A year of work, a month of AI, and a fight over who gets to announce it
The record, in order, from the sources themselves.
Between 6 August and 8 September 2026, the most-watched open problem in mathematics went from a warning about what AI might do to it, to a real result, to a public argument between OpenAI and Anthropic researchers about who got to announce what.
Tristan Buckmaster of NYU and Levent Alpöge of Anthropic proved finite time blowup with smooth forcing for three fluid equations, including 3D incompressible Euler, formalized in Lean. Terence Tao called it a remarkable achievement. Buckmaster published alongside a four-page statement saying he was told OpenAI had gone further, by the same route, starting only after word of his work reached them, and that he was asked twice to drop his co-author because that co-author works at Anthropic.
That evening OpenAI said publicly that it had solved the Millennium Prize problem, congratulated the two of them, and said it had seen none of their work before they published it. It put a write-up of the proof and a Lean formalization on its own site; nobody outside the company has checked either yet.
This page is what each person actually said, in the order they said it.
Before
Five days before any of this, Terence Tao published a case study using Navier-Stokes as his worked example of how an AI-generated solution could damage a field. He was not the only one saying it, and a month earlier three mathematicians had already accused OpenAI of running roughshod over prior work.
This is the precedent, and it is why the argument a month later reads as the second round rather than the first. Joseph Howlett reported for Scientific American that three named mathematicians objected to how OpenAI had presented ten results from its Astra model in August. Stephen Miller teaches at Yeshiva University and says the sphere-packing result leaned on his own 2016 paper. Francesco Fournier-Facio at Cambridge said of a group-theory result that "there is the big PR machine that wants to sound as impressive as possible". OpenAI said it would make updates "consistent with standard academic practice". The complaint on 8 September, that credit for a line of prior work was not carried, is the same complaint about the same company five weeks later.
Stephen Miller, in scientificamerican.com
They are running roughshod over the work of others who came before them.
Tao holds the Fields Medal and is one of the two or three people most often asked about Navier-Stokes. Twelve days before Buckmaster apologises for the state of his own Euler paper and calls it AI slop, Tao sets out why that was going to happen to somebody.
Terence Tao, in mathstodon.xyz
The advent of capable AI tools has highighted a variant of Simpson's paradox: a technological advance can improve the quality and volume of each individual's output, and yet the average quality (signal-to-noise ratio) of the aggregate output can deteriorate as a result. [...] whereas previously 90% of the extant literature was well-written, the situation is now completely reversed: 90% of the literature will now be poorly written.
Duminil-Copin won the Fields Medal in 2022 and works at Geneva and IHES. He says he does not use AI in his own work. His argument is not that machine proofs are worse, it is that solving the problems quickly removes the thing a field is built out of.
Hugo Duminil-Copin, in proofsandprompts.com
These artificial discoveries risk decapitating entire fields before they have had time to develop to their full potential. Even worse, they nuke the mathematical landscape. [...] the current use of AI does not empower us, it petrifies us.
Five days before any of it, and the sentence that names what is at stake in the whole week.
Terence Tao, in mathstodon.xyz
Analogously to pre-atomic steel, open problems in mathematics generated before the AI era have, astonishingly, have become something resembling a non-renewable resource. [...] It may become necessary to declare certain classes of mathematical problems off-limits to automated solvers, in order to preserve their broader value to the mathematical ecosystem.
The entry that reframes the week, and the reason this page starts where it does. On 3 September Tao published a six-part case study using Navier-Stokes as his worked example of how an AI-generated solution could damage a field. Five days later people were arguing about exactly that. The fifth part describes the scenario in advance: an autonomous harness, backed by enormous compute, running the whole search internally while the company keeps the process out of public view, so the problem is technically solved and almost nothing is added to mathematics.
Terence Tao, in mathstodon.xyz
A concrete example of how AI advances in solving key open problems could inihibit the future development of a mathematical field can be found in the global regularity problem for the incompressible Navier-Stokes equations [...] there is now an increasingly realistic scenario in which a primarily AI-generated solution to the problem appears, but in a fashion that contaminates the problem as a source of further advances.
