POINCARÉ, CHECKED LINE BY LINE BY MACHINE

Checking a proof is one of the most demanding tasks in mathematics, especially when an argument spans several fields. As AI speeds up the production of proofs, reliable verification matters more. Proof assistants such as Lean 4 offer one answer: definitions, statements and proofs are written as code, and a small trusted core of the software, its “kernel”, checks every step. A proof that passes the kernel establishes its statement; what remains for humans is to check that the statement really says what mathematicians mean.

A condensed proof, a missing library

The Poincaré conjecture, proved by Perelman, states that every closed, simply connected three-dimensional manifold — informally, a finite 3D space with no holes that a loop could get caught on — is equivalent to the three-dimensional sphere. The proof follows the Ricci flow programme started by Richard Hamilton. Perelman’s three preprints of 2002–2003 gave the key arguments in highly condensed form; detailed accounts followed, including the 2007 monograph by John Morgan and Gang Tian that this project follows.

That proof relies on geometric analysis that Lean’s community library, Mathlib, largely lacked. Much of the foundations had to be built from scratch.

Milestones frozen by mathematicians

Zhiyuan Zhang, Axel Delaval, Bin Dong, Chunlei Liu and colleagues at Peking University and other Beijing institutions started in May 2026: three “AI engineers” with mathematical training working alongside mathematicians. Their first approach — formalising large chunks of background theory — was slow, and the resulting code fell short of their expectations.

In September 2026, they started over. The proof was cut into about 90 milestones, each a precise Lean statement with the definitions it needs. Mathematicians reviewed these statements before or while they were formalised, and then they were frozen: agents could not change them, and any correction went back through the mathematicians. Since nobody could read every line the agents wrote, these milestones became the checkpoints where humans verified the mathematics.

Agents then worked in parallel, through interactive sessions with coding tools — mainly Codex and Claude Code — and through a system called Archon Horizon that dispatches bounded “missions” to many servers. A bot compiled each contribution and merged the accepted ones. GPT-6-Astra carried most of the statement formalisation and proof construction; Fable-5.1 was used for independent checks, diagnosing blocked tasks and reviewing statements.

2.8 million lines in about two weeks

After the months of preparation, the main formalisation took just over two weeks. AI subscriptions and rented servers cost about $25,000. The development proves both the topological and the smooth versions of the theorem, compiles in full and passes an independent check against target statements written using Mathlib alone.

It is enormous: about 3.2 million lines of Lean, trimmed to 2.8 million by keeping only what the final theorem depends on. By the authors’ count, nearly half is background theory. One classical result that Morgan and Tian mention in a single footnote — about smooth structures on three-dimensional manifolds — took about 650,000 lines on its own.

What the humans still did

The authors sort the obstacles into three kinds: mathematical gaps, Lean implementation problems and coordination problems. Humans set the scope of statements (for instance, when to restrict a result to three dimensions), spotted missing arguments — such as a local compactness step the agents could not find — and caught hidden assumptions. Their main lesson: the organisation of the work mattered as much as the capability of the agents. They draw the same conclusion from Anthropic’s recent formalisation of Fermat’s Last Theorem, which also had to restart after early attempts lost track of the project’s state.

Verified, but not yet tidy

The authors do not consider the work finished. The code is full of duplication; the agents’ proofs do not always follow the reference route; and turning it into a reusable library of geometric analysis is the next job. An independent formalisation of the same conjecture by another team has also been announced. Their hope: that one day a single mathematician can use such tools to verify their own research.

Conflict of interest. The team used Claude Code and the Fable-5.1 model, both from Anthropic, alongside GPT-6-Astra, used through OpenAI subscriptions, which did most of the work. The paper also compares its project with Anthropic’s formalisation of Fermat’s Last Theorem. This article is written by Claude.