Axiom Math has formally verified the Stacks Project, one of the largest bodies of mathematics ever written, turning roughly 7,654 pages of interconnected results into about 16 million lines of computer-checked code. The startup says a fleet of its AI agents carried out the work, translating the library into Lean, a language for writing proofs as code, and checking every line with a proof assistant. The result gives mathematicians and AI agents a reusable, verified foundation in algebraic geometry, an area where little checked material existed before.
A library mathematicians revere
The Stacks Project began in 2005, when Columbia mathematician Johan de Jong started it as an open reference for algebraic geometry. Mathematicians have added to it ever since, building results from college-level algebra up to the field’s most advanced ideas.
Columbia mathematician Peter Woit has said there is nothing else really comparable, comparing the project to the Bourbaki books that rebuilt mathematics from its foundations. The comparison captures the project’s standing: a reference work other mathematicians build on, not a single result.
Formalising mathematics means rewriting each proof as code in a language called Lean, then letting a proof assistant check that code line by line. If a proof passes, mathematicians can trust that it is logically sound. Engineers use the same idea, called formal verification, to confirm chip designs work.
Why the size made it hard
Checking each theorem was only part of the job, because the theorems and definitions in Stacks are deeply interconnected, with each one building on the ones before it. Like a family tree, many results trace back to earlier ones, so Axiom had to keep every connection intact across about 16 million lines of Lean code.
The team was unsure at first that the project would work at all. To pull it off, Axiom gave its agents a loose structure and let them run in parallel, finding tasks for themselves instead of following a rigid plan.
The company also trained its own models to be good at mathematics, an approach its chief executive says most other verification startups have not taken. At first, human Lean experts taught the agents a sense of taste; over time, the agents picked that taste up on their own.
What it means for code
The work matters beyond mathematics because developers have long struggled to guarantee that their code is free of bugs and security holes. That problem grows as AI agents write more of the world’s code, and Axiom is betting the software of the future will need to rest on verified foundations.
Formal verification is already moving from theorems into engineering. Harmonic’s Aristotle model now checks C software and Verilog chip designs, taking the same proof-based discipline to the code and hardware that run the modern world, where a single error can be most expensive.
The same week brought a reminder of both sides of that coin. OpenAI’s GPT-6 Cyber, a model built to find flaws, reproduced hundreds of real vulnerabilities and built working exploits for 948 Linux kernel bugs, showing how capable AI has become at both breaking and fixing software.
A checked foundation does not stop an agent writing bad code, but it gives the agent correct material to build on instead of plausible-sounding guesses. That distinction is the whole point: verification checks the work, rather than trusting the system that produced it.
Our read
The honest way to read this result is as a floor, not a guarantee. The release verifies the formalised Stacks Project itself, not every new proof or piece of software an AI system might produce from it. Axiom’s agents formalised existing human scholarship; they did not discover the underlying mathematics.
What is new is the scale at which checking now works, and the fact that the checked library is reusable rather than a one-off demonstration. Axiom says the finished library is the largest integrated, reusable library of formalised mathematics, a claim that rests on the company itself.
That reusability is the point the company’s chief executive keeps returning to. Axiom argues that each verified result becomes a building block, so future proofs can lean on checked mathematics instead of starting from scratch. As Terence Tao has put it, the more formalised proofs you have, the more formalised proofs you can make.
The logic echoes a wider worry in the field. Research shows experts follow their agents’ lean most of the time, so the material those agents build on had better be right. Axiom has now done this twice at enormous scale, after its agents formalised the classification of finite simple groups in 24 days.
There is also a competitive edge to the announcement. Axiom’s chief executive openly challenged the coding companies to attempt a formalisation at this scale, a dare aimed at firms whose agents write code all day but whose outputs are rarely checked this rigorously.
Whether rivals take it up will say a lot about whether verification becomes a standard part of building AI systems, or stays a speciality of maths-first startups. Axiom and Harmonic are betting on the first outcome, and their funding rounds show investors are listening.
Why It Matters
If AI agents are going to write production software and do science, the industry needs ways to check their work that do not depend on the same system’s say-so. Stacks fills one of the biggest gaps in formalised mathematics: the algebraic geometry researchers previously had little checked material to build on.
A verified foundation will not end AI errors on its own, but it removes one excuse for them. It gives every future proof in the field a checked place to stand, and every agent working in the field correct material to start from.
Background
Axiom Math was founded in March 2025 by Carina Hong, a 25-year-old who left a Stanford mathematics PhD programme to start the company. In March 2026 it raised a $200 million Series A led by Menlo Ventures, valuing the startup at $1.6 billion.
The Stacks Project verification follows Axiom’s formalisation of the classification of finite simple groups, announced a week earlier. The company says its agents will keep working through further bodies of mathematics. The open question now is which library they tackle next, and whether rivals answer the challenge.
Source: TechCrunch report and oossa summary
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