Navier–Stokes Priority Controversy

OpenAI claimed a solution to the Navier–Stokes existence and smoothness problem on 8 September 2026, but mathematician Tristan Buckmaster stated roughly 12 hours earlier that his and Levent Alpöge's advances on Euler equations were leaked to OpenAI days before and may have inspired the prompts used by OpenAI's agents. OpenAI denied direct use of the work, citing "significant" differences in proof methods in the Euler case, and both the Buckmaster statement and OpenAI's announcement confirm OpenAI does not plan to claim the Clay Mathematics Institute's US$1 million Millennium Prize, which the Institute has yet to award to any party.

Navier–Stokes priority controversy | Part of a series https://en.wikipedia.org/wiki/Category:Artificial intelligence on | | Artificial intelligence AI https://en.wikipedia.org/wiki/Artificial intelligence | |---| | Glossary https://en.wikipedia.org/wiki/Glossary of artificial intelligence | On 8 September 2026, artificial intelligence https://en.wikipedia.org/wiki/Artificial intelligence company OpenAI https://en.wikipedia.org/wiki/OpenAI claimed a solution to the Navier–Stokes existence and smoothness https://en.wikipedia.org/wiki/Navier–Stokes existence and smoothness problem. \ 1\ cite note-KHbbc-1 <sup \ 2\ cite note-2 </sup However, approximately 12 hours before OpenAI's announcement, mathematician Tristan Buckmaster made a statement on behalf of himself and partly Levent Alpöge https://en.wikipedia.org/wiki/Levent Alpöge , claiming that their recent advances on Euler equations <https://en.wikipedia.org/wiki/Euler equations fluid dynamics a special case of Navier Stokes , reported in the same post,<sup \ 3\ cite note-TBmastodon-3 </sup were leaked to OpenAI a few days earlier and may have inspired OpenAI's prompts for the agents https://en.wikipedia.org/wiki/Prompt engineering in the process that led to the solution. \ 4\ cite note-JKfortune-4 <sup \ 5\ cite note-TBstatement-5 </sup OpenAI denied such direct use of their work, claiming "significant" differences in proof methods in the Euler case. \ 6\ cite note-6 \ 7\ cite note-OAIpost-7 Buckmaster made the statement after analysts tipped him and Alpöge that their "progress had been passed to OpenAI" \ 5\ cite note-TBstatement-5 <sup : 2</sup and intended to narrate the events from his perspective lest "a sequence of announcements say something he knows to be false". \ 8\ cite note-GRusatoday-8 \ 5\ cite note-TBstatement-5 <sup : 4</sup OpenAI researcher Sébastien Bubeck https://en.wikipedia.org/wiki/Sébastien Bubeck later posted on X <https://en.wikipedia.org/wiki/X social network disputing some of the claims thereof, \ 9\ cite note-KNindiatoday-9 <sup \ 10\ cite note-SBtweet-10 </sup and the CEO of OpenAI Sam Altman https://en.wikipedia.org/wiki/Sam Altman retweeted Bubeck's statement and maintained that OpenAI had contacted Buckmaster and Alpöge in good faith https://en.wikipedia.org/wiki/Good faith , \ 9\ cite note-KNindiatoday-9 <sup \ 11\ cite note-SAtweet-11 </sup while Buckmaster's statement highlighted some communication from OpenAI that could be seen as hostile. \ 12\ cite note-RBtc-12 \ 4\ cite note-JKfortune-4 \ 13\ cite note-RHverge-13 \ 5\ cite note-TBstatement-5 <sup : 3</sup Buckmaster also raised concerns about whether OpenAI could have accessed user data from Codex <https://en.wikipedia.org/wiki/OpenAI Codex AI agent , an AI agent developed by OpenAI which Buckmaster and Alpöge used during their research, either directly or as anonymised training data https://en.wikipedia.org/wiki/Training data . \ 14\ cite note-MMaxios-14 \ 15\ cite note-MMTPMWsz-15 The Clay Mathematics Institute https://en.wikipedia.org/wiki/Clay Mathematics Institute listed the Navier–Stokes problem as one of the Millennium Prize Problems https://en.wikipedia.org/wiki/Millennium Prize Problems and pledged to pay US$1 million for a solution to each. \ 16\ cite note-16 <sup \ a\ cite note-18 </sup 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 https://en.wikipedia.org/wiki/Competition in artificial intelligence , questioning the role of Alpöge's employment by rival AI company Anthropic https://en.wikipedia.org/wiki/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. \ 18\ cite note-DCnature-19 \ 19\ cite note-CZACscience-20 \ 14\ cite note-MMaxios-14 \ 20\ cite note-21 <sup \ 13\ cite note-RHverge-13 </sup A declaration published on 11 September and signed by 26 Fields Medalists https://en.wikipedia.org/wiki/Fields Medal , titled A Severe Misalignment of