On 8 September 2026, artificial intelligence company OpenAI claimed a solution to the Navier–Stokes existence and smoothness problem with a smooth external force. However, approximately 12 hours before OpenAI's announcement, mathematician Tristan Buckmaster made a statement on behalf of himself and partly Levent Alpöge, claiming that their recent advances on Euler equations (a special case of Navier–Stokes), reported in the same post, were leaked to OpenAI a few days earlier and may have inspired OpenAI's prompts for the agents in the process that led to the solution. OpenAI denied such direct use of their work, claiming "significant" differences in proof methods in the Euler case.
Buckmaster made the statement after analysts tipped him and Alpöge that their "progress had been passed to OpenAI" and intended to narrate the events from his perspective lest "a sequence of announcements say something [he knows] to be false". OpenAI researcher Sébastien Bubeck later posted on X disputing some of the claims thereof, and the CEO of OpenAI Sam Altman retweeted Bubeck's statement and maintained that OpenAI had contacted Buckmaster and Alpöge in good faith, while Buckmaster's statement highlighted some communication from OpenAI that could be seen as hostile. Buckmaster also raised concerns about whether OpenAI could have accessed user data from Codex, an AI agent developed by OpenAI which Buckmaster and Alpöge used during their research, either directly or as anonymised training data.
The Clay Mathematics Institute listed solving one of four instances of the Navier–Stokes existence and smoothness problem as one of the Millennium Prize Problems and pledged to pay US$1 million for a solution to each. The Buckmaster statement and the OpenAI announcement both confirm that OpenAI does not plan to claim the award, and the Institute has yet to acknowledge any party for a correct solution. Some observers have seen the controversy as part of competition in artificial intelligence, questioning the role of Alpöge's employment by rival AI company Anthropic; OpenAI stated they began large-scale work on all Millennium Prize Problems on 1 September, following rumours that Anthropic had solved two such problems. A declaration published on 11 September and signed by 28 Fields Medalists, titled A Severe Misalignment of AI in Mathematics, warned against "detrimental" impact on mathematics from AI companies treating unsolved problems as benchmarks without enhancing human understanding.
Background Work before Buckmaster and Alpöge
The Navier–Stokes equations are a system of partial differential equations that has the physical interpretation of describing the velocity and pressure of certain fluids in space, potentially in the presence of an external force (e.g. gravity). The Euler equations are a special case of Navier–Stokes equations where the viscosity of the fluid is zero, a property called superfluidity. The Navier–Stokes existence and smoothness problem asks specifically whether "physically reasonable solutions exist" for these equations in the case where the fluid is incompressible, does not have superfluidity (i.e. the viscosity is positive), the space is three-dimensional Euclidean, the initial condition is smooth, and with an external force. There are different variations of the Navier–Stokes existence and smoothness problem, depending on whether the external force is smooth or equal to zero. Here, "physically reasonable" entails that the solutions are also smooth, globally defined, and that their kinetic energy is globally bounded. Charles Fefferman's problem statement for the Clay Mathematics Institute's Millennium Prize Problems asks to prove one of four statements, where the first pair (A) and (B) correspond to a positive answer to the question ("existence and smoothness") in the absence of an external force, and the last pair (C) and (D) to a negative answer to the question ("breakdown") in the presence of a smooth external force. Each pair is in two different settings of the space being unbounded or periodic.
In 2013, Guo Luo and Thomas Hou presented numerical evidence that solutions to Euler equations could reach a blowup singularity in finite time, tilting public opinion about the answer to the Navier–Stokes problem. Later, and introduced new approaches for constructing finite-time blowup singularities in a series of papers from 2023. Ultimately, in collaboration with Fan Zheng, they arrived at an unforced blowup result for 3D incompressible Euler equations and a forced blowup result for so-called hypodissipative Navier–Stokes equations, a variant of the classical Navier–Stokes equations. Here, "unforced" means there is no external force present, and "forced" means the construction of such blowups revolves around deliberate choice of the external force. However, both results have gaps with the original Navier–Stokes problem, which excludes Euler equations and requires the external force to be smooth. Nonetheless, the two results together prompted beliefs that their strategy may be extended to the Millennium setup.
Buckmaster and Alpöge's collaboration Tristan Buckmaster, professor of mathematics at the Courant Institute, revealed in a statement on the night of 7 September 2026 shortly before midnight that he and Levent Alpöge, a mathematician and an employee at the artificial intelligence company Anthropic, had been trying to close the aforementioned gaps for at least a year in secrecy, employing several large language models (LLMs) from both Anthropic and its immediate rival OpenAI in the research process. Buckmaster and Alpöge each stressed that their work is "a strictly personal collaboration" and not associated with Anthropic in any formal or legal way, while Buckmaster mentioned he had a collaboration with Google DeepMind. This collaboration presumably produced the preprint uploaded to arXiv on 17 September 2025 by 22 authors from academic institutions and DeepMind, including Buckmaster, demonstrating new numerical approaches to construct singularities of solutions to incompressible porous media (IPM) and Boussinesq equations, another two partial differential equations in fluid dynamics closely related to Navier–Stokes. Two days later, on 19 September, Alpöge sent an email to Buckmaster expressing predictions that Navier–Stokes was the "only [...] Millennium Problem [that will be] solved in the foreseeable future" and concerns that "corporations [would] claim [...] the deepest pure math work", presumably initiating the year-long collaboration which is "in the traditional mathematical way".
