Axiom Releases Lean Formalization of the 7,500-Page Stacks Project in Mathematics

Formalizing the Stacks Project

Axiom has released its Lean formalization of the Stacks Project, a 7,500-page open-source reference for algebraic geometry. This formalization comprises approximately 16 million lines of code, making it the largest integrated and reusable library of formalized mathematics built to date.

Impact on Research and AI

The complete mathematical development of the Stacks Project is now available as a single, machine-checked library. This allows researchers and AI agents to directly import definitions and theorems into new formal proofs. Each Lean declaration is linked to the canonical Stacks Project text, providing a bridge between traditional and formalized mathematics.

Foundations and Development

The Stacks Project, initiated in 2005 by Johan de Jong, grew through contributions from hundreds of mathematicians over two decades. Axiom's formalization relied on the existing infrastructure and mathematical libraries developed by the Lean and Mathlib communities, which provided the necessary abstractions for organizing such a large corpus.

Formalizing this scale of work was possible due to the years of foundational work by these communities, which built a rich mathematical library and the organizational abstractions before Axiom's project began.

Why Formalize?

Formal verification is becoming essential as AI systems generate mathematical arguments at a pace beyond human inspection. While new results in fields with mature Lean infrastructure can be formalized quickly, modern algebraic geometry previously lacked this capability. A single contemporary proof in this field often involves a long chain of theorems across various mathematical areas before the core argument begins. The Stacks Project contained the mathematics, but the corresponding Lean infrastructure was absent until now.

Engineering and AI Collaboration

Closing this gap involved both mathematical and software engineering challenges. Scaling to millions of lines of verified mathematics required clean abstractions, stable interfaces, and interoperable components. Axiom used coordinated AI agents to translate, prove, and integrate thousands of theorems in parallel. This formalization of the Stacks Project establishes a new starting point for algebraic geometry and related fields.