$. Using $H(\tau)\ge(c/2)\tau^{-1/2}$ and the boundedness of $\tau^2e^{-\tau/20}$, we again find $K/H\le C\sqrt\alpha$. This proves the first bound in \eqref{cg-angular-1}. For the second, suppose first $\alpha<\tau\le4\alpha$. The bound $w^2\le\alpha^2+2\alpha(w-\alpha)_++(w-\alpha)_+^2$ gives $\E w^2\le\alpha^2+2\alpha K(\tau)/H(\tau)+K_2(\tau)/H(\tau)$, where $K_2(\tau)=\int_\alpha^\tau(w-\alpha)^2(\tau-w)^{-1/2}p_\alpha(w)\,dw$. The same substitution and the same split, with $x^2$ in place of $x$, give $K_2(\tau)\le C\alpha^{3/4}(T+1)^{-1/2}$. Then $K_2/H\le C\alpha$ and $\E w^2\le C\alpha^2$. If $\tau\ge4\alpha$, the integral of $w^2(\tau-w)^{-1/2}p_\alpha(w)$ over $[0,\tau/2]$ is at most $(2/\tau)^{1/2}\alpha(\alpha+1)$, and over $[\tau/2,\tau]$ it is at most $C\alpha^{-1/2}\tau^{5/2}e^{-\tau/20}$. Dividing by $H(\tau)\ge(c/2)\tau^{-1/2}$, and using that $\tau^3e^{-\tau/20}$ is bounded, we find $\E w^2\le C\alpha^2$. For (ii), we write $\omega=(\omega_1,(1-\omega_1^2)^{1/2}\eta)$ with $\eta$ in the unit sphere of the coordinates $2,\dots,k$. Under $\sigma^k$, the variable $v=1-\omega_1^2$ has density proportional to $v^{(k-3)/2}(1-v)^{-1/2}$ and is independent of $\eta$, which is uniform. In these coordinates, $\mathcal A(\omega)=v\sum_{i\ge2}a_i\eta_i^2$, and $S\mathcal A=\tau v$ with $\tau=S\sum_{i\ge2}a_i\eta_i^2\ge0$. Given $\eta$, the law of $v$ under $\vartheta_S$ is then $\nu_{\alpha,\tau}$ with $\alpha=(k-1)/2\ge2$. The bounds \eqref{cg-angular-1} are uniform in $\tau$, and integrating them over $\eta$, we obtain \eqref{cg-angular-2}; in the second bound, we bounded the variance by the second moment. The sign of $\omega_1$ plays no role in this argument, and the same bounds hold under $\vartheta_S^+$. \end{proof} \begin{lemma}[A drift inequality for the remaining coordinates]\label{lem:ineq-drift} Assume \eqref{cg-folded-H1}. For $\zeta\in(0,1/2]$, let $\tilde a_j=a_j-\zeta a\ge a/2$ for $j>k$, and \[ Q_\zeta(y)=\sum_{j>k}\frac{y_j^2}{2\tilde a_j},\qquad R_\zeta=\frac{Q_\zeta}S,\qquad D_\zeta=N-\sum_{j>k}\frac1{2\tilde a_j}. \] \begin{enumerate}[label=(\roman*)] \item On $B_N$, \begin{equation}\label{cg-angular-5} L_{\mathrm{rem}}R_\zeta=2-\frac{2D_\zeta}S-4\zeta aR_\zeta+4\langle\mathcal A\rangle_SR_\zeta-\frac{2(k-4)R_\zeta}S . \end{equation} \item There exists $k_0(\zeta)$, depending only on $\zeta$, such that if $k\ge k_0(\zeta)$ and \begin{equation}\label{cg-folded-H2} D_\zeta\ge0, \end{equation} then on $B_N$, \begin{equation}\label{cg-angular-6} L_{\mathrm{rem}}R_\zeta\le2-2\zeta aR_\zeta . \end{equation} \item We have $\mu_{\mathrm{rem}}(L_{\mathrm{rem}}R_\zeta)=0$. For $w=(1+aR_\zeta)^{1/2}$ and every bounded $u\in C^1(B_N)$ with $\int\Gamma_y(u)\,d\mu_{\mathrm{rem}}<\infty$, \[ \int u^2\,\frac{-L_{\mathrm{rem}}w}{w}\,d\mu_{\mathrm{rem}}\le\int\Gamma_y(u)\,d\mu_{\mathrm{rem}} . \] \end{enumerate} \end{lemma} \begin{proof} For (i), we compute $L_{\mathrm{rem}}u=\operatorname{div}(M_y\nabla u)+\nabla\log\rho\cdot M_y\nabla u$, with \[ \nabla\log\rho=-2(a_jy_j)_{j>k}-2y\Big(\frac{k/2-1}S-\langle\mathcal A\rangle_S\Big), \] where we used $\nabla S=-2y$ and $\psi'=-\langle\mathcal A\rangle_S$. For a smooth $u$, we have $\operatorname{div}(M_y\nabla u)=\Delta u-((N-k+1)\,y\cdot\nabla u+y^\T\nabla^2u\,y)/N$. For $u=S$, we have $M_y\nabla S=-2Sy/N$ and $\operatorname{div}(M_y\nabla S)=-2(N-k)+2(N-k+2)|y|^2/N$. For $u=Q_\zeta$, we have $M_y\nabla Q_\zeta=\nabla Q_\zeta-2Q_\zeta y/N$ and $\operatorname{div}(M_y\nabla Q_\zeta)=\sum_{j>k}\tilde a_j^{-1}-2(N-k+2)Q_\zeta/N$. We also use $\sum_{j>k}a_jy_j^2/\tilde a_j=|y|^2+2\zeta aQ_\zeta$, $|y|^2=N-S$ and $\sum_{j>k}\tilde a_j^{-1}=2N-2D_\zeta$. Collecting terms, we obtain, with $V_*=V+S\langle\mathcal