Which AI models did Axiom use to formalize the Classification of Finite Simple Groups?

Axiom says its AxiomProver system ran the CFSG formalization with agent swarms built on a changing mix of models, but its announcement does not name the specific models. Lead agents planned and reviewed statements, other agents searched for results and wrote Lean proofs, and engineers and mathematicians directed the swarms and handled the hardest mathematical obstructions.

Did the Lean formalization of CFSG find mistakes in the published proof?

Yes. Axiom logged 34 noteworthy deviations from the published CFSG proof while formalizing it. Most are minor, such as typos, table errors and dropped hypotheses. The largest concerns the Unbalanced Group Theorem: agents found counterexamples to a closure claim in John Walter's 1986 argument, and Axiom covers that gap in a separate paper.

Can I build the CFSG Lean proof myself?

Yes. Axiom published the CFSG formalization as AxiomMath/CFSG on GitHub under the Apache-2.0 license, built on Lean 4 v4.35.0-rc3, Mathlib and TauCeti. Running `lake build` builds the whole project. Axiom notes the build is very large and says it used changes to Lake and planned optimizations to bring build time under 24 hours.

How long did Axiom take to formalize the Classification of Finite Simple Groups?

Axiom says the CFSG formalization took 24 days. The project Slack channel opened on October 1, 2026, the first agent swarm went live on October 2, the Monster group was proved simple on October 10, and Axiom announced the finished 43.1-million-line Lean proof on October 26, 2026.

Axiom · 2026-10-26 · major

Axiom formalizes the Classification of Finite Simple Groups in Lean

Axiom's AxiomProver agent swarms wrote a complete Lean proof of the Classification of Finite Simple Groups in 24 days: 43.1M lines checked by Lean's kernel, with 34 deviations from the published proof logged.

Axiom's Classification of Finite Simple Groups Lean formalization project card

AI agent swarms turned one of the longest proofs in mathematics into machine-checked Lean code in 24 days.

Quick facts

Maker	Axiom (Axiom Math)
Theorem	Classification of Finite Simple Groups (first-generation proof)
Size	43.1M lines of Lean, 2.05M declarations
Time	24 days (first agent swarm live October 2, 2026)
Errata found	34 deviations from the published record
License	Apache-2.0 (AxiomMath/CFSG)
Lean version	Lean 4 v4.35.0-rc3, on Mathlib and TauCeti

What is it?

The Classification of Finite Simple Groups, often called the Enormous Theorem, now has a complete Lean formalization from Axiom. The theorem says every finite simple group is cyclic of prime order, alternating, of Lie type, or one of 26 sporadic groups. Axiom's version follows the first-generation proof, and every logical step is checked by Lean's kernel with no `sorry` and only Lean's three standard axioms.

How does it work?

Axiom's AxiomProver system directed dozens of AI agents in swarms running on a changing mix of models that the post does not name. Lead agents broke arguments down and reviewed statements, while other agents searched for existing results or wrote Lean proofs, working from a proof map split into 108 milestones and about 380 smaller tasks. Human mathematicians handled hard mathematical obstructions and reviewed the final statement, with group theorists Richard Lyons and Ronald Solomon consulted on the arguments.

Why does it matter?

The original proof is spread over hundreds of papers and books (the project corpus lists 715), so no single person has checked all of it. Formalizing it surfaced 34 deviations from the published record, mostly typos and dropped hypotheses, but also a counterexample to a step in John Walter's 1986 Unbalanced Group Theorem argument. It shows AI agents can now do formal verification at a scale that used to take human teams decades.

Who is it for?

mathematicians, Lean users and formal-methods researchers

Frequently asked questions

Which AI models did Axiom use to formalize the Classification of Finite Simple Groups?: Axiom says its AxiomProver system ran the CFSG formalization with agent swarms built on a changing mix of models, but its announcement does not name the specific models. Lead agents planned and reviewed statements, other agents searched for results and wrote Lean proofs, and engineers and mathematicians directed the swarms and handled the hardest mathematical obstructions.
Did the Lean formalization of CFSG find mistakes in the published proof?: Yes. Axiom logged 34 noteworthy deviations from the published CFSG proof while formalizing it. Most are minor, such as typos, table errors and dropped hypotheses. The largest concerns the Unbalanced Group Theorem: agents found counterexamples to a closure claim in John Walter's 1986 argument, and Axiom covers that gap in a separate paper.
Can I build the CFSG Lean proof myself?: Yes. Axiom published the CFSG formalization as AxiomMath/CFSG on GitHub under the Apache-2.0 license, built on Lean 4 v4.35.0-rc3, Mathlib and TauCeti. Running `lake build` builds the whole project. Axiom notes the build is very large and says it used changes to Lake and planned optimizations to bring build time under 24 hours.
How long did Axiom take to formalize the Classification of Finite Simple Groups?: Axiom says the CFSG formalization took 24 days. The project Slack channel opened on October 1, 2026, the first agent swarm went live on October 2, the Monster group was proved simple on October 10, and Axiom announced the finished 43.1-million-line Lean proof on October 26, 2026.

Try it

git clone https://github.com/AxiomMath/CFSG && cd CFSG && lake build