What did Axiom Math verify in the Stacks Project?
Axiom Math has formally verified the Stacks Project, one of the largest bodies of mathematics ever written by humans, the company announced Wednesday. The work turns a sprawling reference library of roughly 7,654 pages into about 16 million lines of computer-checked code, and Axiom says it is a step toward giving AI agents a sturdy base to build on.
The timing matters. AI systems have been producing a steady flow of mathematical claims, and not all of them hold up. Last month, OpenAI released 719 manuscripts claiming solutions to some of the field’s hardest problems. Even computer checks can slip. Against that backdrop, Axiom is betting that math can stay trustworthy by resting on firmer ground rather than slowing down.
Why is the Stacks Project important to mathematicians?
Mathematicians hold the Stacks Project in unusual regard. Columbia mathematician Peter Woit has said there is “nothing else really comparable,” likening it to “the famous Bourbaki books,” a series that rebuilt math from its foundations in the 20th century. Woit’s Columbia colleague Johan de Jong started Stacks in 2005 as an open-source reference. Mathematicians have contributed to it ever since, building results that run from college-level algebra to the field’s most advanced ideas in algebra, geometry and number theory.
How does formal verification check a mathematical proof?
Formal verification is the method that makes the translation useful. Mathematicians rewrite a proof as code in a programming language called Lean, and a program known as a proof assistant checks that code line by line. If the proof passes, mathematicians can trust that it is logically sound. Engineers use the same idea to confirm that chip designs work, and researchers in France have used it to catch bugs in software that calculates family benefits.
Stacks presented a particular challenge. Its theorems and definitions are tightly interconnected, with each one building on those before it. Many results trace back through the library like a family tree. That made Axiom uncertain at first that the project could be done. The team had to keep every connection intact across about 16 million lines of Lean code.
Axiom CEO Carina Hong described the tension between scale and precision in blunt terms. “Things that are big and things that are delicate in life just don’t go together,” she told TechCrunch. “[Stacks] is a collection of mathematical results that is the foundation of mathematics. You want [theorems] to be consistent and not, like, paradoxical. You want the definitions to not fight with each other.”
Hong, 25, dropped out of a Stanford math PhD program and founded the company in March 2025. In March, Axiom raised a $200 million Series A led by Menlo Ventures. Hong said the team was unsure whether AI-assisted coding could preserve the structure that makes Stacks valuable. “We were not sure if vibe coding would be able to preserve the beauty of Stacks,” she said.
To get the job done, the team gave its agents a loose structure and let them work in parallel. Ashvin Swaminathan, a member of Axiom’s technical staff, said the approach was not a rigid chain of command. “With Stacks, we imposed a hierarchy … [but] it was not, like, a complete hierarchy of dictating what every single agent was doing,” he said. “The agents were finding tasks for themselves and identified things that they could work on in parallel.”
Axiom also trained its own models to be strong at math, an approach Hong said most other verification startups have not taken. At first, one of the company’s in-house Lean experts taught the agents a human sense of mathematical taste. Over time, Hong said, the agents picked up that taste on their own.
The result fills a conspicuous hole. “When you look at what has been formalized in mathematics, there’s this huge black hole. And that black hole covers algebraic geometry,” Swaminathan said. Before Stacks, researchers who wanted to check new algebraic geometry work in Lean had little to build on. “Well, now they do.”
Axiom argues that each verified result becomes a building block for the next. Terence Tao, the UCLA mathematician and Fields Medal winner, has put the general principle simply: “The more formalized proofs you have, the more formalized proofs you can make.”
Mathematicians are not the only audience Axiom has in mind. Developers have long struggled to guarantee that their code is free of bugs and security holes, and the problem grows as AI agents write more of the world’s software. Swaminathan said the company sees mathematics as a training ground for a broader skill. “We believe that mathematics is the domain where you start building these things and that it’s a transferable skill,” he said. “Agents can read our code, understand it, and build new infrastructure on top of it. Our Lean formalization is now the substrate that agents can build upon.”
Hong sees the same pattern shaping AI’s future work. Agents, she argues, will need a well-built, reusable body of knowledge to avoid producing low-quality output. “You want [agents] to have a reusable library for them to build on and keep pushing the frontier,” she said. “If you slop drop in a way that’s bad engineering choices in the beginning, then it’s very hard for them to grow that boundary.”
Hong is willing to put the industry on notice. “I openly challenge all the coding companies to work on formalization at this scale.”
This article was written with the assistance of AI.
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