A\rangle_S$, \[ L_{\mathrm{rem}}S=2k-2S-4S\langle\mathcal A\rangle_S+\frac{4SV_*}N,\qquad L_{\mathrm{rem}}Q_\zeta=2S-2D_\zeta-(2+4\zeta a)Q_\zeta+\frac{4Q_\zeta V_*}N, \] \[ \Gamma_y(S)=4S\Big(1-\frac SN\Big),\qquad\Gamma_y(Q_\zeta,S)=-\frac{4Q_\zeta S}N . \] The quotient rule $L(Q/S)=L Q/S-QLS/S^2-2\Gamma(Q,S)/S^2+2Q\Gamma(S)/S^3$ then gives \eqref{cg-angular-5}; the terms containing $V_*$ and the terms of order $R_\zeta/N$ cancel. For (ii), by \eqref{cg-angular-2}, we have $4\langle\mathcal A\rangle_S-2(k-4)/S\le(6+C_4\sqrt k)/S$ with $C_4=2\sqrt2\,C_1$. Let $L_\zeta=\max\{1,(C_4+6/\sqrt5)/(2\zeta)\}$ and $k_0(\zeta)=\max\{8,\lceil16L_\zeta^2\rceil\}$. If $S\ge L_\zeta\sqrt k/a$, then $(6+C_4\sqrt k)/S\le(6/\sqrt5+C_4)a/L_\zeta\le2\zeta a$. If $S0$ near $\partial B_N$, where $|y|^2$ is close to $N$. Using $\nabla R_\zeta=\nabla Q_\zeta/S+2Q_\zeta y/S^2$ and $M_yy=Sy/N$, we find $|M_y\nabla R_\zeta|\le C/S$ and $\Gamma_y(R_\zeta)\le C/S^3$, and \eqref{cg-angular-5} gives $|L_{\mathrm{rem}}R_\zeta|\le C/S^2$. Integrating $L_{\mathrm{rem}}R_\zeta\,\rho=\operatorname{div}(\rho M_y\nabla R_\zeta)$ over $\{S>\varepsilon\}$, the boundary term is at most $C\varepsilon^{k/2-1}\varepsilon^{-1}=C\varepsilon^{k/2-2}$, which tends to zero because $k\ge5$. The function $S^{-2}$ is $\mu_{\mathrm{rem}}$-integrable, because $\int_0S^{-2}S^{k/2-1}dS<\infty$ for $k\ge5$, and dominated convergence gives $\mu_{\mathrm{rem}}(L_{\mathrm{rem}}R_\zeta)=0$. For $w=(1+aR_\zeta)^{1/2}$, we have $\nabla\log w=a\nabla R_\zeta/(2(1+aR_\zeta))$ and \[ \frac{-L_{\mathrm{rem}}w}{w}=\frac{-aL_{\mathrm{rem}}R_\zeta}{2(1+aR_\zeta)}+\frac{a^2\Gamma_y(R_\zeta)}{4(1+aR_\zeta)^2}. \] Near $\partial B_N$, we have $1+aR_\zeta\ge c/S$, which implies that $|L_{\mathrm{rem}}w/w|$ and $\Gamma_y(\log w)$ are at most $C/S$, a $\mu_{\mathrm{rem}}$-integrable function, and $|M_y\nabla w|/w\le C/S$. As in \cref{lem:ineq-tools}\ref{i:ineq-ground}, on $\{S>\varepsilon\}$ we have \[ \int_{\{S>\varepsilon\}}u^2\,\frac{-L_{\mathrm{rem}}w}w\,d\mu_{\mathrm{rem}}=\int_{\{S>\varepsilon\}}\big(2u\Gamma_y(u,\log w)-u^2\Gamma_y(\log w)\big)\,d\mu_{\mathrm{rem}}+O\big(\varepsilon^{k/2-2}\big), \] where the error is the boundary term, bounded using $\sup|u|$. The integrand on the right is at most $\Gamma_y(u)$, and it is integrable by the Cauchy--Schwarz inequality. Letting $\varepsilon\to0$ proves (iii). \end{proof} The next lemma combines a Poincar\'e inequality on a set $\Omega$ with an inequality of the type in \cref{lem:ineq-drift}(iii), which has a gain outside $\Omega$. This is the Lyapunov-function method; see \cite[Theorem~4.6.2, pp.~202--203]{BakryGentilLedoux2014}. \begin{lemma}[From a local to a global Poincar\'e inequality]\label{cg-local-global-form} Let $\mu$ be a probability measure and $\cE(u)=\int\Gamma(u)\,d\mu$ an energy with $\Gamma(u)\ge0$ and $\Gamma(u+c)=\Gamma(u)$ for constants $c$. Let $\Omega$ be measurable with $\mu(\Omega)>0$, and let $P_{\mathrm{loc}},\lambda>0$ and $B\ge0$. Suppose that, for every $u$ in a class of square-integrable functions of finite energy that is stable under the addition of constants, \[ \Var_{\mu(\cdot\mid\Omega)}(u)\le P_{\mathrm{loc}}\int\Gamma(u)\,d\mu(\cdot\mid\Omega),\qquad \int u^2\big(\lambda\ind_{\Omega^c}-B\ind_\Omega\big)\,d\mu\le\cE(u). \] Then $\Var_\mu(u)\le\big[\lambda^{-1}+(1+B/\lambda)P_{\mathrm{loc}}\big]\cE(u)$ for every such $u$. \end{lemma} \begin{proof} Let $c=\int u\,d\mu(\cdot\mid\Omega)$ and $v=u-c$. The first hypothesis and $\Gamma\ge0$ give $\int_\Omega v^2\,d\mu\le P_{\mathrm{loc}}\int_\Omega\Gamma(u)\,d\mu\le P_{\mathrm{loc}}\cE(u)$. The second hypothesis, applied to $v$, gives $\lambda\int_{\Omega^c}v^2\,d\mu\le\cE(u)+B\int_\Omega v^2\,d\mu$. Adding