Song–Zhang v1: analytic foundations and Appell variance estimates

Part of the first version of Song–Zhang, Chapter Song–Zhang, first version: polynomial estimates and curvature; the reading order is on the full proofs page.

Overview. This reconstructs the polynomial estimate of Song–Zhang, Song & Zhang, 2026, Sections 3–4, with the quadratic estimate Theorem 25.1 as its nonclassical input. All tensor norms sum over ordered indices. The proof constructs the localization globally, keeps the covariance–mean cross variation, and closes a finite induction in the polynomial degree. It also records the analytic foundations needed by the subsequent inverse-operator argument. No eigenfunction or inverse operator is passed through an approximation limit. The source version is arXiv:2610.01447v1.

Appell expansion identifies the derivative means that polynomial testing controls.
Quadratic variance controls every whitened third-moment noise coefficient.
A coupled tensor hierarchy controls the mean and fluctuating derivative energies.
Matrix strong convexity at a short terminal time closes the degree induction.
Cutoff and moment approximation remove all auxiliary regularity assumptions.

Statement and Appell coordinates¶

Definition 120.1 (Appell coefficients and ordered-index norms)

For a probability measure $\mu$ with all polynomial moments define the symmetric tensor polynomials $\mathcal A_k^\mu$ by the formal identity

\frac{e^{z\cdot x}}{\mathbb E_\mu e^{z\cdot X}} =\sum_{k\ge0}\frac{\langle\mathcal A_k^\mu(x),z^{\otimes k}\rangle}{k!}.

(120.1)

The quotient means division of formal power series with denominator constant term one. Set $P_k^\mu[T]=\langle T,\mathcal A_k^\mu\rangle$ and $K_k(\mu)=\sup_{T\text{ symmetric},\ \|T\|_{\mathrm{HS}}=1} \operatorname{Var}_\mu P_k^\mu[T]$ . The Hilbert–Schmidt norm uses the full tensor space, without dividing by a factorial. For a polynomial $q$ , $D^kq$ is its full symmetric derivative tensor.

Theorem 120.1 (Polynomial variance and derivative means)

For every isotropic log-concave probability $\mu$ on $\mathbb R^n$ , every integer $d\ge1$ and every symmetric $d$ -tensor $T$ , the unique polynomial $P$ satisfying $D^dP=d!T$ and $\mathbb E_\mu D^jP=0$ for $0\le j<d$ is $P_d^\mu[T]$ , and

\operatorname{Var}_\mu P_d^\mu[T]\le1024^d(d!)^4\|T\|_{\mathrm{HS}}^2.

(120.2)

For every polynomial $q$ of degree at most $s$ ,

\sqrt{\operatorname{Var}_\mu q} \le\sum_{k=1}^s32^k k!\|\mathbb E_\mu D^kq\|_{\mathrm{HS}}.

(120.3)

Both inequalities hold with the same constants for every centered log-concave probability of covariance at most $I$ , including singular covariance. These are the assertions of Theorem 7.1.

Formal differentiation followed by expectation gives $\mathbb E D^jP_k[T]=0$ for $j<k$ and $D^kP_k[T]=k!T$ . These conditions characterize $P_k[T]$ : the top derivative fixes its homogeneous part, and descending through the degrees fixes each remaining homogeneous part. Applying the same descending argument to an arbitrary polynomial gives the exact identity

q-\mathbb E q=\sum_{k=1}^s\frac1{k!}P_k^\mu[\mathbb E D^kq].

(120.4)

Consequently a bound $K_k(\mu)\le B_k$ implies the derivative-mean estimate with coefficients $\sqrt{B_k}/k!$ . This uses the triangle inequality in $L^2$ , with no orthogonality assertion. It applies to a polynomial valued in any finite-dimensional Hilbert space as well: first sum the scalar bounds in an orthonormal output basis, then apply the triangle inequality in the direct sum. Thus no output dimension is lost.

The quadratic input and global localization¶

For an isotropic log-concave vector $\xi$ set $S_i=\mathbb E[\xi_i\xi\xi^T]$ and $S_z=\sum_i z_iS_i$ . For every symmetric $D$ , Theorem 25.1 and Cauchy–Schwarz give

|\langle D,S_z\rangle| =|\mathbb E[(z\cdot\xi)(\xi^TD\xi-\operatorname{tr}D)]| \le\sqrt8|z|\|D\|_{\mathrm{HS}}.

