Song–Zhang, first version: polynomial estimates and curvature

What to retain. Bounds on the Appell polynomials of a measure control its spectral gap, through a first eigenfunction followed by repeated centered gradients; stochastic localization turns a spectral gap back into better polynomial bounds. Composing the two conversions gives $\CP\lesssim16^{\log^*(n+2)}$, and its end point, one exponential base for the coefficients of every degree, is equivalent to KLS (Proposition 7.1).

This chapter works through the first version of Song and Zhang’s preprint Song & Zhang, 2026 (SZ v1), whose polynomial–curvature iteration gives the dimension-dependent bound Theorem 7.5, and locates its losses. Its spectral criterion is the starting point of two of the three proofs: Bizeul–Klartag–Lehec (BKL) prove its exponential end point directly (Chapter Bizeul–Klartag–Lehec: cumulants and suspension), and the second version of Song–Zhang (SZ v2) makes the losses of the iteration summable (Chapter Song–Zhang, second version: repeated refinement with summable losses) Song & Zhang, 2026. Balasubramanian–Kasiviswanathan (BK) share its Appell normalization but reach the spectral gap through their own integration calculus (Chapter Balasubramanian–Kasiviswanathan: compatible integration). The estimates and limitations below are those of SZ v1.

Why the iteration stalls. Each round of the loop multiplies the profile constant by about 4, and is admissible only above thresholds that grow with its depth; so the constants of the curvature profiles grow like $4^r$ at depth $r$ (Section Feeding the curvature estimate back into the polynomials). The depth needed grows, very slowly, with the dimension, and so does the bound.

Where this argument meets the alternative mechanisms. It shares the first eigenfunction with the fixed eigenfunction and Letwin’s quadratic estimate with the moment map; Section What the three proofs contribute to each mechanism says, mechanism by mechanism, what the proofs built on it give and what still needs an estimate of its own.

Notation and the smallest cases¶

A probability measure $\nu(dx)=e^{-W(x)}dx$ on $\R^n$ is called regular here if $W$ is smooth and $aI\preceq D^2W\preceq bI$ for some $0<a\le b<\infty$; the lower bound $a$ is its curvature. The Bakry–Émery bound gives $\CP(\nu)\le1/a$, and every isotropic log-concave measure is a weak limit of regular isotropic ones (Lemma 7.1). The question of this chapter is how much better than $1/a$ one can do when $a$ is small. A curvature profile is an answer valid in every dimension: a function $F$ with $\CP(\nu)\le F(a)$ for every regular isotropic $\nu$ of curvature $a$.

For a centered real random variable of variance $\sigma^2$, the first three Appell polynomials are 1, $x$, and $x^2-\sigma^2$. Their means vanish in positive degree, and differentiation lowers degree: $(x^2-\sigma^2)'=2x$. In several dimensions the quadratic one is $x^TTx-\operatorname{Tr}(T\Sigma)$, where $T$ is symmetric and $\Sigma$ is the covariance. In every degree $d$, the Appell tensor $\mathcal A_d^\nu$ is the coefficient of $z^{\otimes d}/d!$ in the expansion of $e^{\langle z,x\rangle}/\int e^{\langle z,y\rangle}d\nu(y)$ (see Theorem 7.1), and $P_d^\nu[T]=\langle T,\mathcal A_d^\nu\rangle$ for a symmetric $d$-tensor $T$. These polynomials keep the centering and differentiation identities in every degree; they need not be orthogonal in $L^2(\nu)$. The size of degree $k$ is measured by

In isotropic position $P_1^\nu[T]=\langle T,x\rangle$, so $K_1=1$ and $c_1=1$. For the standard Gaussian the Appell polynomials are the Hermite polynomials, $K_k=k!$, and $c_k=1/\sqrt{k!}$.

Degree two already separates measures. For a standard Gaussian $G$, $\Var(G^2-1)=3-1=2$, so $c_2=\sqrt2/2$. For $X=E-1$ with $E$ a mean-one exponential, $\E X^4=9$, hence $\Var(X^2-1)=8$ and $c_2=\sqrt2$, twice the Gaussian value. Here $\E[X(X^2-1)]=\E X^3=2$: even the first two positive degrees are not orthogonal, so the orthogonality of Hermite polynomials is not available in general, and nothing below uses it.