AI in Mathematics, warned against "detrimental" impact on mathematics from AI companies treating as unsolved problems as benchmarks https://en.wikipedia.org/wiki/Language model benchmark without enhancing human understanding. \ 21\ cite note-economist11-22 \ 22\ cite note-fieldsOpenLetter-23 \ 23\ cite note-RHverge2-24 Background edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=1 Work before Buckmaster and Alpöge edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=2 The Navier–Stokes equations https://en.wikipedia.org/wiki/Navier–Stokes equations are a system of partial differential equations https://en.wikipedia.org/wiki/Partial differential equation that has the physical interpretation https://en.wikipedia.org/wiki/Fluid dynamics of describing the velocity and pressure of certain fluids https://en.wikipedia.org/wiki/Fluid in space, potentially in the presence of an external force e.g. gravity . The Euler equations <https://en.wikipedia.org/wiki/Euler equations fluid dynamics are a special case of Navier–Stokes equations https://en.wikipedia.org/wiki/Navier–Stokes equations where the viscosity https://en.wikipedia.org/wiki/Viscosity of the fluid is zero, a property called superfluidity https://en.wikipedia.org/wiki/Superfluidity . \ 24\ cite note-CFps-25 <sup : 1</sup The Navier–Stokes existence and smoothness https://en.wikipedia.org/wiki/Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems https://en.wikipedia.org/wiki/Millennium Prize Problems , asks specifically whether "physically reasonable solutions exist" for these equations in the case where the fluid is incompressible https://en.wikipedia.org/wiki/Incompressible flow , does not have superfluidity i.e. the viscosity is positive , the space is three-dimensional Euclidean https://en.wikipedia.org/wiki/Three-dimensional space , and the initial condition and the external force are smooth https://en.wikipedia.org/wiki/Smoothness . Here, "physically reasonable" entails that the solutions are also smooth, globally defined, and that their kinetic energy https://en.wikipedia.org/wiki/Kinetic energy is globally bounded. Charles Fefferman https://en.wikipedia.org/wiki/Charles Fefferman 's problem statement for the Clay Mathematics Institute https://en.wikipedia.org/wiki/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. \ 24\ cite note-CFps-25 <sup : 2</sup In 2013, Guo Luo and Thomas Hou https://en.wikipedia.org/wiki/Thomas Hou presented numerical evidence that solutions to Euler equations could reach a blowup singularity <https://en.wikipedia.org/wiki/Singularity mathematics in finite time, tilting public opinion about the answer to the Navier–Stokes problem. \ 25\ cite note-26 \ 15\ cite note-MMTPMWsz-15 <sup \ 26\ cite note-27 </sup Later, Diego Córdoba https://en.wikipedia.org/wiki/Diego Córdoba?action=edit&redlink=1 and Luis Martínez-Zoroa https://en.wikipedia.org/wiki/Luis Martínez-Zoroa?action=edit&redlink=1 introduced new approaches for constructing finite-time blowup singularities in a series of papers from 2023. \ 15\ cite note-MMTPMWsz-15 <sup \ 27\ cite note-28 </sup Ultimately, in collaboration with Fan Zheng, they arrived at an unforced blowup result for 3D incompressible Euler equations<sup \ 28\ cite note-29 </sup and a forced blowup result for so-called hypodissipative Navier–Stokes equations, a variant of the classical Navier–Stokes equations.<sup \ 29\ cite note-30 </sup 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. \ 30\ cite note-TTblog-31 Buckmaster and Alpöge's collaboration edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=3 Tristan Buckmaster, professor of mathematics at the Courant Institute https://en.wikipedia.org/wiki/Courant Institute School of Mathematics, Computing, and Data Science , revealed in a statement on the night of 7 September 2026 shortly before midnight that he and Levent Alpöge https://en.wikipedia.org/wiki/Levent Alpöge , a mathematician and an employee at the artificial intelligence https://en.wikipedia.org/wiki/Artificial intelligence company Anthropic https://en.wikipedia.org/wiki/Anthropic , had been trying to close the aforementioned gaps for at least a year in secrecy, employing several large language models https://en.wikipedia.org/wiki/Large language model LLMs from both Anthropic and its immediate rival https://en.wikipedia.org/wiki/Competition in artificial intelligence \ 13\ cite note-RHverge-13 OpenAI https://en.wikipedia.org/wiki/OpenAI in the research process. \ 31\ cite note-CCft-32 \ 5\ cite note-TBstatement-5 <sup : 1</sup 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 https://en.wikipedia.org/wiki/Google DeepMind . \ 19\ cite note-CZACscience-20 \ 5\ cite note-TBstatement-5 <sup : 2</sup <sup \ 32\ cite note-LAtweet-33 </sup This collaboration presumably produced the preprint uploaded to arXiv https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Porous medium IPM and Boussinesq equations <https://en.wikipedia.org/wiki/Boussinesq approximation buoyancy , another two partial differential equations in fluid dynamics closely related to Navier–Stokes.