(120.5)

Duality on symmetric matrices gives $\|S_z\|_{\mathrm{HS}}^2\le8|z|^2$ . Full symmetry of the third moment gives $\sum_i|S_i z|^2=\|S_z\|_{\mathrm{HS}}^2$ . Therefore

\sum_i S_i^2\preceq8I,\qquad \sum_i\operatorname{tr}(S_i^2)\le8n,\qquad \sum_i(\operatorname{tr}S_i)^2\le8n^2.

(120.6)

The last inequality follows from $(\operatorname{tr}S_i)^2\le n\operatorname{tr}(S_i^2)$ . No estimate on the full third-tensor norm independent of dimension is used.

Lemma 120.1 (Global covariance-adapted localization)

For compactly supported isotropic log-concave $\mu_0$ there is a global process

\mu_t(dx)=Z_t^{-1}e^{\theta_t\cdot x-x^T\Lambda_tx/2}\mu_0(dx),\qquad d\theta_t=A_t^{-1}m_t\,dt+A_t^{-1/2}\,d\beta_t, \qquad d\Lambda_t=A_t^{-1}\,dt,

(120.7)

starting at $(\theta_0,\Lambda_0)=(0,0)$ , where $m_t,A_t$ are its mean and covariance. Its covariance stays positive definite at every finite time. For bounded $q$ , writing $\mathbb E_t$ for integration against $\mu_t$ and $\xi_t=A_t^{-1/2}(x-m_t)$ ,

d\mathbb E_tq=\operatorname{Cov}_t(q,\xi_t)\cdot d\beta_t, \quad dm_t=A_t^{1/2}d\beta_t, \quad dA_t=\sum_iU_{i,t}d\beta_{i,t}-A_tdt, \quad U_i=A_t^{1/2}S_iA_t^{1/2}.

(120.8)

The first process is a true martingale. For $t>0$ , $\mu_t$ is strongly log-concave with curvature matrix $\Lambda_t$ in the extended-valued convex-potential sense.

Proof 120.1

Suppose the support lies in a ball of radius $\rho$ . For finite $(\theta,\Lambda)$ the partition function and its derivatives are smooth, by differentiation under the integral on compact parameter sets. The tilted measure has exactly the same null sets as $\mu_0$ , so its covariance is positive definite. The SDE coefficients are therefore locally Lipschitz and admit a unique solution up to their explosion time $\zeta$ .

Work first before leaving a compact parameter set. For $r_t(x)=e^{\theta_t\cdot x-x^T\Lambda_tx/2}$ , Itô’s formula gives $dr_t/r_t=x^TA_t^{-1}m_tdt+x^TA_t^{-1/2}d\beta_t$ : its quadratic variation cancels the derivative of the quadratic tilt. Integration gives the same identity for $Z_t$ with $x$ replaced by $m_t$ . The quotient rule then gives

d(r_t/Z_t)=(r_t/Z_t)(x-m_t)^TA_t^{-1/2}d\beta_t.

(120.9)

Stochastic Fubini is valid on these stops because the support and all coefficients are bounded. Taking $q=x$ gives the mean equation; taking $q=xx^T$ and subtracting $d(m_tm_t^T)$ gives the covariance equation, including drift $-A_tdt$ . The positive semidefinite tilt preserves log-concavity, so the third-moment estimates above hold for $\xi_t$ until $\zeta$ .

Apply Itô’s formula to the logarithmic determinant. Cyclicity of trace gives

\log\det A_t=M_t-nt-\frac12\int_0^t\sum_i\operatorname{tr}(S_{i,s}^2)ds \ge M_t-5nt,\qquad M_t=\int_0^t\sum_i\operatorname{tr}(S_{i,s})d\beta_{i,s}.

