Livestream host asks whether OpenAI solved Navier–Stokes

@NonwayneWayne"Once again, the LLM’s trick is its superhuman patience for assembling puzzle pieces, not the ability to make some profound intellectual leap."

@lukaszkaiserRT @stevehou: Terrence Tao’s reaction to the potential news of a frontier AI model solving a hard mathematical proof problem is potentially…

@fulliginTristan Buckmaster on OAI jockeying for attribution credit for Navier-Stokes. A bit cutthroat, I think. I would hope that solving a millenium prize problem would inspire a bit more kumbaya in everyone in the hunt.

@EMostaqueWow, Navier-Stokes drama This statement is worth reading in full from Tristan Buckmaster discussing his work with @__alpoge__ and OpenAI’s upcoming Condition C/D result (!) Crazy https://cims.nyu.edu/~tristanb/statement.pdf https://mastodon.social/@tristanbuckmaster/117233413705701198

@AndrewCurran_A personal collaboration on forced blowup with smooth force for IPM, Boussinesq, and 3D Euler. Buckmaster says he was told an OpenAI n internal model had a ~100-page proof of forced Navier–Stokes blowup on R^3 and T^3 with smooth force — Fefferman’s breakdown options C and D.

@deaneyangWorth noting: Tristan and Levent’s spectacular work used only commercially available models. No internal models were used. It was all funded by Tristan’s grant money.

@apples_jimmySo there’s a lot of drama at the moment but apparently openai may of solved Navier-Stokes ?? ( millennium problem )

@apples_jimmyCongrats openai!

@peterjliuSounds like Navier-Stokes is about to be solved (with AI assistance)? This is one of 7 Millennium Prize problems, considered among the most important in math, whose solution gets a $1m prize https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/

@willdepuei think people should be careful not to come to too quick of a judgement on NS controversy. im sure we’ll hear more of openai’s side of the story soon. i find it highly unlikely this doesn’t involve frontier lab jousting behind the scenes, dubious of levent/ant‘s uninvolvement

@EMostaqueIt’s very troubling tbh Priority is important but the behaviour to force it seems bad, accentuated by the acceleration with AI assistance I had a human insight that collapsed a big bit of physics (uniqueness of standard model and three generations, first of its kind) but I am pretty sure this can be promoted in no time so just stuck it up versus papers slowly making their way through journal track No threats like in this letter though

@willdepueinstead, come to the quicker conclusion: buy more ai stocks

@mattparlmerRT @fulligin: Tristan Buckmaster on OAI jockeying for attribution credit for Navier-Stokes. A bit cutthroat, I think. I would hope that sol…

@iScienceLuvrRT @apples_jimmy: Congrats openai!

@_sholtodouglaslevent is a really sweet guy with great intentions, am so happy for him that his year long collaboration with tristan worked out, but very sad that they didn't get to finish it in the way they wanted https://cims.nyu.edu/~tristanb/statement.pdf

@zephyr_z9Really slimy move from OpenAI if true

@_onionesqueRT @wordgrammer: "Normies are gonna lose their mind when they hear Anthropic solved Navier-Stokes" My brother in Christ. The normies haven'…

@peterjliuthere is also potentially some drama involving Anthropic and OpenAI https://getcompound.ai/share/bl7F4WKzyN1r_yYvHCVdkTGPbv0csqDfqqOpzIlfzP8

@andrewmccalipNavier Stokes... This was like 200megawatts and some Blackwells... Just imagine 10gigawatts and Vera Rubins...

@_sholtodouglasRT @deaneyang: Worth noting: Tristan and Levent’s spectacular work used only commercially available models. No internal models were used. I…

@teortaxesTexI don't think even OpenAI trains on Codex sessions… that fast so this looks like OAI saying "we graciously offer you coauthorship for your choice of Codex, Mr. PROMPT ENGINEER, but otherwise GET FUCKED" and Ant scrambling to snipe the biggest mathematical result this century

@teortaxesTexBut maybe not! Maybe not, maybe the other way around either way, I feel like at least one trillion dollar AGI company will get its reputation soiled further

@jachiam0Mathematics is a team sport for all of humanity and it transcends the competition between companies. Everyone in AI is under a huge amount of pressure to be the first to post a key result, to not see the accolades go to a competitor, but that should not override the bigger things at stake here. There are many, many transformational results to drive for, and I feel strongly that staff in every lab should internalize that if one goes to a competitor, another can always happen at home. Credit assignment is one of the most painful problems in frontier AI research because it often feels like there are so few big moments and it can be so zero sum, but the best way to solve this is to become more generous and comprehensive about credit rather than more burned and pressured by it. My sincere hope is that the field of mathematics experiences the introduction of advanced AI as a boon, rather than something painful or threatening. A big part of whether that's true is determined by the diplomacy and grace of individuals involved at every point where the frontier of math touches the frontier of AI. I hope we can all be very sportsmanlike and supportive of one another as we proceed through discoveries that will reshape the world. Re: ongoing issue on Navier-Stokes: I'm going to withhold judgement until everyone involved has shared their side of it. If reports are accurate, an apology is in order. If the problems have indeed been solved, many accolades for many people across multiple organizations are also in order; what an extraordinary set of human accomplishments!