A prediction that Claude had solved Navier-Stokes ran to 3.4 million views over a weekend. Every specific in it was wrong. By Buckmaster's account it is also what put his unpublished work in front of OpenAI.
Where the rumour became public, at 3.4 million views over a weekend. Curran writes about AI to a large audience and is not a mathematician; he later says he had no independent information. Every specific in it is wrong: it was not Anthropic's result, it was not Navier-Stokes, and it was not out for expert review. Buckmaster's statement records that a rumour of this shape is what he wrote to OpenAI about two days later.
It's fun to make predictions. Here's a new one:
Anthropic has solved a Millennium Prize Problem.
And I'll be even more specific.
Claude has solved Navier–Stokes.
It is out for expert review.
And to give myself a hard deadline, they will announce it before the IPO.
Asked about the rumour, Tao says he has not heard anything, then says the thing he had been saying all week in one sentence.
Terence Tao, in mathstodon.xyz
A clarification in response to recent rumors about a possible solution to the Navier-Stokes problem: I am not aware of any significant developments in this regard; the above discussion is hypothetical, but not completely implausible at the current level of development of AI technology.
The point remains that there is a substantial opportunity cost in converting a historically productive and motivating problem (such as Navier-Stokes regularity) into a mere viral social media post advertising some benchmark progress.
The same day, Tao re-runs the 2005 to 2014 story of bounded gaps between primes as if today's labs had existed then. In that version Zhang stays an adjunct lecturer and Maynard leaves the field. The fourth part is his own timeline of the days he was writing in, and the last line of it is the whole page in three characters.
Terence Tao, in mathstodon.xyz
Aug 31, 2026: Stadlmann releases a preprint that develops an equidistribution theorem for numbers with a large smooth factor, and uses this to improve the Polymath8 bound of 246 slightly, to 240.
Sep 3-4, 2026: Various AI companies race to announce their own improvements to the Polymath8 bound via social media.
Glazer runs mathematics at Epoch AI, which builds the FrontierMath benchmark. He spent the weekend trying to source the claim and posted the trace. It is the clearest record there is of what an unsourced rumour looks like from inside: every version came back to an OpenAI employee, none to anybody outside, and the problem being solved changed twice along the way.
>Be me
>Told several days ago thirdhand by OAI employee Anthropic solved a Millennium Prize Problem
>"No way it wouldn't leak in this way"
>Told Wednesday by a different friend at OAI that, actually, he heard it was two.
>"Ok I'm updating against further, obviously it wasn't two."
>OAI source says they'd actually garbled it, friend informs me.
>Goes on Reddit and tells people spreading the rumor it is unfounded
>Terry coincidentally posts a hypothetical about NS getting solved very next day with lots of specifics about the hypothetical proof and its Lean file
>Anthropic formalizes FLT
>Curran's "prediction" seems to align with everything.
>Terry writes follow-up post addressing the rumors, that says he hasn't heard any development on Navier-Stokes, reads sincere to me
>Curran has no independent relevant info here btw
>None of the OAI rumor spreaders can trace their beliefs back to any non-OAI source
>Math prof friend says he heard from a credible source "It was Hodge not NS."
Posted nine minutes after Glazer called the rumour dead. One sentence, and it is the only proposal anybody made all week.
Terence Tao, in mathstodon.xyz
A new proposed competition for AI companies: rather than being the first to announce solutions to unsolved math problems, be the first to announce a new mathematical insight.
What was actually proved, and by whom. Two separate Euler results landed within three hours of each other, by different routes, and only one of them got read.
Tristan Buckmaster is a fluid dynamicist at NYU's Courant Institute. Levent Alpöge is a mathematician at Anthropic. Their statement says the collaboration is personal, with no institutional involvement from either employer, and that Buckmaster pays for the tools out of his own research funds. The papers went up with the Lean formalization alongside them. Half an hour later Deane Yang, a mathematician in Buckmaster's own department, posted the bare link with no comment, and that is how most mathematicians first saw it: 920,000 views for a post that added nothing.
Tristan Buckmaster, in mastodon.social
Today, Levent Alpöge and I have made public three results: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler.
