# Who Checks the Proof? AI Delivered Math’s Biggest Week Yet, and Human Referees Are Still in Charge

*Ember & Oak Institute, LLC · Oct. 27, 2026*

A startup says AI agents turned the longest proof in math history into checkable computer code in 24 days. Two AI labs claim to have toppled a 27-year-old computer science assumption on the same day. The machines are doing the heavy lifting. People still decide what counts as true.

![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.](https://emberandoakinstitute.com/blog/who-checks-the-proof/hero.jpg)

**On Monday, a startup said a swarm of AI agents had turned the longest proof in the history of mathematics into computer code that a machine can check, line by line, in 24 days. The same day, two rival AI labs each claimed their systems had knocked down a 27-year-old pillar of computer science. The machines did the heavy lifting. Deciding whether any of it is true is still a human job.**

Monday’s announcements were the biggest yet in a fast-moving race to point AI at the hardest problems in math. Axiom [said](https://axiommath.ai/research/the-enormous-theorem-formalized/) it had produced “the first complete computer-checked proof” of the Classification of Finite Simple Groups, a result that took about 100 mathematicians decades to write. OpenAI [said](https://openai.com/index/building-scientific-understanding-with-academic-communities) its models had proved three major results, including one its system says shows a widely believed computer science assumption is false. Hours later, Anthropic [posted](https://github.com/anthropics/formal-math/tree/main/eth) a paper by its Claude model claiming the same refutation. The common thread is not that AI is suddenly always right. It is that AI now produces work too big for any one person to read, so the real question has become who checks it, and how.

## The enormous theorem

The Classification of Finite Simple Groups is a kind of periodic table for symmetry. It lists the basic building blocks that every finite group, a mathematical description of symmetry, is made from. Axiom calls its original proof “widely considered to be the longest mathematical proof ever written by humans,” at roughly 15,000 pages in more than 500 articles by about 100 mathematicians, [according to the company’s write-up](https://axiommath.ai/research/the-enormous-theorem-formalized/). Mathematicians announced the classification complete in 1983, but a gap stayed open until Michael Aschbacher and Stephen Smith published a 1,221-page treatment in 2004.

Over 24 days, Axiom said, its engineers and mathematicians “directed swarms of dozens of AI agents” that wrote about 43.1 million lines in Lean, a programming language that checks every logical step of a proof. The team opened a project channel on Oct. 1 and split the job into 108 milestones and about 380 smaller tasks. On Oct. 10, a message appeared: “WE PROVED MONSTER IS SIMPLE,” a reference to the largest of the unusual “sporadic” groups. It was not smooth. Agents had quietly assumed things they had not proved, and the team “deleted tens of thousands of lines built on those assumptions,” the company wrote. On Oct. 13, only one milestone closed, which set off a replanning effort.

![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.](https://emberandoakinstitute.com/blog/who-checks-the-proof/chart-cfsg-timeline.png)

*Source: [Axiom, “Formalizing the Enormous Theorem,” Oct. 26, 2026](https://axiommath.ai/research/the-enormous-theorem-formalized/). Chart: Ember & Oak Institute, LLC.*

The checking found real problems in the human record. Most were small, such as typos and dropped assumptions, but Axiom said it found “34 noteworthy deviations from the published record,” the biggest involving a step known as the Unbalanced Group Theorem. In one example, a lemma in the Aschbacher–Smith volumes said two quantities were equal, but the agents found a case where one was 1 and the other needed to be 2. The work also produced a new result in group theory, which the team [wrote up separately](https://hexagonmath.org/2610.00575), and Axiom said it is making its full list of errors available to group theorists. The work builds on Mathlib, Lean’s community math library, and about 1.2 million lines from an independent effort called [Qiuzhen-CFSG](https://github.com/Qiuzhen-CFSG/CFSG). Axiom posted its Lean code [on GitHub](https://github.com/AxiomMath/CFSG) on Tuesday, so anyone can rerun the check.