the two bounds, we find $\int v^2\,d\mu\le[\lambda^{-1}+(1+B/\lambda)P_{\mathrm{loc}}]\cE(u)$, and $\Var_\mu(u)\le\int v^2\,d\mu$. \end{proof} If the class contains a function $u$ with $\mu(u^2)+\cE(u)<\infty$, then applying the second hypothesis to $u+c$, dividing by $c^2$, and letting $|c|\to\infty$, we find $\lambda\mu(\Omega^c)-B\mu(\Omega)\le0$, that is, $\mu(\Omega)\ge\lambda/(\lambda+B)$. In particular, the lemma does not apply to sets of arbitrarily small mass with fixed $\lambda$ and $B$. We first use it on the unit hemisphere, where it yields the angular part of the estimate. \begin{lemma}[Poincar\'e inequality on a hemisphere with a quadratic potential]\label{lem:ineq-tilted} Let $k\ge5$, let $t_2,\dots,t_k\ge t\ge8k$, and let $\vartheta$ be the probability measure on $S^{k-1}_+$ proportional to $\exp(-\sum_{i\ge2}t_i\omega_i^2)\,\sigma^k_+$, with the reflecting form of the unit hemisphere. Then \begin{equation}\label{cg-folded-7} C_{\mathrm P}(\vartheta)\le\frac{14}t . \end{equation} \end{lemma} No upper bound on the $t_i$ is needed. \begin{proof} Let $z=\omega_1^2$, $F_t(\omega)=\sum_{i\ge2}t_i\omega_i^2\ge t(1-z)$, and let $L_t$ be the generator of $\vartheta$. On the unit sphere, we have $\Delta\omega_1^2=2-2kz$, and the tangential inner product of the gradients of $F_t$ and $z$ is $-4F_tz$. Then \[ L_tz=2-2kz+4F_tz,\qquad\Gamma(z)=4z(1-z). \] Let $\varepsilon=4/t$ and $w=(z+\varepsilon)^{-1}$. Then \[ \frac{-L_tw}{w}=\frac{L_tz}{z+\varepsilon}-\frac{8z(1-z)}{(z+\varepsilon)^2}. \] On $\{z\le1/4\}$, we have $F_t\ge3t/4$ and $2k\le t/4$, so that $L_tz\ge2+\frac52tz$ and \[ \frac{-L_tw}{w}\ge\frac{2\varepsilon-6z}{(z+\varepsilon)^2}+\frac{\frac52tz}{z+\varepsilon}\ge\frac t2 . \] For the last inequality, write $s=tz$; after multiplying by $(s+4)^2/t$, the difference of the two sides is $8-6s+\frac52s(s+4)-\frac12(s+4)^2=2s^2\ge0$. Everywhere, dropping $F_t\ge0$ and using $z/(z+\varepsilon)^2\le1/(4\varepsilon)$, we have $-L_tw/w\ge-2k-t/2\ge-t$. The function $w$ depends on $\omega_1$ only through $\omega_1^2$, and its normal derivative on the boundary $\{\omega_1=0\}$ vanishes. The argument of \cref{lem:ineq-tools}\ref{i:ineq-ground} then shows that $\int u^2(-L_tw/w)\,d\vartheta\le\int\Gamma(u)\,d\vartheta$ for every smooth $u$ on the closed hemisphere. This is the second hypothesis of \cref{cg-local-global-form}, with $\Omega=\{z>1/4\}$, $\lambda=t/2$ and $B=t$. On $\Omega$, we use the coordinates $v=(\omega_2,\dots,\omega_k)$ in the ball $\{|v|^2<3/4\}$, with $\omega_1=(1-|v|^2)^{1/2}$. In these coordinates, the uniform measure has density proportional to $z^{-1/2}$, the metric is $I+vv^\T/z$, and its inverse is $I-vv^\T\succeq zI\succeq\frac14I$. The density of $\vartheta$ is proportional to $e^{-U}$ with $U(v)=\sum_{i\ge2}t_iv_i^2+\frac12\log(1-|v|^2)$, and \[ \nabla^2U=2\diag(t_i)-\frac Iz-\frac{2vv^\T}{z^2}\succeq(2t-4-24)I\succeq t\,I, \] since $z\ge1/4$, $|v|^2\le3/4$ and $t\ge8k\ge40$. A smooth spherical test is smooth on a neighborhood of the closure of this chart ball and, after multiplication by a cutoff outside that neighborhood, admits an ambient smooth extension. By \cref{lem:ineq-tools}\ref{i:ineq-bl}, $\Var_{\vartheta(\cdot\mid\Omega)}(u)\le t^{-1}\int|\nabla_vu|^2\,d\vartheta(\cdot\mid\Omega)\le4t^{-1}\int\Gamma(u)\,d\vartheta(\cdot\mid\Omega)$. \Cref{cg-local-global-form} with $P_{\mathrm{loc}}=4/t$ gives $C_{\mathrm P}(\vartheta)\le2/t+3\cdot4/t=14/t$. \end{proof} \begin{lemma}[Poincar\'e inequality on the hemisphere with a few small precisions]\label{cg-folded-pi} Assume \eqref{cg-folded-H1}, and suppose that \eqref{cg-folded-H2} holds for some $\zeta\in(0,1/2]$. There exist $k_0(\zeta)$ and $C(\zeta)<\infty$, depending