(120.10)

Its quadratic variation is at most $8n^2t$ . Extend the predictable integrand by zero after $\zeta$ ; the resulting integral is a continuous martingale on each deterministic finite interval $[0,T]$ . Its pathwise minimum on that interval is finite. Consequently $\det A_t$ has a positive pathwise lower bound for $t<T\wedge\zeta$ . Since $A_t\preceq\rho^2I$ , the smallest eigenvalue is at least $\det A_t/(\rho^2)^{n-1}$ . Thus $A_t^{-1}$ is pathwise bounded on that interval. Also $|m_t|\le\rho$ . The finite-variation parts of $\theta_t,\Lambda_t$ then have finite limits at a purported finite $\zeta$ , and the martingale part of $\theta_t$ has finite quadratic variation and a finite limit there. This last assertion follows by stopping when the preceding bounds exceed an integer, then taking their union. At the limiting finite parameters the coefficients are locally Lipschitz, allowing continuation, a contradiction. Thus $\zeta=\infty$ .

Every bounded posterior expectation is now a bounded local martingale, hence a true martingale. Its conditional-expectation consistency at different times is exactly the martingale property; no independent construction of an underlying signal is needed. Finally if $\mu_0(dx)=e^{-V_0(x)}dx$ with $V_0$ convex and possibly infinite, its posterior potential is $V_0-\theta_t\cdot x+\log Z_t+x^T\Lambda_tx/2$ . The integral defining $\Lambda_t$ is positive definite for $t>0$ , proving the claim.

Tensor hierarchy with all cross terms¶

Fix a polynomial $f$ of degree $d$ , and put

h_j(t)=\mathbb E_tD^jf,\quad K_j(t)=A_t^{\otimes j},\quad N_j(t)=\mathbb E_{\mathrm{loc}}\langle h_j,K_jh_j\rangle,

(120.11)

L_j(t)=\mathbb E_{\mathrm{loc}}\mathbb E_t \langle D^jf-h_j,K_j(D^jf-h_j)\rangle.

(120.12)

The tensor slots here concern input derivatives; any additional output slots are contracted with their ordinary Euclidean metric.

Proof 120.2

On two tensor slots, for each symmetric $S_i$ ,

2S_i\otimes S_i\preceq S_i^2\otimes I+I\otimes S_i^2

(120.13)

because the difference is $(S_i\otimes I-I\otimes S_i)^2$ . Hence $\sum_iS_i\otimes S_i\preceq8I$ , and congruence gives $\sum_iU_i\otimes U_i\preceq8A_t\otimes A_t$ . The product rule for $K=K_j$ now bounds its drift by $(-j+8\binom j2)K=(4j^2-5j)K$ . Indeed each slot contributes $-K$ and each unordered pair contributes once its product of noise coefficients. The normalized noise on the $i$ th Brownian coordinate is $V_i=K^{-1/2}(dK/d\beta_i)K^{-1/2}=\sum_{r=1}^jS_i^{(r)}$ . For a vector $v$ , $|\sum_rS_i^{(r)}v|^2\le j\sum_r|S_i^{(r)}v|^2$ , so $\sum_iV_i^2\preceq8j^2I$ .

Write $dh_j=\sum_ic_i d\beta_i$ , where $c_i=\mathbb E_t[(D^jf-h_j)\xi_i]$ . The functions $\xi_i$ are orthonormal in $L^2(\mu_t)$ . Bessel’s inequality applied after the deterministic-at-time- $t$ linear map $K^{1/2}$ yields

Q:=\sum_i\langle c_i,Kc_i\rangle \le\mathbb E_t\langle D^jf-h_j,K(D^jf-h_j)\rangle.

(120.14)

The entire drift of $\langle h_j,Kh_j\rangle$ is at most

(4j^2-5j)\langle h_j,Kh_j\rangle+Q +2\sum_i\langle K^{1/2}h_j,V_iK^{1/2}c_i\rangle.

(120.15)

Cauchy–Schwarz over $i$ , followed by $2ab\le a^2+b^2$ , bounds the last term by $\langle h_j,Kh_j\rangle+8j^2Q$ . Consequently its expected drift is bounded by $(4j^2-5j+1)N_j+(1+8j^2)L_j$ , which implies the assertion.