The argument in one loop¶

The chapter follows the loop of Figure Figure 7.1. The first estimate, Theorem 7.1, bounds every coefficient, with factorial growth in the degree. The comparison Theorem 7.2 turns coefficient bounds into a spectral gap for a regular measure of curvature $a$; fed the factorial bound, it gives a first curvature profile of order $\log(e+1/a)^2$, already far below the Bakry–Émery $1/a$. Applied to the measures produced by stochastic localization, that profile improves the coefficients, and the comparison then gives the next profile, one logarithm deeper: this is Theorem 7.3. Finally Theorem 7.4 evaluates a profile at curvature of order $1/\log(en)$ to reach every isotropic log-concave measure, which gives Theorem 7.5.

Controlling polynomial coefficients¶

The proof starts with Letwin’s quadratic estimate Theorem 25.1, which controls the whitened third moments driving covariance noise. Localization with covariance-dependent noise then follows all derivative means of a degree-$d$ polynomial, weighted by the matching tensor powers of the covariance. The lower derivative means of an Appell polynomial start at zero. The drift inequalities couple degree $j$ only to higher derivatives, allowing induction on $d$; retaining the covariance–mean cross variation is essential. After a time of order $d^{-2}$, the accumulated Gaussian curvature bounds the remaining variance. Conditioning on growing balls and convergence of finitely many moments remove compact support. The Appell expansion then gives the estimate for a general polynomial by the triangle inequality.

In the notation above, this first estimate gives $c_k\le32^k k!$. The dimension has disappeared, but the factorial remains: fed into the comparison below, it gives only a bound depending on the curvature, of order $\log(e+1/a)^2$.

The analytic setting¶

The passage to the spectral gap uses inverse powers of the diffusion operator. The next lemma supplies their domains, a first eigenfunction, and the approximation needed to return to general measures: the analytic prerequisites for applying the polynomial estimate to a function selected by the measure.

The proof uses cutoffs and local elliptic regularity to extend the Bochner identity from compact smooth functions to the operator domain. Conjugation by $e^{-W/2}$ gives a Schrödinger operator with a confining quadratic lower bound, so its resolvent is compact. Spectral calculus then defines the inverse square root on centered functions. For approximation, Gaussian convolution gives a smooth convex potential with bounded Hessian; a small quadratic tilt supplies positive curvature, and whitening restores isotropy. Only scalar Poincaré inequalities pass through the final weak limit. The eigenfunction and its inverse-operator iterates are used at a fixed regular measure.

From coefficients to every test function¶

The proof follows a first eigenfunction of eigenvalue $\lambda=\CP(\nu)^{-1}$ through repeated application of a centered gradient and an inverse square root of the diffusion operator. Normalization preserves the size of each family, while centering removes a nonnegative amount of mass. Bochner’s identity charges positive curvature against the energy of every surviving family. Polynomial tests estimate the mass lost to centering; the difficulty is that iterated derivative tensors are only approximately symmetric. A two-block tensor recovery inequality and a dyadic decomposition bound the defects, and a convolution estimate sums their overlapping contributions without a factor depending on the number of iterations. If $\lambda$ were too small, most mass would survive while curvature used up more energy than was initially available. The resulting contradiction gives the displayed comparison.

This is a direct argument with an extremal function. Its all-degree polynomial hypothesis controls the centering losses; no density argument with a degree-dependent Poincaré constant is involved. The threshold $R\ge2^{40}\epsilon^{-2}$ pays for the initial small degrees and the absorption of normalization errors.

Feeding the curvature estimate back into the polynomials¶

Start with the factorial coefficient estimate and choose a dyadic degree comparable to $\log(e+a^{-1})$ in Theorem 7.2. This gives the first logarithmic curvature profile. To return from a profile to coefficients, apply it to localized measures after an affine change of variables. Their covariance and accumulated curvature enter together as a matrix weight; treating them as unrelated scalar bounds would lose the required control. The derivative hierarchy used in the first polynomial estimate then improves the coefficient growth. Keeping its zero initial values below the top degree makes the extra coefficient loss approach one at large depth. A further use of the comparison advances the logarithmic depth, and the constants can be chosen with

above one fixed initial depth. The summability of $r^{-2}$ gives one envelope $C_0 4^r$ valid for all finite depths.