only on $\zeta$, such that if $k\ge k_0(\zeta)$, then \begin{equation}\label{cg-folded-P} C_{\mathrm P}(\mu_+)\le C(\zeta)\exp\Big\{C(\zeta)k^2+\frac{8k\bar a}{a_2}\Big\}\Big(\frac ka+\frac1{a_2}\Big). \end{equation} The same bound holds for the coordinate-even sector. \end{lemma} The lemma assumes no upper bound on the coefficients $a_j$, $j>k$. It does not control the full sphere: functions that are odd in $x_1$ are treated in \cref{cg-full-sphere-bridge}. \begin{proof} We decompose the variance of $f$ into the variance of its conditional mean given $y$ and the conditional variance given $y$. The marginal law $\mu_{\mathrm{rem}}$ of $y$ satisfies a Poincar\'e inequality by \cref{cg-local-global-form}: on a region where $S$ is not small, a convexity argument applies after a bounded modification of the density (Steps~2 and~3), and \cref{lem:ineq-drift} controls the complement of this region (Step~4). The angular law is controlled by \cref{lem:ineq-tilted} and a bounded perturbation (Step~1). Step~5 combines the two, using an exact formula for the gradient of the conditional mean. \Cref{tab:ineq-folded} lists the constants, in the order in which they are chosen. By density, it suffices to consider $f$ smooth on the closed hemisphere $\{x_1\ge0\}$. \begin{table}[ht] \centering \caption{Constants in the proof of \cref{cg-folded-pi}, in the order of their choice.}\label{tab:ineq-folded} \begin{tabular}{lll} \toprule Constant & Choice & Depends on\\ \midrule $C_A,C_V$ & \cref{lem:ineq-angular} & nothing\\ $L_\zeta$, $k_0(\zeta)$ & \cref{lem:ineq-drift}(ii) & $\zeta$\\ $K_0$ & $2/\zeta$ & $\zeta$\\ $C_T$ & $\max\{1,4C_A/\zeta,32C_VK_0\}$ & $\zeta$\\ $\lambda_0$, $T$ & $a/(8K_0)$, $C_Tk^2/a$ & $\zeta$, $k$, $a$\\ $\Theta$ & $\exp(8k\bar a/a_2)$ & $k$, $\bar a/a_2$\\ \bottomrule \end{tabular} \end{table} \paragraph{Step 1: the angular law.} By \cref{lem:ineq-angular}(ii) and $\langle\mathcal A\rangle_S\le\bar a$, \begin{equation}\label{cg-folded-1} \langle\mathcal A\rangle_S\le\min\Big\{\bar a,\frac{C_Ak}S\Big\},\qquad\Var_{\vartheta_S}(\mathcal A)\le\frac{C_Vk^2}{S^2}. \end{equation} Let $\Theta=\exp(8k\bar a/a_2)$. If $Sa_2\ge8k$, \cref{lem:ineq-tilted} with $t_i=Sa_i$ and $t=Sa_2$ implies that $C_{\mathrm P}(\vartheta_S^+)\le14/(Sa_2)$. If $Sa_2<8k$, then $S\mathcal A$ has oscillation at most $S\bar a\le\log\Theta$, and \cref{lem:ineq-tools}\ref{i:ineq-perturb},\ref{i:ineq-sphere} give $C_{\mathrm P}(\vartheta_S^+)\le\Theta/(k-1)$. In both cases, for every $S>0$ and every smooth $g$ on the closed unit hemisphere, \begin{equation}\label{cg-folded-8} \Var_{\vartheta_S^+}(g)\le\frac{2\Theta}k\int|\nabla_\omega g|^2\,d\vartheta_S^+,\qquad \Var_{\vartheta_S^+}(g)\le\frac{14\Theta}{a_2}\int S^{-1}|\nabla_\omega g|^2\,d\vartheta_S^+ . \end{equation} For the first bound we used $14/(Sa_2)\le14/(8k)$ and $1/(k-1)\le2/k$; for the second, $\Theta/(k-1)\le(8k/(Sa_2))\Theta/(k-1)\le10\Theta/(Sa_2)$ when $Sa_2<8k$ and $k\ge5$. \paragraph{Step 2: a modification of the marginal density.} Recall that $\psi=\log Z$ is convex with $-\bar a\le\psi'\le0$. Let $T=C_Tk^2/a$, and let $\tilde\psi$ be the $C^1$ convex function equal to $\psi$ on $[T,\infty)$ and to its tangent line at $T$ on $[0,T)$; if $T>N$, only the tangent line is used on $[0,N]$. For $Sk}\Big(\frac1{2\tilde a_j}+\frac{K_0}a\Big)y_j^2<\frac{K_0N}a\Big\}, \] with $R_\zeta$, $Q_\zeta$ and $\tilde a_j$ as in \cref{lem:ineq-drift}. This is a nonempty convex ellipsoid with smooth boundary, and its closure lies in $B_N$, since $Q_\zeta\le K_0S/a$ on $\bar\Omega$ forces $S>0$. Similarly, $\Omega'=\{R_\zeta<2K_0/a\}$ is an open neighborhood of $\bar\Omega$ whose closure is a compact subset of $B_N$. The function $\tilde U$ belongs to $C^{1,1}_{\mathrm{loc}}(B_N)$, and in particular to $C^{1,1}(\Omega')$. Almost everywhere, \begin{equation}\label{cg-folded-3} \nabla^2\tilde U=2\diag(a_j)_{j>k}+\Big(\frac{k-2}S+2\tilde\psi'\Big)I+\Big(\frac{2(k-2)}{S^2}-4\tilde\psi''\Big)yy^\T . \end{equation} We claim that $\nabla^2\tilde U\succeq\lambda_0M_y^{-1}$ almost everywhere on $\Omega'$, where $M_y^{-1}=I+yy^\T/S$. First, $\tilde\psi'\ge-C_Ak/T$ on $(0,N]$ by \eqref{cg-folded-1}, and $\tilde\psi''=0$ below $T$ while $0\le\tilde\psi''\le C_Vk^2/S^2$ above $T$. The diagonal part of $\nabla^2\tilde U-\lambda_0M_y^{-1}$ is \[ \Lambda_y=2\diag(a_j)+\Big(\frac{k-2}S+2\tilde\psi'-\lambda_0\Big)I\succeq2\diag(\tilde a_j), \] because $2C_Ak/T+\lambda_0\le\zeta a/(2k)+\zeta a/16\le2\zeta a$ by the choice of $C_T$. The remaining part is $\xi\,yy^\T$ with $\xi=2(k-2)/S^2-4\tilde\psi''-\lambda_0/S$. If $\xi\ge0$, there is nothing to prove. If $\xi<0$, then $\Lambda_y+\xi yy^\T\succeq0$ as soon as $|\xi|\,y^\T\Lambda_y^{-1}y\le1$, by the Cauchy--Schwarz inequality $(y^\T e)^2\le(y^\T\Lambda_y^{-1}y)(e^\T\Lambda_ye)$. We have $y^\T\Lambda_y^{-1}y\le Q_\zeta$ and $|\xi|\le\lambda_0/S+4\tilde\psi''$. On $\Omega'$ with $S\le T$, this gives $|\xi|y^\T\Lambda_y^{-1}y\le\lambda_0R_\zeta<2\lambda_0K_0/a=1/4$. On $\Omega'$ with $S>T$, it gives at most $(\lambda_0+4C_Vk^2/T)R_\zeta<1/4+8C_VK_0/C_T\le1/2$. This proves the claim. The matrix field $M_y^{-1}$ is continuous and positive definite on $B_N$. Applying \cref{lem:ineq-tools}\ref{i:ineq-bl} with $\Lambda=\lambda_0M_y^{-1}$, we obtain, for ambient $C^1$ tests $u$, \[ \Var_{\tilde\mu_{\mathrm{rem}}(\cdot\mid\Omega)}(u)\le\frac1{\lambda_0}\int(\nabla u)^\T M_y\nabla u\,d\tilde\mu_{\mathrm{rem}}(\cdot\mid\Omega)=\frac{8K_0}a\int\Gamma_y(u)\,d\tilde\mu_{\mathrm{rem}}(\cdot\mid\Omega). \] By \eqref{cg-folded-2} and \cref{lem:ineq-tools}\ref{i:ineq-perturb}, applied to the conditioned measures, \begin{equation}\label{cg-folded-4} \Var_{\mu_{\mathrm{rem}}(\cdot\mid\Omega)}(u)\le P_{\mathrm{loc}}\int\Gamma_y(u)\,d\mu_{\mathrm{rem}}(\cdot\mid\Omega),\qquad P_{\mathrm{loc}}=\frac{8K_0}a\,e^{C_Tk^2}. \end{equation} \paragraph{Step 4: the Poincar\'e inequality for the marginal.} Let $w=(1+aR_\zeta)^{1/2}$. By \eqref{cg-angular-6} and $\Gamma_y(R_\zeta)\ge0$, we have \[ \frac{-L_{\mathrm{rem}}w}w\ge\frac{-aL_{\mathrm{rem}}R_\zeta}{2(1+aR_\zeta)}\ge a\,\frac{\zeta aR_\zeta-1}{1+aR_\zeta}. \] The right side is at least $-a$ everywhere. It is increasing in $aR_\zeta$, and for $aR_\zeta\ge K_0=2/\zeta$ it is at least $a\zeta/(\zeta+2)\ge\zeta a/4$. With $\lambda=\zeta a/4$, this means that \begin{equation}\label{cg-folded-5} \frac{-L_{\mathrm{rem}}w}w\ge\lambda\ind_{\Omega^c}-a\ind_\Omega . \end{equation} By \cref{lem:ineq-drift}(iii), the second hypothesis of \cref{cg-local-global-form} holds with $B=a$ for every bounded $u\in C^1(B_N)$ with finite energy, and \eqref{cg-folded-4} is the first hypothesis. Applying \cref{cg-local-global-form}, we obtain, for every such $u$, \begin{equation}\label{cg-folded-6} \Var_{\mu_{\mathrm{rem}}}(u)\le P\int\Gamma_y(u)\,d\mu_{\mathrm{rem}},\qquad P=\frac1\lambda+\Big(1+\frac a\lambda\Big)P_{\mathrm{loc}}\le\frac{C'(\zeta)}a\,e^{C_Tk^2}, \end{equation} where $C'(\zeta)$ depends only on $\zeta$. \paragraph{Step 5: the conditional mean and the conclusion.