Navier–Stokes priority controversy

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 class of Navier–Stokes equations where the viscosity of the fluid is zero, namely superfluidity.^[21]: 1 The Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems, 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, and the initial condition and the external force are smooth. 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 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"), and the last pair (C) and (D) to a negative ("breakdown"). Each pair is in two different settings of the space being unbounded or periodic.^[21]: 2

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.^[22][15][23] Later, Diego Córdoba [eu] and Luis Martínez-Zoroa [eu] introduced new approaches for constructing finite-time blowup singularities in a series of papers from 2023.^[15][24] Ultimately, in collaboration with Fan Zheng, they arrived at an unforced blowup result for 3D incompressible Euler equations^[25] and a forced blowup result for so-called hypodissipative Navier–Stokes equations, a variant of the classical Navier–Stokes equations.^[26] 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.^[27]

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, have 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^[13] OpenAI in the research process.^[28][5]: 1 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.^{[19][5]: 2 [29]} 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.^[30] Two days later, on 19 September, Alpöge sent an email to Buckmaster expressing predictions that Navier–Stokes is 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".^[8][31]

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,^[3] which were obtained on 15 August 2026 and verified a week later by 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.^[19][5]: 1 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".^[27][32] Buckmaster recognises the importance of work of Córdoba and Martínez-Zoroa, saying the latter deserves a Fields Medal, often considered the highest honour in mathematics.^{[5]: 1 [15]}

OpenAI announcement

After an 88-hour run that started on 1 September 2026, an internal multiagent system of OpenAI produced a 166-page (earlier 165) paper,^[33] along with a computer-verified Lean formalisation,^[7] that are claimed to have established Fefferman's alternatives (C) and (D) in an announcement 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 4.9 million messages and spent 130 billion output tokens.^[7] It was estimated that the electricity cost alone would be in the order of millions,^{[34][1][35][4]} which was confirmed by OpenAI researcher Noam Brown.^[36]

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;^[7] on the same date, Brown said that OpenAI had already attempted Millennium Prize Problems but failed to produce any result.^[37] 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,^[7] with OpenAI researcher Sébastien Bubeck later further specifying that the rumours explicitly mentioned Anthropic resolved the two and were started by "viral [tweets]".^[10][38] It is not clear which tweets he is referring to, but Alpöge's tweet "Augustus Mirabilis" (miraculous August) on 31 August^[39] was cited as what led OpenAI to believe that, by extension, Anthropic has solved at least one Millennium problem.^[40] However, traces of such rumours and their developments can be found on online forums like Reddit later on 3–5 September.^[41][42][43] OpenAI further explained that they later "realised" the rumours were related to Alpöge and Buckmaster.^[7][18]

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,^[29] 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.^[10][5]: 2 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".^{[7][10][5]: 2}

A meeting was held later in the afternoon between Buckmaster, Bubeck and the unnamed mathematician; Alpöge did not attend.^[5]: 2 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 have been working on, namely the Córdoba–Martínez-Zoroa programme.^[38][5]: 3 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 intepreted this as contradicting the "very little human input" Bubeck earlier told Alpöge,^[5]: 3 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".^[40] 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.^[14] 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.^[7][28]

OpenAI then proposed to Buckmaster that he first report their Euler result before OpenAI announces their Navier–Stokes result, or, if he is willing, that he rewrites OpenAI's proof on behalf of the company, either as the sole author (per Buckmaster) or lead author (per Bubeck); but in any case the prize pledged by the Clay Mathematics Institute for a correct solution to a Millennium problem, one million US dollars, belongs to Buckmaster and Alpöge as they are the "closest humans to the problem".^{[44][4][5]: 3} In the second proposal, both Buckmaster and Bubeck confirmed that the latter asserted against Alpöge being listed as author of the human-rewritten paper, citing conflict of interest.^{[5]: 3 [10]} Buckmaster declined both offers, rejecting that he (in the first proposal) or Alpöge (in the second) be removed from authorship of work initiated by them.^[8] He pushed further that if OpenAI went ahead and announced their proof, he would make the dispute public;^[12] according to Bubeck, at this point Buckmaster further threatened to "go to the press with [...] unfounded accusations", to which Bubeck asked why would one risk to ruin their career.^{[45][10][5]: 3} Buckmaster in turn understood this as a threat,^[45] responded with confusion as to why going public would be career suicide and claimed to have received the reply "if you don't want me to be nice, then I don't have to be nice."^[13] Bubeck apologised for his choice of words in his tweet after the OpenAI announcement.^[15][10]

