Navier–Stokes: Buckmaster's Accusations Against OpenAI and OpenAI's Response

Tristan Buckmaster, a professor at NYU, accuses OpenAI of launching a massive effort to solve the Navier–Stokes problem after learning about his work with Levent Alpöge. OpenAI claims to have started its research on September 1 with a new model, denies any access to specific user data, and mentions an expenditure of 300 billion tokens. The controversy revolves around the timeline, methods, and potential use of data from Codex.

Accusations of Pressure, Contested Timeline, and Fear of Recycling via Codex

Tristan Buckmaster reports that when he attempted to make the dispute with OpenAI public, Bubeck allegedly told him, "Why would you want to ruin your career?" and then, "If you don't want me to be nice, then I don't have to be." According to Buckmaster, it was eventually admitted that the first prompt sent by OpenAI was made in the last days, after information about the work with Levent Alpöge had circulated. He finds it suspicious that OpenAI adopted the same approach at the same time and believes that such a direction cannot be improvised in just a few days by merely providing the problem statement to a model. Buckmaster also expresses concern that the intensive use of Codex to assemble his project may have, indirectly, fed into OpenAI's effort. OpenAI reserves the right to train its models on interactions with Codex, with an opt-out option for users. It remains plausible that if these interactions had been included, a model could regurgitate elements in response to a similar problem. Buckmaster believes that the best response is to publish as much information as possible about the research.

OpenAI Claims Evidence, Details Its Timeline, and Denies Access to Work

OpenAI has published comprehensive evidence of the existence and regularity of Navier–Stokes, attributed to an unpublished next-generation model that addressed several unresolved problems over the course of a week. The organization places the launch of this effort on September 1, motivated by rumors of solving two Millennium Prize problems, and confirms ongoing exchanges with Tristan Buckmaster and Levent Alpöge. OpenAI downplays the hypothesis of regurgitation and asserts that no researcher or agent saw Buckmaster and Alpöge's work before its public release, nor accessed specific user data to solve this problem. However, the organization concedes that it is not excluded that de-identified data derived from the use of its products may have contributed to improving its models, and argues that its evidence differs significantly, particularly in the case of Euler (forced vs. non-forced). OpenAI estimates the week's effort at 300 billion output tokens, an expenditure valued at $22.5 million based on current Astra rates.

Path of Independent Researchers and Rare Methodological Choices

Tristan Buckmaster, a professor at NYU, and Levent Alpöge, a mathematician at Anthropic, announced three proofs along with a preliminary discovery on a major theoretical problem, relying on the Codex and Claude AI models. Alpöge was not undertaking this work on behalf of Anthropic, and the duo primarily mobilized Codex in their approach. Buckmaster specifies that he followed a path via a smooth force, corresponding to options c and d of the Fefferman statement, a route opened by Luis and Diego and discreetly chosen with Alpöge. He claims that very few people he knows were working on this approach, which he describes as uncommon.

A Million-Dollar Problem and Persistent Uncertainties

The existence and regularity of Navier–Stokes are among the seven Millennium Prize problems, each endowed with a million dollars by the Clay Institute. While the equations are ubiquitous in fluid mechanics, their theoretical understanding remains lacking, and a solution would mark a significant advancement in mathematical physics. This episode could reignite the debate on the role of AI in mathematical research and the incentives specific to OpenAI. According to Buckmaster and Alpöge, information about their progress circulated to OpenAI while they were finalizing their results; OpenAI indicated that it already had complete evidence, but the chronological details and human intervention remained vague. Buckmaster adds that an entire team seemed mobilized and that substantial computational resources were engaged, while the hypothesis of a computational advantage applied to their approach remains a possibility raised.