--- id: "2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim" url: "https://postcutoff.com/e/2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim/" as_of: "2026-10-30T13:42:00+01:00" date: "2026-10-29" date_precision: day category: science importance: 4 confidence: medium status: [Awaiting review] verification: Expert-checked sources: 7 editor: human editor human_review: null version: "2026-10-29" --- As of: 2026-10-30 13:42 CET. Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/ Canonical page: https://postcutoff.com/e/2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim/ # Sharp Berry–Esseen constant claimed: C = Esseen's 0.4097 Full title: Kołodziejek and Rodziewicz claim the sharp i.i.d. Berry–Esseen constant: Esseen's 1956 lower bound 0.40973… is the optimal constant, by analytic reductions plus a computer-checked certificate family built with ChatGPT, Codex and Gemini "The sharp iid Berry–Esseen inequality: Analytic reductions and exact certificate proof" (Hexagon 2610.00590, dated 27 Oct, posted 29 Oct 2026) by Bartosz Kołodziejek and Kacper Rodziewicz claims that sup_n C_n = C_E = (3+√10)/(6√(2π)) ≈ 0.4097321837, Esseen's 1956 two-point lower bound, and that C_n ≤ C_E at every n ≥ 1. That would settle the sharp-constant problem usually attributed to Zolotarev (1966–67). The argument reduces the problem to finitely many cases checked by exact rational certificates; ChatGPT, Codex and Gemini/Antigravity helped with exploration, checking programs, cloud verification and drafting. It follows a 2026 run of AI-assisted bounds: Xiao–Li 0.4395 and Yongxi Lin 0.423 (both Lean, unverified per Tao's list) and He–Cheng's "eventually true" theorem. ## Key facts - Claim (abstract, verbatim): 'We prove that the optimal constant in the Berry–Esseen inequality for sums of i.i.d. random variables with finite third moment equals Esseen's constant C_E = (3 + √10)/(6√(2π)), the limiting value for one fixed two-point law, and that C_n ≤ C_E holds at every sample size n ≥ 1.' - Before: Esseen's lower bound C_E ≈ 0.4097 (1956); published upper bounds 0.7655 (Shiganov), 0.4785 (Tyurin), 0.4748 and 0.4690 (Shevtsova, 2011–2013). Tao's Optimization Constants page (Problem 19a) lists Xiao–Li 0.4395 and Lin 0.423 as unverified: Lean developments whose finite certificates rely on native_decide, not run through the comparator. Sharpness was already known for Bernoulli summands (Schulz, Trier thesis 2016) - Method (abstract): a maximising law exists at each n and satisfies a contact condition; integrating it against the zero-bias density gives an identity linking the error to a Fourier remainder and a moment defect that vanishes only at Esseen's two-point law; a one-step smoothing recurrence controls the first failure index (n ≥ 42), with finite certificates for 1 ≤ n ≤ 48 and n ≥ 49 ('thirteen-role' certificate family, full_pass: 'ESTABLISHED') - AI statement (paper, verbatim): 'AI systems were used extensively in the research, verification, and preparation of this manuscript. ChatGPT, Codex, and Gemini/Antigravity assisted with mathematical exploration, programs for exact checks, distributed cloud verification of the full certificate family, source and bibliography comparisons, and drafting and revising the exposition. The authors take responsibility for the mathematical content and presentation.' - Hexagon labels: 'Primarily AI-generated' text; human understanding 'some parts'. Provenance: 'All mathematical reductions, certificate schemas, and verification executions were directed and verified by the human authors.' It builds on their earlier Hexagon draft 2610.00147 (v3, 17 Oct) - Verification status: preprint plus self-run certificate checks (code at github.com/rodziewiczk/berry-esseen-certificates and kolodziejek/berry-esseen-certificates, created 27 Oct, single-command multi-core Docker replay); no Lean proof of the full theorem; Tao's page