Why optimizing the multiplier alone does not give a bounded profile. With $\epsilon=r^{-2}$ and $R=(1+r^{-2})\Gamma_r$, the comparison requires $R\ge2^{40}r^4$. Initializing the improved coefficient induction also requires $\Gamma_r\ge Kr^2$ for a universal $K$. Exponentially growing constants can meet both thresholds simultaneously; a bounded sequence cannot. Replacing the factor 4 by a fixed number greater than one still gives an unbounded product, and replacing it by one does not remove these growing admissibility costs. Within this iteration, a bounded profile needs both a smaller multiplier and a replacement for these growing thresholds. These are obstructions to retaining the existing estimates, not lower bounds on every possible comparison argument.

The two thresholds pay different bills. The coefficient induction uses the factorial bound below degree $\lceil32r^2\rceil$; the nearly lossless hierarchy starts above that degree. The comparison must also absorb centering losses along the whole inverse-gradient sequence, including its initial terms. Improving only the final tensor estimate or only the high-degree tail leaves these earlier costs in place. A useful replacement would supply low-degree coefficient control conditional on the current curvature profile, together with a comparison whose multipliers have bounded cumulative product and whose admissibility is uniform in depth. The initialization bound now follows from BKL (Corollary 8.1), though not independently of KLS (Section Dependencies). SZ v2 avoids the question altogether: it iterates a common coefficient radius instead and pays the fixed spectral conversion only once (Section What separates the arguments).

The exact coefficient growth demanded by KLS¶

The coefficient hierarchy gives an exact reformulation of the dimension-free bound, now obtained by BKL. The two assertions below have the same strength; the second identifies the coefficient growth established by Theorem 8.3.

For the forward direction, differentiation lowers an Appell polynomial by one degree. A Poincaré constant $C$ therefore gives $K_k\le Ck^2K_{k-1}$, starting from $K_1=1$ in isotropic position, and the factorials cancel in $c_k$. For the converse, exponential growth fits Theorem 7.2 with $\ell=1$, $\epsilon=1$, and $R=\max\{4A,2^{40}\}$. At one fixed regular measure, let the dyadic degree tend to infinity: its positive curvature $a$ stays fixed, so $a^{-1/(d+1)}$ tends to one. This gives a scalar bound uniform over all regular measures, which then passes to arbitrary isotropic log-concave measures by approximation.

The order of quantifiers is the substantive demand: one $A$ must work for every degree, dimension and regular measure, regardless of its curvature bounds. Neither the factorial estimate of Theorem 7.1 nor constants chosen afresh at each logarithmic depth meet it.

The iterated-logarithm bound¶

A curvature profile valid in every dimension bounds the Poincaré constant of every isotropic log-concave measure: it suffices to evaluate it at curvature of order $1/\log(en)$.

This statement is where ordinary Gaussian stochastic localization enters. A bounded Lipschitz test detects the original Poincaré constant: after variance normalization, its supremum is universally bounded and its Lipschitz constant is of order $\CP(\mu)^{-1/2}$ Klartag & Lehec, 2025, Theorem 20. Stop the posterior covariance at its first exit from $(I/2,2I)$. Letwin’s quadratic estimate controls the exit probability on a time interval of order $1/\log(en)$, while the stopped variance decays by at most a fixed factor. The boundedness of the test pays for the exceptional paths, leaving one posterior with controlled covariance and substantial variance. Whitening that posterior gives curvature at least half the elapsed time, so evaluating $F$ at this deterministic bound controls the original Poincaré constant. No comparison of nearby values of $F$ is used. Regular approximation passes the same scalar bound to arbitrary measures with its argument unchanged. Any improved curvature profile can be substituted here.

Feeding the profiles of Theorem 7.3 into this transfer gives Song–Zhang’s main theorem, stated in their first-version preprint (arXiv:2610.01447v1, 1 October 2026) as Theorem 7.1.

Insert the profile of Theorem 7.3 into Theorem 7.4, absorbing the universal rescaling of its argument into the outer constant, and pass to the Cheeger scale by (0.3). Uniformity in the depth $r$ is essential, because the depth is chosen depending on $n$: near $\log^*(n+2)$, the iterated logarithm $\ell_r(\log(en))$ is universally bounded and only the factor $16^r$ remains. That factor is all the dimension dependence left in this particular iteration. SZ v2 removes it by refinements with summable costs (Chapter Song–Zhang, second version: repeated refinement with summable losses).

Whitening an arbitrary positive definite covariance contributes $\|\Sigma\|_{\mathrm{op}}$ to the gradient energy, which turns the isotropic bound into the affine form.