} For $y\in B_N$, let $m(y)=\int f(\sqrt S\omega,y)\,\vartheta_S^+(d\omega)$. Differentiating the conditional law, whose density is proportional to $e^{-S\mathcal A}$, and using $\nabla S=-2y$, we find \[ \nabla_ym=\int\nabla_yf\,d\vartheta_S^++2y\Cov_{\vartheta_S^+}(f,\mathcal A). \] By Jensen's inequality for the quadratic form $M_y$, the Cauchy--Schwarz inequality, $y^\T M_yy=|y|^2S/N\le S$, \eqref{cg-folded-1}, and the first bound in \eqref{cg-folded-8}, \begin{equation}\label{cg-folded-9} \begin{aligned} \Gamma_y(m)&\le2\int\Gamma_y(f)\,d\vartheta_S^++8S\Var_{\vartheta_S^+}(\mathcal A)\Var_{\vartheta_S^+}(f)\\ &\le2\int\Gamma_y(f)\,d\vartheta_S^++16C_Vk\Theta\int S^{-1}|\nabla_\omega f|^2\,d\vartheta_S^+ . \end{aligned} \end{equation} The function $m$ is bounded and $C^1$ on $B_N$, and $\Gamma_y(f)+S^{-1}|\nabla_\omega f|^2=\Gamma(f)$ is bounded. Further, $m$ has finite energy by \eqref{cg-folded-9}, and \eqref{cg-folded-6} applies to it. By the conditional variance formula, the second bound in \eqref{cg-folded-8}, \eqref{cg-folded-6} and \eqref{cg-folded-9}, \[ \begin{aligned} \Var_{\mu_+}(f)&=\int\Var_{\vartheta_S^+}(f)\,d\mu_{\mathrm{rem}}+\Var_{\mu_{\mathrm{rem}}}(m)\\ &\le2P\int\Gamma_y(f)\,d\mu_++\Big(\frac{14\Theta}{a_2}+16C_Vk\Theta P\Big)\int S^{-1}|\nabla_\omega f|^2\,d\mu_+ . \end{aligned} \] By \eqref{eq:ineq-energy-split}, the two integrals add up to $\cE_{\mu_+}(f)$. Using $\Theta\ge1$, $2+16C_Vk\le(2+16C_V)k$ and the bound on $P$ in \eqref{cg-folded-6}, we see that the right side is at most $\big((2+16C_V)C'(\zeta)\Theta e^{C_Tk^2}k/a+14\Theta/a_2\big)\cE_{\mu_+}(f)$. This proves \eqref{cg-folded-P} with $C(\zeta)=\max\{C_T,14,(2+16C_V)C'(\zeta)\}$. Finally, since $\mu_+$ is invariant under $R_2,\dots,R_N$, the argument of \cref{lem:ineq-reflection} bounds the constant of the coordinate-even sector by $C_{\mathrm P}(\mu_+)$. \end{proof} \begin{lemma}[From the hemisphere to the full sphere]\label{cg-full-sphere-bridge} Assume the hypotheses of \cref{cg-folded-pi}, and suppose in addition that \[ D_0=N-\sum_{j>k}\frac1{2a_j}\le\frac{M_0k}a \] for some $M_0\ge0$. Let $M_1=M_0+1/\zeta$ and $T_1=2M_1k/a$. Then \[ C_{\mathrm P}(\mu)\le5C_{\mathrm P}(\mu_+)+\frac{4T_1e^{\bar aT_1}}{k-1}. \] \end{lemma} \begin{proof} We first show that $\mu(S\le T_1)\ge1/2$. Let $R_0=\sum_{j>k}y_j^2/(2a_jS)$. For $j>k$, we have $\partial_j\log\rho=-2a_jy_j-(k-2)y_j/S+2\langle\mathcal A\rangle_Sy_j$, and the divergence of the vector field $\rho(y)y_je_j/(2a_j)$ is $\rho(1+y_j\partial_j\log\rho)/(2a_j)$. We integrate it over $\{S>\varepsilon\}$. The boundary term is $O(\varepsilon^{k/2-1})$, and the integrand is $O(1/S)$ near $\partial B_N$, which is integrable. Letting $\varepsilon\to0$ and summing over $j>k$, we obtain \[ \sum_{j>k}\frac1{2a_j}=\E_{\mu_{\mathrm{rem}}}[N-S]+(k-2)\E_{\mu_{\mathrm{rem}}}[R_0]-2\E_{\mu_{\mathrm{rem}}}\big[S\langle\mathcal A\rangle_SR_0\big], \] so that $\E S\le D_0+(k-2)\E R_0$. We have $R_0\le R_\zeta$, because $a_j\ge\tilde a_j$, and \cref{lem:ineq-drift}(iii) and \eqref{cg-angular-6} give $0=\E[L_{\mathrm{rem}}R_\zeta]\le2-2\zeta a\E R_\zeta$. We obtain $\E R_0\le1/(\zeta a)$ and $\E S\le M_0k/a+k/(\zeta a)=M_1k/a$. By Markov's inequality, the set $B_0=\{S\le T_1\}$ satisfies $\mu(B_0)\ge1/2$. For $S\le T_1$, the conditional law $\vartheta_S$ on the full unit sphere has potential $S\mathcal A$ with oscillation at most $\bar aT_1$. By \cref{lem:ineq-tools}\ref{i:ineq-perturb},\ref{i:ineq-sphere}, for every smooth $g$ on $S^{k-1}$, \[ \Var_{\vartheta_S}(g)\le\frac{e^{\bar aT_1}}{k-1}\int|\nabla_\omega g|^2\,d\vartheta_S\le J_1\int S^{-1}|\nabla_\omega g|^2\,d\vartheta_S,\qquad J_1=\frac{T_1e^{\bar aT_1}}{k-1}. \] Let $f$ be smooth on $\Sph$, and let $f_{\mathrm e}$ and $f_{\mathrm o}$ be its even and odd parts under $R_1$. Since $\mu$ and $\Gamma$ are invariant under $R_1$, the cross terms are odd and integrate to zero, so that $\Var_\mu(f)=\Var_\mu(f_{\mathrm e})+\mu(f_{\mathrm o}^2)$ and $\cE_\mu(f)=\cE_\mu(f_{\mathrm e})+\cE_\mu(f_{\mathrm o})$. As in \cref{lem:ineq-reflection}, $\Var_\mu(f_{\mathrm e})\le C_{\mathrm