Tao had read the papers before they were public and spoke to Buckmaster on the phone about them, which he notes made a refreshing change from AI-based communication modalities. This is the assessment everything else that day is arguing around. In the second half he adds that nothing in principle seems to stop the method reaching Navier-Stokes, and that battering out such an extension with enormous compute does not particularly hold his interest.
Terence Tao, in mathstodon.xyz
A remarkable achievement: Alpöge and Buckmaster have managed to push one of the major promising approaches towards constructing blowup solutions to fluid equations --- as developed by Cordoba and Martınez-Zoroa --- to establish finite time blowup for many key fluid equations, including 3D incompressible Euler, with a smooth forcing term. The arguments have been formalized in Lean.
A second Euler result landed within three hours of the first, from a different group by a different route, and got almost none of the attention because it arrived in the middle of an argument. Anandkumar is at Caltech; the paper is with Ganeshram and Duruisseaux. Their case is the harder one, without forcing. Tao read it the same morning and called it relatively AI-light, using models mostly for literature review and Lean formalization, while noting that the rigorous demonstration of stability is still lacking and may need enormous further effort.
Our starting point is a physics-informed neural network (PINN) to come up with an approximate answer, and then to argue stability around that to complete the proof.
The challenge so far has been that PINNs have not been successful in discovering singularities on the current problem. A common failure mode is PINNs converging to a trivial solution. We take special care to nudge our PINN to interesting regions through a combination of constraints.
We believe that such physics-informed and physics-centric AI are critical ingredients across many areas of research involving physical systems, and LLMs lack such physical grounding.
We have been working on this problem for much of this year, and we just saw the announcement by Tristan on Euler with forcing. In contrast, we consider without forcing and use a PINN formulation.
Both accounts of the same two calls on 6 September, and what people at each lab said in public afterwards. Nobody outside OpenAI has seen the proof at the centre of it.
The four-page statement Buckmaster published alongside the papers, because a rumour had been running since Friday and, as he puts it, the alternative was to let a sequence of announcements say something he knew to be false. It sets out two calls on 6 September, with Sébastien Bubeck on both and Alpöge on neither. He says he was told an internal OpenAI model had produced a roughly 100-page proof of forced Navier-Stokes blowup, by the same route he and Alpöge had quietly chosen; that the first prompt was sent only after word of their work reached OpenAI; and that he was asked twice to leave Alpöge off a paper because Alpöge works at Anthropic. He also states the limits of his own claim, and they belong beside everything above: "I have not seen OpenAI's proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything."
Tristan Buckmaster, in cims.nyu.edu
I was told that an internal OpenAI model had produced a proof of finite time blowup for the forced Navier-Stokes equations. I was told the proof is about 100 pages. I have not seen it. [...] I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.
Bubeck leads OpenAI's mathematics team and is the person named throughout the account of the calls. He addresses no specific allegation and says a fuller account is coming. At the time of writing it has not arrived, OpenAI has published nothing as an organisation, and nobody outside the company has seen the 100-page proof.
A series of false and inflammatory allegations against me are currently circulating on social channels. To clarify, I came into the discussion following academic norms, and I'm disappointed that it has come to this. Anyone who knows me knows that academic standards are of the highest importance to me. Will have more to say tomorrow.
Brown is a researcher at OpenAI and co-created the o-series reasoning models. Read this one and the next together, and with the timestamps: they are twelve minutes apart.
Seb 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. More to say tomorrow.
Douglas is a researcher at Anthropic. This is the post above with the names and the durations changed. Printed one after the other, the pair is the clearest record of the two labs' postures anywhere in the week, and neither needs a word of explanation: week long against year long is the whole argument in two adjectives.
levent 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
Six hours after the statement, and the last thing Buckmaster said. It is also what his own statement said it had wanted to be about: a mathematician and a model did in a month what he expected to take years, and he calls it a Deep Blue and Kasparov moment for how the field trains students, assigns credit and decides what is worth a life's attention.
Tristan Buckmaster, in mastodon.social
I want to again express my profound gratitude to the outpouring of support from the greater mathematical community. I also want to remark that there is a far bigger story here than the one in my statement: the sheer magnitude of what frontier models can now do, and what that means for us all. I hope the labs can see this and set the petty posturing aside.
