Song–Zhang v2: the dimension bound of the inner iteration

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 new input is the improved curvature profile of Theorem 9.1. The Gaussian localization transfer already proved in Theorem 7.4 accepts exactly this input. This reconstruction of the implication in Section 7 of Song & Zhang, 2026 therefore needs no new stochastic calculation. The proof below makes the uniformity in depth and the finite stopping depth explicit.

Dependencies. The argument uses Theorem 7.4, the curvature assertion Theorem 9.1, and the established Cheeger–Poincaré comparison (0.3) Klartag & Lehec, 2025. The curvature assertion is an unresolved input until its own reconstruction is independently certified. No BKL result, consequence of BKL, or dimension-free KLS assertion is used.

Theorem 138.1 (Dimension estimates from the improved curvature profile)

Set $g(x)=\log(e+x)$ , $\ell_r=g^{\circ r}$ , and $L_n=\log(en)$ . Assume the curvature profile in Theorem 9.1. There are universal constants $C,C'>0$ such that, for every $n\ge1$ , every isotropic log-concave probability measure $\mu$ on $\mathbb R^n$ , and every integer $r\ge1$ ,

C_P(\mu)\le C(r+1)^{1/3}\ell_r(L_n)^2, \qquad \Psi_{\mathrm{KLS},n}\le C'(r+1)^{1/6}\ell_r(L_n).

(138.1)

In particular,

C_P(\mu)\le C(1+\log^*(n+2))^{1/3}, \qquad \Psi_{\mathrm{KLS},n}\le C'(1+\log^*(n+2))^{1/6}.

(138.2)

Here $\log^*x$ is the number of natural logarithms needed to reach a number at most one. Constants may be enlarged between the two displays. This is the implication needed for Theorem 9.2.

Proof 138.1

Let $A$ denote the universal constant in the curvature input. Fix any finite $r\ge1$ and define

F_r(a)=A(r+1)^{1/3}\ell_r(a^{-1})^2\qquad(a>0).

(138.3)

Every isotropic regular measure is centered and has covariance at most $I$ . Thus the assumed curvature estimate applies to every measure and every admissible lower Hessian bound required by the transfer theorem. Write $c_T,C_T>0$ for that theorem’s universal constants. It gives

C_P(\mu)\le C_T A(r+1)^{1/3}\ell_r(c_T^{-1}L_n)^2.

(138.4)

This transfer already includes arbitrary nonsmooth isotropic log-concave laws and all locally Lipschitz tests of finite energy. In particular no regularity of the initial law needs to be imposed here.

To absorb the change of argument uniformly in $r$ , put $D=\max(1,c_T^{-1})$ . Concavity of $g$ implies

g(x)=g(D^{-1}Dx+(1-D^{-1})0) \ge D^{-1}g(Dx)+(1-D^{-1})g(0)\ge D^{-1}g(Dx).

(138.5)

Consequently $g(Dx)\le Dg(x)$ . Monotonicity and induction give $\ell_r(Dx)\le D\ell_r(x)$ for every integer $r\ge1$ . Thus the first asserted estimate holds with $C=C_TAD^2$ , independent of both $n$ and $r$ . The comparison $\psi_\mu^2\le\pi C_P(\mu)$ followed by the supremum over isotropic laws gives the asserted estimate for $\Psi_{\mathrm{KLS},n}$ .

For completeness, choose a finite depth without making a limit in $r$ . Put $x_0=n+2$ , and let $m$ be the first index at which the ordinary logarithmic iterates $x_{j+1}=\log x_j$ satisfy $x_m\le4$ . The sequence is defined until that index and $m\le\log^*(n+2)$ . For $y\ge4$ ,

g(y+1)=\log y+\log(1+(e+1)/y)\le\log y+1,

(138.6)

because $e+1<4$ and $\log(1+t)\le t$ . When $m\ge1$ , $L_n\le1+x_1$ ; while $j<m$ , $x_j>4$ , so induction gives $\ell_{m-1}(L_n)\le1+x_m\le5$ . When $m=0$ , $n+2\le4$ and $L_n<5$ directly. Since $g([0,5])\subset[0,5]$ , the choice $r=\max(1,m-1)$ in either case satisfies

\ell_r(L_n)\le5,\qquad r\le\log^*(n+2).

(138.7)

Substitute this single depth in the preceding estimates and absorb the factors 25 and 5 in the constants. This proves both optimized estimates.

Fences respected. No new assertion discharges any CMH, occupation or trace-upgrade antecedent. The dimension bounds retain dimension dependence. The small-degree and uniform-admissibility issues are delegated to the explicit curvature premise, not assumed solved by this transfer.

References¶

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