{"slug": "to-serve-man-ai-math-and-navier-stokes", "title": "To Serve Man: AI, Math, and Navier–Stokes", "summary": "OpenAI announced on September 8, 2026, that its AI model has produced a proof of finite-time blowup for the forced three-dimensional Navier-Stokes equations, a Millennium Prize Problem worth $1 million. The announcement, made at 1:20 p.m. ET, follows earlier AI-assisted results in mathematics, including a May disproof of the Erdős unit distance conjecture, and has prompted reactions from mathematicians like Terence Tao.", "body_md": "By now, most people have heard that OpenAI has likely [solved](https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/) [a](https://www.wired.com/story/openai-navier-stokes-math-discovery-academics/) [Millennium](https://www.scientificamerican.com/article/ai-may-have-just-solved-a-million-dollar-math-problem-the-field-will-never-be-the-same/) [Prize](https://www.nytimes.com/2026/09/08/science/openai-proof-millennium-problem.html) [Problem](https://www.semafor.com/article/09/08/2026/openai-agents-find-proof-to-1-million-millennium-prize-problem); specifically, a proof\n                    of finite-time blowup for the forced three-dimensional\n                    Navier-Stokes equations, announced yesterday afternoon\n                    at 1:20 p.m. ET.[@OpenAI](https://x.com/OpenAI/status/2097374640582668336), September 8, 2026,\n                        1:20 p.m. ET As someone who has followed the story\n                    somewhat closely, along with the broader debate around AI\n                    in mathematics, I wanted to write a short blog post that\n                    lays out a timeline of events, and (hopefully) contextualizes\n                    this result with developments in the past year.\n\n## Trust me bro\n\nTowards the beginning of 2026, AI in mathematics was\n                    still treated as somewhat of a novelty. There were some\n                    interesting results, mostly on Erdős problems, mostly\n                    with commercially available AI models, and mostly from a\n                    few peoplePrimarily [@AcerFur](https://x.com/AcerFur)\n                        and [@Liam06972452](https://x.com/Liam06972452). on\n                    Twitter. Mathematicians were paying attention, though\n                    without much alarm. For example, after one such problem\n                    was solved in January, Terence Taothe Fields Medal-winning,\n                        MacArthur-recognized, field-defining\n                        mathematician,\n                    who kept a page tracking AI-assisted mathematical\n                    contributions,[AI contributions to Erdős\n                            problems](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems)\n                    remarked that he was less interested in the solution\n                    itself than in \"the emerging AI-powered capability to\n                    rapidly write and rewrite expositions of the solution.\"[@tao](https://mathstodon.xyz/@tao/115855840223258103), January 7,\n                        2026.\n\nThe results being produced began to look like less of a\n                    novelty in May, when OpenAI announced that an internal\n                    model had disproved the Erdős unit distance conjecture,[An OpenAI model has disproved a\n                            central conjecture in discrete geometry](https://openai.com/index/model-disproves-discrete-geometry-conjecture/), May 20,\n                        2026. an open problem in discrete geometry. This\n                    result was supplemented by a companion paper, written by\n                    nine mathematicians, that digested the argument and\n                    simplified it.Alon et al., [Remarks on the disproof of the unit\n                            distance conjecture](https://arxiv.org/abs/2605.20695), May 20, 2026. This\n                    seemed like an encouraging model for AI-assisted\n                    mathematics: the model produced a result, while\n                    mathematicians worked together to make it legible and\n                    useful to the field. The proof was\n                    treated not as the end of the process, but as the beginning\n                    of one. This sort of human-centered focus, however, did not\n                    last very long.\n\nBy the summer, the character of the results being posted\n                    had changed quite a bit. On July 19, Levent Alpöge (a\n                    member of technical staff at Anthropic) posted a\n                    counterexample to the Jacobian conjecture:[@__alpoge__](https://x.com/__alpoge__/status/2079028340955197566), July 19,\n                        2026.\n\nhello there the jacobian conjecture is false\n                    thanx to my close friend akhil for asking about it and\n                    my other close friend fable for working during the world\n                    cup final\n\n((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \\C^3\\to \\C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)\n\nNotably, this was a black-box proof: although the\n                    counterexample itself was easy to verify as correct, no\n                    paper accompanied it, nor was there any explanation of how\n                    it had been found. A better understanding emerged only\n                    afterward, when other mathematicians (with assistance from\n                    AI models)[@davikrehalt](https://x.com/davikrehalt/status/2079175065695035442), July 20,\n                        2026.\n                    worked to find one. Subsequently, Alpöge posted a resolution\n                    of the Hadamard matrix conjecture on August 12,[@__alpoge__](https://x.com/__alpoge__/status/2087504790435840207), August 12,\n                        2026.