did not list the claim on 29 Oct; no HN thread or expert comment found - Related AI-era work cited by the paper: He and Cheng, 'The Berry–Esseen constant conjecture is eventually true' (arXiv 2609.06358, v3 30 Sept), sharp for all n ≥ 2⌈exp(10^17)⌉, with the Lean project BerryEsseenLean (UCLA Trustworthy AI Lab); Logan Bell's bounds for non-identically distributed summands (below 0.44988) ## What happened The Berry–Esseen theorem (1941–42) says the distribution of a normalised sum of n i.i.d. variables is within Cρ/√n of the normal law, where ρ is the third absolute moment. Esseen showed in 1956 that no constant below C_E ≈ 0.4097 works, using a fixed two-point law. Upper bounds came down slowly for 70 years, to Shevtsova's 0.4690 in 2013. In 2026 the problem drew AI-assisted work: Lean-checked claims of 0.4395 (Xiao–Li) and 0.423 (Yongxi Lin), which Tao's optimization-constants list keeps marked unverified, and He–Cheng's proof that Esseen's constant is sharp for astronomically large n. On 29 Oct Kołodziejek and Rodziewicz posted a claimed proof that C_E is the exact answer for every n. The paper reduces the question to the first index where the sharp bound could fail and closes the remaining cases with exact rational certificates in 13 ranges of n. Their programs and a Docker image replay the whole check. They say ChatGPT, Codex and Gemini/Antigravity were used extensively, including for the distributed verification runs. ## Why it matters It is one of the best-known constants in probability. If the certificates and the reductions hold, the problem is closed exactly, not just improved. The proof is computer-assisted but not formalised in Lean, so trust rests on the authors' checkers and on specialists reading the analytic part. As of 29 Oct we found no such reading. ## What is disputed or not yet verified - Verification: Unrefereed Hexagon preprint with author-run exact certificates and a public Docker replay; no Lean proof of the full theorem; no independent check found ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 182 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 121 days after its cutoff - Gemini 4 Argon (training cutoff March 2026): 212 days after its cutoff - Grok 4.7 (training cutoff May 2026): 151 days after its cutoff ## Sources 1. [Hexagon 2610.00590: The sharp iid Berry–Esseen inequality (Kołodziejek, Rodziewicz)](https://hexagonmath.org/abs/2610.00590v1) (hexagonmath.org, paper) 2. [Certificates and Docker replay (rodziewiczk/berry-esseen-certificates)](https://github.com/rodziewiczk/berry-esseen-certificates) (github.com, code) 3. [Earlier draft: Hexagon 2610.00147, Moment defects and sharp iid Berry–Esseen bounds](https://hexagonmath.org/2610.00147v3) (hexagonmath.org, paper) 4. [He and Cheng: The Berry–Esseen constant conjecture is eventually true (arXiv 2609.06358)](https://arxiv.org/abs/2609.06358) (arxiv.org, paper) 5. [Xiao and Li: a 0.4395 upper bound (GitHub, Lean)](https://github.com/haonan-xiao/iid-berry-esseen) (github.com, code) 6. [Yongxi Lin: Berry–Esseen bounds in Lean (0.423)](https://github.com/CoolRmal/BerryEsseen) (github.com, code) 7. [Tao and contributors, Optimization Constants in Mathematics: 19a, the Berry–Esseen constant](https://github.com/teorth/optimizationproblems/blob/main/constants/19a.md) (github.com, discussion) ## Changes - 2026-10-29 (filed): Created (Hexagon API record and PDF of 2610.00590, Tao's 19a page, GitHub API for the certificate repos; HN search found no thread) ## Related - 2026-10-29: [AI-assisted results, 29 Oct (part 3): Hexagon's evening batch](https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-3/index.md) - 2026-10-06: [Hexagon launches: a new arXiv-style repository for LLM-assisted mathematics papers](https://postcutoff.com/e/2026-10-06-hexagon-repository-llm-assisted-math/index.md) - People: [Terence Tao](https://postcutoff.com/person/terence-tao/)