<sup \ 33\ cite note-34 </sup 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". \ 8\ cite note-GRusatoday-8 \ 34\ cite note-35 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,<sup \ 3\ cite note-TBmastodon-3 </sup which were obtained on 15 August 2026 and verified a week later in Lean <https://en.wikipedia.org/wiki/Lean proof assistant , an interactive theorem prover https://en.wikipedia.org/wiki/Proof assistant based on type theory https://en.wikipedia.org/wiki/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\ cite note-CZACscience-20 \ 5\ cite note-TBstatement-5 <sup : 1</sup Shortly after the post, mathematician Terence Tao https://en.wikipedia.org/wiki/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". \ 30\ cite note-TTblog-31 <sup \ 35\ cite note-36 </sup Buckmaster recognises the importance of the work of Córdoba and Martínez-Zoroa, saying the latter deserves a Fields Medal https://en.wikipedia.org/wiki/Fields Medal , often considered the highest honour in mathematics. \ 5\ cite note-TBstatement-5 <sup : 1</sup <sup \ 15\ cite note-MMTPMWsz-15 </sup He left his office at 2:30 am on 8 September 2026. \ 36\ cite note-KCnyt-37 Dispute between OpenAI and Buckmaster–Alpöge edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=4 OpenAI announcement edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=5 After an 88-hour run that started on 1 September 2026, an internal multiagent system of OpenAI produced a 166-page paper,<sup \ 37\ cite note-OAIpaper-38 </sup along with a computer-verified Lean formalisation,<sup \ 7\ cite note-OAIpost-7 </sup 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 https://en.wikipedia.org/wiki/Large language model Tokenization .<sup \ 7\ cite note-OAIpost-7 </sup It was estimated that the electricity cost alone would be in the order of millions, \ 38\ cite note-39 \ 1\ cite note-KHbbc-1 \ 39\ cite note-40 <sup \ 4\ cite note-JKfortune-4 </sup which was confirmed by OpenAI researcher Noam Brown. \ 40\ cite note-41 OpenAI acknowledged that Buckmaster and Alpöge's input to models of OpenAI via their use of Codex <https://en.wikipedia.org/wiki/OpenAI Codex AI agent might have improved the model used for the run, whose training started on 28 August;<sup \ 7\ cite note-OAIpost-7 </sup on the same date, Brown said that OpenAI had already attempted Millennium Prize Problems but failed to produce any result.<sup \ 41\ cite note-42 </sup 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,<sup \ 7\ cite note-OAIpost-7 </sup with OpenAI researcher Sébastien Bubeck https://en.wikipedia.org/wiki/Sébastien Bubeck later further specifying that the rumours explicitly mentioned Anthropic resolved the two and were started by "viral tweets ". \ 10\ cite note-SBtweet-10 <sup \ 42\ cite note-ACconvo-43 </sup It is not clear which tweets he is referring to, but Alpöge's tweet "Augustus Mirabilis" miraculous August on 31 August<sup \ 43\ cite note-44 </sup was cited as what led OpenAI to believe that, by extension, Anthropic had solved at least one Millennium problem.<sup \ 44\ cite note-SBtweet2-45 </sup OpenAI further explained that they "later realized" the rumours were related to Alpöge and Buckmaster. \ 7\ cite note-OAIpost-7 \ 18\ cite note-DCnature-19 Buckmaster and Alpöge's contact with OpenAI before the announcement edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=6 According to both Buckmaster's statement and Alpöge's tweet the next day on 9 September,<sup \ 32\ cite note-LAtweet-33 </sup 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\ cite note-SBtweet-10 \ 5\ cite note-TBstatement-5 <sup : 2</sup 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\ cite note-OAIpost-7 \ 10\ cite note-SBtweet-10 \ 5\ cite note-TBstatement-5 <sup : 2</sup A meeting was held later in the afternoon between Buckmaster, Bubeck and the unnamed mathematician; Alpöge did not attend. \ 5\ cite note-TBstatement-5 <sup : 2</sup 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. \ 42\ cite note-ACconvo-43 \ 5\ cite note-TBstatement-5 <sup : 3</sup 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, \ 5\ cite note-TBstatement-5 <sup : 3</sup 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".