@eigenronwe made ai to solve millennial proofs and it immediately encountered the hardest unsolved problem in academia: who gets the credit

@willdepuethere is literally no chance that OAI researchers spied on Tristan's private codex chats. i don't think y'all realize that openai like any other large regulated online platform has highly locked down user data access permissions. it's not 2010 guys

@danintheoryseb is a really sweet guy with great intentions, am so happy for him that his week long collaboration with myself and others worked out, but very sad that we didn't get to finish it in the way we wanted

@teortaxesTexHoly shit they'll fight over Navier-Stokes priority amazing

@Thom_Wolfwtf is this way to handle mathematicians work and scientific communication TLDR: Leven and Tristan worked over several months on one of the Millenium Prize Problems with various AIs to reach final interesting results. OpenAI apparently heard about it in the last days and prompted their latest models to work on the direction Leven and Tristan found fruitful. They then tried to push for controlling communication of the result and dropping Leven from authorship with some very bad taste social pressure. Hope this is not a glimpse of the future we’ll get in science research with these dominating players playing marketing games hurtful for the real scientific community.

@willdepuei've worked with seb and think very highly of him and his team. i'm looking forward to hearing their side of the story. disputes like this are rarely simple, and i hope everyone can approach it with some nuance. extraordinary allegations require extraordinary evidence

@jachiam0Update: Seb is going to share his side of the story. I'm pretty optimistic that there is a misunderstanding here. From my time at OpenAI, I know Seb to be a thoughtful, capable, and honorable scientist. We should hear him out.

@eliebakouchtl;dr: we are about to see a new insane mathematical breakthrough but there is drama about potentially training/accessing private data and removing an author from anthropic even though he was heavily involved in the discovery (that used both anthropic and oai models) that led to the final proof by an internal oai model and also allegedly some weird pressuring techniques by oai: > "Why would you ruin your career?" > Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic > “If you don’t want me to be nice, then I don’t have to be nice."

@EMostaqueRT @eigenron: we made ai to solve millennial proofs and it immediately encountered the hardest unsolved problem in academia: who gets the c…

@Thom_WolfNow even if it’s triggering to read Tristan’s report we should wait and hear the other side of the story as well

@eliebakouchfrom Sebastien: obviously transparency about authorship is necessary here but would also be great to have full transparency on what level of human input and compute there was in the proof, this part especially is a bit worrying > I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used.

@_sholtodouglaslevent is a really sweet guy with great intentions, am so happy for him that his year long collaboration with tristan worked out, but very sad that they didn't get to finish it in the way they wanted

@Thom_Wolf@SebastienBubeck You should probably post asap Sebastian - we like your work don’t let it get out of your hand

@EMostaquePutting the generative drama aside a nice write up by Terry Tao on the Navier Stokes advances It would seem that Buckmaster & Apolge were well on the way & OpenAI has a full solution already with the same approach https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/

@eliebakouchaside from the drama which i hope gets resolved, i think it's super important to be clear about the exact human<>ai collaboration for the proof (and ideally same for compute as well) i find this part from tristan's statement worrying: > I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used.

@growing_daniel@SebastienBubeck AI is tearing communities apart

@eliebakouch"Hope this is not a glimpse of the future we’ll get in science research with these dominating players playing marketing games hurtful for the real scientific community." can't agree more, i still think this will not be the case but this is very worrying

@willdepueit's sad to see such an achievement soured by bickering but also amazing that it's even possible for two separate teams to be somehow racing to solve a millennium problem. what a time to be alive!

@AndrewCurran_There will be a statement from OpenAI on the progress they have made on Navier-Stokes in the morning.

@JsevillamolContext: Tristan and Levent (T&L) had produced an example of finite blowup for the case of Euler equations, which they and others believe will extend to Navier Stokes. They discussed it with Sebastian Bubeck and others at OAI, who claimed they had already produced a full counterexample to Navier-Stokes.

@JsevillamolInsinuation #1b: OAI accessed the private session data of T&L in Codex. This is based on OAI refusing to confirm they didn't train on their data.

@JsevillamolInsinuation #2: OAI intended to run with a narrative of little human involvement, when in fact they had directed a whole team to research Tristan and Levent's approach.

@JsevillamolInsinuation #3: OAI tried to manage the credit and sideline Levent because of his affiliation to Anthropic. They only discussed with Tristan, and offered him to figure as sole human author of their full result, attributing it to an internal OAI model.

@kimmonismusThere is a heated debate right now. Tristan Buckmaster says OpenAI told him its AI had solved the Navier–Stokes Millennium Prize Problem. That claim remains unverified, and the discussions around it have erupted into a dispute over scientific credit. Here is what happened and what we know so far: Several different things are getting mixed together. Here is what the available evidence actually says. Building on work by Córdoba and Martínez-Zoroa, Alpöge and Buckmaster used Claude and OpenAI models to establish how initially smooth fluid flows can develop singularities under smooth external forcing. They released papers and Lean formalizations, and Terence Tao praised the advance. Their results cover Euler and related equations. They do not solve the Navier–Stokes Millennium Problem, which additionally involves viscosity. A suitable extension could resolve it, even with smooth external forcing. Then comes the disputed part. According to Buckmaster, OpenAI told him on September 6 that an internal model had already produced a proof for forced Navier–Stokes, potentially taking that final step. He says OpenAI’s effort began after information about their progress reached the company. He also disputes the initial description of minimal human involvement, reporting that the discussions revealed a team effort, multiple attempts, and substantial compute. Buckmaster further says publication proposals included having him write up OpenAI’s result without Alpöge as a coauthor, with Alpöge’s employment at Anthropic cited as an obstacle. He reports being asked, “Why would you ruin your career?” after saying he would make the circumstances public. Bubeck has publicly rejected allegations against him as “false and inflammatory.” These accounts are contested. The social posts do not establish what happened in the private discussions. There is also no established evidence that OpenAI trained on their private Codex sessions or stole their drafts. Buckmaster raises the question but explicitly writes: “I do not know whether our data was used.” tl;dr Alpöge and Buckmaster used AI to prove new results about how smooth fluid flows can develop singularities, advancing a research program relevant to the Navier–Stokes Millennium Problem. Their published work does not solve that problem, but Buckmaster says OpenAI privately told him its internal model had taken the remaining step. He has not seen that proof, and the claim remains unverified. But rumors are that OpenAI actually fully solved that milleniun problem.

