Kannan–Lovász–Simonovits (KLS) conjecture proved in two AI-assisted preprints: Song–Zhang's O(1) bound, then Bizeul–Klartag–Lehec ('most proofs … found by ChatGPT')
The KLS conjecture (1995), which says isoperimetry in convex bodies is governed up to a universal constant by half-spaces, was claimed resolved twice in three days. Zhao Song and Xinzhi Zhang (arXiv 2610.01447, Oct 2; v2 Oct 4) prove an O(1) bound on the KLS constant with GPT-6 Astra, GPT-5.6 Sol and Claude Fable 5/5.1 after exploring 100+ approaches. Pierre Bizeul, Boaz Klartag and Joseph Lehec (arXiv 2610.05474, Oct 4) "present a proof" built on Song–Zhang's criterion, saying most proofs and ideas in it were found by ChatGPT.
Key facts
- Song & Zhang, 'An O(1) Bound for the KLS Constant', arXiv 2610.01447 (math.PR), Oct 2, 2026, v2 Oct 4: 'We prove that ψ_n ≤ C for a universal constant C > 0, resolving the KLS conjecture'; also a universal Poincaré constant for isotropic log-concave measures
- Song–Zhang disclosure: 'The AI tools used in this work were GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5, and Fable 5.1.' From July 28, 2026 they explored 'more than 100 approaches in collaboration with AI tools' to improve a log^(1/4) bound (from a paper that appeared on July 27, 2026) to log^(1/6); on Sept 30 the authors found a route to an f(log* n) bound, then sharpened it to a constant. 'All of the proofs have been carefully verified by the authors and several AI tools.'
- Bizeul, Klartag & Lehec, 'Presenting a proof of the Kannan-Lovasz-Simonovits conjecture', arXiv 2610.05474, Oct 4: 'The crucial step in the proof is the very recent criterion from Song and Zhang involving high-order derivatives of tilt averages'
- Bizeul–Klartag–Lehec disclosure: 'Most proofs and mathematical ideas in this paper were found by ChatGPT; a notable exception is the idea to use suspension which was suggested by the authors. The role of the authors has been mostly to understand these proofs and improve their exposition.' Their introduction says the conjecture now 'poses little challenge for contemporary AI tools'
- Background: Klartag and Lehec had earlier proved a polylog bound and (with others) resolved Bourgain's slicing problem; KLS implies the slicing and thin-shell results
- Verification status: two preprints, not peer-reviewed, no formal proof; Klartag–Lehec's independent write-up is a strong expert signal (confidence medium until reviewed)
Science result
- Field
- mathematics / high-dimensional convex geometry / probability
- Problem
- Kannan–Lovász–Simonovits conjecture (universal Cheeger/Poincaré constant for isotropic log-concave measures) (open since 1995)
- Result
- Universal O(1) bound on the KLS constant ψ_n (Song–Zhang); a second, AI-generated proof via Song–Zhang's criterion (Bizeul–Klartag–Lehec)
- AI system
- GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5, Claude Fable 5.1, ChatGPT
- Human role
- AI-assisted (Song–Zhang: human-chosen approaches, AI co-developed); largely AI-generated (Bizeul–Klartag–Lehec: authors mainly understood and exposited ChatGPT's proofs)
- Verification
- Two preprints; expert authors; not peer-reviewed or formalized
- Status
- pending
- Why surprising
- A central conjecture of asymptotic convex geometry fell within days to two AI-assisted teams, and leading experts wrote that it 'poses little challenge for contemporary AI tools'.
What happened
KLS asks for a dimension-free isoperimetric constant for convex bodies. Since Eldan's stochastic localization (2013), bounds improved from polynomial to polylogarithmic (Chen; Klartag–Lehec; Klartag). Song and Zhang, working with OpenAI and Anthropic models since late July, posted a proof of about 138 pages of an O(1) bound on Oct 2. Two days later Bizeul, Klartag and Lehec, among the area's leading experts, posted a shorter proof built on Song–Zhang's criterion, written mostly by ChatGPT.
Why it matters
KLS was among the best-known open problems in high-dimensional geometry, with consequences for sampling algorithms on convex bodies. The same week saw AI-originated proofs of the linear Hadwiger conjecture and a refutation of the 3SUM/APSP hypotheses. Confidence is medium only because neither paper has been refereed or formalized yet.
Changelog
- 2026-10-06: created (arXiv scan; both PDFs read with pdftotext)
People
Related events
- Klartag and Moshe prove the ε-Dvoretzky conjecture (polynomial dependence on ε); the key probabilistic idea came from a ChatGPT discussion ★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
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Sources (3)
- paperarXiv 2610.01447: An O(1) Bound for the KLS Constant (Song, Zhang)
- paperarXiv 2610.05474: Presenting a proof of the Kannan-Lovasz-Simonovits conjecture (Bizeul, Klartag, Lehec)
- discussionETH Randomstrasse101: The KLS Conjecture (problem 30)
id: 2026-10-02-kls-conjecture-proved-ai-assisted · updated 2026-10-06 · open in the interactive timeline