All preceding identities initially hold with compact parameter stops. They pass to expectations without a residual local-martingale term: on fixed spatial support the derivatives of $f$ , $h_j$ and $A_t$ are uniformly bounded; the Bessel bound controls $K^{1/2}c_i$ in square sum, and the bound for $V_i$ controls all normalized metric noise. These bounds also bound the actual scalar stochastic integrand in square sum, since $K^{1/2}h_j$ is bounded. The drifts are absolutely bounded by the same arguments (use absolute values for each pair term). Global existence and dominated convergence remove the stops and give local absolute continuity of $N_j$ .

Closing the polynomial induction¶

Proof 120.3 (Proof of the polynomial theorem)

For degree one isotropy gives $\operatorname{Var}(T\cdot X)=|T|^2$ . Induct on $d\ge2$ , first for compactly supported isotropic laws. Each whitened posterior is again in that class. Applying the already established Appell inequality through degree $d-j$ to the Hilbert-valued polynomial

y\longmapsto(A_t^{1/2})^{\otimes j}D^jf(m_t+A_t^{1/2}y)

(120.16)

and then using the triangle inequality in the localization probability space gives

\sqrt{L_j(t)}\le\sum_{k=1}^{d-j}32^k k!\sqrt{N_{j+k}(t)} \quad(1\le j<d).

(120.17)

The new $k$ derivative slots supply the extra $A_t^{1/2}$ factors. This explains exactly why the right metric is $A_t^{\otimes(j+k)}$ and why no output dimension appears.

Take $f=P_d^{\mu_0}[T]$ . The zero tensor case is trivial; otherwise set

Q=(d!)^2\|T\|_{\mathrm{HS}}^2,\quad w_j=1024^{d-j}((d-j)!)^2,\quad M(t)=\max_{1\le j\le d}\frac{N_j(t)}{Qw_j}.

(120.18)

Initially the lower derivative means vanish, while $N_d(0)=Q$ , so $M(0)=1$ . Writing $s=d-j\ge1$ , cancellation of the powers of 32 gives

\frac{\sqrt{L_j(t)}}{\sqrt{Qw_j}} \le\sqrt{M(t)}\sum_{k=1}^s\binom sk^{-1}\le2\sqrt{M(t)}.

(120.19)

For $s=1$ the sum is one; for $s\ge2$ its first $s-1$ terms are at most $1/s$ and its last is one. Also $L_d=0$ . The hierarchy lemma therefore gives

\frac{N_j(t)}{Qw_j}\le\frac{N_j(0)}{Qw_j} +40d^2\int_0^tM(u)du.

(120.20)

Taking the maximum and applying the scalar integral Gronwall inequality shows $M(t)\le e^{40d^2t}$ . In particular, for $r=\nabla f$ ,

G_1(t):=\mathbb E_{\mathrm{loc}}\mathbb E_t[r^TA_tr] =N_1(t)+L_1(t) \le5(d!)^2\|T\|^2 1024^{d-1}((d-1)!)^2e^{40d^2t}.

(120.21)

The martingale identities for $f$ and $f^2$ give $V'(t)=-\mathbb E_{\mathrm{loc}}|\operatorname{Cov}_t(f,\xi_t)|^2\ge-V(t)$ , where $V(t)=\mathbb E_{\mathrm{loc}}\operatorname{Var}_t f$ and the inequality is Bessel’s. Hence $V(\tau)\ge e^{-\tau}\operatorname{Var}_{\mu_0}f$ .

The matrix strong-convexity Poincaré inequality gives $\operatorname{Var}_{\mu_\tau}f\le\mathbb E_\tau[r^T\Lambda_\tau^{-1}r]$ . Here we use the established Brascamp–Lieb inequality Brascamp & Lieb, 1976; its smooth matrix form is also stated in Bakry et al., 2014, Theorem 4.9.1. We use its constant-curvature form: if the potential is $V+x^TBx/2$ , $V$ extended-valued convex and $B\succ0$ , then $\operatorname{Var}q\le\mathbb E\nabla q^TB^{-1}\nabla q$ . Its applicability to nonsmooth convex supports can also be obtained by replacing $V$ by smooth convex approximations increasing to it (Moreau envelopes followed by mollification), applying the smooth inequality, and passing compact smooth tests to the limit; truncation and spatial cutoff extend it to finite-energy locally Lipschitz tests. A supporting affine lower bound for $V$ supplies Gaussian domination under the fixed positive quadratic term in this approximation.