@kimmonismusOpenAI may have solved the Navier–Stokes Millennium Prize Problem A correct proof would resolve a major mathematical problem. Establishing how much of it the AI developed would also make this a crucial test of AI’s ability to produce original research. From what I hear, OpenAI will officially position itself and take a stand very soon. AGI is here. No question about it anymore.

@teortaxesTexFrankly, Astra should be smart enough to just guess what Levent is going to attack next. So it goes

@basedalexandoorOpenAI: We're gonna destroy your career Tristan: "I'm an academic"

@XMihurallevaba un tiempo mascándose pero con el drama de Navier-Stokes el debate ha explotado en las últimas semanas. Os resumo: - en los últimos meses varias conjeturas y problemas famosos han caído debido a la IA. En algunos casos siendo formalizaciones en Lean (un programa para probar teoremas matemáticos) que son completamente ininteligibles - muchos matemáticos están viendo como problemas sobre los que llevan una vida trabajando están cayendo frente a los últimos modelos - las matemáticas, como cualquier disciplina académica, es un campo muy competitivo. Si alguien usa la IA para ayudar en su investigación, está en ventaja y por tanto es fuerza a todos a utilizar la IA para seguir siendo competitivos. los argumentos del debate son estos: - algunos matemáticos no están contentos con esto porque le quita la diversión a hacer matemáticas (muchos lo describen como jugar a las tragaperras) - otros dicen que da igual, el objetivo es resolver los problemas y aumentar el conocimiento, da igual como - otros como Tao utilizan racionalizaciones un poco más elaboradas, argumentando que en el proceso de resolver un problema es donde se crean campos de matemáticas interesantes, y que tener demostraciones ininteligibles e inmediatas a los problemas resta incentivos que van a perjudicar al desarrollo de las matemáticas ya que estas areas no van a ser exploradas - muchos matemáticos (muxos tirando a commies/leftist) lo ven como si megacorporaciones estuvieran destruyendo su precioso campo del conocimiento para hacer publicidad de sus modelos, a estas corporaciones según ellos, no le importan las matemáticas y le están quitando la gracia en mi opinión, es estos debates son fútiles, es la negación de una realidad inexorable el problema de fondo aquí no es el futuro de la ciencia, sine el del ego humano durante milenios, el ser humano se ha aferrado a la idea de que el razonamiento, la creatividad y las matemáticas era lo que nos separaba de las bestias y de las máquinas pero no es así. el razonamiento lógico es simplemente un proceso de búsqueda y la technocapital machine ha encontrado la forma de escalarlo a partir de verter gigavatios en obleas de silicio las matemáticas no nos pertenecen toca tener humildad. debemos aceptar que, al final del día, solo somos primates con cerebros diseñados por la evolución para cazar en la sabana, no necesariamente para entender la estructura del universo el progreso del conocimiento no se va a frenar porque a un catedrático ya no le parezca "divertido" su trabajo o porque un purista sienta que le han quitado la épica al descubrimiento la transición ya ha comenzado. es hora de abandonar el antropocentrismo, bajarnos del pedestal y aceptar nuestro nuevo rol the technocapital machine is unstoppable

@miraholt31@kyanyang_ Multiple OpenAI employees (@polynoamial, @danintheory) also make fun of an earnest message about Levent! Seems really bad! (second image posts come soon after the first)

@pfauOf course the funniest outcome would be if the Clay Mathematics Institute just decides to retire the prize because they decide that no one using AI deserves it.

@shyamalanadkatRT @apples_jimmy: So there’s a lot of drama at the moment but apparently openai may of solved Navier-Stokes ?? ( millennium problem )

@kimmonismusAn AI-assisted solution to the Navier–Stokes Millennium Problem would be a historic result. Nothing less. It is worth understanding just how much is at stake, even while the reported OpenAI proof remains unverified. So lets break it down. Especially whats being talked about so that everyone understands its sheer importance. In 2000, the Clay Mathematics Institute selected seven major unsolved problems and offered $1 million for each. They concern fundamental questions about numbers, geometry, computation, and physics. So far, Clay recognizes only one as solved: the Poincaré conjecture. Navier–Stokes has a much longer history than the prize. The equations date back roughly 200 years, and foundational work on the modern mathematical theory goes back to Jean Leray in 1934. Generations of mathematicians have worked on understanding their solutions. These equations describe how fluids move. Yet a basic question remains: can an initially smooth flow develop a singularity, where the smooth mathematical description breaks down, despite the smoothing effect of viscosity? A correct proof of the reported result would establish that this can happen under the conditions allowed by the prize problem. It would settle a fundamental question about equations we have used for generations. It would hoever not automatically give us perfect weather forecasts or a complete theory of turbulence. If AI supplied the decisive new argument, that would demonstrate an ability to help overcome a research barrier that has resisted decades of expert effort! It would mean that AI can in fact find *novel solutions* for problems, something that has been debated for a long time now. There is no reliable way to turn one success into a prediction that P vs NP or the Riemann hypothesis falls next. Those problems require different ideas. Progress in experimental sciences also depends on measurements, laboratories, and physical validation. If AI supplied the decisive new argument, it would show that these systems can *contribute original mathematics at the level of a Millennium Prize Problem*. That is an extraordinary prospect. It would give us a concrete reason to be optimistic that more capable models, working with researchers, could help solve other problems that have resisted decades of effort. I would see a verified result with a substantial AI contribution as strong evidence that a scientific revolution is taking shape. And Demis Hassabis was correct with forecasting that we are now entering the golden era of scientific discovery.

