Song–Zhang v2: the uniform Poincaré bound - The KLS theorem and its methods

Part of the second version of Song–Zhang, Chapter Song–Zhang, second version: repeated refinement with summable losses; the reading order is on the full proofs page.

Overview. The last composition in the Song–Zhang v2 argument is short once the bounded-amplitude profiles are available. This text gives that composition with the order of quantifiers and the approximation step explicit. Its crucial upstream input, Proposition 9.4, is not proved by this composition.

Dependencies. Use Proposition 9.4, Proposition 9.1, and Lemma 7.1, with the scalar threshold and burn-in calculations accompanying Proposition 9.3. No use is made of BKL, of its coefficient corollaries, or of the already proved KLS statement.

Proof 142.1 (Composition of the reconstructed bounded profiles)

First fix one centered regular measure with covariance at most $I$ and curvature at least $aI$ , $a>0$ . Set $x=\max\{1,a^{-1}\}$ . Choose a finite $i$ with $W_i(x)=4$ , possible by Lemma 10.13. For a fixed sufficiently large universal integer $C_*$ , put

r=R_i+\lceil C_*t(x)\rceil.

(142.2)

This integer lies in the domain of the profile at index $i$ ; $i$ and $r$ are chosen after fixing the measure. The burn-in part of Lemma 10.13 gives a universal constant $M$ with $\mathcal L_r(a^{-1})\leq M$ , and Lemma 10.14 gives $W_i(r)\leq4+b2^{-i}$ . Hence the bounded-profile assertion yields

\mathcal A(\mu)\leq A_0e^{C_A/8}(5+b+2B)^{1/3}M^2=:C_1.

(142.3)

Every quantity on the right is independent of $a,n,\mu,i,r$ . The radius comparison gives $C_P(\mu)\leq2^{85}C_1=:C_2$ . The fixed comparison factor has been used once, after all refinements.

For an arbitrary isotropic log-concave $\mu$ , the approximation assertion of Lemma 7.1 gives regular isotropic probabilities $\mu_j$ converging weakly to $\mu$ . For $\phi\in C_c^\infty$ , both $\phi^2$ and $|\nabla\phi|^2$ are bounded continuous. Passing the common inequality to the limit gives

\operatorname{Var}_\mu\phi\leq C_2\int|\nabla\phi|^2\,d\mu.

(142.4)

For completeness, the extension to finite-energy locally Lipschitz functions retains its integrability conclusion. First apply clipping and compact cutoffs to a bounded locally Lipschitz test; mollify on each compact set, where the log-concave density is locally integrable, as in the stated analytic-foundations lemma. Cutoff errors vanish because the test is bounded and gradients of cutoffs tend uniformly to zero. This gives the same variance inequality for the clipped tests $f_N=\max\{-N,\min\{f,N\}\}$ , with $\int|\nabla f_N|^2\,d\mu\leq\int|\nabla f|^2\,d\mu=:E$ .

Choose a ball $B$ of positive $\mu$ -mass on which $|f|\leq M_B$ ; local Lipschitz regularity supplies such a bound. For $N\geq M_B$ , $f_N=f$ on $B$ . If $m_N=\int f_N\,d\mu$ , then

\mu(B)(|m_N|-M_B)_+^2 \leq\int_B|f_N-m_N|^2\,d\mu \leq C_2E.

(142.5)

Thus $(m_N)$ is bounded, as is $\int f_N^2\,d\mu\leq C_2E+m_N^2$ . Since $f_N^2\uparrow f^2$ , monotone convergence proves $f\in L^2(\mu)$ . Now $f_N\to f$ in $L^2$ , their means converge, and the variance inequality passes to the limit with the same constant. Taking the supremum of $C_P(\mu)$ over dimensions and isotropic laws concludes the composition.

No supremum over polynomial degrees or internal profile parameters was passed through weak convergence. Each approximant uses its own finite parameters, whereas the final scalar constant is common to all of them.

The input Proposition 9.4 is reconstructed in its own dossier through finite-chain blocks and near-unit profile induction. The present composition uses that input and the analytic foundations; no BKL statement or already established KLS assertion enters its proof.

Fences respected. The statement makes no sharper constant claim and discharges no unrelated structural conjecture. Uniform admissibility and the finite-versus-uniform quantifier order are kept explicit.

References¶

Song, Z., & Zhang, X. (2026). An O(1) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447v2