\n                    and, on August 23, a proof that $S^6$ admits a complex\n                    structure.[@__alpoge__](https://x.com/__alpoge__/status/2091639597193368014), August 23,\n                        2026.\n\nThe labs followed a similar trend. On August 1, OpenAI\n                    published ten results attributed to an internal version\n                    of Astra,[Ten advances in mathematics and\n                            theoretical computer science](https://openai.com/index/ten-advances-in-mathematics/), August 1,\n                        2026.\n                    among them the first explicit construction of a\n                    non-sofic group. This construction (and the nine other\n                    results) did not come with a companion paper written by\n                    mathematicians with domain expertise, the way the unit\n                    distance result had been. Instead, the only proofs\n                    provided were entirely AI-written. Again,\n                    understanding had to come from outside. Francesco\n                    Fournier-Facio, a mathematician at Cambridge,\n                    independently worked out the criterion underneath the\n                    OpenAI results and used that to produce a broader family\n                    of examples, including torsion-free ones[A torsion-free non-sofic group](https://arxiv.org/abs/2608.02025),\n                        August 3, 2026. Similarly, on August 10,\n                    Anthropic announced that an unreleased model had raised\n                    the lower bound on the proportion of zeros of the zeta\n                    function lying on the critical line from 41.6% to 67.2%,\n                    after a member of technical staff asked it to \"take a\n                    real stab\" at the Riemann hypothesis and then mostly\n                    sent it words of encouragement.[Learning more about Claude's\n                            mathematical capabilities](https://www.anthropic.com/research/riemann-zeta), August 10,\n                        2026.\n                    Like several of the other recent announcements, this one\n                    came without any companion paper aimed at helping\n                    mathematicians understand the argument beyond simply\n                    assessing its correctness.\n\n## Hype moments and aura\n\nAround this time, some mathematicians started pushing\n                    back. Their issue wasn't that the proofs were wrong; in\n                    fact, most were accompanied by (compiling) Lean\n                    certificates. One complaint was the writing. In a talk\n                    at IPAM,[Accelerating Math and Theoretical\n                            Physics with AI](https://www.youtube.com/watch?v=AXlif6z9a2A), IPAM, UCLA. Tao\n                    described\n                    how AI-generated proofs would spend pages on trivial\n                    details and then a single line on the key idea. The\n                    larger complaint was that mathematics was being used as\n                    a benchmark for AI companies, strip-mined[@tao](https://mathstodon.xyz/@tao/117204930249967695), September 2, 2026.\n                    for hype moments and aura by people with no particular\n                    interest in the field itself. One such critique, written\n                    by Hugo Duminil-Copin,A recipient of the 2022 Fields\n                        Medal. in a blog post sarcastically titled \"Care\n                    for a little more AI?\", lamented that:[Care for a little more AI?](https://proofsandprompts.com/2026/08/30/care-for-a-little-more-ai/),\n                        Proofs and Prompts, August 30, 2026.\n\n...the current use of AI does not empower us, it\n                    petrifies us. These artificial discoveries risk\n                    decapitating entire fields before they have had time to\n                    develop to their full potential. Even worse, they nuke\n                    the mathematical landscape, making it increasingly\n                    difficult to inhabit after each blast.\n\nTerence Tao (previously a vocal proponent of AI in math)[How Terry Tao Became an Evangelist\n                            for AI in Math](https://www.quantamagazine.org/how-terry-tao-became-an-evangelist-for-ai-in-math-20260608/), Quanta, June 8, 2026.\n                    soon\n                    began making a similar case. In a sequence of posts on\n                    Mathstodon, written over four days from the evening of\n                    September 2 to the afternoon of September 5, he\n                    developed his argument.\n\nIn the first of these posts, Tao described the idea of\n                    contamination,A term from AI evaluation, where a\n                        problem whose solution is already public is no longer\n                        useful as a test. cited Duminil-Copin's article,\n                    and suggested that:[@tao](https://mathstodon.xyz/@tao/117204930249967695), September 2, 2026, 11:15 p.m.\n                        ET.\n\nIt may become necessary to declare certain\n                    classes of mathematical problems off-limits to automated\n                    solvers, in order to preserve their broader value to the\n                    mathematical ecosystem (for instance, through the\n                    training of future mathematicians). Admittedly this can\n                    be hard to enforce when such tools are both powerful and\n                    widely available.\n\nIn a second post (whose importance will be obvious later)\n                    he gave a concrete example. Using the famous\n                    Navier-Stokes equations, Tao explained what could be\n                    lost if an important problem like this were solved\n                    without any insight into the process that produced the\n                    answer:[@tao](https://mathstodon.xyz/@tao/117207849921390904), September 3, 2026, 11:37 a.m.