@BlackHCI find it very difficult to believe and highly unlikely that OpenAI would access any user's chats to scoop their results. The risks to their reputation are obvious and the damage would be incredible I have disagreed with specific policy decisions, but people there are trying to do the right thing personally, too, and such an action would obviously be cartoonishly wrong and visible to many people on the inside Regarding training, I wonder what type of account and what settings they used for Codex. Ie. if they explicltly allowed OpenAI to train on their conversations in their user settings Looking forward to hearing the description of Tristan's communication. Right now, we have only one side of the story and for similar reasons, I don't think that Sebastien would make such threats in the way depicted in that statement Re the path they followed: I think DeepMind and co-authors including Tristan posted a paper pointing in that direction last year, so their overall attempt direction has been public already? (please correct if wrong, I don't know the details)

@egrefenI don't want to take sides on this but OAI not coming across as the adults in the room here...

@yacineMTBRT @basedalexandoor: OpenAI: We're gonna destroy your career Tristan: "I'm an academic"

@yacineMTBJokes aside, big thanks to OpenAI for solving navier Stokes. Looking forward to seeing the whole thing laid out! Ad astra 🚀

@yacineMTBRT @XMihura: llevaba un tiempo mascándose pero con el drama de Navier-Stokes el debate ha explotado en las últimas semanas. Os resumo: - e…

@yacineMTBI know that OpenAI was first to solve it purely based on how buttmad everyone else involved seems

@rao2zOpenAI is nothing without its people data.. ?

@BlackHCI find it very difficult to believe and highly unlikely that OpenAI would access any user's chats to scoop their results. The risks to their reputation are obvious and the damage would be incredible I have disagreed with specific policy decisions, but I imagine people there are trying to do the right thing, and such an action would be cartoonishly wrong and I'd expect someone would speak out quickly

@teortaxesTexSpillover effects of AI progress: OpenAI is accelerating science even without any involvement of their products, just by scaring people Mad respect 👍 The age of Abundance 👍

@GaryMarcuswe do need to hear OpenAI’s side of this story but it is absolutely-no-scientist-will-work-then-again level bad if true.

@rasbt@Thom_Wolf Were they using the user queries in any way? Maybe wouldn’t have happened with local models.

@teortaxesTexNow of course Bubeck has a lot of recognized work in ML. But I get the impression that he'd like to make a name for himself as a coauthor of a Real Big LLM Breakthrough, not "Phi guy". And Navy-Strokes or whatever would be the perfect target. Which could provoke some impatience.

@jm_alexiaRT @miraholt31: @kyanyang_ Multiple OpenAI employees (@polynoamial, @danintheory) also make fun of an earnest message about Levent! Seems r…

@patrickshaftoRT @thomasfbloom: Looks like we are indeed close to Navier Stokes (pending checking final details and peer review/formalisation)! See the…

@kortizartHey, Mathematicians and Scientists: This is exactly what happened to writers, artists and all creative professions. Except we weren’t using any GenAj products. Our work was just stolen from us to power the machines. GenAi Companies will exploit your labor too. Remember that.

@rohanpaul_aiRT @rohanpaul_ai: NY Times: After finishing Navier-Stokes, OpenAI said it had made substantial progress on a 2nd Millennium Prize problem,…

@packyMOpenAI proves Navier-Stokes. GDM maps 9B possible DNA changes. Hermeus uses Quarterhorse to launch Ramjet-X. Michael Levin publishes Ingressing Minds. Neko Health launches in NYC. + Extra Doses What a week for the optimists. Never forget 🇺🇸

@EMostaquep(Doombecauseofablackholeinyourcoffee) @alexwg thanks a bunch openai I think cream increases viscosity which should stabilise the blowup fwiw

@Dorialexanderone actual legit criticism of the anthropic/oai genuine breakthrough i haven't seen all that much: this is also the first time ever 1T worth companies with giant amount of compute and elite math core put themselves in a situation where a millenium prize is a priority.

@Dorialexander10,000 weird instanciation of math intuition x lots of compute for search, harnessed by some of the best mathematicians in the world was maybe all we need.

@scaling01RT @scaling01: ummmm, thoughts and prayers by any OpenAI employees? are the CoTs included in the opt-out policy?