Axiom was blunt about what the machine does and does not do. “Humans inspect that statement and the definitions it uses to confirm that it expresses the intended classification; Lean checks the formal deductions establishing it,” the company [wrote](https://axiommath.ai/research/the-enormous-theorem-formalized/). It also quoted Aschbacher, who wrote in 2005 that “it is impossible to write out a very long and complicated argument without error.”

## A pillar of computer science, wobbling

OpenAI’s announcement went further: new results, not a check of old ones. The company [said](https://openai.com/index/building-scientific-understanding-with-academic-communities) its models had proved Christopher Sogge’s 1991 local smoothing conjecture in every dimension, a result about how waves spread out; produced a proof that certain physics theories, including quantum chromodynamics, which describes the force holding atomic nuclei together, exist in a mathematically rigorous sense; and found an algorithm showing that the Exponential Time Hypothesis, or ETH, is false. OpenAI wrote that “our system has recently proved that the ETH is false,” a result “subsequently verified in Lean.”

ETH, proposed in 1999, says a classic logic puzzle called 3-SAT can’t be solved much faster than by trying a huge number of possibilities. OpenAI notes it has been used in hundreds of papers to argue that other problems are hard. MIT computer scientist Ryan Williams, one of the outside experts reviewing the claim, [put it simply on X](https://x.com/rrwilliams/status/2114891835403633038): the claimed algorithm “would be surprisingly faster than brute force.” OpenAI said its model found the result on Oct. 16 and it was checked in Lean the next day. Anthropic’s [paper](https://hexagonmath.org/2610.00577v1) says an internal Claude model found its version “on October 16” too, “with 14M output tokens and no human input,” and that “independently and concurrently” OpenAI’s model claimed a similar result. Both are theory, not a speed-up for real software. Anthropic’s write-up says its result “says nothing about solving formulas in practice.”

## The referees

Neither lab is asking anyone to take its word for it. OpenAI asked outside experts to act as “shepherds” for each result and said it seeks no academic credit. For ETH, those shepherds are Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who proposed the hypothesis, and Williams. In [an open letter to colleagues](https://ethdiscussion.blog/), the three wrote: “Our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny.” Williams [said on X](https://x.com/rrwilliams/status/2114833321675112826) the group signed no nondisclosure agreements and will not be paid, and asked OpenAI to release “all their raw data regarding ETH, including PDFs and Lean source.” OpenAI [posted the paper](https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf) and agreed to fund a conference, [the company said](https://openai.com/index/building-scientific-understanding-with-academic-communities).

The physics group, led by Nobel laureate David Gross, chose a slower path and is holding back the AI’s output while it writes up the work itself, [OpenAI said](https://openai.com/index/building-scientific-understanding-with-academic-communities). The labs also warn that a machine check has limits. Lean confirms that each step follows, but not that the formal statement means what the paper claims. Anthropic’s [repository](https://github.com/anthropics/formal-math/tree/main/eth) says it plainly: “Lean checks the proofs, but no machine checks that this file says what the paper says.” It keeps that file short so a person can read it.

OpenAI’s own record shows why the referees matter. Its public math collection describes the AI’s manuscripts as “raw research data, rather than polished academic papers,” and warns that “some of the unformalized results could have issues,” [according to the repository](https://github.com/openai/math). In its [Oct. 7 update](https://github.com/openai/math/blob/main/history.md), OpenAI withdrew three manuscripts after a sign error broke a key argument and revised 14 others. At that point, about 42% of its top-line results had a Lean proof.