People associated with the two parties

After OpenAI's announcement, Buckmaster explicitly accused them on social media of "us[ing] customer's data to try to scoop their customer", citing OpenAI's admission to using user data for model training.^[46] Alpöge made the same accusation more implicitly in a tweet approximately 1 hour later.^[47][28] Buckmaster also criticised an earlier version of OpenAI's paper for having no references to work of Córdoba and Martínez-Zoroa, which were later added in a new 166-page version.^[48][33] Professor at the University of Erlangen–Nuremberg Emil Wiedemann also pointed out that a longer list of references was expected for such a "monumental proof".^[15]

Bubeck called the meeting with Buckmaster "incredibly difficult".^[10] Sam Altman, CEO of OpenAI, retweeted Bubeck's statement and maintained that OpenAI has contacted Buckmaster and Alpöge in good faith. He emphasised that the methods of Buckmaster and Alpöge's work and of OpenAI's solution "appear to be different" and hence the two are independent^[11]—so did Ven Chandrasekaran, a mathematician at the company^[49]—but Alpöge refuted this argument, saying OpenAI's proof "looks more along the lines of another Euler blowup proof we had". Alpöge also expressed disappointment in OpenAI's unwillingness to include him in the potential human-rewritten paper of OpenAI's solution and in their offering of the Millennium Prize, and ultimately in the failure of two AI labs to cooperate on science.^[47] A colleague at Anthropic, Sholto Douglas, shared the latter sentiment^[50] and offered sympathy in Buckmaster and Alpöge not being able to bridge their gaps to Navier–Stokes in the way they intended.^[51] Mark Chen, chief research officer of OpenAI, reiterated that "no people or AI systems searched through user data to solve this problem".^[14]

Wider public

Terence Tao raised general concerns on his social media about the future of mathematical collaboration between humans; he worries that rumours about a mathematician's own work might prompt AI companies to invest huge efforts into pushing it instead of allowing the mathematician to develop their research to "full potential", whether the companies know about the details of the human mathematician's work or not.^[52] He also expressed dissatisfaction with the current conduct of AI companies "using press releases or social media posts to communicate mathematical results" instead of uploading preprints or arrange talks at conferences,^[18] not taking the mathematical obligation to push broader understanding of their results.^[38] Mathematician Michael Harris however said such high-level financial investments would not continue as "there is no profit in it"; however, he also worries that practices of AI companies related to mathematics would "convince decision makers that human mathematicians are obsolete, and [...] young people that their passion for mathematics has no future", hence "extremely damaging to mathematics".^[19] Professor at Queen Mary University of London Abhishek Saha agreed that there are few other such "high-caliber" problems that AI companies are willing to invest "such vast sums solving".^[13] Professor at Leibniz University Hannover Joachim Escher [de] said asking every mathematical problem to OpenAI was simply impossible due to energy and capacity constraints, hence mathematics had not "reached its end".^[53] On the other hand, Wiedemann worries that mathematics will no longer be a competition between intelligence and diligence, but material resources.^[15]

Although Martin Bridson, the president of the Clay Mathematics Institute, called the series of announcements "exciting", he stressed that the Institute requires a solution to a Millennium problem to be in the form of peer-reviewed publications, which would then be further scrutinised before the prize is awarded.^[18] Ravi Vakil and John Meier, on behalf of the American Mathematical Society, congratulated both Alpöge–Buckmaster and OpenAI for "a milestone advance in human knowledge".^[54]

Joseph Perla, the founder of model-routing company TrustedRouter, staunchly stands against OpenAI's decision and way of announcing their solution, saying "this is what all the labs do every day: read your chats and copy ideas".^[14] Mathematician Andreas Thom also pointed to the general "unanswered question" of lack of transparency between OpenAI and mathematicians, suggesting that models of the former have been "hoovering up unpublished human work and presenting it as AI generated".^[55][38] Lecturer at Monash University Melissa Lee raised similar concerns and warned researchers "to know very clearly what happens to unpublished work they enter into AI systems".^[44] Professor at the University of Sydney Zsuzsanna Dancso accused OpenAI of failing to observe two fundamental ethics in science: "one is we acknowledge our sources, and two is we acknowledge our collaborators".^[44]