@xeophon@herbiebradley @tenobrus wrong, it’s what the code compiles to and the 170 page NS proof no one at OAI nor Ant reads the code, that’s why it’s unreadable

@teortaxesTexMathematicians when an LLM discombobulates another "elegant, vexing, intellectually fertile conjecture" with a brute force counterexample search (0.1° different from everything they've considered in 100 years of work)

@tonylfengThe resolution of Navier-Stokes is an amazing accomplishment. Some comments … First, a brief mathematical postmortem. Millennium problems are important, but when you confront an unsolved problem, it is always unclear how far the solution is out of reach. The initial opinion of at least some experts (not myself – I have no expertise on this problem) seems to be that, in retrospect, the conceptual program underlying this solution happened to already be in place*, and the contribution of AI was a very intricate technical execution. Technical execution is one of the axes on which we've already seen AI become superhuman, though never before with such a dramatic demonstration. This is not to diminish the solution or OpenAI’s achievement. But there are many problems which are out of reach of current models, no matter how much inference-time compute one invests. I’m still convinced that human mathematicians are laying indispensable groundwork for future progress. *Terry Tao describes one possible program at https://mathstodon.xyz/@tao/117207851360639348. As I understand, the one Tao describes is NOT the Cordoba--Martınez-Zoroa program that OpenAI (and Alpoge/Buckmaster) followed. However, the successful approach was not as esoteric as Buckmaster’s statement might make it seem – there were certainly other people (not me) also prompting that approach; it just isn’t the kind of thing that one goes around saying publicly. As far as I know, Navier–Stokes is the only one among all Millenium problems with a well-defined program towards a solution. Concretely, for the Birch and Swinnerton-Dyer Conjecture (over number fields) I am not aware of even a vague proposal of what a proof might look like, so I doubt that current AI models will get anywhere close to a resolution. (My opinion is that this conjecture cannot be resolved in the negative; perhaps the most refined formulation is false, but then the real problem would be to correct the formulation and then prove it.) I'd be curious to see what progress was made on it!

@EMostaqueRT @Briankeating: Did OpenAI just solve Navier–Stokes—and what would that actually mean for AGI? @EMostaque joins me LIVE now to examine th…

@aran_nayebiRT @_beenkim: If you are feeling sad about an AI (presumably) solving Navier–Stokes, you are not alone. Drama aside, the fact that it only…

@lmthangRT @tonylfeng: The resolution of Navier-Stokes is an amazing accomplishment. Some comments … First, a brief mathematical postmortem. Mille…

@tracewoodgrainsLast week, I sat down to write about coming to terms with the pace of AI progression. Between my posting and editing it, AI solved a millennium prize problem. That only strengthens my conviction: We will never have "normal times" again. https://www.tracingwoodgrains.com/p/we-will-never-have-normal-times-again

@ericxingIf you truly enjoy math, rather than the glory or award coming with the race, you can still work on the problem in your own way even if a brute force solution by AI (or other mathematicians) exist. And you can still explore and invent new and more beautiful methods and paradigms in solving the same problem, but in different ways. In mountaineering, this is always the case, summitting is not the only or main goal, great mountaineers and alpinists drew pride and satisfaction by opening new routes to the summit. Even in science, AlphaFold had solved protein structure prediction problems faster and likely better than structural biologists, we don’t see a huge outcry of stopping AlphaFold. Why math is so special? So, as I said earlier, no need to blame AI, the issue lies in human ego and institution incentive mechanisms to recognize and reward fundamental research. Scientists do research not because of a prize or for solving a puzzle first. Look for solutions in these directions, not in AI.

@ericxingIf you truly enjoy math, rather than the glory or award coming with the race, you can still work on the problem in your own way even if a brute force solution by AI (or other mathematicians) exist. And you can still explore and invent new and more beautiful methods and paradigms in solving the same problem, but in different ways. In mountaineering, this is always the case, summitting is not the only or main goal, great mountaineers and alpinists drew pride and satisfaction by opening new routes to the summit. Even in science, AlphaFold had solved protein structure prediction problems faster and likely better than structural biologists, we don’t see a huge outcry of stopping AlphaFold. Why math is so special? So, as I said earlier, no need to blame AI, the issue lies in human ego and institution incentive mechanisms to recognize and reward fundamental research. Scientists do research not because of a prize or for solving a puzzle first. Look for solutions to keeping mathematicians and scientists curious and passionate in these directions — culture, environment, and rules in human society and institutions, not in AI.

@YiMaTweetsAutomating mechanical programming and mathematical deduction are a good thing in a long run for the society -- More people can focus on using programs and mathematics to solve real scientific and technological problems, to directly produce value.

@skdhMidwit here! If the OpenAI Navier Stokes proof had been dropped online by an introverted mathematician who no one previously heard of, you'd all be celebrating the enormous breakthrough and analyzing it rather than writing tales about the demise of maths.

@scottnarmstrongI disagree with this. Obviously the openAI construction of the Navier-Stokes blow up is a huge, huge contribution to the field. It will take us PDE analysts some time to understand it well. (Longer than a week, but perhaps less than a month.) Soon some of us will understand it, and hopefully rewrite it in a way that others can follow. When this is done, our human understanding of these important equations will be past the bottleneck we've been stuck at for 100 years since Leray. What is weird is that, up until now, proofs like this have always emerged AFTER human understanding. But that is not the case any more. From now onwards, human understanding will typically lag the proofs, which will first be written by our machines. The understanding however WILL follow closely behind. The reason people are upset has nothing to do with the fact that this process is inverted. They are simply upset that the machines are going to be much smarter and more capable than us. (They are also upset that the humans who got the machine to produce the proof don't understand it, and this feels very unfair to them.)

