On September 8 OpenAI published a preprint, an announcement and a public Lean 4 repository claiming that an unreleased internal model had produced a proof about the three-dimensional Navier–Stokes equations, one of the Clay Mathematics Institute’s seven Millennium Prize Problems. The claim has a precise scope, and as of September 11 no independent mathematician or body had published a verification of it.
What the proof claims
The official problem, written by Charles Fefferman, asks for a proof of any one of four statements. Two of them, (A) and (B), would show that smooth solutions exist forever when no external force acts on the fluid. The other two, (C) and (D), would show the opposite: that there are smooth initial conditions and a smooth external force for which no well-behaved solution exists for all time.
OpenAI claims (C) and (D). Its construction says that for every positive viscosity there are smooth initial data at rest and a smooth force such that the velocity becomes unbounded in finite time while the energy stays bounded. As the problem is written, proving one of the four statements counts as a resolution. It leaves (A) and (B), the unforced case, untouched, and whether a forced construction is what the problem was meant to capture is part of the current debate. OpenAI’s own homepage card says the model “proposes” a solution.
How it was produced, by OpenAI’s account
According to OpenAI, the model that found the proof is internal and more capable than GPT-6 Astra. It ran as roughly 10,000 concurrent agents and reached the result about 88 hours after launch, and GPT-6 Astra then took 17 more hours to formalize and check it in Lean. These are OpenAI’s own figures, and no evaluation method for the model has been published.
What the Lean formalization shows, and what it does not
The repository reports zero unproven steps and only Lean’s standard axioms, and it ships configurations so that outsiders can re-check the proofs with Comparator, a tool that confirms a proof matches the stated theorem. The challenge statements were adapted from Google DeepMind’s Formal Conjectures project. Two limits remain: a machine check confirms the proof of the Lean statement, not that the Lean statement captures what mathematicians mean by the problem, and the repository’s own metadata lists its review status as “self-assessed.”
Where the prize and the community stand
The Clay Institute’s page still lists the problem as active. Its rules require publication in a refereed outlet, at least two years and general acceptance in the mathematics community before any prize is considered, and OpenAI says it does not intend to claim it. The European Mathematical Society called the announcement “a milestone in the history of mathematics” because “a major open problem may finally have been resolved,” and noted that the work builds on an existing strategy by other mathematicians.
A priority dispute alongside
The mathematics comes with a conflict. Tristan Buckmaster of NYU and Levent Alpöge released their own results with smooth forcing for related equations hours before OpenAI’s announcement. In a public statement, Buckmaster describes contacts with OpenAI in the days before, including two co-release proposals, and writes: “I have not seen OpenAI’s proof. I do not know what their model did, or how.” OpenAI says it contacted the two only after finishing its own work, offering a joint release and recognition of their priority.
What to read as settled
Settled: what OpenAI published and claims, which statements it targets, and that its Lean build reports no gaps. Not settled: whether the proof is correct and whether the Lean statements match the problem as the community reads it. The honest headline is a claim under review, and the first independent checks are what to watch.
Related reading
- Official problem description by Charles Fefferman — Clay Mathematics Institute
- NavierStokesAndEuler repository — OpenAI
- GPT-6 Astra ships as a product — IA al Día