Bryan is an associate professor of strategy at Toronto's Rotman School and chief economist at CDL Toronto. By late morning the reading that OpenAI had trained on private Codex sessions had taken over X, and this is the most substantial public pushback on it: he accepts that OpenAI pursued the route after hearing the rumours and still calls the training insinuation unfair. It is the strongest version of the account that does not assume the worst, which is why it sits here rather than among the reactions.
1) Navier-Stokes result will be shown soon, by humans+AI. This is "Nobel-level" result. 2) OAI has an internal proof also. 3) They had AI pursue a path partly based on rumors of a strategy that might work. 4) OAI did not "train on the humans' GPT input"- insinuation is unfair
The widest-read account of the day at 533,000 views, and one of the few at that reach that states it correctly: major progress, and a claim by somebody else that has not been shown.
SITUATION DETECTED: Mathematicians Tristan Buckmaster and Levent Alpöge have made major progress toward solving the Navier-Stokes existence and smoothness problem, one of the most important problems in mathematics, and say OpenAI may have solved it fully.
347,000 views, linking the statement, which says the opposite. Navier-Stokes has not been publicly solved by anyone. This is here because correcting it is most of what a page like this is for.
It appears OpenAI is claiming its AI just solved Navier-Stokes, the 3rd most famous unsolved math problem in the world.
It's one of the 7 Millennium Prize Problems, with a $1 million bounty.
DRAMA: The mathematician who spent a year on this (with AI) says OpenAI raced to the finish after his work leaked to them, then pressured him to drop his co-author, who works at Anthropic
To be clear, what's actually verified is the mathematician's result on Navier-Stokes next-door neighbor problem. OpenAI's proof of the real thing hasn't been revealed yet, and OpenAI hasn't responded to the accusations
Gerko founded and runs XTX Markets and funds the AI Mathematical Olympiad prize, so he pays for this kind of work himself. He goes on to argue that the same result was a few months away on consumer hardware and an ordinary budget; the rest is at the link.
On the topic of Navier-Stokes mess: what if, hear me out, Anthropic and OpenAI just completely stopped competing on who can deploy obscene amounts of compute into a non-public model to solve some long-standing math problem?
The most careful summary written by anybody outside mathematics, and the only widely shared one that carries Buckmaster's own disclaimer rather than dropping it.
1. The authors - Alpöge and Buckmaster - showed how to achieve a blowup with smooth forcing for 2D Boussinesq equation. Then they also worked it out for 3D incompressible Euler equations. These are Navier-Stokes but without viscosity.
Now, Tarrence Tao says that this approach may likely be extended to actual Navier-Stokes.
BUT there's a big drama behind it:
- OpenAI had learned about their progress before it was published and pushed their AI to quickly come up with a similar solution.
- However, OpenAI's solution was based on the same smooth-forcing approach that Buckmaster/Alpöge had quietly been developing. Buckmaster said that he does not know if their data was used, so he is “not accusing anyone of anything.”
- However, Sébastien Bubeck (a research lead at OpenAI) didn’t want to see Alpöge as the author because Alpöge works at Anthropic.
Bubeck from OpenAI called these allegations “false and inflammatory”. As of now, there are no further comments.
By late morning this was the highest-engagement account of the day at 5,300 likes and 263,000 views, and it is the version most people will have read. Every line of it is in the statement. What is not in the statement is the conclusion a reader draws from the last two: Buckmaster asked about training and got no answer, and he is explicit that he does not know what the answer is.
>math professor spends a year on one of the hardest unsolved problems in math
>"Navier-Stokes"
>his drafts for the whole project went through codex sessions
>openai had his logs in codex
>suddenly rumor leaks AI solved it
>he emails openai to ask what's going on
>openai calls within days
>"funny story, our model also solved it"
>using his exact approach
>asks openai: did you train on my sessions?
>no answer
On the evening of 8 September, OpenAI said it had solved the Millennium Prize problem, and put a write-up and a Lean formalization on its own site. Nobody outside the company has checked either yet, and the argument about how it got there was already running before the announcement went out.