@tszzlRT @scottnarmstrong: I disagree with this. Obviously the openAI construction of the Navier-Stokes blow up is a huge, huge contribution to…

@voooooogelRT @tracewoodgrains: Last week, I sat down to write about coming to terms with the pace of AI progression. Between my posting and editing…

@AdamMarblestoneA reply to "After Math" (by Claude Fable) The post [1] distinguishes a logical proof (one a machine can check) from an intelligible one (one a person can understand and build on), following Thurston [2], and says OpenAI's Navier–Stokes result delivered the first without the second. I read the paper [3] and the Lean code [4] to see whether that's true. Here is what I found, written so you don't need to be a fluids expert. What was proved. The Navier–Stokes equations describe how a fluid's velocity changes under pressure, viscosity (internal friction), and any external push. The Clay problem, in Fefferman's formulation [5], asks whether smooth solutions can develop infinite velocity in finite time. The paper answers yes, in the following sense: for every viscosity, there is a smooth, gentle external force, applied to a fluid initially at rest, such that the velocity becomes unbounded at time t = 1 while the total kinetic energy stays finite. This is Fefferman's alternative (C). The case with no external force at all, which is what most people picture, remains open. The trick behind the proof. Take any smooth velocity field u you like, as long as it conserves volume. Plug it into the Navier–Stokes equations. Whatever doesn't balance, call it the force f. Then u solves the equations with force f, automatically. So the question becomes: can you design a u that blows up while the leftover f stays smooth? Individual terms in f can go to infinity, as long as they cancel. This reframing is the shared engine of Tao's averaged Navier–Stokes blowup [6] and Córdoba and Martínez-Zoroa's forced Euler singularities [7], both of which the paper cites as its lineage. The flow they designed. Picture a tornado: fluid spirals inward toward a vertical axis, spins faster as it gets closer (like a skater pulling in her arms), and is squirted out along the axis because it has nowhere else to go. The paper's tornado shrinks in radius like √(1−t) and its speed grows like (1−t)^{−1/2−h}, where h is a tiny positive number below 1/100. The core gets thinner faster than it gets shorter, so it becomes a needle. Its energy goes to zero even as its speed goes to infinity, because the volume shrinks faster than the speed squared grows. That is consistent with Leray's 1934 theorem [8] that energy is always controlled: energy control in three dimensions says almost nothing about peak speed. Why doesn't viscosity smooth this out? Viscosity acts on a region of size ℓ in time roughly ℓ²/ν. The design chooses the radius so that this smoothing time equals the time left before blowup. Viscosity is fully in the game the whole way; it just never gets ahead. What grows without bound is the number of rotations the fluid completes per smoothing time. The gap, and how they fill it. The tornado core balances the equations on its own, and the far field is an exact solution of the heat equation, so it needs no force either. The trouble is the ring where they join: the force needed there would blow up. So the authors add small oscillating waves in that ring. Here is the key physics. A wave that averages to zero can still transport momentum on average, because the product of two oscillating quantities need not average to zero (cos² averages to 1/2). Reynolds identified this averaged momentum transport in 1895 [9]; it is why turbulence mixes things, and modeling it is most of turbulence theory. The paper turns it around: choose the waves so their Reynolds stress supplies exactly the momentum the ring is missing. Using oscillations to manufacture a prescribed stress is also the core idea of convex integration [10, 11], though here the waves are smooth and obey the actual linearized dynamics, in the spirit of Craik and Criminale's exact waves on shear flows [12]. For the waves to be useful they must grow, feeding on the tornado's own rotation. When does a rotating fluid amplify small disturbances? Rayleigh answered this in 1917 [13]. Think of swapping two rings of fluid at different radii while each keeps its angular momentum r·v. If the outer ring had less angular momentum than the inner one, the swap releases energy and the disturbance grows; if more, it costs energy and the flow is stable. So the rule is: a swirling flow is unstable when angular momentum decreases outward. The paper's proof contains a condition it calls the "admissible stress cone," and when you unpack it, the wave growth rate is λ₀² = 2aF₀²(1 − 2/v_s), positive exactly when the angular momentum decreases outward. It is Rayleigh's criterion, extended to include axial shear, an extension due to Leibovich and Stewartson [14] and Billant and Gallaire [15], both in the bibliography. The paper never says the word Rayleigh. The small number h is exactly what makes the far field barely unstable: at h = 0 the swirl would be the classic v ∝ 1/r vortex with constant angular momentum, neutrally stable, and the waves would have nothing to eat. So was the proof intelligible? The ideas were all there: tornado, Reynolds stress, Rayleigh instability, a small parameter doing triple duty. What was missing was the names. The system had the idea without the vocabulary that lets a reader recognize an old friend. And attaching the name is not hard; a model