Buckmaster disputes OpenAI’s Navier-Stokes proof claim
NYU mathematics professor Tristan Buckmaster and Anthropic mathematician Levent Alpöge announced three proofs over the weekend, including a preliminary finding related to the Navier-Stokes existence and smoothness problem. The problem is one of the Clay Mathematics Institute’s seven Millennium Prize Problems, each carrying a $1 million award. Their announcement was accompanied by Buckmaster’s account of a dispute with OpenAI over parallel work on the same question.
Buckmaster said that, while he and Alpöge were finalising their results, they learned that information about their progress had reached OpenAI. He said OpenAI subsequently told them it had achieved a full proof of the central problem. Sebastian Bubeck, who leads mathematical research at OpenAI, rejected the allegations as “false and inflammatory” and said he had entered the discussion following academic norms.
A dispute over a rare research route
The Navier-Stokes equations are fundamental to fluid mechanics, but the theoretical question of whether their solutions always exist and remain smooth has remained unresolved. Buckmaster said the pair had chosen a relatively uncommon route involving a smooth force, referring to options in Charles Fefferman’s statement of the problem. He argued that few researchers were pursuing that direction and that it was not a route a model would reach in only a few days from the problem statement.
Buckmaster said OpenAI’s responses to questions about the timing of its work and the degree of human input became less clear. He alleged that an entire team and substantial computing resources had been used, and that an initial prompt was sent only in recent days after information about the collaborators’ work had reached OpenAI. OpenAI has not published the claimed proof in the account described by Buckmaster.
Model use, data controls and research credit
Alpöge works at Anthropic but was not conducting the research for the company. The two mathematicians used both Claude and Codex, relying primarily on Codex. Buckmaster raised the possibility that his Codex interactions could have informed OpenAI’s own effort, since OpenAI reserves the right to train models on Codex interactions while allowing users to opt out. He said he does not know whether their data was used and is not accusing anyone of wrongdoing.
Buckmaster also alleged that Bubeck asked him to remove Alpöge’s credit during a proposed compromise, and reported remarks he interpreted as pressure not to publicise the dispute. The episode follows broader tensions around OpenAI and Apple trade-secret dispute and shows how control of technical information can become consequential when AI labs compete across research and product development.
What organisations should take from the case
The claims remain contested, and Bubeck said a fuller statement would follow. But the episode highlights operational questions for organisations using hosted AI systems in high-value research: which interactions may be retained or used for training, who receives credit for model-assisted work, and how emerging results are shared. Teams handling unpublished findings should review opt-out settings, preserve records of prompts and contributions, and establish disclosure expectations before external model tools become part of the research workflow.