Var(|X|²) ≤ 8n for isotropic log-concave vectors, with a proof ‘found by GPT-5.6 Pro’
Event confirmedAwaiting review
The takeaway
On 25 July 2026 Yuansi Chen and Boaz Klartag posted arXiv 2607.23307, which determines the optimal constant in the thin-shell theorem: for every isotropic log-concave random vector X in R^n, Var(|X|²) ≤ 8n, with equality for independent centered exponential coordinates.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 13 October 2026
Your AI and this story
- GPT-6 Astra86 days after its cutoff
- Claude Opus 5.525 days after its cutoff
- Gemini 4 Argon116 days after its cutoff
- Grok 4.755 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 25 days before it.
Key facts
- Paper: ‘Digesting the proof of the sharp thin-shell inequality’, Yuansi Chen and Boaz Klartag, arXiv 2607.23307 [math.MG], 25 July 2026, 23 pages; arXiv comment: ‘Statement of AI use included. Chat log is in the ancillary files as a pdf’
- Theorem 1.1: for any isotropic, log-concave random vector X in R^n, E(|X|² − n)² ≤ 8n. ‘The constant 8 is optimal: equality is attained when X_1, …, X_n are independent, identically distributed, standard, centered exponential random variables’
- Among isotropic random vectors uniform on convex bodies in R^n, Var(|X|²) is maximized by the uniform distribution on a regular simplex; the paper also gives a sharp bound on the Hilbert–Schmidt norm of the third-moment tensor
- Background: the thin-shell (variance) conjecture of Anttila–Ball–Perissinaki and Bobkov–Koldobsky asked for a universal constant; it ‘was recently proven by Klartag and Lehec’ (arXiv 2507.15495, 2025) after bounds by Fleury, Guédon–Milman, Lee–Vempala, Jambulapati–Lee–Vempala and Guan. The new paper gives the exact constant by a different method
- Method: a weighted Riemannian manifold associated with log-concave moment measures and the Monge–Ampère equation, studied by Klartag in 2014; ‘The main improvement … comes from a concise yet effective analysis of the 3rd-derivatives tensor of the potential’. Earlier arguments used concentration on the sphere or Eldan’s stochastic localization
- AI disclosure (abstract): ‘The proof was found by GPT-5.6 Pro in response to prompts supplied by the first-named author, following general discussions between the two authors concerning log-concave moment measures. The prompts referred to the paper “Logarithmically-concave moment measures I” and suggested bootstrapping a bound on the second trace moment.’
- AI use statement: ‘GPT-5.6 Pro produced the initial proof in response to prompts from the first-named author. Both authors verified and revised the argument, and rewrote it to make it more accessible. GPT-5.6 was then used to polish the writing.’ Section 3 contains ‘the 3rd-moment computations by GPT-5.6 Pro’
- Acknowledgement: Chen ‘thanks Ronen Eldan for providing access to GPT Pro and for encouraging its use’
Show 1 more
- Reception: Gil Kalai’s 22 Sept 2026 essay on AI and mathematics uses the paper as his example of AI simplifying mathematics. No refereed version or independent written check found as of 13 Oct 2026
What happened
A log-concave random vector in high dimension, once normalized to mean zero and identity covariance, has most of its mass in a thin spherical shell of radius about √n. How thin was a central question of asymptotic convex geometry for two decades, tied to the central limit theorem for convex bodies and to Bourgain’s slicing problem. Klartag and Lehec proved in 2025 that the width is bounded by a universal constant. Chen and Klartag’s paper gives the exact constant: the variance of |X|² is at most 8n, and independent exponential coordinates attain it.
The authors are open about where the proof came from. Chen prompted GPT-5.6 Pro, pointing it to Klartag’s 2014 paper on log-concave moment measures and suggesting a bootstrap of a bound on a second trace moment; the model produced the initial proof. The two authors then checked it, revised it and rewrote it for readers, which is what the title’s “digesting” refers to. The chat log is included with the arXiv submission.
Why it matters
This is a sharp result in a mature area, signed by people who built that area, with the model credited for the proof in the abstract. It is also an early example of a practice that became common in the summer of 2026: specialists publish an AI-found proof after rewriting it, and attach the transcript.
What is disputed or not yet verified
| Verification | Unrefereed preprint; proof verified and rewritten by the two authors, both specialists in the field; chat log published with the paper |
|---|
The ancillary chat log was not read for this entry. Eldan’s role is taken from the acknowledgement only; his affiliation was not checked. The Kalai link is cited from our archived summary of his post.
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 2 primary, 1 reaction
Primary
- arXiv 2607.23307: Digesting the proof of the sharp thin-shell inequalityarxiv.org, paper
- arXiv 2507.15495: Klartag & Lehec, Thin-shell bounds via parallel coupling (2025)arxiv.org, paper
Reactions
- Gil Kalai: “Be a Grothendieck!”—On AI and Mathematics (22 Sept 2026)gilkalai.wordpress.com, discussion
Changes
- Filed from a lead (Kalai’s 22 Sept post); abstract and AI statement read in the arXiv PDF