<sup \ 44\ cite note-SBtweet2-45 </sup 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.<sup \ 14\ cite note-MMaxios-14 </sup 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\ cite note-OAIpost-7 \ 31\ cite note-CCft-32 OpenAI then proposed to Buckmaster that he first reports 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 https://en.wikipedia.org/wiki/United States dollar , belongs to Buckmaster and Alpöge as they are the "closest humans to the problem". \ 45\ cite note-DMJBabc-46 \ 4\ cite note-JKfortune-4 \ 5\ cite note-TBstatement-5 <sup : 3</sup 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 https://en.wikipedia.org/wiki/Conflict of interest . \ 5\ cite note-TBstatement-5 <sup : 3</sup <sup \ 10\ cite note-SBtweet-10 </sup 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.<sup \ 8\ cite note-GRusatoday-8 </sup He pushed further that if OpenAI went ahead and announced their proof, he would make the dispute public;<sup \ 12\ cite note-RBtc-12 </sup to which Bubeck asked why one would risk to ruin their career over "unfounded accusations". \ 46\ cite note-MFtelegraph-47 \ 10\ cite note-SBtweet-10 \ 5\ cite note-TBstatement-5 <sup : 3</sup Buckmaster in turn understood this as a threat,<sup \ 46\ cite note-MFtelegraph-47 </sup 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."<sup \ 13\ cite note-RHverge-13 </sup Bubeck apologised for his choice of words in his tweet after the OpenAI announcement. \ 15\ cite note-MMTPMWsz-15 \ 10\ cite note-SBtweet-10 Timeline of OpenAI assertions concerning reuse of user data edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=7 - 8 September morning, initial press release: "We the researchers and the agents did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."<sup \ 7\ cite note-OAIpost-7 </sup - 8 September morning, chief research officer Mark Chen: "No people or AI systems searched through user data to solve this problem or any specific problem that we were trying. And I'm a little bit disappointed by the allegations of the huge breach of user trust. We did not do that."<sup \ 47\ cite note-CFvb-48 </sup - 8 September afternoon, official tweet: "We the researchers and the agents did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. However, our proofs differ significantly and even the precise results proved are different in the Euler case forced vs. unforced ."<sup \ 48\ cite note-49 </sup - 9 September evening, spokesperson Laurance Fauconnet: "We can say categorically that it is impossible for Dr. Buckmaster's Codex prompts over the last two months to have influenced the system in any way, including training."<sup \ 36\ cite note-KCnyt-37 </sup <sup \ 23\ cite note-RHverge2-24 </sup - 10 September late evening, updated press release: "Following an investigation, we have confirmed that Buckmaster's Codex prompts over the two months preceding this announcement and paper on September 8, 2026, could not have influenced the system in any way, including through training. The OpenAI internal model used for this result was developed through large-scale reinforcement learning on top of a previously pretrained model."<sup \ 7\ cite note-OAIpost-7 </sup - 13 September, Fauconnet: "After investigating, we can say with full confidence that no user inputs past July 3rd could have influenced this system in any way."<sup \ 49\ cite note-50 </sup Further responses edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=8 People associated with the two parties edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=9 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.<sup \ 50\ cite note-51 </sup Alpöge made the same accusation more implicitly in a tweet approximately one hour later. \ 51\ cite note-LAtweet2-52 <sup \ 31\ cite note-CCft-32 </sup Buckmaster also criticised an earlier version of OpenAI's paper for having no references to the work of Córdoba and Martínez-Zoroa, which were later added in a new 166-page version. \ 52\ cite note-53 <sup \ 37\ cite note-OAIpaper-38 </sup Professor at the University of Erlangen–Nuremberg https://en.wikipedia.org/wiki/University of Erlangen–Nuremberg Emil Wiedemann also pointed out that a longer list of references was expected for such a "monumental proof". \ 15\ cite note-MMTPMWsz-15 Bubeck called the meeting with Buckmaster "incredibly difficult". \ 10\ cite note-SBtweet-10 Sam Altman https://en.wikipedia.org/wiki/Sam Altman , CEO of OpenAI, retweeted Bubeck's statement and maintained that OpenAI has contacted Buckmaster and Alpöge in good faith https://en.wikipedia.org/wiki/Good faith .