Axiom has released a Lean formalization of the entire Stacks Project, the 7,500-page open-source reference for modern algebraic geometry. Spanning 16 million lines of code, it is the largest integrated, reusable library of formalized mathematics ever built. This milestone makes the full mathematical development of the Stacks Project available as a single, machine-checked library, allowing researchers and AI agents to import its definitions and theorems directly into new formal proofs.
Initiated by Johan de Jong in 2005, the Stacks Project has grown through the contributions of hundreds of mathematicians over two decades. The formalization rests on the foundations of the Lean and Mathlib communities, which spent years building a rich mathematical library. Formalizing a corpus of this magnitude was only possible because of those communities' prior work.
The release addresses the growing need for formal verification as AI systems begin generating mathematical arguments at a pace beyond human inspection. In fields with mature Lean infrastructure, new results can often be formalized shortly after discovery, but modern algebraic geometry has been harder to approach this way. A single contemporary proof may invoke a long chain of theorems across commutative algebra, schemes, sheaves, cohomology, and algebraic stacks.
The mathematics existed in the Stacks Project, but the corresponding Lean infrastructure did not. Closing that gap required coordinated swarms of AI agents capable of translating, proving, and integrating thousands of theorems in parallel. The formalization establishes a new starting point for algebraic geometry and its neighboring fields.
Source: Hacker News · Summarized by HeadlinesBriefing