10,000 agents, 88 hours: what the Navier–Stokes proof means for checking

Two announcements landed four days apart this month. On 4 September Anthropic said Claude had produced the first complete computer-checked proof of Fermat's Last Theorem. On 8 September OpenAI said an unreleased internal model had produced a proof for the Navier–Stokes problem, one of the Clay Mathematics Institute's seven Millennium Prize Problems, and had it machine-verified.

Neither is construction news. However, both matter to anyone whose job is producing calculations that other people sign off. The maths is explained in two sentences: The Navier–Stokes equations describe how fluids move, and they sit underneath the CFD runs engineers use for wind loading, HVAC and flood modelling. Lean is a programming language in which a proof is written out step by step so a computer can check every link in the chain.

The Millennium problem asks whether smooth three-dimensional fluid flow can break down. OpenAI says its system showed that it can: a fluid starting at rest, pushed by a smooth external force, develops a 'singularity' where speeds grow without bound in finite time while its energy stays finite. That settles statements C and D of the official problem. OpenAI says it will not claim the $1m prize. The Clay Institute has not accepted the result; as of 11 September, its website lists Navier–Stokes under a new heading, 'Active', separate from both its unsolved and its solved problems. This is OpenAI's claim, checked by a computer and not yet by the field.

There’s one caveat though. The singularity lives in the idealised equations, not in water; as Quanta's Konstantin Kakaes noted, it has no immediate practical consequences. Your CFD model will behave next week exactly as it did last week.

OpenAI says it heard rumours on 1 September that two Millennium problems had fallen and pointed a model it had been training since 28 August at all of them. Agents worked in groups, each on a different variant of the problem, with tools to read a cached internet and run code. The group that cracked Navier–Stokes was 'on the order of 10,000 concurrent agents'. It arrived at its answer about 88 hours after launch; formalising the proof in Lean took another 17 hours with GPT-6 Astra. The whole effort used about 300 billion output tokens, and Sébastien Bubeck of OpenAI told Quanta it cost several million dollars.

Anthropic's Fermat work has the same shape at a smaller scale. Claude wrote 13 million lines of Lean and proved 29,500 intermediate theorems in 11 days, with dozens of agents working in parallel. Kevin Buzzard, Professor of Pure Mathematics at Imperial College London, who has led the community's own Fermat formalisation effort since 2024, reviewed the finished proof.

Until now the constraint on AI-generated mathematics was that humans had to check it. The checking is now being automated too, so the pace is set by compute budgets rather than referees.

Tristan Buckmaster, Professor of Mathematics at NYU's Courant Institute, published a statement late on 7 September. He and Levent Alpöge, a mathematician employed by Anthropic, had spent most of a year on the same family of problems as a personal collaboration using Claude and Codex, and had Lean-verified proofs for the related Euler equations by 22 August. Buckmaster says OpenAI took the same unusual route within days of learning of their work, and that Bubeck twice asked for Alpöge to be dropped from authorship.

He adds that he is "not accusing anyone of anything”.

OpenAI's position moved over the week. On 8 September it said "while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models". By 10 September the post had been updated to say an investigation confirmed Buckmaster's Codex prompts "could not have influenced the system in any way, including through training". Both sides credit Diego Córdoba and Luis Martínez-Zoroa, whose approach both proofs build on; Córdoba told Scientific American: "We're a little bit in shock."

Strip out the maths, and the pattern points at generating candidates at an enormous scale, then running every one through a checker that cannot be argued with. That is a verification architecture, and it's the most significant cost and revenue consideration in our industry.

An engineering equivalent of Lean does not exist yet for most of these, but the barrier turned out to be effort rather than possibility, and effort is what two labs just showed being bought with tokens. Axios's Madison Mills asked what happens when the company supplying your AI tools can mobilise vastly more resources to compete with you. Any practice building a proprietary method on a vendor's tools should check its data-training settings this week.

The result changes nothing in your fluid models. The method changes the economics of checking. If you run a team whose value is verification, write down which of your outputs are checked by a rule and which by judgement. The first list is where this is heading, sooner than most of us budgeted for. The open question is whether you automate those checks or wait for someone else to.

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