Alternative mechanisms after KLS

The three proofs of KLS (Chapter The proofs of KLS compared) obtain a universal constant through estimates on polynomials of every degree. This part develops three further mechanisms, each through a sufficient condition of its own: the moment-Hessian inequality, an occupation estimate for one eigenfunction, a gap for resampling along lines. Each condition implies KLS, and no converse is known. Proved, each would give a further proof together with something the three proofs do not provide: a constant, an object followed, or a kind of argument. A fourth method, the fixed cut, is kept as an archive for what its obstructions teach. This chapter compares the four; the obstacles they face are explained in Section Obstacles for alternative arguments.

The mechanisms, side by side¶

What each would add. The moment map has an implication towards KLS that loses no constant, exact values on the line and on products, and the sharp bound 4 on Dirichlet laws; it would give a deterministic proof, with no stochastic localization, and the constant 4, which the exponential on the line attains (Section The question of the constant). The fixed eigenfunction would give a localization mechanism that ignores covariance spikes in directions the eigenfunction does not use. Conditional fibers would give an elementary mechanism whose only analytic input is one-dimensional. The order is that of what each has established; it is not a forecast.

The constraint every mechanism must meet¶

The covariance spike (Proposition 0.1, explained in Section The covariance spike, and why the direct repair fails) cuts in two directions. A direct “bound $\norm{A_t}_\op$ better” approach cannot work, since the statement it needs is false; and rare spikes can be harmless, so a successful potential must recognize them rather than charge the full top eigenvalue whenever one occurs. The working criterion is the tensorization test of Section The tensorization test: evaluate the proposed quantity on a product of independent copies, and reject it if it charges $n$ independent coordinates $n$ times. It is the first thing to check on each direction below.

What each mechanism gives, and what blocks it¶

The table above says what each mechanism tries; this one says where each has got to. The middle column is the mechanism’s own contribution, the right column the statement whose absence stops it; the full account of both is the opening summary of each entry chapter.

The moment map: from fixed matrices to a nonlinear estimate¶

Brascamp–Lieb in moment-map coordinates already gives (4.30), so KLS would follow from

The identity $\E\tau_\mu=I$ is insufficient, because $\tau_\mu(X)$ may correlate with $\nabla f(X)$. Letwin’s fixed-matrix inequality controls a deterministic matrix $B$; an extension of this kind would need to handle an $X$-dependent direction or matrix field.

The moment-map chapters (Chapter The moment map: the deterministic inequality) pursue a precise form of this idea: the inequality $\mathrm{CMH}(4)$ (Definition 16.1), which bounds the affine Poincaré constant by Theorem 16.1. Two facts developed there change how the extension should be read. $\mathrm{CMH}(4)$ also charges a solenoidal excess, so it is a sufficient condition rather than a known reformulation of (14.1) (Chapter The moment map: the deterministic inequality). And its cheapest necessary consequence, the linear test of Section The linear test: the necessary linear-sector condition (Conjecture 16.1), is itself an average-versus-uniform statement, which Proposition 16.3 shows no fixed-matrix argument supplies: the moment map relocates the difficulty rather than escaping it. How hard even the linear test is, Corollary 16.2 calibrates: its sharp form implies $\kappa_n\le2$, so proving it is at least as hard as a sharp directional third-moment bound.

The fixed eigenfunction: localization adapted to one function¶

For a fixed test function set $M_t(f)=\E_{p_t}f$, so that

A covariance-based estimate bounds the integrand by the worst case,

which discards essentially all information about $f$. A direct estimate of

for a near-extremizing eigenfunction would avoid the top-eigenvalue entropy cost of (2.11) and would respect tensorization, since (14.4) factorizes over independent blocks in a way that $\norm{A_t}_\op$ does not.

This is exactly Conjecture 21.1, the central problem of the fixed-eigenfunction chapter (Chapter The fixed eigenfunction: following one eigenfunction through localization). Its analogue for a fixed cut is Conjecture 29.1, the operator-to-trace upgrade of Chapter The fixed cut: remaining problems.