P}(\mu_+)\cE_\mu(f_{\mathrm e})$. For the odd part, for each $y$, the function $\omega\mapsto f_{\mathrm o}(\sqrt S\omega,y)$ is odd in $\omega_1$, and its mean under $\vartheta_S$ vanishes. By the last display and \eqref{eq:ineq-energy-split}, we have \[ \int_{B_0}f_{\mathrm o}^2\,d\mu=\int_{\{S\le T_1\}}\Var_{\vartheta_S}(f_{\mathrm o})\,d\mu_{\mathrm{rem}}\le J_1\,\cE_\mu(f_{\mathrm o}). \] Let $m=\mu_+(f_{\mathrm o})$. The restriction of $f_{\mathrm o}$ to the hemisphere is a function of the reflecting form, with no boundary condition, and $f_{\mathrm o}^2$, $\Gamma(f_{\mathrm o})$ and $\ind_{B_0}$ are even. Using $\mu_+(B_0)=\mu(B_0)\ge1/2$, we find \[ \frac{m^2}2\le m^2\mu_+(B_0)\le2\int_{B_0}f_{\mathrm o}^2\,d\mu_++2\int_{B_0}(f_{\mathrm o}-m)^2\,d\mu_+\le2J_1\cE_\mu(f_{\mathrm o})+2C_{\mathrm P}(\mu_+)\cE_\mu(f_{\mathrm o}). \] We conclude that \[ \mu(f_{\mathrm o}^2)=\mu_+(f_{\mathrm o}^2)=\Var_{\mu_+}(f_{\mathrm o})+m^2\le\big(5C_{\mathrm P}(\mu_+)+4J_1\big)\cE_\mu(f_{\mathrm o}). \] Combining the two parts proves the lemma. Note that the indicator of $B_0$ is only integrated, never differentiated. \end{proof} \section{The critical window}\label{sec:ineq-critical} We now prove the Poincar\'e and logarithmic Sobolev inequalities used in the critical window. In this section, $r_1\le r_2\le\dots\le r_N$ are positive half-precisions with $\sum_i(2r_i)^{-1}=N$, and $\pi_r$ is the probability measure proportional to $e^{-\sum_ir_ix_i^2}\nu_N$. We abbreviate $b=N^{2/3}$. In the application to the GOE, $r_i=(z_N-c_N\lambda_i)/2$ with $c_N>0$ and $\lambda_1\ge\dots\ge\lambda_N$. These half-precisions are ordered, and on the events of \cref{cg-canonical-weyl} they satisfy \eqref{cg-weyl-hypotheses}. \begin{lemma}[Poincar\'e and logarithmic Sobolev constants for half-precisions of order $(i/N)^{2/3}$]\label{lem:ineq-critical-stationary} Let $00$. There exist $N_0$ and $C$, depending only on $c_*$, $C_*$ and $\delta_*$, such that the following holds for $N\ge N_0$. Suppose that $00$. This is \eqref{cg-folded-H1}. From \eqref{eq:ineq-weyl-type} and the choice of $J$, \[ \frac{c_*(J+1)^{2/3}}{2b}\le\frac{c_*(J+1)^{2/3}-C_*}b\le a\le r_{J+1}\le\frac{2C_*J^{2/3}}b,\qquad\bar a\le r_J\le\frac{C_*J^{2/3}}b . \] We check \eqref{cg-folded-H2}. By the choice of $\zeta$ and $c_*\le C_*$, we have $\zeta a\le c_*J^{2/3}/(48b)$, and $r_1\le C_*/b\le c_*(J+1)^{2/3}/(2b)$. For $j>J$, it follows that $r_1+\zeta a\le\frac34c_*(J+1)^{2/3}/b\le\frac34r_j$, that is, $\tilde a_j=r_j-r_1-\zeta a\ge r_j/4$. By the trace condition $\sum_i(2r_i)^{-1}=N$, we have \[ D_\zeta=\sum_{i\le J}\frac1{2r_i}-\sum_{j>J}\frac{r_1+\zeta a}{2r_j\tilde a_j}. \] The first sum is at least $J/(2r_J)\ge bJ^{1/3}/(2C_*)$. Using $\sum_{j>J}j^{-4/3}\le3J^{-1/3}$ and $r_1+\zeta a\le(C_*/b)(1+2\zeta J^{2/3})$, we see that the second sum is at most \[ 2(r_1+\zeta a)\sum_{j>J}\frac1{r_j^2}\le\frac{6C_*b}{c_*^2}\big(J^{-1/3}+2\zeta J^{1/3}\big)\le\frac{bJ^{1/3}}{4C_*}, \] where the last inequality uses $J^{2/3}\ge48C_*^2/c_*^2$ for the first term and $\zeta\le c_*^2/(96C_*^2)$ for the second. Then $D_\zeta>0$. Moreover, since $a_j\le r_j$ and $\tilde a_j\le a_j$, \[ 0\le D_\zeta\le D_0\le\sum_{i\le J}\frac1{2r_i}\le\frac b{2c_*}\sum_{i\le J}i^{-2/3}\le\frac{3bJ^{1/3}}{2c_*}\le\frac{3C_*}{c_*}\,\frac Ja, \] so the hypothesis of \cref{cg-full-sphere-bridge} holds with $M_0=3C_*/c_*$. We have $8J\bar a/a_2\le8C_*J^{5/3}/\delta_*$, $J/a\le2J^{1/3}b/c_*$ and $1/a_2\le b/\delta_*$, and \cref{cg-folded-pi} implies that $C_{\mathrm P}(\pi_{r,+})\le Cb$, where $\pi_{r,+}=\pi_r(\cdot\mid x_1\ge0)$. In \cref{cg-full-sphere-bridge}, we have $T_1=2M_1J/a\le4M_1J^{1/3}b/c_*$ and $\bar aT_1\le4M_1C_*J/c_*$, and the lemma implies that $C_{\mathrm P}(\pi_r)\le Cb$. For the entropy, the variances $v_i=1/(2r_i)$ satisfy $\max_iv_i=v_1\le b/(2c_*)$ and $v_i/v_1=r_1/r_i\ge c_*/(3^{2/3}C_*)$ for $i\le3$. \Cref{cg-three-soft-lsi} implies that $C_{\mathrm{LS}}(\pi_r)\le C(v_1+C_{\mathrm P}(\pi_r))\le Cb$. The bounds for hemispheres and for the coordinate-even sector follow from \cref{lem:ineq-reflection}. \end{proof} We now state the result in the form used later. Part (ii) concerns the Bingham comparison law of \cref{def:heat-covariance} for the matrix with half-precisions $r_i$. With $H=-4\diag(r_1,\dots,r_N)+\frac2N\ind\ind^\T$ and $h(t)=e^{tH}\ind$, as in \eqref{eq:heat-objects} with $p_i=2r_i$, its Gaussian variance parameters are, by \eqref{eq:heat-errors}, \[ v_i(t)=\frac{h_i(t)}{\bar h(t)},\qquad\bar h(t)=\frac1N\sum_jh_j(t), \] and $\mu_t=\mu_{v(t)}$ is proportional to $e^{-\sum_ir_i(t)x_i^2}\nu_N$ with $r_i(t)=1/(2v_i(t))=\bar h(t)/(2h_i(t))$. Since $\sum_iv_i(t)=N$, these are the half-precisions of $\mu_t$ in the sense of \cref{sec:ineq-setting}, and $\mu_0=\nu_N$. We emphasize that $\mu_t$ is a comparison law; in general, it need not coincide with the law $\nu_NP_t$ of the diffusion. \begin{proposition}[Poincar\'e and logarithmic Sobolev constants in the critical window]\label{cg-critical-lsi} Let $r_1\le\dots\le r_N$ satisfy \eqref{cg-weyl-hypotheses} with constants $c_0,C_0,\delta$, that is, $\sum_i(2r_i)^{-1}=N$, $c_0(i/N)^{2/3}\le r_i\le C_0(i/N)^{2/3}$ for all $i$, and $r_2-r_1\ge\delta N^{-2/3}$. \begin{enumerate}[label=(\roman*)] \item There exist $N_0$ and $C$, depending only on $c_0$, $C_0$ and $\delta$, such that for $N\ge N_0$, \[ C_{\mathrm P}(\pi_r)\le CN^{2/3},\qquad C_{\mathrm{LS}}(\pi_r)\le CN^{2/3}. \] \item For every $K<\infty$, there exist $N_0$ and $C_K$, depending only on $K$, $c_0$, $C_0$ and $\delta$, such that for $N\ge N_0$ and $0\le t\le KN^{2/3}$, \begin{equation}\label{cg-moving-lsi} C_{\mathrm P}(\mu_t)+C_{\mathrm{LS}}(\mu_t)\le C_K(1+t). \end{equation} \end{enumerate} These bounds hold for the full sphere, for every coordinate hemisphere with its reflecting form, and for the coordinate-even sector. \end{proposition} \begin{proof} Part (i) is \cref{lem:ineq-critical-stationary} with $(c_*,C_*,\delta_*)=(c_0,C_0,\delta)$. For part (ii), we may enlarge the time horizon by replacing $K$ with $\max\{K,1\}$, and hence assume $K\ge1$. The hypotheses \eqref{eq:heat-weyl} of \cref{cd-heat-profiles} hold; that lemma uses only the trace condition and the two-sided bounds on the $r_i$, for every $N\ge2$. It gives constants $00$ depending only on $K$, $c_0$ and $C_0$. Also $\mathfrak h_t(r)\le\mathfrak h_t(r_1)=h_1(t)\le C_K(1+t)^{1/2}$ and $\mathfrak r_t(r)\ge\mathfrak r_t(r_1)=r_1(t)\ge c_K/(1+t)$. Combining these bounds, we find $\mathfrak r_t'(r)\ge c't^2/(1+t)^2\ge c'/4$, since $t\ge1$. By the mean value theorem, $r_2(t)-r_1(t)\ge(c'/4)(r_2-r_1)\ge c'\delta/(4b)$. We have shown that the half-precisions $r_i(t)$ satisfy \eqref{eq:ineq-weyl-type}, and the trace condition, with \[ c_*=\frac{c_Kc_0}2,\qquad C_*=C_K\big(\varepsilon_0^{-1}+C_0\big),\qquad\delta_*=\frac{c'\delta}4, \] which depend only on $K$, $c_0$, $C_0$ and $\delta$. For $N\ge N_0(c_*,C_*,\delta_*)$, \cref{lem:ineq-critical-stationary} gives $C_{\mathrm P}(\mu_t)+C_{\mathrm{LS}}(\mu_t)\le Cb\le(C/\varepsilon_0)\,t$. Taking $N_0$ to be the largest of the thresholds above proves \eqref{cg-moving-lsi}. The bounds for hemispheres and for the coordinate-even sector follow from \cref{lem:ineq-reflection}. \end{proof} The restriction $N\ge N_0$ is used only to ensure that the fixed numbers $J$ and $k$ of eigencoordinates are smaller than $N$ and that $\varepsilon_0N^{2/3}\ge1$; in the applications $N\to\infty$.