Song–Zhang v1: Gaussian transfer, the all-depth bound and its affine form

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. We reconstruct Section 7 of Song & Zhang, 2026, including the bounded test function used to transfer a curvature estimate to an arbitrary isotropic law. Gaussian observations produce a regular posterior; Letwin’s quadratic inequality controls its covariance on a short interval. One bounded Lipschitz test retains a fixed amount of variance on a posterior with controlled covariance. This proves a transfer for any curvature profile, not only the iterated-logarithm profiles. We then apply Theorem 7.3, select a finite depth, and whiten to obtain the affine assertion. Approximation passes only scalar Poincaré inequalities to limits; no continuity of spectral objects or of the profile is required.

A bounded test detecting the Poincaré constant¶

The established bounded-witness theorem of Klartag & Lehec, 2025, Theorem 20, in the version arXiv:2406.01324v2, states that for a log-concave law there is a 1-Lipschitz $g$ with

\operatorname{Var}_\mu(g)\ge c_1\psi_\mu^2, \qquad \|g\|_\infty^2\le C_1\operatorname{Var}_\mu(g),

(123.1)

where $c_1,C_1>0$ are universal and $\psi_\mu$ is the reciprocal Cheeger constant. Corollary 21 and its following remark give

C_P(\mu)\le4\psi_\mu^2,\qquad \psi_\mu^2\le\pi C_P(\mu). \tag{1}

(123.2)

These are literature inputs, not consequences of the new preprint. If the bound for $g$ is essential, clip to its essential range; this preserves its almost-everywhere values and Lipschitz constant and makes its bound global. For $0<k=C_P(\mu)<\infty$ , put $\sigma^2=\operatorname{Var}_\mu g$ and $f=(g-\mathbb E g)/\sigma$ . Since $\sigma^2\ge c_1k/4$ , there is a universal $B_0\ge1$ such that

\mathbb Ef=0,\quad \mathbb Ef^2=1,\quad \sup|f|\le B_0,\quad \operatorname{Lip}(f)\le B_0/\sqrt k. \tag{2}

(123.3)

For instance take $B_0=\max(1,2\sqrt{C_1},2/\sqrt{c_1})$ . The global bound in (2) is what pays for the exceptional covariance paths.

Gaussian observations and posterior martingales¶

Take a regular isotropic initial law $\mu=e^{-W}dx$ with $aI\preceq D^2W\preceq bI$ . Let $X\sim\mu$ be independent of standard Brownian motion $B_t$ , set $Y_t=tX+B_t$ , and let $\mathcal F_t$ be the usual augmentation of its observation filtration. The conditional distribution is

\mu_t(dx)=Z_t^{-1}e^{Y_t\cdot x-t|x|^2/2}\mu(dx),\qquad Z_t=\int e^{Y_t\cdot x-t|x|^2/2}\mu(dx). \tag{3}

(123.4)

For fixed $x$ , multiplying likelihood ratios of independent Gaussian increments on any finite partition gives the factor $e^{x\cdot Y_t-t|x|^2/2}$ . Cylinder events generate the path sigma-field, so Bayes’ formula proves (3) for the entire observation filtration. In particular $\mathbb E_tq=\mathbb E[q(X)\mid\mathcal F_t]$ . For every Borel $q$ of polynomial growth these are square-integrable martingales: strong convexity of $\mu$ gives all polynomial moments, and conditional Jensen applies.

Write $m_t=\mathbb E_tX$ , $A_t=\operatorname{Cov}_tX$ , and $\mathcal W_t=Y_t-\int_0^tm_sds$ . For $s<t$ , the future increments of $B$ are independent of $X$ and the past, while $\mathbb E[m_u\mid\mathcal F_s]=m_s$ for $u\ge s$ . Consequently $\mathbb E[\mathcal W_t-\mathcal W_s\mid\mathcal F_s]=0$ . All these quantities are integrable by $\mathbb E|m_u|^2\le\mathbb E|X|^2$ . The process is continuous and has quadratic covariation $tI$ , so Lévy’s characterization makes it Brownian in the observation filtration.