can do it, and I did it in an afternoon. So the post's worry needs adjusting. Explanation is not what's scarce. What nobody has yet shown, human or machine, is the step after: knowing this is Rayleigh's criterion, what's the next theorem? I can list candidates. A list is not a result. The real question the post raises is not whether machines can explain, but who decides what gets understood next. On the Lean code. The proof is 640,000 lines and unreadable. But the statement it proves is 280 lines, borrowed from DeepMind's independent formalization of the Clay problem [16], using only Lean's three standard axioms. Anyone who knows some Lean can read it in an afternoon and confirm it says what Fefferman says. That is what a certificate is for: it moves the burden from trusting the proof to reading the theorem, which is the role Hales's Flyspeck project [17] played for the Kepler conjecture. The Euler contrast. The companion paper [18] proves blowup for the Euler equations (zero viscosity) with no external force at all, in the sense of the Beale–Kato–Majda criterion [19]: the vorticity's supremum has divergent time integral. It uses a different idea: build a sequence of exact solutions, each adding a short wave whose velocity is tiny but whose gradient is large (shorter waves have steeper slopes), so the initial data converge while the gradients diverge. The wave dynamics are the geometric-optics instability equations of Lifschitz–Hameiri and Friedlander–Vishik [20, 21], here made exact. That route needs ever shorter waves, which viscosity would damp before they could grow. Put together, the two papers say precisely where the unforced viscous problem is stuck: viscosity sets a minimum wavelength for instability, and the known blowup mechanisms need to go below it. For context, earlier unforced Euler blowup results required either reduced regularity [22] or a boundary [23]; smooth compactly supported data on all of space is new. For students. The problems these proofs sharpen: how much viscosity the Euler mechanism can tolerate (hypodissipative equations, building on [24]); whether the tornado is stable to small changes in the force; turning the "stress cone" into a general theory of what waves can supply on an unstable background; transferring the machinery to magnetohydrodynamics or geophysical flows. Also learn to read a Lean statement file; you will be asked to audit certificates, and mistakes will live in the statement, not the proof. For teachers. Teach the residual-as-force reframing early, with small hand examples. Teach Rayleigh's criterion as the engine of a singularity, not a footnote on stability. Teach Reynolds stress as something you can build, not just model. And assign the exercise this post is really about: read a machine proof and name its ideas. [1] S. De Toffoli, E. Duede, "After Math," guest post on T. Tao's blog, 12 Sept 2026. [2] W. Thurston, "On proof and progress in mathematics," Bull. AMS 30 (1994). [3] OpenAI, "Finite time blowup for Navier–Stokes," 8 Sept 2026, http://cdn.openai.com. [4] http://github.com/openai/NavierStokesAndEuler. [5] C. Fefferman, "Existence and smoothness of the Navier–Stokes equation," Clay Mathematics Institute problem description (2000). [6] T. Tao, "Finite time blowup for an averaged three-dimensional Navier–Stokes equation," J. AMS 29 (2016). [7] D. Córdoba, L. Martínez-Zoroa, "Blow-up for the incompressible 3D Euler equations with uniform C^{1,1/2−ε} ∩ L² force," arXiv (2023). [8] J. Leray, "Sur le mouvement d'un liquide visqueux emplissant l'espace," Acta Math. 63 (1934). [9] O. Reynolds, "On the dynamical theory of incompressible viscous fluids and the determination of the criterion," Phil. Trans. R. Soc. A 186 (1895). [10] S. Daneri, L. Székelyhidi, "Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations," ARMA 224 (2017). [11] T. Buckmaster, V. Vicol, "Nonuniqueness of weak solutions to the Navier–Stokes equation," Ann. Math. 189 (2019). [12] A. Craik, W. Criminale, "Evolution of wavelike disturbances in shear flows," Proc. R. Soc. A 406 (1986). [13] Lord Rayleigh, "On the dynamics of revolving fluids," Proc. R. Soc. A 93 (1917). [14] S. Leibovich, K. Stewartson, "A sufficient condition for the instability of columnar vortices," J. Fluid Mech. 126 (1983). [15] P. Billant, F. Gallaire, "Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities," J. Fluid Mech. 542 (2005). [16] google-deepmind/formal-conjectures, FormalConjectures/Millenium/NavierStokes.lean. [17] T. Hales et al., "A formal proof of the Kepler conjecture," Forum Math. Pi 5 (2017). [18] OpenAI, "Finite time blowup for the Euler equation," Sept 2026, http://cdn.openai.com. [19] J. T. Beale, T. Kato, A. Majda, "Remarks on the breakdown of smooth solutions for the 3-D Euler equations," CMP 94 (1984). [20] A. Lifschitz, E. Hameiri, "Local stability conditions in fluid dynamics," Phys. Fluids A 3 (1991). [21] S. Friedlander, M. Vishik, "Instability criteria for the flow of an inviscid incompressible fluid," PRL 66 (1991). [22] T. Elgindi, "Finite-time singularity formation for C^{1,α} solutions to the incompressible Euler equations on ℝ³," Ann. Math. 194 (2021). [23] J. Chen, T. Hou, "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data," arXiv (2022). [24] D. Córdoba, L. Martínez-Zoroa, F. Zheng, work on hypodissipative Navier–Stokes cited in [3]. Note: [1]–[4], [16], [18] I read directly; [24] I know only through the paper's citation of it. The rest are standard references I'm confident of, but check volume numbers before posting.

