Axiom CFSG Lean proof: what it is and how to check it

The Axiom CFSG Lean proof is a machine-checked version of the Classification of Finite Simple Groups, one of the longest proofs in mathematics, produced in about 24 days by swarms of AI agents with human mathematicians steering. You can check it yourself: the code is public on GitHub under Apache-2.0, and an interactive proof map shows how the 108 milestones connect.

@axiommathai announced it on October 27 with a big line: "The largest math proof in human history is now computer-verified." The post describes the original proof as about 15,000 pages, built by more than 100 mathematicians over several generations. A couple of minutes later Axiom's Vasily Ilin, who also runs the UW Math AI Lab, posted that the team had "formalized the ENOURMOUS theorem," finishing on October 26. He said they found hundreds of errors in the literature, and the proof still held.

What the theorem says

Axiom calls the classification the periodic table of symmetry. It says every finite simple group belongs to one of these:

• cyclic groups of prime order
• alternating groups on five or more symbols
• sixteen infinite families of groups of Lie type
• 26 sporadic groups, with the Tits group sometimes counted as a 27th

The human proof took decades and is spread across hundreds of papers and books. Ilin's post points out that most of the people who truly understood it have retired. That is why a computer check is worth having: Lean confirms each step follows from the ones before it, and nobody has to take the experts' word for it anymore.

How the agents did it

Axiom's write-up is the useful part for anyone running agents at work. The team built a proof map from the literature and split the job into 108 milestones and about 380 smaller tasks. Agents claimed milestones through GitHub issues and reported finished work in Slack. Lead agents broke down hard arguments, other agents looked up existing results or wrote Lean, and integration agents traced the final theorem back through its prerequisites and filled the gaps.

The swarm went live on October 2. By October 13, only one milestone had closed. Early milestones were hiding large prerequisites the agents had assumed without proving, so the team deleted tens of thousands of lines, split the unfinished work into smaller, precisely stated theorems, and kept going. Some milestones had been closed with conditional proofs that rested on unproven assumptions. Lean accepts those, so humans had to check that each proved statement matched the intended target.

Anthropic hit the same wall on Fermat. As we wrote in Claude's Fermat run failed until the agents got a to-do list, what got that run moving was a shared task list. A better model alone didn't do it.

What it found, and what is still unchecked

Axiom reports 34 noteworthy deviations from the published record. Most are minor: typos, table errors, dropped hypotheses. The big one is in John Walter's 1986 proof of the Unbalanced Group Theorem, where the agents found explicit counterexamples to a closure claim. The team took a different route, and the new argument is written up in a 35-page paper (arXiv 2610.00575). Group theorists Richard Lyons and Ronald Solomon were consulted.

My read: this looks like a real milestone, but for now the claim is Axiom's own. People connected to the project reviewed the formal statement, the errata page says the list is still being compiled with group theorists, and I couldn't find an independent outside audit yet. Lean checks the logic. What outsiders will poke at next is whether the top-level statement really says what the classification says.

How to check it yourself

The statement to read is ClassificationStatement, which says every finite simple group is isomorphic to one on the list indexed by CFSGIndex. At about 43.1 million lines, a full local build is a job for a large machine. Most people will want the proof map and the axiom list. For a smaller example of how Lean checking works, see our post on the OpenAI Exponential Time Hypothesis disproof, or Anthropic formal verification for the same idea applied to software.

What it means for a business

You won't be formalizing group theory, but the setup carries over to ordinary agent work. Axiom only got moving once the job was cut into small, precisely stated tasks on a board everyone could see, with a checker at the end that ignored how confident an agent sounded. If you want help working out which of your own processes could run that way, our free process audit is a good place to start.