For any $v$ , expansion of the square yields the exact identity

\int_0^\tau v^TA_sv\,ds-\tau^2v^T\Lambda_\tau^{-1}v =\int_0^\tau|A_s^{1/2}v-\tau A_s^{-1/2}\Lambda_\tau^{-1}v|^2ds.

(120.22)

Thus $\Lambda_\tau^{-1}\preceq\tau^{-2}\int_0^\tau A_sds$ . For fixed $s\le\tau$ , the entries of $A_s$ are $\mathcal F_s$ -measurable and $\mathbb E_t(r_ir_j)$ is a true martingale, so

\mathbb E_{\mathrm{loc}}\mathbb E_\tau[r^TA_sr] =\mathbb E_{\mathrm{loc}}\mathbb E_s[r^TA_sr]=G_1(s).

(120.23)

All tests are bounded on the fixed support; Fubini is legitimate. Combining these facts proves

\operatorname{Var}_{\mu_0}f\le\frac{e^\tau}{\tau^2}\int_0^\tau G_1(s)ds.

(120.24)

Take $\tau=(40d^2)^{-1}$ . Since $d\ge2$ , $e^{1+\tau}<3$ , and bounding the integral by its interval length times its maximum gives

\operatorname{Var}_{\mu_0}f \le600d^2\cdot1024^{d-1}((d-1)!)^2(d!)^2\|T\|^2 \le1024^d(d!)^4\|T\|^2.

(120.25)

This completes the induction for compact support. For arbitrary isotropic log-concave $\mu$ , condition on growing centered Euclidean balls, then center and whiten. All polynomial moments of a finite-dimensional log-concave law are finite; hence all conditional moments through order $2d$ converge. Conditional means tend to zero and covariances to $I$ , so the affine normalizations tend to the identity. The coefficients of each Appell polynomial are polynomial expressions in moments up to degree $d$ (obtain the inverse generating series recursively). Their variances therefore converge using moments up to degree $2d$ . This passes the same coefficient bound to $\mu$ . The exact Appell expansion proves the stated derivative-mean inequality.

Finally let a centered law have covariance $0\preceq\Sigma\preceq I$ . On its linear support choose an isometric embedding $J:\mathbb R^m\to\mathbb R^n$ and positive covariance $B\preceq I_m$ , so $X=JB^{1/2}Y$ with $Y$ isotropic. The formal identity gives $P_k^\mu[T](JB^{1/2}y)=P_k^{\mathcal L(Y)}[((JB^{1/2})^T)^{\otimes k}T](y)$ . The tensor map has norm at most one, proving the coefficient bound. The exact Appell expansion, valid for singular laws as a formal identity, proves the polynomial bound too. Rank zero means a point mass and both variances vanish.

Analytic foundations for the curvature comparison¶

The following two lemmas record the precise analytic claims of Lemma 7.1. They are included here so that applying polynomial testing to inverse operators does not leave an implicit domain assumption.

Lemma 120.3 (Operator domain, Bochner identity and spectral gap)

Let $\mu(dx)=e^{-W(x)}dx$ be a probability with $W$ smooth and $0<aI\preceq D^2W\preceq bI<\infty$ . The nonnegative self-adjoint operator $H=-\Delta+\nabla W\cdot\nabla$ associated with the closed Dirichlet form has form domain $H^1(\mu)=\{g\in L^2(\mu):\nabla g\in L^2(\mu)\}$ , constants as kernel, and $C_c^\infty$ as graph core. For every $g\in\operatorname{Dom}H$ , $D^2g\in L^2(\mu)$ , its mixed weak derivatives commute, and

\|Hg\|_2^2=\mathbb E\|D^2g\|_{\mathrm{HS}}^2+ \mathbb E\langle\nabla g,D^2W\nabla g\rangle.

(120.26)

In particular each $\partial_i g$ belongs to the form domain. Polynomials belong to that domain, and $\langle\phi,Hg\rangle=\mathbb E\nabla\phi\cdot\nabla g$ for every form-domain $\phi$ . The resolvent is compact. The smallest positive eigenvalue is $\lambda=C_P(\mu)^{-1}>0$ , with a real normalized centered eigenfunction. On the centered subspace $H^{-1/2}$ is bounded. If centered $h\in\operatorname{Dom}H^{1/2}$ and $g=H^{-1/2}h$ , then $g\in\operatorname{Dom}H$ , $\|Hg\|_2^2=\|H^{1/2}h\|_2^2$ and $\|\nabla g\|_2^2=\|h\|_2^2$ .

