Song–Zhang, second version: repeated refinement with summable losses

What to retain. Instead of the Poincaré constant, the second version of Song–Zhang (SZ v2) iterates one number per measure, a common radius for the Appell coefficients of every degree, which controls the Poincaré constant up to a fixed factor paid once. Each refinement of that radius passes through curvature profiles and localization, as in the first version (SZ v1), but with margins $\alpha_i=2^{-i}/16$ whose costs have a bounded product, so infinitely many refinements leave a universal bound.

This chapter works through the second version of Song and Zhang’s preprint, An O(1) Bound for the KLS Constant Song & Zhang, 2026; Chapter Song–Zhang, first version: polynomial estimates and curvature concerns SZ v1 Song & Zhang, 2026, and the preceding chapter the proof of Bizeul, Klartag and Lehec (BKL). Why bounded costs per repetition do not suffice, and summable ones may, is the calibration of Section Song–Zhang, second version: summable losses.

Versions and the two closing mechanisms¶

SZ v2 and BKL are two distinct closing mechanisms on a shared spectral foundation. SZ v2 improves the iteration producing the coefficients; BKL estimate the exponential coefficient end point directly (Chapter Bizeul–Klartag–Lehec: cumulants and suspension). The argument of this chapter uses nothing from BKL; Chapter The proofs of KLS compared compares both with the proof of Balasubramanian and Kasiviswanathan (BK).

The argument in outline¶

The quantity improved throughout the proof is a single radius for the whole coefficient hierarchy, and four improvements bring it to a universal bound.

1. Polynomial cost in the logarithmic depth. Theorem 9.1 replaces the factor $16^r$ of SZ v1 by $(r+1)^{1/3}$, because blocks of inverse-gradient iterates retain centering and symmetry information over many steps instead of charging a loss at each one; via the transfer, $\CP\lesssim(1+\log^*(n+2))^{1/3}$ (Theorem 9.2).

2. Height reduction. Repeating the improvement replaces $\log^*$ by $\log^*\circ\log^*$, and so on, at a fixed cost $C_*$ per repetition (Proposition 9.2), which still grows with the number of repetitions.

3. Multipliers close to one. With a margin $\delta$, $m$ refinements cost $e^{C\delta m}$ (Proposition 9.3), at the price of a degree cutoff of order $\delta^{-2}$ and a starting depth of order $\delta^{-12}$; earlier coefficient bounds are kept on the degree ranges where they are stronger.

4. Summable costs. The margins $\alpha_i=2^{-i}/16$ give a multiplicative cost at most $e^{C_A/8}$ and, once the growing starting depths are absorbed into the improving height profile, an additive cost of order $2^{-i}$ (Proposition 9.4).

The main estimates appear first, ending with the universal bound. The technical part then explains how one family of inverse-gradient iterates supplies the centering, symmetry and energy estimates needed throughout. Full proofs are linked beside the statements.

One radius for all degrees¶

For the standard Gaussian, $c_d=1/\sqrt{d!}$, so the coefficient radius below equals one. For a general regular law it is finite for each fixed measure; this does not initially give a bound uniform over measures. Finiteness must precede any argument using the supremum over all degrees.

The operator $\mathcal T$ differentiates after solving the diffusion Poisson equation and removes the mean. The removed linear component has norm at most one, which explains the comparison of $R$ with $\CP$. Iterating the Appell derivative identity tests finite powers of $\mathcal T$ against every higher degree. Low degrees are covered by $G(m)$; the remaining degrees are covered by the same operator power. The final spectral comparison gives $\CP\le2^{85}\mathcal A$. That fixed factor is paid once, after refining $\mathcal A$, and does not accumulate with the number of refinements.

Polynomial cost in the logarithmic depth¶

Section 6 of the preprint replaces the exponential depth cost of SZ v1 by a polynomial cost. Its inverse-gradient blocks retain centering and symmetry information over many steps instead of charging a fixed loss at each step.

The power $(r+1)^{1/3}$ improves the factor $16^r$ in Theorem 7.3, but still grows with the depth. The proof estimates the accumulated errors in the inverse-gradient hierarchy before applying the spectral comparison. This yields a curvature profile valid uniformly in $r$. The existing transfer Theorem 7.4 evaluates that profile at a curvature inverse of order $\log(en)$, giving the following consequence.

