The proofs of KLS compared

This chapter compares the three proofs of Conjecture 0.1: Bizeul–Klartag–Lehec (BKL), Chapter Bizeul–Klartag–Lehec: cumulants and suspension Bizeul et al., 2026; Song–Zhang, second version (SZ v2), Chapters Song–Zhang, second version: repeated refinement with summable losses and Song–Zhang, second version: technical estimates Song & Zhang, 2026; and Balasubramanian–Kasiviswanathan (BK), Chapter Balasubramanian–Kasiviswanathan: compatible integration Balasubramanian & Kasiviswanathan, 2026. The idea of each is in Section How KLS was proved. All three control the exponential growth of Appell coefficients; they differ in the closing estimate, the spectral conversion and the constant obtained, and the table below is the reference for these differences.

The three proofs side by side¶

Shared coefficients, distinct spectral reductions¶

The Appell coefficients $c_d(\mu)$ (Section Notation and the smallest cases) have the same convention in all three arguments, and their exponential growth, uniform in degree, dimension and measure, is equivalent to KLS (Proposition 7.1). That criterion describes what all three achieve; it is not the identical last lemma in each proof. BKL close through the tilt-average criterion Theorem 8.1, whose operator norm is exactly $c_d$ (Proposition 8.1). SZ v2 close through the single conversion $\CP\le2^{85}\mathcal A$, after all refinements; the curvature transfer Theorem 7.4 of the first version of Song–Zhang (SZ v1) serves only their intermediate dimension-dependent bound. BK close through the integration calculus itself: at one fixed regular measure, the observation degree tends to infinity in Corollary 11.1, and approximation removes regularity.

What separates the arguments¶

Cumulants, refinement, or a degree recurrence. BKL estimate cumulants inductively and use suspension to reach general functions. SZ v2 instead improve a curvature profile repeatedly, controlling every round’s admissibility and making the losses summable. BK close an induction straight on $c_d$: lower degrees control integrations under curved laws, and localization transfers those integration bounds to degree $d$. For its large-degree step the integration length is $q=\lfloor d/2\rfloor$, while the observation degree is $D=\lfloor\sqrt d\rfloor$. Keeping one prefactor for the whole integration chain is essential; multiplying separate one-step bounds discards the gain.

What localization transports. BKL and BK both normalize the noise by the inverse covariance. BKL follow cumulants in a moving metric; BK control covariance tensor products and the variance of a polynomial whose Appell normalization changes with the law. SZ v2 use Gaussian localization to convert curvature profiles to coefficient bounds. A shared noise normalization does not make the objects or estimates interchangeable.

Where the constants are spent. The SZ v1 spectral comparison incurred a fixed cost at every round. SZ v2 make the radius-refinement costs summable and spends the final conversion once. BKL avoid that refinement altogether. BK’s finite observation bound supplies a factor independent of the number of integrations, allowing a direct recurrence with a numerical majorant. The resulting constant is explicit but far from the lower bound 4 supplied by the exponential law (Chapter KLS after its proofs).

Two different Hodge comparisons. BK’s Hodge estimate, for the potential of the original measure, is not the moment-Hessian comparison Corollary 16.1; the moment-Hessian comparison is set out in Section The Hodge content: the affine channel and the solenoidal excess, and the harmonic-mean computation behind BK’s estimate is in Chapter Balasubramanian–Kasiviswanathan: compatible integration.

Dependencies¶

The three arguments share earlier polynomial and localization ideas. BK’s quadratic initialization explicitly uses Letwin’s Theorem 25.1, and its Appell convention agrees with Theorem 7.1. Distinct proofs here means distinct closing arguments, not independence from the earlier literature.^[1]

The BKL argument uses no SZ v2 estimate or KLS conclusion as an input. The SZ v2 argument uses no BKL cumulant or tilt bound, including their consequence Corollary 8.1. The BK induction uses neither of those proofs’ conclusions nor KLS itself. Feeding the BKL initialization into the SZ v1 iteration would recover KLS through BKL, not by another argument.

What KLS itself gives, and the question of its best constant, follow in Chapter KLS after its proofs; what the alternative mechanisms would add to these proofs is in Chapter Alternative mechanisms after KLS.