Proof 120.4

Choose smooth cutoffs $\chi_R$ equal to one on the radius- $R$ ball, zero beyond radius $2R$ , with derivative bounds $C/R,C/R^2$ . They approximate every weighted Sobolev function in the form norm: the extra gradient term has norm at most $C\|g\|_2/R$ , and the other terms tend to zero by integrability. On compact sets the positive smooth density is bounded above and below, so ordinary mollification proves density and identifies the form domain. The same cutoff approximates the constant one; a zero weak gradient on connected $\mathbb R^n$ gives a constant, identifying the kernel.

For $g\in\operatorname{Dom}H$ the weak equation and interior elliptic regularity give $g\in H^2_{\mathrm{loc}}$ . The product rule reads

H(\chi_Rg)=\chi_RHg-2\nabla\chi_R\cdot\nabla g+gH\chi_R.

(120.27)

Since $|\nabla W(x)|\le|\nabla W(0)|+b|x|$ , the functions $H\chi_R$ are uniformly bounded for $R\ge1$ and supported in the cutoff annuli. Every displayed term belongs to $L^2(\mu)$ , so the weak operator characterization puts $\chi_Rg$ in its domain. The last two terms tend to zero in $L^2$ , proving graph convergence. Mollification on a fixed compact neighborhood of each support converges in $H^2$ ; bounded coefficients there give graph convergence. This proves the graph-core claim. Subtracting means gives the centered graph core as well.

For a compact smooth function differentiate $\partial_iHg=H\partial_i g+\sum_jW_{ij}\partial_jg$ and integrate by parts to obtain the Bochner identity. For differences $u$ of graph-core approximants, $\|\nabla u\|_2^2=\langle u,Hu\rangle\le\|u\|_2\|Hu\|_2$ . The gradients are Cauchy, and Bochner with $D^2W\succeq0$ makes their Hessians Cauchy. Their limits are distributional derivatives of $g$ . Boundedness of $D^2W$ passes the curvature term to the limit. Mixed weak derivatives commute, and the form-domain characterization applies to $\partial_i g$ . The testing formula is the definition of the associated operator. Strong convexity gives Gaussian tails, so polynomials and their gradients lie in $L^2$ .

The unitary map $g\mapsto e^{-W/2}g$ conjugates $H$ to the Friedrichs operator $-\Delta+V$ , where $V=|\nabla W|^2/4-\Delta W/2$ , first on its graph core and then by closure. If $x_*$ minimizes $W$ , strong monotonicity gives $|\nabla W(x)|\ge a|x-x_*|$ , while $\Delta W\le nb$ . Consequently $V\ge a^2|x-x_*|^2/4-nb/2$ . After adding $nb/2+1$ , the form norm controls $\|u\|_2$ , $\|\nabla u\|_2$ , and $\||x-x_*|u\|_2$ . Rellich compactness on balls and the uniform tail bound $\int_{|x-x_*|>R}|u|^2\le R^{-2}\||x-x_*|u\|_2^2$ imply compact embedding of the form domain in $L^2$ . Thus the resolvent is compact.

The kernel is one dimensional, so its first positive eigenvalue $\lambda$ exists. The Rayleigh inequality and the truncation argument in the next lemma identify its reciprocal with the Poincaré constant for all locally Lipschitz finite-energy tests. The reverse inequality follows by approximating a first eigenfunction in form norm by compact smooth functions. Real coefficients allow a real eigenfunction. On the centered spectral subspace the spectrum lies in $[\lambda,\infty)$ . Spectral calculus gives $Hg=H^{1/2}h$ , $H^{1/2}g=h$ , and all remaining claims.

