{ "slug": "who-checks-the-proof", "title": "Who Checks the Proof? AI Delivered Math’s Biggest Week Yet, and Human Referees Are Still in Charge", "url": "https://emberandoakinstitute.com/blog/who-checks-the-proof.html", "date": "2026-10-27", "category": "AI news", "tags": [ "artificial intelligence", "mathematics", "formal verification", "Lean proof assistant", "AI research", "computer science", "AI coding", "AI news" ], "summary": "Axiom says AI-agent swarms produced the first complete Lean (computer-checked) proof of the Classification of Finite Simple Groups in 24 days (~43.1M lines), finding 34 noteworthy deviations in the ~15,000-page human record, while stressing humans must still check the final statement; OpenAI and Anthropic each claimed on Oct. 26 that internal models refuted the Exponential Time Hypothesis (both found Oct. 16, both with Lean proofs), with the original ETH proposers and Ryan Williams organizing an open, unpaid, non-endorsing review; OpenAI's own repo shows ~42% of its AI math results Lean-checked and 3 withdrawn; Anthropic's Claude Code team moves review to machine-run evidence while keeping required human approvals.", "images": { "hero": { "url": "https://emberandoakinstitute.com/blog/who-checks-the-proof/hero.jpg", "alt": "Painted storybook scene at a kitchen table in a Mat-Su Valley log home on an autumn evening: a mother and her son and daughter look over a large paper puzzle map and a small laptop whose screen faces them, while a friendly little robot holds up a magnifying glass beside a stack of notebooks. Snowy mountains glow through the window." }, "charts": [ { "url": "https://emberandoakinstitute.com/blog/who-checks-the-proof/chart-cfsg-timeline.png", "alt": "Two-part timeline. Top: the human proof of the Classification of Finite Simple Groups, from the 255-page Feit–Thompson paper in 1963, through Gorenstein’s 16-step program in 1972, the 1983 announcement that the classification was complete, the 1,221-page Aschbacher–Smith quasithin treatment in 2004 and a gap closed by Harada and Solomon in 2008, totaling about 15,000 pages in more than 500 articles by about 100 mathematicians. Bottom: Axiom’s October 2026 formalization, from the project channel opening Oct. 1 and the first AI-agent swarm Oct. 2, to the “Monster is simple” milestone Oct. 10, a slow day and replanning Oct. 13, and release Oct. 26, producing about 43.1 million lines of Lean over 24 days.", "source": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "url": "https://emberandoakinstitute.com/blog/who-checks-the-proof/chart-openai-math-status.png", "alt": "Horizontal bar chart: of 719 top-line results in OpenAI’s public math collection as of its Oct. 7, 2026 update, 300 (about 42%) had a Lean formalization and 419 did not. A note says the same update withdrew 3 manuscripts after a sign error and revised 14 others, and that on Oct. 26 OpenAI added two manuscripts with Lean proofs, including its claimed refutation of the Exponential Time Hypothesis.", "source": [ "https://github.com/openai/math/blob/main/history.md", "https://github.com/openai/math" ] } ] }, "key_facts": [ { "fact": "Axiom said on Oct. 26, 2026 it released the first complete computer-checked (Lean) proof of the Classification of Finite Simple Groups.", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "fact": "The original CFSG proof spans roughly 15,000 pages in more than 500 articles by about 100 mathematicians (per Axiom).", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "fact": "Over 24 days, Axiom engineers and mathematicians directed swarms of dozens of AI agents that produced about 43.1 million lines of Lean; project channel opened Oct. 1, first swarm Oct. 2; 108 milestones and about 380 tasks.", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "fact": "Oct. 10 channel message: 'WE PROVED MONSTER IS SIMPLE.' On Oct. 13 only one milestone closed, prompting replanning; tens of thousands of lines built on unproved assumptions were deleted.", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "fact": "Axiom found 34 noteworthy deviations from the published record, the largest involving the Unbalanced Group Theorem; an Aschbacher–Smith lemma had an SL2(5) counterexample (1 where equality required 2). A new result was written up on Hexagon (2610.00575).", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/", "https://hexagonmath.org/2610.00575" ] }, { "fact": "Axiom: 'Humans inspect that statement and the definitions it uses to confirm that it expresses the intended classification; Lean checks the formal deductions establishing it.'", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "fact": "Axiom's formalization builds on Mathlib and ~1.2 million lines from Qiuzhen-CFSG; its Lean code is public on GitHub (AxiomMath/CFSG, created Oct. 27, 2026 UTC).", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/", "https://github.com/AxiomMath/CFSG", "https://github.com/Qiuzhen-CFSG/CFSG" ] }, { "fact": "Feit–Thompson's 1963 proof ran 255 pages; classification announced complete in 1983; Aschbacher and Smith published a 1,221-page quasithin treatment in 2004; Aschbacher wrote in 2005 that 'it is impossible to write out a very long and complicated argument without error.'", "sources": [ "https://axiommath.ai/research/the-enormous-theorem-formalized/" ] }, { "fact": "OpenAI said on Oct. 26, 2026 its models proved Sogge's 1991 local smoothing conjecture in every dimension, produced an existence proof for QCD and pure Yang–Mills, and that 'Our system has recently proved that the ETH is false,' verified in Lean (found Oct. 16, Lean-checked Oct. 17).", "sources": [ "https://openai.com/index/building-scientific-understanding-with-academic-communities", "https://github.com/openai/math/blob/main/history.md" ] }, { "fact": "ETH shepherds are Russell Impagliazzo and Ramamohan Paturi (UC San Diego) and Ryan Williams (MIT); their open letter says 'Our involvement should not be taken as certification of the claim.'", "sources": [ "https://ethdiscussion.blog/", "https://openai.com/index/building-scientific-understanding-with-academic-communities" ] }, { "fact": "Williams said the group signed no NDAs, will receive no compensation, and asked OpenAI to release all raw ETH data including PDFs and Lean source; OpenAI says it agreed and will fund a conference.", "sources": [ "https://x.com/rrwilliams/status/2114833321675112826", "https://openai.com/index/building-scientific-understanding-with-academic-communities" ] }, { "fact": "The physics group led by David Gross is holding back the AI-produced material while it writes up the work.", "sources": [ "https://openai.com/index/building-scientific-understanding-with-academic-communities" ] }, { "fact": "Anthropic's paper says an internal Claude model found a 2^o(n)-time 3-SAT algorithm refuting ETH on Oct. 16 'with 14M output tokens and no human input,' independently of OpenAI; formalized in Lean 4; the README says 'no machine checks that this file says what the paper says' and the result 'says nothing about solving formulas in practice.'", "sources": [ "https://hexagonmath.org/2610.00577v1", "https://github.com/anthropics/formal-math/tree/main/eth" ] }, { "fact": "OpenAI's math repo calls its manuscripts 'raw research data'; its Oct. 7 update withdrew 3 manuscripts after a sign error, revised 14, and put Lean coverage at 300 of 719 top-line results (~42%).", "sources": [ "https://github.com/openai/math", "https://github.com/openai/math/blob/main/history.md" ] }, { "fact": "Anthropic's Claude Code team says Claude writes about 90% of Claude Code's code; June–September merged PRs more than doubled while the revert share fell by more than 40%; agent approvals never count as a human's.", "sources": [ "https://claude.dev/blog/solving-code-review-at-anthropic/" ] } ] }