\n                        ET.\n\n[T]here is now a scenario in which an autonomous\n                    AI harness, backed by an enormous amount of\n                    computational resources, performs this entire iteration\n                    internally, and ends up producing the final ansatz, and\n                    thence the solution to the Navier-Stokes regularity\n                    problem, while the AI company running the harness keeps\n                    the process to arrive at that ansatz almost completely\n                    out of public view. Technically, one of the most\n                    prominent open problems in mathematics would now be\n                    solved; but there would be almost no value added to\n                    mathematics as a consequence.\n\nLater that day, he put the concern more generally:[@tao](https://mathstodon.xyz/@tao/117208617602946453), September 3, 2026, 2:53 p.m.\n                        ET.\n\nBut the currently fashionable practice of\n                    pointing a powerful AI tool at the task of answering a\n                    problem $X$, unguided by any human expert in the field\n                    $X$ resides in, has created an unprecedented divergence\n                    between the production of answers, and the production of\n                    insight, to the point where the two questions have\n                    become *negatively correlated*.\n\nHe followed this up with a case study that had unfolded a\n                    few days earlier. On August 31, Julia StadlmannJ.L. Doob Research Assistant\n                        Professor at the University of Illinois. released\n                    a preprint lowering the best known bound on gaps between\n                    primes.[arXiv:2608.31126](https://arxiv.org/abs/2608.31126), August 31,\n                        2026. Tao described how, immediately after\n                    this:[@tao](https://mathstodon.xyz/@tao/117208617602946453), September 3, 2026, 2:53 p.m.\n                        ET.\n\n...we were treated to the unedifying spectacle\n                    of no fewer than three separate AI companies racing to\n                    announce their own improvement on the bounded gaps\n                    between prime result that I mentioned yesterday. These\n                    results are numerically stronger than Stadlmann's\n                    improvement, but I am very glad that Stadlmann was able\n                    to complete her analysis just in time before the problem\n                    became contaminated [...]\n\n## It's fun to make predictions\n\nUnbeknownst to Tao, prior to his post on Navier-Stokes,\n                    rumors had already been circulating about an AI lab\n                    solving two Millennium Prize Problems. Motivated by\n                    these rumors, and Tao's post, Andrew Curran tweeted the\n                    following:[@AndrewCurran_](https://x.com/AndrewCurran_/status/2096062392442724805), September 4,\n                        2026, 10:26 p.m. ET.\n\nIt's fun to make predictions. Here's a new one:\n                    Anthropic has solved a Millennium Prize Problem. And\n                    I'll be even more specific. Claude has solved\n                    Navier-Stokes. It is out for expert review. And to give\n                    myself a hard deadline, they will announce it before the\n                    IPO.\n\nThe next morning, Tao added a clarification to his own\n                    thread:[@tao](https://mathstodon.xyz/@tao/117219101339291693), September 5, 2026, 11:19 a.m.\n                        ET.\n\n[I]n response to recent rumors about a possible\n                    solution to the Navier-Stokes problem: I am not aware of\n                    any significant developments in this regard; the above\n                    discussion is hypothetical, but not completely\n                    implausible at the current level of development of AI\n                    technology.\n\nThe rumors continued regardless.\n\n## Let ten thousand agents bloom!\n\nAt 11:58 p.m. ET on Monday, September 7, Tristan BuckmasterProfessor of mathematics at NYU's\n                        Courant Institute. posted three results, in a\n                    joint collaboration with Levent Alpöge: finite-time\n                    blowup with smooth forcing for incompressible porous\n                    media, for Boussinesq, and for 3D incompressible Euler,\n                    with Lean formalizations for each.[@tristanbuckmaster](https://mathstodon.xyz/@tristanbuckmaster@mastodon.social/117233413735526010), September 7,\n                        2026, 11:58 p.m. ET. Alongside them, he posted a\n                    four-page statement,[statement.pdf](https://cims.nyu.edu/~tristanb/statement.pdf). explaining\n                    the following.\n\nOn September 3, with the rumors circulating, Buckmaster emailed a mathematician at OpenAI to correct the record. Over the following days, he was asked with increasing urgency to meet. On a call on Sunday, September 6, he was told that an internal OpenAI model had proved finite-time blowup for forced Navier-Stokes:\n\nI asked when the first prompt was sent by them.