For completeness define $I_q(t,y)=\int q(x)e^{y\cdot x-t|x|^2/2}\mu(dx)$ . Gaussian tails permit differentiation locally in $(t,y)$ , including around $t=0$ . We have $\partial_tI_q+\Delta_yI_q/2=0$ and $\nabla_yI_q=I_{qx}$ . Itô’s formula followed by the quotient rule therefore gives

d\mathbb E_tq=\operatorname{Cov}_t(q,X)\cdot d\mathcal W_t. \tag{4}

(123.5)

These computations can first be stopped on $|Y_t|\le R$ . To remove the stop, conditional Cauchy--Schwarz, ordinary Cauchy--Schwarz and conditional Jensen bound, for each finite $T$ ,

\mathbb E\int_0^T|\operatorname{Cov}_t(q,X)|^2dt \le T\big(\mathbb E q(X)^4\,\mathbb E|X|^4\big)^{1/2}<\infty.

(123.6)

Indeed the conditional squared covariance is at most $\mathbb E_tq^2\,\mathbb E_t|X|^2$ . Continuity of $Y$ and Itô isometry then justify (4) without the stop. Bounded Borel tests, including the witness $f$ and $f^2$ , are included.

Taking $q=x_i$ and $q=x_ix_j$ in (4) yields

dm_t=A_t\,d\mathcal W_t,\qquad dA_t=\sum_jU_{j,t}\,d\mathcal W_{j,t}-A_t^2dt, \tag{5}

(123.7)

where $U_{j,t}=\mathbb E_t[(X_j-m_{j,t})(X-m_t)(X-m_t)^T]$ . The negative drift is $-dm_tdm_t^T$ ; expanding $X=m_t+(X-m_t)$ in the remaining martingale terms leaves exactly the displayed third centered moment. Every posterior is regular, with potential Hessian $D^2W+tI$ .

Covariance exit from a fixed interval¶

Proof 123.1

For any centered log-concave $Z$ of positive covariance $A$ , the whitened form of Theorem 25.1 gives

\operatorname{Var}(Z^TDZ)\le8\|A^{1/2}DA^{1/2}\|_{\rm HS}^2 \le8\|A\|_{\rm op}^2\|D\|_{\rm HS}^2

(123.9)

for symmetric $D$ . For $S_z=\mathbb E[(z\cdot Z)ZZ^T]$ , trace duality and Cauchy--Schwarz give

|\langle D,S_z\rangle| =|\mathbb E[(z\cdot Z)(Z^TDZ-\operatorname{tr}(DA))]| \le\sqrt{8z^TAz}\|A\|_{\rm op}\|D\|_{\rm HS}.

(123.10)

Thus $\|S_z\|_{\rm HS}^2\le8\|A\|_{\rm op}^2z^TAz$ . Symmetry in all three indices of the third moment yields $\sum_j|U_jz|^2=\|S_z\|_{\rm HS}^2$ , so before $\tau$

\sum_jU_{j,t}^2\preceq64I. \tag{6}

(123.11)

Set $M_u=\int_0^{u\wedge\tau}\sum_jU_{j,s}d\mathcal W_{j,s}$ . For symmetric matrices $M,T$ and $\theta>0$ ,

D^2\operatorname{tr}(e^{\theta M})[T,T] \le\theta^2\operatorname{tr}(e^{\theta M}T^2). \tag{7}

(123.12)

Diagonalize $M$ . The coefficient of $T_{ij}^2$ on the left is $\theta^2$ times the logarithmic mean of $e^{\theta m_i},e^{\theta m_j}$ . This mean equals $\int_0^1e^{\theta[(1-s)m_i+sm_j]}ds$ and is at most their arithmetic mean by convexity. Summing the coefficients proves (7). Itô’s formula and (6) now show that $Z_u=e^{-32\theta^2u}\operatorname{tr}(e^{\theta M_u})$ is a nonnegative local supermartingale, hence a supermartingale. At the first crossing of $\lambda_{\max}(M_u)\ge\rho$ before time $t$ , its value is at least $e^{\theta\rho-32\theta^2t}$ . Optional stopping, with $Z_0=n$ , gives the one-sided probability bound $n e^{-\theta\rho+32\theta^2t}$ . Optimize at $\theta=\rho/(64t)$ , repeat for $-M$ , and take the union:

\mathbb P\left(\sup_{u\le t}\|M_u\|_{\rm op}\ge\rho\right) \le2n e^{-\rho^2/(128t)}. \tag{8}

(123.13)

The stopped integrand is square integrable by (6), and all optional stopping arguments can first be made with bounded localizing stops; nonnegativity and Fatou then pass to the displayed estimates.

