Welcome

The Kannan–Lovász–Simonovits (KLS) conjecture, Conjecture 0.1, asked whether every isotropic log-concave measure has a Poincaré constant bounded independently of the dimension. For a Gaussian that constant is one; the question was whether a universal bound survives without symmetry or product structure. It is now a theorem: three preprints of October 2026 prove it, by Bizeul–Klartag–Lehec (BKL) Bizeul et al., 2026, by Song–Zhang in the second version of their preprint (SZ v2) Song & Zhang, 2026, and by Balasubramanian–Kasiviswanathan (BK) Balasubramanian & Kasiviswanathan, 2026.

This site adds three things the preprints do not: a complete account of each of the three arguments, a comparison of them, and an account of the methods around them, including three alternative mechanisms for the Poincaré bound with exact computations, conditional reductions and counterexamples of their own. Each argument is rewritten so that every step is a complete statement with a complete proof, or a citation of an established published result, and each of these proofs has been checked against its statement by a separate reviewer agent; no person has yet reviewed or accepted them.

Where to start¶

Four ways in, depending on what you came for:

• Discover. The overview, Chapter The KLS theorem and its methods; what the theorem gives and the question of its constant, Chapter KLS after its proofs; then the map of alternative mechanisms, Chapter Alternative mechanisms after KLS.

• Read the proofs. The first version of Song–Zhang, whose spectral criterion underlies BKL and SZ v2, Chapter Song–Zhang, first version: polynomial estimates and curvature; Bizeul–Klartag–Lehec, the shorter argument from that criterion, Chapter Bizeul–Klartag–Lehec: cumulants and suspension; the second version of Song–Zhang, Chapter Song–Zhang, second version: repeated refinement with summable losses, with its technical estimates in Chapter Song–Zhang, second version: technical estimates; Balasubramanian–Kasiviswanathan, Chapter Balasubramanian–Kasiviswanathan: compatible integration; then their comparison, Chapter The proofs of KLS compared.

• Read the alternative mechanisms. The moment map, Chapter The moment map: the deterministic inequality; the fixed eigenfunction, Chapter The fixed eigenfunction: following one eigenfunction through localization; conditional fibers, Chapter Conditional fibers: inverse-variance frames of line resamplings. The fixed cut, an earlier localization argument kept for its obstructions and counterexamples, opens the archive, Chapter The fixed cut: approach and lessons.

• Contribute. The problems for someone who might take them up, Section Problems for someone who might take them up, then how results are checked, below.

Recurring terms and their normalizations are collected in the glossary.

What this manuscript contributes¶

The three proofs.

• BKL bound cumulants of every order uniformly in the dimension, then encode an arbitrary test function as one extra coordinate (suspension): Chapter Bizeul–Klartag–Lehec: cumulants and suspension.

• SZ v2 refines one coefficient radius repeatedly, with losses whose product stays bounded: Chapters Song–Zhang, second version: repeated refinement with summable losses and Song–Zhang, second version: technical estimates.

• BK control every power of an integration operator on compatible tensor fields with one common factor, and close a direct induction in the polynomial degree, with an explicit constant: Chapter Balasubramanian–Kasiviswanathan: compatible integration.

• The first version of Song–Zhang, whose spectral criterion the first two proofs use, is worked through as preparation (Chapter Song–Zhang, first version: polynomial estimates and curvature); the three proofs are compared in Chapter The proofs of KLS compared.

Results proved in this manuscript.

• The moment-Hessian inequality Definition 16.1 bounds the affine Poincaré constant with no loss (Theorem 16.1).

• Its exact value on the line (Theorem 17.1) and on products (Theorem 17.2), and the bound 4, sharp over the family, on every log-concave Dirichlet law (Theorem 17.3, Corollary 17.3).

• Its linear test reduced to a third-moment tensor (Lemma 16.2), and a countermodel for fixed-matrix arguments (Proposition 16.3).

• A reduction of KLS to an occupation estimate for one eigenfunction (Proposition 21.1).

• A reduction of KLS to a gap for resampling along lines (Lemma 22.1), and an obstruction on the simplex (Proposition 22.1).

• In the fixed-cut archive: a bootstrap (Theorem 33.1), its ceiling (Proposition 33.1) and a product counterexample (Proposition 29.1).

Each is explained with the idea of its proof in Section The main results in short.

How results are checked, and how to contribute¶

Every labelled statement is fixed text, and its status, shown next to its title, is kept apart from the prose:

• Not settled here: this manuscript does not settle the statement; it says nothing about the literature.

• Preprint, not yet checked here: a recent source’s announced result, not yet checked by this project.

• Proved, or Proved (from a preprint) for a source’s result checked by this project: a written proof, checked against the statement by a reviewer, linked from the status, which names who checked it and when.

• Established in the literature: a result of the field, cited and not reproved.

• Refuted by: the statement that refutes it.

A reviewer agent’s check is distinct from journal peer review and from a person’s review or acceptance, which are recorded separately; computations never count as proof. Since KLS is proved, a counterexample to a statement not settled here refutes that statement, not KLS; the alternative mechanisms aim at other proofs, a sharp constant, or properties that imply KLS. What each certification means is explained on the full proofs page.

How to contribute. The site is maintained by Nicolas Brosse and open to collaboration; contributors are credited in the history of the repository. A proof, a counterexample, a partial result, a missed reference or a correction is welcome:

• On a statement: next to its title are its label, for instance conj:gate-zero-sharp, and links that open a form on the project repository with the label filled in — Idea or Counterexample on a statement not settled here, Correction on any other.

• In discussion: ask a question (Q&A), think out loud (Ideas) or point at a reference (Literature) in the project’s GitHub Discussions, naming a statement by its label.