<sup \ 9\ cite note-KNindiatoday-9 </sup 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<sup \ 11\ cite note-SAtweet-11 </sup —so did Ven Chandrasekaran, a mathematician at the company<sup \ 53\ cite note-54 </sup —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.<sup \ 51\ cite note-LAtweet2-52 </sup A colleague at Anthropic, Sholto Douglas, shared the latter sentiment<sup \ 54\ cite note-55 </sup and offered sympathy in Buckmaster and Alpöge not being able to bridge their gaps to Navier–Stokes in the way they intended.<sup \ 55\ cite note-56 </sup Mark Chen, chief research officer of OpenAI, reiterated that "no people or AI systems searched through user data to solve this problem". \ 14\ cite note-MMaxios-14 In a recounting of the events on the afternoon of 8 September, Buckmaster asserted that he did not have an interest in the prize money or even winning the race to the answer per se; he simply wished himself and Alpöge to be credited since they were "part of the mathematical story". He called the intention of OpenAI to remove Alpöge's authorship "the big sticking point", and considered himself to be an innocent bystander of the "petty games" or childish "petty drama"<sup \ 23\ cite note-RHverge2-24 </sup that is the rivalry between OpenAI and Anthropic, with the former on the evening of 9 September going further in their denying of using Buckmaster's Codex prompts, calling it "categorically impossible".<sup \ 36\ cite note-KCnyt-37 </sup On the general issue of changes to mathematics brought about by AI, Buckmaster considers the race for more papers and credit now "pointless" and calls for discussion within the mathematical community to re-evaluate the nature of the discipline<sup \ 56\ cite note-57 </sup and operational aspects including credit assignment and hiring. \ 57\ cite note-58 Wider public edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=10 Terence Tao https://en.wikipedia.org/wiki/Terence Tao raised general concerns on his social media about the future of mathematical collaboration between humans; he worried 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.<sup \ 58\ cite note-59 </sup 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 arranging talks at conferences,<sup \ 18\ cite note-DCnature-19 </sup not taking the mathematical obligation to push broader understanding of their results.<sup \ 42\ cite note-ACconvo-43 </sup Mathematician Michael Harris <https://en.wikipedia.org/wiki/Michael Harris mathematician 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".<sup \ 19\ cite note-CZACscience-20 </sup Professor at Queen Mary University of London https://en.wikipedia.org/wiki/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".<sup \ 13\ cite note-RHverge-13 </sup Professor at Leibniz University Hannover https://en.wikipedia.org/wiki/Leibniz University Hannover Joachim Escher https://en.wikipedia.org/wiki/Joachim Escher said asking every mathematical problem to OpenAI was simply impossible due to energy and capacity constraints, hence mathematics had not "reached its end".<sup \ 59\ cite note-60 </sup On the other hand, Wiedemann worries that mathematics will no longer be a competition between intelligence and diligence, but material resources.<sup \ 15\ cite note-MMTPMWsz-15 </sup Mathematician Shing-Tung Yau https://en.wikipedia.org/wiki/Shing-Tung Yau also raised concerns that the disparity of resources held between AI companies and early career researchers could discourage the latter "to pursue ambitious questions". \ 23\ cite note-RHverge2-24 Although Martin Bridson https://en.wikipedia.org/wiki/Martin Bridson , 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.<sup \ 18\ cite note-DCnature-19 </sup Later on 11 September, the Institute released a statement acknowledging that "the Navier–Stokes problem has apparently been settled" but maintained the process for verification and credit assignment "is deliberately unhurried";<sup \ 60\ cite note-61 </sup the status of the Navier–Stokes problem is now "active" instead of unsolved or solved on the Institute's website.<sup \ 61\ cite note-62 </sup Meanwhile, Ravi Vakil https://en.wikipedia.org/wiki/Ravi Vakil and John Meier, on behalf of the American Mathematical Society https://en.wikipedia.org/wiki/American Mathematical Society , congratulated both Alpöge–Buckmaster and OpenAI for "a milestone advance in human knowledge". \ 62\ cite note-63 Joseph Perla, 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".