A further direction: parallel coupling beyond linear tilts¶

Parallel coupling, the tool behind the thin-shell theorem, controls the finite-dimensional family of exponential tilts $e^{\inner\theta x}\mu(\dd x)$ (Chapter Family 6: parallel coupling of exponential tilts). A coupling for perturbations $(1+\eps f)\mu$, with cost controlled by $\int\abs{\nabla f}^2\dd\mu$, would address arbitrary spectral directions directly rather than one linear family. No statement of this manuscript formulates such an extension, and none of the mechanisms above carries it.

What is already ruled out¶

Negative results are the most reusable part of a search, and four of them constrain everything above. Each is stated and explained in its own section; they are collected here so that a reader does not rediscover a dead variant.

• No uniform operator-norm covariance bound. Proposition 0.1: the statement a direct “bound $\norm{A_t}_\op$ better” approach would need is false, for a measure that satisfies KLS (Section The covariance spike, and why the direct repair fails).

• Independent spectators and global covariance weights. Proposition 29.1 supplies the product-cylinder witnesses relevant to Assumption 28.3 and Conjecture 29.2.

• No matrix-moment argument supplies the linear test (Section The linear test: the necessary linear-sector condition). Proposition 16.3: the fixed-matrix estimate does not supply it, so the moment-map mechanism relocates the average-versus-uniform difficulty rather than escaping it.

• The natural conditional-fiber root frame fails. The most obvious implementation of the conditional-fiber mechanism is ruled out by Proposition 22.1, which is why the question is stated for all frames, Conjecture 22.1.

Two further cautions are advisory rather than established: the two-tail obstruction (Remark 25.3) and the circularity warning of Section Excess propagation. They are recorded as remarks for exactly that reason: they guide work, but no statement here is excluded on their strength.

What the three proofs contribute to each mechanism¶

For each mechanism, the table records what the three proofs contribute towards its sufficient condition, and what remains to be proved.

For the archived fixed cut, what remains is Conjecture 29.1, or a replacement for the weighted package that independent coordinates leave unchanged; Proposition 29.1 still applies.

Scope of the quadratic input¶

Several mechanisms start from Letwin’s quadratic estimate Theorem 25.1 and its directional consequence Proposition 26.1 (sources in Section Sources). Their limits are precise. The covariance consequence Corollary 26.1 bounds fixed-time moments only up to time $c/\log n$, and Letwin’s general bound Theorem 4.6 depends on the dimension (Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels). Neither supplies a universal-time occupation bound, control of orientation, an adaptive matrix estimate or the linear test, and the implications Theorem 30.2, Theorem 30.1 and Theorem 28.1 keep their Carleson or centroid hypotheses. The three proofs of KLS use the same input through conversions on polynomials; they do not supply the moment-Hessian inequality, its sharp linear test or the occupation estimate either (Section What the three proofs contribute to each mechanism).

Suggested external reading order¶

For a reader coming to the subject rather than to this manuscript: Kannan et al., 1995 for the conjecture and deterministic localization; Eldan, 2013 for stochastic localization and the third-moment parameter; Lee & Vempala, 2018Lee & Vempala, 2024 for the cleanest entry to the covariance SDE and the set-transfer argument; Klartag & Lehec, 2025 for the best conceptual exposition of localization, filtering, Bochner, and the operator-norm obstruction; Klartag & Lehec, 2022 for heat flow, spectral measures, and $H^{-1}$; Klartag, 2023 for improved Lichnerowicz and the published bound; Letwin, 2026 for the quadratic input and its localization consequences; Song & Zhang, 2026 for Song and Zhang’s polynomial–curvature iteration, developed in Chapter Song–Zhang, first version: polynomial estimates and curvature; Song & Zhang, 2026 for its second version, by repeated refinement, in Chapter Song–Zhang, second version: repeated refinement with summable losses; Bizeul et al., 2026 for cumulants and suspension in Chapter Bizeul–Klartag–Lehec: cumulants and suspension; Balasubramanian & Kasiviswanathan, 2026 for compatible integration in Chapter Balasubramanian–Kasiviswanathan: compatible integration; and Klartag & Lehec, 2025Chen & Klartag, 2026 for the strongest related tools and the clearest illustration of the quadratic-to-all-functions gap.