Choose enough logarithms to bound $\ell_r(\log(en))$ universally. Only $(r+1)^{1/3}$ remains, with $r$ of order $1+\log^*(n+2)$. This bound also supplies an initial coefficient estimate for the next improvement; it is not yet the final dimension-free estimate.

Reducing the height at a fixed cost¶

There are now two kinds of iteration. The index $r$ counts logarithms of the inverse curvature. The index $j$ below counts repetitions of the improvement of that profile. For example, $t_1(Q)=t(Q)$ grows like $\log^*Q$, whereas $t_2(Q)=t(t(Q))$ applies the same reduction to that already slow growth. The averaged functions $W_m$ will let the proof control changing starting depths quantitatively.

The proof keeps a static coefficient bound throughout all inner depths. Finite operator blocks improve that bound, and localization converts the improved curvature estimate into a new coefficient bound. Repetition replaces $t_j$ by $t_{j+1}$ without changing the admissible starting depth $r_Q$. The block estimates of Chapter Song–Zhang, second version: technical estimates explain how actual functions realize these bounds. The cost $A_j=A_1C_*^{j-1}$ still grows with the number of repetitions, so this estimate alone cannot give a universal constant by taking $j$ large.

Refinements with a multiplier close to one¶

The final refinement of the preprint uses a degree cutoff of order $\delta^{-2}$ and a starting depth of order $\delta^{-12}$. The smaller multiplier is useful only if those growing thresholds remain admissible. Earlier coefficient estimates are retained on the degree ranges where they are stronger; the finite-chain estimate has constants independent of how many such estimates are kept. One outer round turns a radius bound into a new coefficient cap and then into an improved radius bound, with a cost $e^{16\delta}$ for forming the new cap and $e^{24\epsilon j}$ for repeating the inner improvement; this is Lemma 10.12, stated with the technical estimates in Chapter Song–Zhang, second version: technical estimates. Its explicit dependence on both margins gives the following profile improvement.

For a prescribed finite number $m$ of refinements, choose the margin $\delta$ before iterating. The multiplier is $e^{C\delta m}$, and $S_m$ stays bounded because its increments form a geometric series. The retained floor $\bar t(\delta^{-1})$ and the threshold $R_\delta$ keep the cost of that small margin visible. Taking $m$ large with $\delta$ fixed would still lose a uniform constant; the next estimate varies the margin from one repetition to the next.

Summable costs and the universal bound¶

The choice $\alpha_i=2^{-i}/16$ has $\sum_{i\ge0}\alpha_i=1/8$. Thus the multipliers can have a bounded product. The additional task is to absorb $R_i$, which grows like $2^{12i}$, into the improving height profile $W_i$. The proof first uses how $W_i$ behaves under fixed powers of its argument, then its contraction near four. Applying only a Lipschitz estimate directly to $R_i$ would leave a growing error. With the two estimates combined, the additive cost is of order $2^{-i}$ and is summable too.

The induction retains every earlier valid coefficient cap but uses each on a disjoint range of degrees. It therefore pays no factor equal to the number of retained caps. The finite-chain construction supplies one starting family with matched energy and centering losses; the outer profile estimate applies to that family. Threshold absorption accounts for the next starting depth through the increment of $S_i$, while the increment of $\log A_i$ pays the multiplicative cost.

Fix one regular measure and put $x=\max\{1,a^{-1}\}$. Choose a finite $i$ with $W_i(x)=4$, then choose $r=R_i+\lceil C_*t(x)\rceil$ with $C_*$ universal and sufficiently large. The normalized logarithm $\mathcal L_r(a^{-1})$ is bounded universally, and threshold absorption gives $W_i(r)\le4+b2^{-i}$. The bounded profiles therefore give a universal radius bound, and Proposition 9.1 converts it to a Poincaré bound once. Pass this scalar inequality to regular approximants using Lemma 7.1; no supremum over degrees passes through weak convergence. Clipping and truncation give both $L^2$ integrability and the finite-energy formulation above.