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Axiom Says Its Lean Formalization Covers the Entire Stacks Project
Axiom says it has released a Lean formalization of the entire Stacks Project, an open-source reference for algebraic geometry spanning roughly 7,500 pages. The company says its formalization spans some 16 million lines of code and can be us
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Axiom says it has released its Lean formalization of what it describes as the entire Stacks Project, the roughly 7,500-page open-source reference for modern algebraic geometry. Axiom says its formalization spans some 16 million lines of code and is the largest integrated, reusable library of formalized mathematics ever built. Axiom says researchers and AI agents can import its definitions and theorems into new formal proofs. Axiom says permanent Stacks tags link each Lean declaration back to the canonical text.
Axiom says the roughly 7,500-page Stacks Project is formalized in Lean
The Stacks Project was initiated by Columbia University mathematician Johan de Jong in 2005. Hundreds of mathematicians have contributed proofs, examples and refinements over two decades. Axiom says its formalization rests equally on the foundations of the Lean and Mathlib communities, and that the project was only possible because those communities spent years building a rich mathematical library and the abstractions needed to organize it.
Axiom presents the release as a single machine-checked library. Its account describes the Stacks Project as a foundation for algebraic geometry, with a long chain of prerequisites across commutative algebra, schemes, sheaves, cohomology and algebraic stacks. Axiom says modern algebraic geometry had been far harder to approach this way because the corresponding Lean infrastructure did not exist. Axiom says the formalization puts the project’s definitions and theorems into a single codebase that researchers and AI agents can reuse.
Axiom says coordinated swarms of AI agents worked across a shared dependency graph
Axiom says formalizing millions of lines of verified mathematics was as much a software engineering challenge as a mathematical one. The company deployed coordinated swarms of AI agents to translate, prove and integrate thousands of theorems in parallel. Axiom says it designed a hierarchical system of AI agents with specialized responsibilities and strict coordination protocols. Their work included translating statements, searching Mathlib for existing abstractions, constructing proofs and resolving prerequisites.
Axiom says agents worked across the project’s dependency graph while keeping definitions, typeclass interfaces and mathematical abstractions aligned across chapters. The company argues that independently verified proofs can still be difficult to combine if they use incompatible definitions or duplicate infrastructure already present in Mathlib. Its release describes checks for statement faithfulness that include multi-agent auditing, semantic cross-checks against the source text and a strict requirement that definitions and lemmas successfully interface with downstream theorems. Axiom also says it conducted targeted audits of foundational definitions and landmark theorems.
Axiom says its formalization exposed dozens of statements needing refinement
Axiom says its system surfaced dozens of Stacks Project statements that needed local refinement. The company lists missing hypotheses, edge cases in positive characteristic and claims that did not hold in the full generality stated. Its agents constructed counterexamples to show where the original statements broke down. Axiom says its agents identified the exact missing hypotheses and formalized corrected theorems alongside the counterexamples.
Where a false supporting lemma created a gap in the proof of a deeper theorem, Axiom says agents found alternative routes through the literature to establish the main result under its original hypotheses. On October 27, 2026, Axiom released a formalization of the Classification of Finite Simple Groups, which Axiom describes as organized around a single landmark theorem. Axiom says the Stacks Project’s value lies in thousands of interlocking definitions, constructions and lemmas designed to be reused across future proofs.
Axiom says its Lean formalization is an independent open-source project
In an October 21, 2026 announcement, Johan de Jong requested that AI-generated comments and pull requests not be submitted directly to the upstream Stacks Project repository for the time being. Axiom says the Stacks Project remains a living text under the stewardship of Johan de Jong and its community of contributors, while the Lean formalization is an independent open-source project maintained by Axiom. The company says it keeps the formalization, counterexamples and refinements in its own repository. Axiom says readers can view the original Stacks Project text alongside the Lean code and explore the dependency graph at stacks.axiommath.ai.