Over the Labor Day weekend, Tristan Buckmaster, a mathematician at New York University, found himself at the center of a contentious dispute involving OpenAI and a longstanding mathematical challenge known as the Navier-Stokes problem. This puzzle, which addresses the behavior of fluid flow described by the Navier-Stokes equations, is one of seven Millennium Prize problems established by the Clay Mathematics Institute, each carrying a $1 million reward for a verified solution.
Throughout August, Buckmaster and his collaborator Levent Alpöge, who is affiliated with Anthropic, another artificial intelligence company, made considerable progress on the problem. However, OpenAI, spurred in part by rumors that Anthropic had solved one or more Millennium problems, accelerated its own efforts and announced last week that it had produced a proof related to the Navier-Stokes equations, potentially beating Buckmaster and other human mathematicians to the milestone.
The situation quickly escalated into a dispute involving questions about intellectual property and attribution. Buckmaster alleged that OpenAI’s proof may have been influenced by his and Alpöge’s recent work, which they had not yet publicly disclosed but had partially processed using OpenAI’s Codex tool. Codex is typically used to assist in writing and managing computer code, and since OpenAI’s systems are trained on user inputs, Buckmaster expressed concern that the company’s AI agents might have incorporated his unpublished insights.
OpenAI categorically denied any such influence, stating that it was impossible for Buckmaster’s recent Codex prompts to have affected their system’s output. The company also emphasized that they had thoroughly investigated the claim and found no evidence that user inputs from July onward played any role in training the relevant AI models.
The dispute also revealed tensions between the two companies over co-authorship and collaboration. Sébastien Bubeck, who leads OpenAI’s mathematical research effort, proposed several overtures to Buckmaster, including allowing him to publish his results first, lead the authorship of OpenAI’s paper, or receive computational resources to continue his work. OpenAI also indicated that it would support Buckmaster and Alpöge receiving the $1 million prize if their results were validated.
However, complication arose because Alpöge is employed by Anthropic, a rival AI firm founded by former OpenAI leaders amid concerns over AI safety and ethics. OpenAI officials expressed reservations about including Alpöge on the paper, leading to a breakdown in negotiations. Buckmaster described the impasse as a refusal by OpenAI to acknowledge his collaborator, a position he found unacceptable.
“I just wanted to be part of the story,” Buckmaster said. “They couldn’t understand why I wouldn’t take this bargain—why I wouldn’t throw Levent under the bus.”
The exchange between Buckmaster and OpenAI also highlighted broader frictions between the competitive, fast-paced culture of technology companies and the traditionally collaborative world of academic mathematics. Mathematicians generally rely increasingly on AI tools but maintain a collegial ethos that contrasts with the business-driven priorities of tech firms.
Buckmaster characterized the experience as intimidating, remarking that going up against a “trillion-dollar company” was “super scary and frightening.” He suggested OpenAI was attempting to “buy him off” and threatened to make the dispute public after a conversation with Bubeck became tense. Bubeck denied any threats, stating his offers were intended to be helpful and that he was surprised by Buckmaster’s uncooperative response.
The larger mathematical community has watched with interest. Peter Sarnak of the Institute for Advanced Study in Princeton praised Buckmaster for adhering to longstanding academic standards in the face of unusual commercial pressures.
While OpenAI has presented its proof as a significant breakthrough affirming that Navier-Stokes equations can produce singularities—cases where fluid velocity accelerates to infinity—Buckmaster emphasized the contributions of other researchers, including Luis Martínez-Zoroa and Diego Córdoba, for their earlier work on strategies to attack related problems.
As the debate continues, Buckmaster has posted detailed manuscripts outlining his and Alpöge’s results and a history of the ongoing research. The controversy underscores the evolving and sometimes uneasy relationship between human mathematicians and AI-driven research efforts, particularly when high stakes and corporate interests are involved.