<sup \ 14\ cite note-MMaxios-14 </sup Mathematician Andreas Thom <https://en.wikipedia.org/wiki/Andreas Thom mathematician 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". \ 63\ cite note-64 <sup \ 42\ cite note-ACconvo-43 </sup Lecturer at Monash University https://en.wikipedia.org/wiki/Monash University Melissa Lee raised similar concerns and warned researchers "to know very clearly what happens to unpublished work they enter into AI systems". \ 45\ cite note-DMJBabc-46 Professor at the University of Sydney https://en.wikipedia.org/wiki/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".<sup \ 45\ cite note-DMJBabc-46 </sup Mathematician Peter Sarnak https://en.wikipedia.org/wiki/Peter Sarnak commended Buckmaster for "behaving the way mathematicians behave",<sup \ 36\ cite note-KCnyt-37 </sup while Alpöge previously also called him "a mathematician's mathematician" and praised him for his integrity.<sup \ 32\ cite note-LAtweet-33 </sup Martínez-Zoroa thanked Buckmaster for crediting him and Córdoba and "mak ing it clear that the ideas came from their group".<sup \ 64\ cite note-PFdLpais-65 </sup Córdoba, initially in shock about the news and having been notified by neither of the two parties, acknowledged the current tendency to incorporate AI in mathematical research, but considered human-written proofs still essential and AI unable to find new original ideas in the foreseeable future; he stressed that "if their work had not existed, AI would not have solved the problem".<sup \ 64\ cite note-PFdLpais-65 </sup Mathematician Steven Strogatz https://en.wikipedia.org/wiki/Steven Strogatz called Buckmaster "a real gentleman" and wished the Millennium prize money to go to Córdoba and Martínez-Zoroa; however, he feared that machines could learn and be skilled in even aspects of mathematics that are perceived as intrinsically human, including interpretation and communication for advancing human understanding, or appreciation and arbitration of aesthetic values in mathematical work. \ 65\ cite note-66 On 11 September 2026, 24 Fields Medalists https://en.wikipedia.org/wiki/Fields Medal later cosigned by two more published an open letter titled A Severe Misalignment of AI in Mathematics , warning that the human foundation of interaction and student nurture and the core value of "conceptual understanding and insight" that form a "fertile ground" for mathematics could be eroded by the failure to hinder AI companies from continuing the "push" to resolve mathematical problems en masse.<sup \ 22\ cite note-fieldsOpenLetter-23 </sup Such concerns have been put in the larger context of cognitive offloading https://en.wikipedia.org/wiki/Cognitive load and the possibility of humans becoming more reliant on AI tools for information processing and critical thinking https://en.wikipedia.org/wiki/Critical thinking . \ 21\ cite note-economist11-22 See also edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=11 Notes edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=12 1. ↑ cite ref-18 The only time this pledge was invoked was in 2010 when Grigori Perelman https://en.wikipedia.org/wiki/Grigori Perelman 's proof of the Poincaré conjecture https://en.wikipedia.org/wiki/Poincaré conjecture was accepted, but Perelman rejected the prize, considering previous work of Richard S. Hamilton https://en.wikipedia.org/wiki/Richard S. Hamilton more fundamental and hence making the latter more entitled to the award.<sup \ 17\ cite note-17 </sup References edit https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes priority controversy&action=edit§ion=13 1. 1 cite ref-KHbbc 1-0 2 cite ref-KHbbc 1-1 Hays, Kali 8 September 2026 . "OpenAI says it cracked 90-year-old maths problem in 88 hours" https://www.bbc.co.uk/news/articles/cy7zygy3rl2o . BBC News https://en.wikipedia.org/wiki/BBC News . Retrieved 9 September 2026. 2. ↑ cite ref-2 Sample, Ian; Milmo, Dan 8 September 2026 . "OpenAI claims to have solved maths problem that stumped humans for decades" https://www.theguardian.com/science/2026/sep/08/openai-claims-to-have-solved-maths-problem-that-stumped-humans-for-decades . The Guardian https://en.wikipedia.org/wiki/The Guardian . Retrieved 9 September 2026. 3. 1 cite ref-TBmastodon 3-0 2 cite ref-TBmastodon 3-1 Buckmaster, Tristan 8 September 2026 . 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