KLS Conjecture for Quadratic Forms

Precise statement

Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is $\operatorname{Var}\langle MX, X\rangle \le C\, \mathbb{E}|\nabla\langle MX, X\rangle|^2$ for every symmetric $M$?

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

ChatGPT-5.6 Pro

The key argument was developed in collaboration with ChatGPT-5.6 Pro and checked by the author.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Author-checked arXiv preprint proving the quadratic-form case with constant 2 and deriving the global estimate $\psi_n \le C \log^{1/4} n$. Not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-27

   arXiv:2607.24164 - The KLS constant is O(log^(1/4) n)

   with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.