This collaboration eventually gave rise to smooth forcing results for Euler, IPM, and Boussinesq, reported together in the same post that contains the statement by Buckmaster, which were obtained on 15 August 2026 and verified a week later in Lean, an interactive theorem prover based on type theory. According to the statement, they originally intended to delay the announcement since they wanted to produce human-written and readable papers that are the result of their own understanding of the proofs. Shortly after the post, mathematician Terence Tao congratulated the two in a blog post and on social media, calling the results "a remarkable achievement" and claiming "there does not seem to be anything in principle preventing the methods from extending all the way to Navier–Stokes". Buckmaster recognises the importance of the work of Córdoba and Martínez-Zoroa, saying the latter deserves a Fields Medal, often considered the highest honour in mathematics. He left his office at 2:30 am on 8 September 2026.
Dispute between OpenAI and Buckmaster–Alpöge OpenAI announcement After an 88-hour run that started on 1 September 2026, an internal multiagent system of OpenAI produced a 166-page paper, along with a computer-verified Lean formalisation, that are claimed to have established Fefferman's alternatives (C) and (D) in an announcement at noon on 8 September, approximately 12 hours after Buckmaster's post. At least 10,000 agents were employed, which, in the immediate process that led to the Navier–Stokes solution, sent 2.7 million messages and spent 130 billion output tokens. It was estimated that the electricity cost alone would be in the order of millions, which was confirmed by OpenAI researcher Noam Brown.
OpenAI acknowledged that Buckmaster and Alpöge's input to models of OpenAI via their use of Codex might have improved the model used for the run, whose training started on 28 August; on the same date, Brown said that OpenAI had already attempted Millennium Prize Problems but failed to produce any result. OpenAI explained that it was rumours about two Millennium problems being solved that motivated them to initiate an attempt at all six open problems using the model, with OpenAI researcher Sébastien Bubeck later further specifying that the rumours explicitly mentioned Anthropic resolved the two and were started by "viral [tweets]". It is not clear which tweets he is referring to, but Alpöge's tweet "Augustus Mirabilis" (miraculous August) on 31 August was cited as what led OpenAI to believe that, by extension, Anthropic had solved at least one Millennium problem. OpenAI further explained that they "later realized" the rumours were related to Alpöge and Buckmaster.
Buckmaster and Alpöge's contact with OpenAI before the announcement According to both Buckmaster's statement and Alpöge's tweet the next day on 9 September, Alpöge received tips that his work with Buckmaster had been leaked to OpenAI. Prompted by this, the two mathematicians contacted OpenAI independently—Alpöge on the night of 2 September, and Buckmaster on 3 September (to an unnamed mathematician at OpenAI), expressing his confidence in potential advances on Navier–Stokes and his wishes against rushing an AI write-up and a Lean certificate. He was then later contacted by Bubeck, who insisted on meeting with him as soon as possible, still under the impression that Buckmaster and Alpöge had resolved Navier–Stokes instead of smooth forced Euler. Bubeck contacted both again on 6 September, first messaging Alpöge at 11:48 about OpenAI's resolution of Navier–Stokes which used "very little human input" and that they were "ready to share everything [...] including the human prompts" since he wanted "to be maximally open", and then later at 12:45 asking Buckmaster if they could meet "at any point today".
A meeting was held later in the afternoon between Buckmaster, Bubeck and the unnamed mathematician; Alpöge did not attend. Buckmaster, being told some more details about OpenAI's solution, made mathematical judgements that led him to the conclusion that OpenAI did pursue the path he and Alpöge had been working on, namely the Córdoba–Martínez-Zoroa programme. He also learned that a large team within OpenAI had been working on the problem and initially tried easier problems, including ones concerning Euler equations; he interpreted this as contradicting the "very little human input" Bubeck earlier told Alpöge, but Bubeck later explained that this meant the team did not contain experts in Navier–Stokes-related fields and hence there was no one from the team who could "meaningfully contribute to the mathematical content". Buckmaster then asked if OpenAI had direct access to files that he and Alpöge had input into Codex, including entire drafts of papers, or if OpenAI used sessions to train their models. OpenAI rejected the first during the meeting but admitted the second in the later post, albeit in a downplaying manner, saying that the data are "de-identified" and "derived" from their use.