OpenAI says an internal model resolved a Navier–Stokes problem, amid a credit dispute

OpenAI reported that an internal model “significantly more capable than GPT‑6 Astra” had produced, and formally verified in Lean, a proof bearing on the Navier–Stokes existence-and-smoothness problem — one of the seven Clay Mathematics Institute Millennium Prize Problems, open for roughly 90 years. The result establishes that a three-dimensional incompressible fluid, starting smoothly and at rest under a smooth applied force, can develop a singularity — speeds growing without bound — in finite time. OpenAI said this establishes statements “C” and “D” of the official formulation, but that it did not intend to claim the $1m prize, framing the work as “a snapshot in time” of AI progress rather than a culmination.

The construction ran a system of coordinating agents — on the order of 10,000 concurrent agents for the Navier–Stokes group — with access to a cached copy of the internet and a code sandbox. The agents reached their resolution about 88 hours after launch, with a further 17 hours of Lean formalisation via GPT‑6 Astra; across all the problems attempted they exchanged 4.9 million messages and roughly 300 billion output tokens. OpenAI said the effort had begun on 1 September after it heard a rumour that two Millennium problems might have been solved.

That rumour is the seed of a dispute. OpenAI later realised it concerned work by NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge, who — using an internal Anthropic model — had resolved the related forced Euler problem. OpenAI said it had offered a concurrent release recognising their priority, and that its own system saw none of their work before publication. Buckmaster and Alpöge disputed the framing: as reported by Fortune, Buckmaster alleged that OpenAI’s Sébastien Bubeck had pressured him to remove Alpöge’s name over his Anthropic affiliation, and quoted Bubeck asking “Why would you ruin your career?” OpenAI said an investigation confirmed Buckmaster’s own Codex prompts could not have influenced its model.

Mathematicians quoted by Quanta and Fortune noted the result concerns a forced construction — widely regarded as a distinct and, to some, less significant question than the unforced Clay problem — which is part of why OpenAI declined the prize. The episode arrived days after OpenAI’s ten formally verified mathematical advances and its chief scientist’s warning that progress may be entering a new regime, and stands as both a capability claim and a case study in how priority and credit are contested when the work is done by machines.