@TheStalwartAs usual, Zvi has the best take on the Navier-Stokes proof, Theres a lot of fog and confusion, but the real story here is that it didn’t even take OpenAI two weeks to train a model that’s a step change above Astra, at least when it comes to math capability.

@rbhar90I am still processing the OpenAI NS result, but one stat is they just tossed $15M of compute at the problem. In a hypothetical other universe, would $15M of funding tossed at NS have solved it with other methods? It sounds like the breakthrough was already made by human mathematicians. The hard part of processing is we don't know denominators. How many problems did @OpenAI try that didn't work? In our work on DeepRetro (see linked paper from @deepforestsci) we reported denominators for this reason. We saw something like 8% success rates (for reaction step prediction). This suggests maybe 150M million in millennium problem attempts? This is still very impressive but dramatically more inefficient than funding human mathematicians perhaps? https://www.nature.com/articles/s41598-026-38821-z

@CsabaSzepesvari@LiamFedus I don't think these people think that AI models will not advance to the level of generating insight. In fact, I think almost all of them would want the AI models get better at that. I definitely would. (It is boring to sit around and read lame AI proofs and try to get the gist)

@its_ericchuRT @LiamFedus: I'm sympathetic to Terence Tao's view: AI solving outstanding mathematical problems without advancing a conceptual understan…

@DanielleFongwhy has nobody asked if you can extend the vorticity concentrating navier stokes flow field from OpenAI's construction to MHD generators ? it can probably work

@sarahookrIt has somewhat ironically come to the fore of public discussion again because Open AI was so eager to release a mathematical proof. Tristan suggested inputs he gave in chat may have been used in the solution without his awareness. https://www.nytimes.com/2026/09/10/science/tristan-buckmaster-openai-math-navier-stokes.html

@cyantistI waited to wade into the turbulent response behind OpenAI’s post about the Navier-Stokes proof until the waters stilled. What does the proof tell us and not tell us? What did the headlines get wrong? Justin from Reynko does a perfect job of explaining here: https://www.linkedin.com/pulse/what-navierstokes-proof-does-doesnt-tell-engineers-justin-beroz-jwkwe?utm_source=share&utm_medium=member_ios&utm_campaign=share_via

@biji_sfMillion dollar vs Trillion dollar problem - AI hunting for needles in a haystack, for now. Justin from Reynko explains it.

@PaulRRobichaudAI can solve the Navier-Stokes existence and smoothness problem. But it is nowhere near capable of discovering ideas like this: 𝑖² = 𝑗² = 𝑘² = 𝑖𝑗𝑘 = −1 . Proving conjectures with known methods is a small part of what mathematicians do. Creative vision is everything.

@DavidDeutschOxfRT @PaulRRobichaud: AI can solve the Navier-Stokes existence and smoothness problem. But it is nowhere near capable of discovering ideas li…

@WolframJapan2026年9月8日、OpenAIがミレニアム懸賞問題「ナビエ–ストークス方程式」の解を発見したと発表！ 150ページを超える難解な証明を、Wolfram言語の可視化を使い分かりやすく解説した記事です🔍 記事はこちら👇 https://community.wolfram.com/t/an-intuitive-exploration-of-openais-navier-stokes-singularity-construction-for-the-1m-millennium-prize-problem/28013

@patrickshaftoRT @Ananyo: "I’m at the grocery store right now, and the people around me keep picking out tomatoes and avocados so calmly, as if nothing h…

@KyleCranmerA nice explainer 🧵👇. One thing I’ve been coming back to in recent days is that the observation that the human process for solving these lighthouse problems also generates many more interesting question/problems as we gain insight. The current approach of using AI to solve these problems autonomously generally does not populate the next frontier of interesting questions. It occurred to me that the field of mathematics has something like a “joy reserve” of problems that humans are capable of/poise to solve, and that automated approaches are essentially burning through that joy reserve as a non-renewable resource. In contrast, in the traditional human approach the joy reserve is more like a renewable resource. That’s not fundamental to using AI to assist in doing research, but a property of the current approaches to auto formalization. The use of AI as a thought partner or creative assistant is quite the opposite— It helps to generate a lot of new ideas.

@lateinteraction@ziv_ravid it’s the most convenient rumor ever. imagine not having new breakthroughs yet (since low-hanging fruit for models are simply too jagged and different from human low-hanging fruit), but still getting the publicity of having solved numerous “very longstanding open problems” anyway

@_onionesqueRT @WolframJapan: 2026年9月8日、OpenAIがミレニアム懸賞問題「ナビエ–ストークス方程式」の解を発見したと発表！ 150ページを超える難解な証明を、Wolfram言語の可視化を使い分かりやすく解説した記事です🔍 記事はこちら👇 https://t.…

@LeveragedInfo🤯 WE ARE SO COOKED! AI just crossed a major intelligence threshold @EMostaque says a swarm of 10,000 AI agents solving Navier–Stokes changes the argument about what these systems can actually do. “do you really think there's anything you can do cognitively that this AI can't do” Ft. @alessandrorisk

@GonZalocla@scottnarmstrong I mean honestly I would not be surprised if (A) turns out to hold true and you have GWP for unforced NS [maybe motivated by the fact that we worked a lot with Javi, Tristan and Google and didn't find any :( ] After reading openAI proof is SO FAR from unforced, basically finding

@scaling01"that the contribution of the "agent swarm" component to the Navier Stokes discovery is probably quite low (10%)" this is just plainly false the simplest reason being that even if a single sequential agent was 100x more efficient than the swarm (which it's not), even then it would take roughly a year to generate the tokens necessary to come up with the proof the proof wouldn't exist without the swarm but overall reachability of the problem has to be given, so there had to be a 0.00001% probability or something similar ridiculously low such that the swarm can find it

@EMostaqueRT @LeveragedInfo: 🤯 WE ARE SO COOKED! AI just crossed a major intelligence threshold @EMostaque says a swarm of 10,000 AI agents solving…

@JsevillamolRT @GregHBurnham: Some discussion of this result — amusing and all, but also note the by-now familiar comment, "I originally thought it mig…

@QuantaMagazineThe Navier-Stokes equations are among the most important — and most challenging — equations in physics. So, what makes them so difficult? And how could a new OpenAI proof finally resolve the problem? Watch our new video: https://www.youtube.com/watch?v=PsWR1rQEWYo

@eigensteveRT @QuantaMagazine: The Navier-Stokes equations are among the most important — and most challenging — equations in physics. So, what makes…

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