{"schema":"postcutoff/event@1","as_of":"2026-10-16T12:39:00+02:00","url":"https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/","md":"https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra autonomously disproves Smale's mean value conjecture in Lean","deck":"HKU mathematicians build on it to show Smale's constant exceeds 1.1018","takeaway":"A Smale problem from 1981 fell to an unattended model run in an Epoch AI benchmark, with a Lean proof; six weeks later two complex analysts turned the construction into a quantitative bound M > 1.1018.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"GitHub: tadamcz/mean-value-problem (Lean 4 disproof found by GPT-6 Astra)","url":"https://github.com/tadamcz/mean-value-problem","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"arXiv 2610.21472: Smale's mean value constant and admissible univalent Blaschke pairs (Jatar & Ng)","url":"https://arxiv.org/abs/2610.21472","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"MathWorld: Smale's Mean Value Conjecture","url":"https://mathworld.wolfram.com/SmalesMeanValueConjecture.html","type":"docs","group":"primary","domain":"mathworld.wolfram.com"},{"n":4,"title":"Epoch AI: LeanOpenProblems harness","url":"https://github.com/epoch-research/LeanOpenProblems","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"Formal Conjectures: MeanValueProblem.lean","url":"https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/MeanValueProblem.lean","type":"code","group":"primary","domain":"github.com"}],"official":5,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Epoch AI","University of Hong Kong"],"title":"GPT-6 Astra autonomously disproves Smale's mean value conjecture (constant 1) in Lean; HKU mathematicians build on it to show Smale's constant exceeds 1.1018","summary":"In an Epoch AI LeanOpenProblems evaluation run in September 2026, a pre-release GPT-6 Astra autonomously produced a Lean 4 proof that Smale's mean value conjecture with constant 1 is false: there is a polynomial p with p(0) = 0, p'(0) = 1 and |p(c)/c| > 1 at every critical point c. Tom Adamczewski published the proof (GitHub repository created 3 Sep 2026); MathWorld lists it as a proposed counterexample. On 15 Oct 2026 Aneesh Jatar and Tuen-Wai Ng (University of Hong Kong) posted arXiv 2610.21472, which builds on the Astra construction, recasts Smale's question in univalent function theory and proves Smale's constant M > 1.1018, with polynomials of arbitrarily large degree beating 1.1018.","key_facts":["Conjecture: Smale (1981) proved that every polynomial has a critical point c with |p(z) − p(c)|/|z − c| ≤ 4|p'(z)|, and asked whether 4 can be replaced by 1 (or by (d−1)/d); one of Smale's problems for the 21st century. Known for degree d ≤ 4 and special classes","Claim: the K = 1 conjecture is false; the witness has 'very large, unspecified degree and violates the bound by a small margin' (repository README)","How found (README): Epoch AI evaluation run 'wikipedia-vega-1000usd' of its LeanOpenProblems harness; a pre-release GPT-6 Astra attempted all 222 research-open statements of the Wikipedia collection of Google DeepMind's Formal Conjectures, once each, with $1,000 and 96 hours per statement, in a network-isolated container; 'No human saw or steered the proof search'","Cost (README): about 2.2M output and 68M cache-read tokens, $265 at GPT-6 Astra list prices; 15.0 hours of agent working time","Verification: Lean 4 + Mathlib proof checked by Comparator against the Formal Conjectures statement, using only propext, Quot.sound and Classical.choice; the README's prose account of the proof is machine-generated and 'has not been checked by a human mathematician'","Construction (README): a conformal map of the four-petal set {|z⁴ + 64i| ≤ √4097}, a Blaschke factor, polynomial approximation, then adding the primitive of q²ⁿ pushes all critical points into a thin collar where |g| > 1 + δ","MathWorld (last updated 2 Oct 2026): 'Adamczewski (2026) released a proposed counterexample to the constant 1, constructed autonomously by GPT-6 Astra'; independent review 'had not been reported as of Sep. 7, 2026'","Human follow-up: Jatar & Ng, 'Smale's mean value constant and admissible univalent Blaschke pairs', arXiv 2610.21472 (15 Oct 2026): 'Motivated and based upon the GPT-6 Astra construction'; they show M = A = B (two new extremal constants) and 1.1018586788… = √(8081/6656) ≤ B₁ < √(7/5), so M > 1.1018. The original argument can give S(P) > 1 + 2⁻³⁵ but is 'hard to push' further","AI use in the follow-up: 'GPT-6Astra and Grok 4.6 were utilized to conduct numerical experiments and generate preliminary draft proofs via targeted prompting. The authors then collaboratively verified, significantly simplified, and rewrote all proofs in their own words.' Tuen-Wai Ng co-authored the 2002 Beardon–Minda–Ng bound on the same problem"],"key_numbers":[],"tags":["math","complex-analysis","polynomials","smale","gpt-6-astra","lean","formal-conjectures","counterexample","autonomous"],"science":{"field":"mathematics","subfield":"complex analysis (geometric function theory)","problem":"Smale's mean value conjecture (1981), constant-1 form","result":"Disproved: a polynomial with p(0) = 0, p'(0) = 1 and |p(c)/c| > 1 at every critical point exists (Lean). Follow-up: Smale's constant M > 1.1018.","open_since":"1981","ai_system":["GPT-6 Astra (pre-release)","Grok 4.6"],"human_role":"Autonomous (benchmark run, no human steering); follow-up bound by Jatar & Ng with AI drafts rewritten by the authors","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (Comparator-checked); follow-up is an unrefereed preprint by specialists"},"status":"pending","shock":"A named Smale problem was refuted by an unattended benchmark run costing about $265, and the counterexample went largely unnoticed until specialists published on it six weeks later."