Proof 120.5

For the stability assertion first pass each compact smooth test by weak convergence, since its square and squared gradient are bounded continuous. For locally Lipschitz $q$ truncate its values to $q_M=\max(-M,\min(q,M))$ . Then $|\nabla q_M|\le|\nabla q|$ almost everywhere. Multiply by $\chi_R$ from the preceding proof. For fixed $M,R$ the product is a compactly supported Lipschitz function, so mollifications and their gradients converge Lebesgue almost everywhere, are uniformly bounded, and live in a common compact set. Absolute continuity and dominated convergence pass the inequality to $\chi_Rq_M$ . As $R\to\infty$ , its value converges in $L^2$ to $q_M$ and its gradient to $\nabla q_M$ , since the cutoff error is bounded by $CM/R$ . Thus $\operatorname{Var}_\nu q_M\le K\mathbb E_\nu|\nabla q|^2$ . With independent $X,X'$ of law $\nu$ , Fatou gives

\frac12\mathbb E(q(X)-q(X'))^2\le K\mathbb E_\nu|\nabla q|^2.

(120.28)

Fubini gives some $y$ with $\int(q(x)-q(y))^2\nu(dx)<\infty$ ; since $q(y)$ is finite this implies $q\in L^2(\nu)$ . The independent-copy expression is then its variance, proving stability without assuming square integrability in advance.

For approximation let $Y_\delta=X+\sqrt\delta Z$ with $Z$ standard Gaussian independent of the isotropic log-concave $X$ . Its smooth positive log-concave density $e^{-V_\delta}$ satisfies

D^2V_\delta(y)=\delta^{-1}I-\delta^{-2} \operatorname{Cov}(X\mid Y_\delta=y),\qquad 0\preceq D^2V_\delta\preceq\delta^{-1}I.

(120.29)

Differentiate the Gaussian kernel twice; Gaussian damping justifies differentiation locally uniformly in $y$ . The lower bound follows from preserved log-concavity and the upper bound from positivity of the conditional covariance. Multiply this density by $e^{-\epsilon|y|^2/2}$ and normalize. Its Hessian lies between $\epsilon I$ and $(\delta^{-1}+\epsilon)I$ . For fixed $\delta$ , dominated convergence as $\epsilon\downarrow0$ gives total variation convergence and convergence of its first and second moments to those of $Y_\delta$ . Choose $\delta_j\downarrow0$ and then $\epsilon_j\downarrow0$ so that these errors tend to zero. The resulting laws converge weakly to $\mu$ , with means $m_j\to0$ and covariances $A_j\to I$ . The laws of $A_j^{-1/2}(Y-m_j)$ are isotropic and still regular: their Hessians are congruent by $A_j^{1/2}$ to those just constructed. The transformations tend to the identity uniformly on compact sets, so tightness proves the asserted weak convergence. No uniform lower curvature bound is asserted.

Source mapping. The formal Appell identity is source Section 2.3. The global SDE, its logarithmic determinant, and the tensor hierarchy reconstruct source Lemmas 4.2 and 4.3; the coefficient induction reconstructs Theorem 4.1. The two analytic lemmas reconstruct Section 3, separating moment convergence for polynomials from weak scalar Poincaré stability. Standard finite-dimensional SDE existence, Itô calculus, interior elliptic regularity, Rellich compactness and spectral calculus are the classical tools used here. The only nonclassical mathematical dependency is Theorem 25.1.

Fences respected. The new nodes have no proposed bounded_by edges. The projection-only ceiling Remark 25.4 is respected by using all tensor slots. The two-tail boundary Remark 25.3 is untouched: the conclusion is an intrinsic polynomial estimate, not a bound on an unwhitened cut source. The occupation and bootstrap boundaries Remark 33.6, Remark 33.5, and Remark 32.6 are untouched; no universal-time occupation or moving-competitor estimate is claimed. Growing factorials prevent treating this polynomial theorem alone as a dimension-free Poincaré inequality. No comparison between members of the trace-upgrade cluster is asserted. There are no open antecedents or unclosed mathematical steps claimed in this dossier; its independent examination is a separate task.

References¶

Song, Z., & Zhang, X. (2026). An O(4\log^* n) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447v1
Brascamp, H. J., & Lieb, E. H. (1976). On Extensions of the Brunn–Minkowski and Prékopa–Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation. Journal of Functional Analysis, 22(4), 366–389. 10.1016/0022-1236(76)90004-5
Bakry, D., Gentil, I., & Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators (Vol. 348). Springer. 10.1007/978-3-319-00227-9