OpenAI put its account on its own site before posting the announcement below, and this is Buckmaster reading that document twenty-four minutes earlier. The sentence he quotes is OpenAI's own. His statement that morning had said he did not know whether their data was used and that he was not accusing anyone of anything; this is the first thing he has put a question mark on directly.
Tristan Buckmaster, in mastodon.social
"Since August 28 we have been training a new internal model that has exhibited unprecedented performance in our benchmarks, including mathematics. This model's training is ongoing and its performance continues to improve."
Note that they are openly admitting they used training data from a period after we found our result. Is it ethical to use customer's data to try to scoop their customer?
The announcement, from the company's own account. The post itself links to nothing and carries an illustration of a spiralling, stretching fluid; the write-up and the Lean formalization are on OpenAI's own announcement page. Read it against what is not in it: no mention of Diego Córdoba and Luis Martínez-Zoroa, whose programme the route belongs to, or of Buckmaster and Alpöge, who published on it that morning. Being certified in Lean is a claim about the argument holding together, not about who got there first.
We're sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.
The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra.
The problem concerns whether the description of smooth three-dimensional fluid motion modeled by the Navier-Stokes equations can break down. It has remained unresolved for roughly 90 years.
Three minutes after the announcement, the direct answer to the allegation, and the reason this part carries both posts rather than only the first. Read the third paragraph as closely as the second: the denial is that no specific user data was accessed, and the sentence after it, that de-identified data from their usage cannot be ruled out, is OpenAI's own hedge rather than anybody's inference. On the last point their account and the reporting below sit differently. OpenAI says the proofs differ significantly and that the Euler results are not even the same statement, forced against unforced; Scientific American's account of Bubeck's briefing the same evening has him granting that the Navier-Stokes route does follow a similar method to Buckmaster and Alpöge's.
We congratulate Levent Alpöge and Tristan Buckmaster on their remarkable mathematical work.
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).
Joseph Howlett reported the announcement the same evening, for the magazine that ran the misconduct piece in August. Bubeck gave a press briefing and emphatically denied that Buckmaster and Alpöge's work influenced OpenAI, saying "We did not use their prompt or models or proof", that the model solved the Euler problem by totally different means, and that the Navier-Stokes proof was produced over the weekend; he also granted that the Navier-Stokes route does follow a similar method to theirs. The quotation is Diego Córdoba, who with Luis Martínez-Zoroa opened the route everybody has been arguing about, hearing that it may have been walked to the end without him.
Diego Córdoba, in scientificamerican.com
We're a little bit in shock... If it's done, that will be a big surprise for us.
The co-author, at Anthropic, quoting OpenAI's post back at itself. He had been silent all day: he was on neither of the calls, and everything said about him until now had been said by other people. Three things in it. He picks out the same hedge as the sentence worth reading and credits them for it. He has started reading the proof, which is the first public word from anybody outside OpenAI on what is actually in it, and his read is that it looks closer to a different Euler blowup proof he and Buckmaster already had than to the work they published. And he adds a detail about the calls that was not in Buckmaster's statement, which is his own account and nobody else has confirmed it: that a millennium prize was offered if he would be taken off the paper. What he does not do is take the fight, which is worth noticing on a page this long.
“we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”
i mean props to them for straight coming clean.
(so far the proof looks more along the lines of another euler blowup proof we had, off of whose ansatz naming we were making really stupid puns like “smooth criminale”, unlike the much better “ideal fluids explode”, Tristan)
so i’ll now give a bit on my thinking here. i actually woulda been pumped to collaborate on this, there are a lot of people at oai i like (ok, clearly some were indirectly dicks to me because of being part of the whole situation, but im a big boy, i still like them), idgaf about authorship on that step anyway, coulda been me Tristan and every fte at oai for all i care (on that Tristan would disagree:p). but on hearing the loud convo in the hallway, especially the part where a millennium prize was offered if i’d just be removed from the paper, it was kinda clear the die had been cast and things were locked. pretty wacky, unstrategic, and unnecessary, since on my side things were mostly me and claude having a good time yoloing random stuff in the corner rather than anything institutional. i also like the idea of the labs cooperating, and even better on scientific progress. it’s a shame!