On $\{\tau\le t\}$ , continuity gives $M_\tau=A_\tau-I+\int_0^\tau A_s^2ds$ . An upper exit supplies a unit vector with quadratic form at least 1. A lower exit supplies one with quadratic form at most $-1/2+4t\le-1/4$ . Hence this event implies the event in (8) with $\rho=1/4$ , which proves the result.

A transfer for an arbitrary curvature profile¶

Proof 123.2

First take regular isotropic $\mu$ and let $k=C_P(\mu)$ , finite and positive by Lemma 7.1. Choose the witness (2) and form the stopped Gaussian posteriors above. Put $V_t=\operatorname{Var}_{\mu_{t\wedge\tau}}f$ and $S(t)=\mathbb E_{\rm loc}V_t$ . Applying Itô’s formula to the bounded martingales $\mathbb E_tf$ and $\mathbb E_tf^2$ gives

S(t)=1-\int_0^t\mathbb E_{\rm loc} [\mathbf1_{\{s<\tau\}}|\operatorname{Cov}_{\mu_s}(X,f)|^2]ds.

(123.15)

Before exit, directional Cauchy--Schwarz yields $|\operatorname{Cov}_{\mu_s}(X,f)|^2\le\|A_s\|_{\rm op}V_s\le2V_s$ . Thus $S'(t)\ge-2S(t)$ almost everywhere and

S(t)\ge e^{-2t},\qquad 0\le V_t\le B_0^2. \tag{9}

(123.16)

The second assertion follows from the global supremum bound of $f$ .

Set

c_0={1\over2048\log(16B_0^2)},\qquad t={c_0\over\log(en)}.

(123.17)

Then $0<t<1/16$ . With $q=1/(2048c_0)=\log(16B_0^2)>1$ , the exit lemma gives

\mathbb P(\tau\le t)\le2n e^{-1/(2048t)} =2e^{-q}n^{1-q}\le{1\over8B_0^2}.

(123.18)

Since $e^{-2t}\ge1-2t\ge7/8$ , subtracting the exit contribution in (9) gives $\mathbb E_{\rm loc}[\mathbf1_{\{\tau>t\}}\operatorname{Var}_{\mu_t}f]\ge3/4$ . There is consequently a realization with $\tau>t$ and $\operatorname{Var}_{\mu_t}f>1/2$ . Fix it, write $m,A$ for its mean and covariance, and let $\eta$ be the law of $A^{-1/2}(X-m)$ under this posterior. This is regular isotropic and its potential Hessian is

A^{1/2}(D^2W+tI)A^{1/2}\succeq(t/2)I.

(123.19)

It also has a finite upper Hessian bound. The transformed witness $z\mapsto f(m+A^{1/2}z)$ has variance greater than $1/2$ and gradient energy at most $\|A\|_{\rm op}\operatorname{Lip}(f)^2\le2B_0^2/k$ . The definition of $C_P(\eta)$ therefore gives

k\le4B_0^2 C_P(\eta)\le4B_0^2 F(t/2). \tag{10}

(123.20)

This uses the hypothesis at the particular admissible lower curvature bound $t/2$ ; it never compares values of $F$ at nearby arguments.

For arbitrary isotropic log-concave $\mu$ , choose the regular isotropic approximation of Lemma 7.1. The regular-law argument bounds every approximant by the same number $K=4B_0^2F(c_0/(2\log(en)))$ . The isotropic limit is full-dimensional and absolutely continuous. The scalar stability part of that lemma passes the inequality to the limit and all locally Lipschitz finite-energy tests. Taking $c=c_0/2$ and $C=4B_0^2$ proves the claim. No regularity of $F$ enters this limit either, since its argument is unchanged along the approximants.

All depths, finite stabilization, and affine coordinates¶

Theorem 123.2 (Song–Zhang bound and its affine consequence)

There are universal $C,C'>0$ , independent of $r,n$ , such that every isotropic log-concave measure $\mu$ on $\mathbb R^n$ satisfies, for every integer $r\ge1$ ,

C_P(\mu)\le C16^r\ell_r(\log(en))^2, \qquad \Psi_{\mathrm{KLS},n}\le C'4^r\ell_r(\log(en)).