![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.](https://emberandoakinstitute.com/blog/who-checks-the-proof/chart-openai-math-status.png)

*Source: [OpenAI, openai/math repository history, Oct. 7 and Oct. 26, 2026](https://github.com/openai/math/blob/main/history.md). Chart: Ember & Oak Institute, LLC.*

## Not just for mathematicians

The same pattern is showing up in everyday software. Anthropic says Claude writes about 90% of the code for its Claude Code tool, and in a [post this month](https://claude.dev/blog/solving-code-review-at-anthropic/) its engineers described how they review that much work. They don’t take the AI’s word that a task is done. They have it run tests, builds or screenshot comparisons first. “We didn’t remove review. But we did move most of the reading to machines,” they wrote. An AI approval never stands in for a required human one, and “the person directing the agent counts as one of the required reviewers.” From June to September, the team said, merged changes more than doubled while the share that had to be undone fell by more than 40%.

## What it adds up to

Whether the ETH claims hold up is now up to unpaid academics working in the open. The Axiom proof shows the other side: when the steps are machine-checked, people can stop rereading 15,000 pages and focus on one question, whether the final statement really says what we meant. In both cases, the AI did the building. People set the goal, chose the referees and decided what to trust.

For families in the Mat-Su Valley, that is the useful lesson. The skill that grows in value is not grinding through every step by hand. It is knowing what you are trying to prove, asking for evidence and checking the claim before you believe it.

At Ember & Oak Institute, LLC, learners don’t hand-code. They direct AI tools through a simple loop: idea, build, test, improve. This week’s news is that loop at the scale of a whole field. The machines are racing through the build. The test, and the judgment about what the test should prove, still belong to people.

**Try it at home:** Ask your learner to direct an AI helper to build a simple “proof checker” game for a puzzle like a sudoku or a magic square, where every move must follow the rules or the game flags it. Then have them write, in one plain sentence, what a “solved” puzzle means. Test the game with a deliberately wrong move, and improve it until it catches the mistake. Talk about which job was harder: checking each move, or saying exactly what “solved” means.

## Sources

- [Axiom, “Formalizing the Enormous Theorem” (Oct. 26, 2026)](https://axiommath.ai/research/the-enormous-theorem-formalized/)
- [Axiom, CFSG Lean formalization (GitHub)](https://github.com/AxiomMath/CFSG)
- [Axiom, CFSG project site](https://cfsg.axiommath.ai/)
- [Axiom, Unbalanced Group Theorem paper, Hexagon 2610.00575 (Oct. 2026)](https://hexagonmath.org/2610.00575)
- [Qiuzhen-CFSG, independent CFSG formalization (GitHub)](https://github.com/Qiuzhen-CFSG/CFSG)
- [OpenAI, “Working with academic communities to build scientific understanding” (Oct. 26, 2026)](https://openai.com/index/building-scientific-understanding-with-academic-communities)
- [OpenAI, openai/math repository README (GitHub)](https://github.com/openai/math)
- [OpenAI, openai/math repository history (Oct. 7 and Oct. 26, 2026 entries)](https://github.com/openai/math/blob/main/history.md)
- [OpenAI, “The Exponential Time Hypothesis is False” (preprint, Oct. 25, 2026)](https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf)
- [Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, “ETH Refutation: Examination and Discussion” open letter (Oct. 26, 2026)](https://ethdiscussion.blog/)
- [Ryan Williams on X, thread opener (Oct. 26, 2026)](https://x.com/rrwilliams/status/2114833319431209352)
- [Ryan Williams on X, on NDAs, pay and raw data (Oct. 26, 2026)](https://x.com/rrwilliams/status/2114833321675112826)
- [Ryan Williams on X, plain-language explanation (Oct. 26, 2026)](https://x.com/rrwilliams/status/2114891835403633038)
- [Anthropic, formal-math/eth README and Lean formalization (GitHub, Oct. 26, 2026)](https://github.com/anthropics/formal-math/tree/main/eth)
- [Claude (Anthropic), “A 2^o(n) Time Algorithm for 3-SAT,” Hexagon 2610.00577v1 (Oct. 27, 2026)](https://hexagonmath.org/2610.00577v1)
- [Addy Osmani, Morgan Westlee Lunt and Qing Wang, “How the Claude Code team does code review” (Claude blog, Oct. 15, 2026)](https://claude.dev/blog/solving-code-review-at-anthropic/)