Author verified Collaborative

Kannan–Lovász–Simonovits conjecture proved in two preprints posted the same day

Two preprints posted on 4 October 2026, one by Bizeul, Klartag and Lehec and one by Song and Zhang, prove the Kannan–Lovász–Simonovits conjecture that every isotropic log-concave measure satisfies a Poincaré inequality with a dimension-free constant, and both say AI systems produced much of the mathematics.

Lab: Independent
Model: ChatGPT, version not stated (Bizeul, Klartag and Lehec); GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5 and Fable 5.1 (Song and Zhang)
Field: Mathematics
Date: 2026-10-04
Human collaborators: Pierre Bizeul, Boaz Klartag, Joseph Lehec, Zhao Song, Xinzhi Zhang
Problem posed: 1995 · open 31 yrs

Sources

Original work

arXivBizeul, Klartag and Lehec, Presenting a proof of the Kannan-Lovász-Simonovits conjecture↗arXivSong and Zhang, An O(1) Bound for the KLS Constant↗

What was found

The conjecture, formulated in 1995, says that isoperimetry in a convex body is saturated by half-spaces up to a universal constant; equivalently $C_P(\mu) \le C$ for every isotropic log-concave $\mu$ on $\mathbb{R}^n$ , so that the Poincaré constant is determined by testing linear functions. It sits above Bourgain's slicing problem and the thin-shell conjecture, both resolved by Klartag and Lehec in the past two years. The bound on the KLS constant $\psi_n$ , the reciprocal Cheeger constant, had fallen from Kannan, Lovász and Simonovits's $\sqrt n$ through Lee and Vempala's $n^{1/4}$ , Chen's subpolynomial bound, Klartag and Lehec's polylogarithm and Klartag's $\sqrt{\log n}$ to Letwin's $(\log n)^{1/4}$ of July 2026. Song and Zhang's first version, posted on 1 October, gave an iterated-logarithm bound. Bizeul, Klartag and Lehec (posted 19:30 UTC on 4 October) complete the proof using Song and Zhang's criterion on high-order derivatives of tilt averages and a suspension idea they credit to themselves; their acknowledgements say most proofs and mathematical ideas were found by ChatGPT and that the authors' role was mostly to understand the proofs and improve the exposition. Song and Zhang's second version (21:21 UTC the same day, 140 pages) reaches a universal constant by applying their log-star refinement repeatedly; their disclosure describes exploring more than 100 approaches with AI tools since 28 July, with candidates generated collaboratively and the authors choosing which to pursue.

Novelty check

Read both papers' sources on arXiv: arXiv:2610.05474 v1 and arXiv:2610.01447 v2, with the latter's submission history. Each introduction gives the same ladder of prior bounds, ending at Letwin's, and neither cites an earlier resolution. The two are not independent of each other: the Bizeul, Klartag and Lehec proof takes Song and Zhang's first-version criterion as its crucial step, and Song and Zhang's second version finishes the same programme by a different route.

Caveats and known objections

Neither preprint is refereed, and the two together are under ten days old at entry. Verification is author-verified: Klartag and Lehec are among the people who built this area, which carries weight, but their own account is that ChatGPT found most of the proof and they checked and rewrote it, while Song and Zhang write that the proofs were verified by the authors and several AI tools. No outside reader is on record for either. Autonomy is graded collaborative on the weaker of the two disclosures: the Bizeul, Klartag and Lehec wording on its own reads as ai-led, but the decisive criterion is Song and Zhang's, whose disclosure describes a human-steered search and credits the log-star and height-reduction ideas to the authors' earlier work. The Bizeul, Klartag and Lehec introduction opens by saying the conjecture 'poses little challenge for contemporary AI tools', given recent progress, which is a striking thing for two of the field's leading figures to put in print. The ChatGPT version used is not stated.

The open problem

MathDBKannan–Lovász–Simonovits conjecture

363363

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Also recorded at

vibemathedkannan-lovasz-simonovits-conjecture

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History of this entry

2026-10-16CorrectedRecorded the vibemathed record for this result and the MathDB record for the conjecture as registrations. No grade moved.
2026-10-12AddedEntered the registry graded Author verified and Collaborative.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded author verified for verification and collaborative for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Kannan–Lovász–Simonovits conjecture proved in two preprints posted the same day. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-04-kls-conjecture

BibTeX

@misc{whataifound-independent-2026-conjecture,
  title        = {Kannan–Lovász–Simonovits conjecture proved in two preprints posted the same day},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
  url          = {https://whataifound.org/finding/2026-10-04-kls-conjecture}
}

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