(123.21)

In particular $C_P(\mu)\le C16^{\log^*(n+2)}$ and $\Psi_{\mathrm{KLS},n}\le C'4^{\log^*(n+2)}$ . For a log-concave $\mu$ with positive definite covariance $\Sigma$ , the Poincaré estimates remain valid after multiplying their right sides by $\|\Sigma\|_{\rm op}$ . These are Theorem 7.5 and Corollary 7.1.

Proof 123.3

For each fixed finite $r$ , Theorem 7.3 supplies the profile $F_r(a)=\Gamma_r^2\ell_r(a^{-1})^2$ , with $\Gamma_r\le C_0 4^r$ . The transfer theorem with its constants just constructed gives

C_P(\mu)\le4B_0^2\Gamma_r^2 \ell_r((2/c_0)\log(en))^2.

(123.22)

The elementary scaling inequality $\ell_r(cx)\le c\ell_r(x)$ for $c\ge1$ was proved in the iteration dossier. Hence this is bounded by

4B_0^2(2/c_0)^2C_0^2\,16^r\ell_r(\log(en))^2. \tag{11}

(123.23)

All constants here are independent of both $r$ and $n$ . By (1), $\psi_\mu\le\sqrt{\pi C_P(\mu)}$ . Taking the supremum over isotropic laws in dimension $n$ gives the asserted reciprocal-Cheeger estimate.

We now choose a finite depth. For $y\ge4$ ,

g(y+1)=\log y+\log(1+(e+1)/y)\le\log y+1, \tag{12}

(123.24)

since $e+1<4$ and $\log(1+u)\le u$ . Let $x_0=n+2$ and $m$ be the least nonnegative integer for which its $m$ th ordinary logarithmic iterate $x_m$ is at most 4. When $m\ge1$ , $\log(en)\le1+x_1$ . For $1\le j<m$ we have $x_j>4$ , so (12) inductively yields $\ell_{m-1}(\log(en))\le1+x_m\le5$ . When $m=0$ , $n+2\le4$ and $\log(en)<5$ directly. Also $g$ maps $[0,5]$ into $[0,5]$ . Therefore

r=\max(1,m-1)\quad\hbox{satisfies}\quad \ell_r(\log(en))\le5,\qquad r\le\log^*(n+2). \tag{13}

(123.25)

Indeed reaching 4 occurs no later than reaching 1, and $\log^*(n+2)\ge1$ for $n\ge1$ . Inserting (13) into (11) proves the $16^{\log^*}$ bound after absorbing 25 into the universal constant, and the $4^{\log^*}$ reciprocal-Cheeger bound follows likewise. Each dimension uses one finite depth; the argument takes no infinite-depth limit.

Finally, let $\mu$ have mean $m$ and positive definite covariance $\Sigma$ , and let $\rho$ be the law of $\Sigma^{-1/2}(X-m)$ , which is isotropic and log-concave. For a locally Lipschitz finite-energy test $f$ , put $h(y)=f(m+\Sigma^{1/2}y)$ . Then

\operatorname{Var}_\mu f=\operatorname{Var}_\rho h, \qquad \int|\nabla h|^2d\rho =\int\nabla f^T\Sigma\nabla f\,d\mu \le\|\Sigma\|_{\rm op}\int|\nabla f|^2d\mu.

(123.26)

The isotropic inequality, which includes $L^2$ membership of finite-energy tests, applies to $h$ and proves both affine Poincaré estimates.

Dependencies and fences. The generic transfer uses only Lemma 7.1, Theorem 25.1, and the established bounded-witness and Cheeger comparison from Klartag & Lehec, 2025. The all-depth result adds Theorem 7.3; its affine corollary only uses the all-depth result. There are no bounded_by edges on these nodes. These estimates retain dimension dependence and prove no CMH statement or universal-time occupation bound. The transfer has a profile hypothesis within its universally quantified implication; applying it here discharges that hypothesis separately at every finite depth using the iterated-curvature theorem.

References¶

Song, Z., & Zhang, X. (2026). An O(4\log^* n) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447v1
Klartag, B., & Lehec, J. (2025). Isoperimetric Inequalities in High-Dimensional Convex Sets. Bulletin of the American Mathematical Society, 62(4), 575–642. 10.1090/bull/1869