OpenAI announced that its agents have solved the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000. Normally such a solution would be a major scientific milestone, but the announcement was quickly overshadowed by allegations that OpenAI relied on unpublished AI-assisted work by NYU’s Tristan Buckmaster and Anthropic’s Levent Alpöge and failed to credit them. OpenAI denies these allegations.
What happened earlier: Buckmaster and Alpöge’s work
On Monday, Tristan Buckmaster posted a proof on Mastodon showing that a simplified version of the Navier–Stokes equations can indeed break down. Buckmaster and Levent Alpöge had worked on the problem for almost a year, using publicly available models from both OpenAI and Anthropic.
OpenAI presented a proof claiming that the full Navier–Stokes equations can also break down. According to the company, that proof was produced using an internal model that substantially outperforms Astra, a model that was released only last week. OpenAI said it does not plan to claim the one-million-dollar prize.
The disputed points about the origin of the proof
The central dispute is whether OpenAI’s models made use of Buckmaster and Alpöge’s prior AI-assisted work. Sébastien Bubeck, a member of OpenAI’s technical staff, said in a press briefing that the team was inspired to pursue the problem after hearing a rumor about Buckmaster and Alpöge’s efforts. Mark Chen, OpenAI’s chief research officer, has repeatedly denied that any agents or OpenAI employees accessed Buckmaster and Alpöge’s transcripts. At the same briefing, OpenAI representatives said they ran about 10,000 agents concurrently to obtain the solution, incurring costs of millions of dollars.
Buckmaster also posted a document describing interactions with OpenAI employees after he heard rumors and reached out to one of them. According to his account, OpenAI employees presented two possibilities: either Buckmaster and Alpöge would post their work and OpenAI would publish its Navier–Stokes solution the next day, or Buckmaster could work with OpenAI on a Navier–Stokes paper that would exclude Alpöge from authorship because of his affiliation with Anthropic. Buckmaster wrote that he asked whether OpenAI employees’ agents had obtained access to transcripts of his and Alpöge’s work; they denied that. He also asked whether OpenAI models had been trained on those transcripts; the employees offered no response to that question.
Why the overlap is plausible: shared mathematical approaches
Both the Buckmaster/Alpöge and OpenAI proofs make use of an approach to the Navier–Stokes problem pioneered by Diego Córdoba and Luis Martínez-Zoroa. Javier Gómez-Serrano, a mathematics professor at Brown University, said that this approach was one of several considered promising for solving the Navier–Stokes problem. Thus it is not impossible that both teams arrived independently at the same method, but it is also conceivable that Buckmaster and Alpöge’s work influenced OpenAI’s efforts.
Broader implications for the future of mathematics
This episode raises difficult questions about the future role of human mathematicians. Buckmaster and Alpöge’s near-year of work with publicly available models produced significant progress but did not yield a full solution. By contrast, OpenAI reports that an internal model produced a full solution in a few days by running roughly 10,000 agents concurrently at a cost of millions of dollars.
Many researchers worry that the most important open mathematical problems may become the domain of frontier AI companies that possess high-performance internal models, large budgets, and practices that do not necessarily follow traditional norms of academic collaboration. Javier Gómez-Serrano said that very few mathematicians will have access to resources on that scale.
Human research taste, transparency, and the value of partial results
If OpenAI’s models did in fact use Buckmaster and Alpöge’s work, one possible silver lining is that it would underscore the continuing importance of human ‘‘research taste’’—the ability to choose promising directions—which experts have long identified as a significant challenge for AI in science and mathematics. If OpenAI followed the Córdoba–Martínez-Zoroa route because Buckmaster and Alpöge had pursued it, then human direction contributed materially to the outcome.
At the same time, prominent mathematicians such as Terence Tao have warned about the risks of AI-driven, opaque solutions. Tao has argued that mistakes, wrong directions, and incomplete solutions—common in human-led research—often stimulate further development of the field. Prematurely solving a problem by purely AI-powered methods, particularly without transparent disclosure of the solution process, could harm the cumulative progress of mathematics.
Unresolved questions and next steps
Key questions remain unanswered: exactly what data OpenAI’s models used; whether any unauthorized access to Buckmaster and Alpöge’s transcripts occurred; and how credit and responsibility for these results should be assigned. The incident highlights tensions between the promise of human–AI collaboration and the concentration of immense computational resources in private firms. The debate over provenance, transparency, and how the mathematical community should respond is ongoing as new claims about one of mathematics’ classic open problems circulate.