},"body_md":"## What happened\n\nSmale proved in 1981 that every complex polynomial p of degree at least 2 has, for every point z, a critical point c with\n|p(z) − p(c)|/|z − c| ≤ 4|p'(z)|. He asked whether 4 can be replaced by 1, or even by (d−1)/d, which the polynomial\nz − z^d/d attains. In 45 years the constant came down only to 4 − O(1/d). The problem is on Smale's list of problems for the\n21st century.\n\nIn September 2026 Epoch AI ran a pre-release GPT-6 Astra over the 222 open statements in the \"Wikipedia\" collection of Google\nDeepMind's Formal Conjectures, once each, with $1,000 and 96 hours per statement. For the mean value problem the model chose the\nnegation and proved it in Lean: there is a polynomial with p(0) = 0 and p'(0) = 1 whose value ratio |p(c)/c| exceeds 1 at every\ncritical point. Tom Adamczewski published the proof with a Comparator check (repository created 3 Sep 2026; the README says the\nwhole repository was machine-written by AI assistants at his direction). The witness has huge, unspecified degree and beats the\nbound only slightly, which fits all known partial results. MathWorld added it as a \"proposed counterexample\" and noted that no\nindependent review had been reported as of 7 Sep.\n\nOn 15 Oct 2026 Aneesh Jatar and Tuen-Wai Ng of the University of Hong Kong posted [arXiv 2610.21472](https://arxiv.org/abs/2610.21472).\nThey describe the Astra construction step by step, say their work is \"motivated and based upon\" it, and turn Smale's question\ninto an extremal problem for pairs of a univalent map and a Blaschke product. They prove that Smale's constant M equals the new\nconstants A and B, and that M > 1.1018 (from B₁ ≥ √(8081/6656)), with polynomials of arbitrarily large degree beating 1.1018.\nThey used GPT-6 Astra and Grok 4.6 for numerical experiments and draft proofs, then \"verified, significantly simplified, and\nrewrote all proofs in their own words\".\n\nNo Hacker News thread on the disproof was found (Algolia, 16 Oct); X was not searched.\n\n## Why it matters\n\nThe conjecture of constant 1 had been the expected answer for decades. Here it was refuted by an unattended benchmark run at\nabout $265 of tokens, with a machine-checked proof, and the result sat in a GitHub repository for six weeks before specialists\n(one of them an author of a classic bound on the problem) built on it. The Lean proof settles correctness of the formal\nstatement; the follow-up paper is the first expert treatment and is not yet refereed. The stronger (d−1)/d form is also refuted,\nsince it implies the constant-1 form.\n\nUnverified: the exact public release date of the repository (3 Sep is its creation date on GitHub) and whether Epoch AI or OpenAI\nannounced the result.","disputed":[{"label":"Verification","text":"Formal proof in Lean (Comparator-checked); follow-up is an unrefereed preprint by specialists","sources":[]}],"related":[{"id":"2026-09-10-astra-leanopenproblems-september-results","url":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/","date":"2026-09-10","date_precision":"day","short_title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results","deck":"Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","takeaway":"Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1","url":"https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/","date":"2026-09-03","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves Erdős's 'first serious problem' (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch's FrontierMath Erdős runs","deck":null,"takeaway":"Problem #1 is probably the longest-standing open Erdős problem.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":6,"filed":"2026-09-30","updated":"2026-10-14","orgs":["OpenAI","Epoch AI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-16","type":"filed","text":"Created from the Jatar–Ng arXiv paper, the GitHub repository and MathWorld; the September result had been missed"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-16","run":null,"sources_read":"The Jatar–Ng arXiv paper, the GitHub repository and MathWorld; the September result had been missed","updated":"2026-10-16","human_review":null,"version":{"date":"2026-10-16"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":126,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":65,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":156,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":95,"in_training_data":false}],"short_url":"https://postcutoff.com/s/gpt-6-astra-autonomously"}