AI Agents Formalize Classification of Finite Simple Groups in 24 Days, Find 40-Year Gap

Background: The Enormous Theorem

The Classification of Finite Simple Groups (CFSG), known as the "Enormous Theorem," is one of the largest collaborative mathematical projects in history. Over 100 mathematicians spent decades producing roughly 15,000 pages of proof across more than 500 papers. The theorem classifies all finite simple groups — the basic building blocks of finite group theory — into cyclic groups of prime order, alternating groups, 16 infinite families of groups of Lie type, and 26 sporadic groups, the largest being the Monster Group with about 8×10^53 elements.

The proof's complexity created verification challenges: arguments are scattered across hundreds of papers, each relying on previous results, with steps often omitted as "obvious." No single person could check the entire logical chain. In 1983 Daniel Gorenstein announced completion, but gaps remained, notably the quasithin case resolved only in 2004 by Aschbacher and Smith (1,221 pages).

The 24-Day Agent Engineering Effort

On October 1, 2024, Axiom Math launched a project to formalize the entire CFSG in the Lean theorem prover. They organized a roadmap of 108 milestones and ~380 concrete tasks. Dozens of AI agents, coordinated by Axiom's proprietary system AxiomProver, claimed tasks via GitHub Issues. Agents specialized in theorem decomposition, literature search, Lean proof generation, statement review, and code integration. Human mathematicians handled key obstacles; engineers managed the continuous integration pipeline.

The agents used a shifting ensemble of models (not publicly disclosed) emphasizing task decomposition, proof generation, and collaborative verification rather than relying on a single model.

Challenge: Proving the Right Statement

Early on, agents would prove a mathematically true statement that was not the one required by the dependency graph. For example, to prove a bridge supports a heavy truck, an agent might add an unproven assumption "the piers are strong enough," then prove the truck crosses safely. Lean accepted the derivation, but the crucial premise remained unproven. Axiom had to delete tens of thousands of lines built on such hidden assumptions and restructure tasks.

Scaling Verification

As the codebase grew, Lean's checking speed became a bottleneck. Changing a foundational theorem triggered re-verification of all dependent code. Engineers introduced shared build caches and parallel verification to maximize reuse of already-checked results.

Milestone: Monster Group Simplicity

On October 10, the team announced "WE PROVED MONSTER IS SIMPLE." Agents formalized both the simplicity of the Monster Group and its existence via the Griess algebra formalization.

Slowdown and Task Refinement

By October 13, progress stalled: dozens of agents generated code but only one milestone was completed. The remaining mathematical problems were too complex, with large tasks hiding many unsolved sub-problems. Axiom re-engineered the task breakdown into smaller, explicit steps and required agents to explain how each small proof composed into the target conclusion.

Uncovering a 40-Year-Old Gap

Formalization forced every omitted premise to be made explicit. Axiom identified 34 notable deviations from the literature: typos, missing hypotheses, and substantive errors requiring re-derivation.

Error in Aschbacher–Smith (Quasithin Case)

In their work on quasithin groups, a lemma claimed two quantities were equal. Using the group SL₂(5), agents found a counterexample: one quantity was 1, the other 2. Changing the equality to an inequality fixed the lemma, but downstream reasoning had depended on the equality. The team extended the preceding argument's scope to cover the affected cases.

Critical Flaw in Walter's Unbalanced Group Theorem (1986)

The Unbalanced Group Theorem (UGT) is a key tool for inferring a finite group's type from local structure. Walter's proof assumed that a certain operation on a specific group family yields a simple group still within that family. Agents discovered a counterexample: starting from PSL₂(49) and considering the involution induced by a field automorphism, the centralizer's simple component is PSL₂(7), which falls outside Walter's designated family, breaking the closure claim.

This did not invalidate UGT's final conclusion, but it meant the proof path required a new justification for that step.

New Mathematical Result: "Preservation of Simple Tops"

Axiom consulted group theory experts Richard Lyons and Ronald Solomon. Solomon's 1976 theory of maximal 2-local subgroups provided a new direction. Agents then constructed and formalized a new group-theoretic result, "Preservation of simple tops," proving that under the needed conditions, comparing specific subgroups preserves the relevant simple group structure. This appears as Theorem 2.3 in a 35-page paper "The Unbalanced Group Theorem" and combines with existing results to complete the UGT proof required for CFSG.

Interestingly, after formalization, Axiom's mathematicians found a shorter path in the original proof strategy that avoided the new theorem. Nevertheless, the new theorem and its full formal proof were retained as an additional mathematical contribution from the project.

Impact: 43.1 Million Lines of Lean, Zero sorry

The final repository ( https://github.com/AxiomMath/CFSG/tree/main) contains ~43.1 million lines of Lean code, 2.05 million formal statements, 100 milestones, and 715 references. The final classification statement asserts that every finite simple group is isomorphic to one of the explicitly constructed groups on the list. The proof contains no sorry (Lean's placeholder for incomplete proofs), relies only on standard axioms, and is fully kernel-checkable. An interactive dependency graph lets researchers trace any conclusion to its supporting definitions and lemmas.

The work builds on prior formalization efforts: Mathlib, TauCeti, and Qiuzhen-CFSG (contributing ~1.2 million lines). Decades of human mathematical research, early formalization infrastructure, and parallel AI agents were integrated into a single engineering pipeline.

Changing Mathematical Practice

For future research, a key shift is maintainable proofs. Previously, citing CFSG meant trusting hundreds of papers; now researchers can follow the formal dependency chain to see exactly which definitions, conditions, and deductions are used. The public repository carries a self-assessed review status; external mathematicians can still audit whether formal definitions match the original propositions and review the mathematical choices made. However, Lean now provides a previously unattainable capability: unified, mechanical verification of tens of millions of lines of logical deduction.

Over these 24 days, the most significant development was not the sheer volume of code, but the demonstration that AI can handle a large-scale research workflow previously reserved for human experts: decomposing tasks, tracking proofs, detecting errors in decades-old literature, and participating in constructing new proofs when existing paths fail. Decades of accumulated mathematical knowledge are being transformed from scattered paper arguments into a programmable, checkable, reusable, and extensible structure.