admin The company said its internal model, described as significantly more capable than GPT-6 Astra, produced a proposed solution after an intensive effort involving as many as 10,000 AI agents. OpenAI released a technical write-up and a formalised proof in Lean on September 8, arguing that smooth three-dimensional fluid motion can develop a singularity in finite time.
The result has not been formally recognised as a solution to the Millennium Prize Problem. The Clay Mathematics Institute still lists Navier-Stokes among its unsolved problems, and its rules require any proposed solution to be published in a qualifying outlet, survive at least two years of rigorous scrutiny and gain general acceptance in the global mathematics community before the institute considers awarding the $1 million prize.
OpenAI said the agents reached their result on September 5, about 88 hours after work began. The Navier-Stokes effort generated about 2.7 million agent messages and roughly 130 billion output tokens. Verification and formalisation in Lean took another 17 hours using GPT-6 Astra, according to the company.
The controversy centres on parallel work by New York University mathematician Tristan Buckmaster and Levent Alpöge, a mathematician employed by Anthropic. The pair had been pursuing related questions in fluid dynamics and used models from several AI laboratories, including OpenAI, during their research.
Buckmaster has questioned whether OpenAI’s system arrived at its approach independently after the company learned that he and Alpöge were preparing to disclose their work. He has argued that their route to the problem was highly specialised and raised the possibility that material generated during their use of OpenAI products may have influenced the company’s models.
OpenAI denies that its researchers or agents accessed the pair’s unpublished work to solve the problem. The company said no specific user data was accessed for the project, while acknowledging that it could not rule out the possibility that de-identified data derived from their use of its products had contributed to model improvements.
That qualification has broadened the dispute beyond mathematical priority to questions about how research conducted with commercial AI systems may later feed into model development. For academics, the case raises difficult issues over confidentiality, attribution and whether unpublished reasoning entered into AI services can indirectly benefit systems operated by the same companies.
Accounts also differ over discussions about coordinating publication and assigning credit. OpenAI said it contacted Buckmaster and Alpöge after determining that the rumours which prompted its own September 1 effort related to their work. The company said it proposed coordinating the release and recognising their priority.
Buckmaster has disputed aspects of that account, including conversations with OpenAI researcher Sébastien Bubeck over how Alpöge would be credited. Bubeck has rejected the claim that he sought to remove Alpöge from authorship of his own work. OpenAI chief executive Sam Altman has said the company preferred coordination and rejected accusations that it improperly appropriated the competing research.
The mathematical claim itself is substantial. The Navier-Stokes equations describe the motion of fluids such as air and water, but mathematicians have lacked a proof establishing whether smooth three-dimensional solutions always remain smooth. OpenAI says its proof constructs a case in which a vortex intensifies without bound in finite time while satisfying the required mathematical conditions.
OpenAI said it initially deployed agents across all six unsolved Millennium Prize Problems after hearing rumours of progress elsewhere. After about 50 hours, researchers redirected greater computing resources towards Navier-Stokes when the system showed promising progress.
The company has said it does not intend to seek the $1 million award and is presenting the work as evidence of advances in AI-assisted scientific research. Only the Poincaré conjecture has previously been removed from the original list of seven Millennium Prize Problems after a solution gained acceptance.