\n                    This question was not answered directly by OpenAI for\n                    some time. Eventually it was agreed that it had been\n                    sent in the past few days, after information about our\n                    work had reached OpenAI. I asked whether the model had\n                    been trained on, or had access to, our sessions in\n                    Codex, into which we had been putting all our drafts for\n                    the whole of this project. I was told the model did not\n                    look up user data. I asked again, about training, and I\n                    did not get an answer.OpenAI says its researchers \"did not\n                        see any of their work through any means until they\n                        released it publicly,\" but adds that \"while unlikely, we\n                        cannot rule out that de-identified data derived from\n                        their usage of our products helped improve our models.\"\n                        [On the Navier-Stokes Millennium\n                            Prize Problem](https://openai.com/index/navier-stokes-solution/).\n\nHe was then offered two proposals:\n\nTwo proposals were offered to me. The first was\n                    that we post our Euler result, and that OpenAI post its\n                    Navier-Stokes result the next day. The second was that,\n                    after posting Euler, I alone write a paper presenting\n                    the Navier-Stokes result, acknowledging that an internal\n                    OpenAI model had resolved it. Sebastien twice asserted\n                    that he wanted Levent removed from authorship, and said\n                    it would all be simple if only it were not the case\n                    that, and it was so annoying that, Levent works at\n                    Anthropic. [...] I declined both offers.\n\nOpenAI's announcement came thirteen hours later. Using\n                    ten thousand agents on an internal model, running for 88\n                    hours, they had produced a proof and a Lean formalization\n                    that a fluid can develop a singularity in finite time.\n                    In other words, a \"solution\" to Navier-Stokes.[@OpenAI](https://x.com/OpenAI/status/2097374646148481532), September 8,\n                        2026.\n                    Shortly after, Sébastien Bubeck posted their account of\n                    how the effort had started:[@SebastienBubeck](https://x.com/SebastienBubeck/status/2097379415747342689), September 8,\n                        2026, 1:39 p.m. ET.\n\nWe began working on the Millennium problems due\n                    to viral twitter rumors that Anthropic had resolved 2\n                    Millenium [sic] problems.\n\nHe said they saw none of Buckmaster and Alpöge's work until it was released, and that they had assumed Anthropic was behind the rumors because Alpöge's posts use Anthropic models. He made sure to emphasize that:\n\nalthough a group of people was involved in our\n                    efforts, we collectively had no research-level expertise\n                    in fluid dynamics and the Navier-Stokes problem, and\n                    therefore were unable to meaningfully contribute to the\n                    mathematical content.\n\nThe difference from May is shocking. The unit distance result had been posted alongside a companion piece written by nine outside domain experts, invited to review the proof and develop a more comprehensive understanding of it. The Navier-Stokes result, rather than the result of collaboration, came out of direct competition with two people already working on the problem. Instead, the focus was on producing and verifying the result as quickly as possible, in an \"unedifying\" rush similar to the one Tao observed with the bounded-gaps result. Little to no attention was paid to developing a broader, digestible mathematical account of why the construction works.\n\n⁂\n\nTao's post, ironically enough, turned out to be a self-fulfilling prophecy. By choosing Navier-Stokes as his example of how an AI-generated solution could inhibit the development of a field, he inadvertently fueled rumors about the problem being solved, which, in almost comically short order, contributed to the problem being solved, more or less in the way he described.\n\nAfter the announcement from OpenAI, he made a new post,\n                    reflecting on the state of mathematics:[@tao](https://mathstodon.xyz/@tao/117237320796901560), September 8, 2026, 4:32 p.m.\n                        ET.\n\n We have now seen that even the rumor of someone\n                    working on a problem can trigger a massive amount of\n                    AI-powered effort to flatten it before the original\n                    research project has time to reach its full potential.\n                    The incentives may now be pointing in the direction of\n                    no longer sharing any promising research directions with\n                    the broader community, which would reverse centuries of\n                    traditions of open science and do serious long-term\n                    damage to the future of the field.\n\nIf this is all a bit too depressing, I recommend reading [The 92-Year-Old Mathematician and\n                        the Teenage Apprentice](https://www.nytimes.com/2026/09/06/science/92-year-old-mathematician-apprentice.html), published a few days ago,\n                    about Joan Birman and the fifteen-year-old she took on\n                    as a mentee at the age of 92.", "url": "https://wpnews.pro/news/to-serve-man-ai-math-and-navier-stokes", "canonical_source": "https://ml5885.github.io/writing/navier-stokes.html", "published_at": "2026-09-09 16:28:57+00:00", "updated_at": "2026-09-09 16:50:24.998439+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["OpenAI", "Terence Tao"], "alternates": {"html": "https://wpnews.pro/news/to-serve-man-ai-math-and-navier-stokes", "markdown": "https://wpnews.pro/news/to-serve-man-ai-math-and-navier-stokes.md", "text": "https://wpnews.pro/news/to-serve-man-ai-math-and-navier-stokes.txt", "jsonld": "https://wpnews.pro/news/to-serve-man-ai-math-and-navier-stokes.jsonld"}}