Water seeping through sand, air rising over warm plains and the swirl in a stirred cup are all described by the same family of equations. And for ninety years, mathematicians have been unable to say whether those equations always sustain.
On Tuesday, two claimed answers were announced within hours of each other. The first from an American mathematician who published his research under protest, apologising for the state of the papers and explaining that outside pressure had forced him out before the work was ready. The second came from OpenAI, which said around 10,000 of its AI agents, coordinating with one another, had gone further in 88 hours.
Tristan Buckmaster, a professor at New York University’s Courant Institute, published the first of the two claims, and in a four-page statement alongside his papers, he said OpenAI turned to this particular problem only after hearing about his work, pressed him to accept publication arrangements he refused, and twice asked him to remove his co-author from a paper because the co-author is employed by Anthropic, a competitor. That co-author is Levent Alpöge, a number theorist who joined Anthropic’s technical staff after a junior fellowship at Harvard, and who was working with Buckmaster in a personal capacity with no involvement from either employer.
Sébastien Bubeck, the OpenAI researcher named in the statement, has called the allegations “false and inflammatory” and said he came into the discussion following academic norms. “Anyone who knows me knows that academic standards are of the highest importance to me,” Bubeck wrote, promising a fuller response.
90 years of uncertainty
The Navier–Stokes equations, written down in the first half of the nineteenth century by Claude-Louis Navier and George Gabriel Stokes, apply Newton’s second law to a fluid treated as continuous stuff rather than as a swarm of molecules. They are the foundation of the models used to forecast weather, design aircraft wings and simulate blood moving through an artery, and engineers have built a century of technology on them without waiting for the mathematics to catch up.
Jean Leray found the limit of what can be proved in 1934, and nobody has been able to dispute it since.
He showed that a fluid starting off calm and smooth can be followed faithfully by the equations for at least a short stretch of time. He also showed that if one accepts a rougher, blurrier description — one that tracks the fluid patch by patch rather than point by point, and permits kinks the equations cannot handle directly — then a description of some sort survives for all time. What he could not show, and what nobody has shown in the ninety years since, is that the sharper description lasts. The equations either go on making sense for the rest of time, or they reach a moment at which they say a part of the fluid is moving infinitely fast — impossible in nature, and a sign that the description has fallen apart. Mathematicians call the second outcome a finite-time blow-up.
Neither has been ruled out.
Fluid motion feeds on itself, gathering energy into ever smaller and faster swirls, while friction spreads that energy out again and calms it. Nobody has been able to prove which of the two wins as the swirls shrink, because the available mathematical tools haven’t been able to match up.
A proof that the equations break down doesn’t translate into water in a pipe moving any differently. It would mean that treating a fluid as one unbroken substance, rather than as the countless separate particles it is really made of, is an assumption that can fail, and that anyone modelling a flow close to that point of failure is using a tool outside the range where it is known to work.
The same question underlies turbulence, which physicists have failed to explain for as long as they have had the equations. The Clay Mathematics Institute made it one of seven Millennium Prize problems in 2000, each carrying a million dollars. Only one of the seven, the Poincaré conjecture, has been settled in the past twenty-six years.
Different theorems, and what OpenAI has
When the Clay Institute set the problem in 2000, it commissioned Charles Fefferman, a Princeton mathematician and Fields medallist, to write the official statement of what would count as an answer.
Fefferman set out four versions of the problem. In two of them the fluid may be pushed from outside — stirred, in effect, by a force written into the equations — provided that push is gentle and smooth everywhere and fades away far from the action. The push is not allowed to cheat, i.e. nobody may break the fluid by shoving it.
Most mathematicians ignore that term and study a fluid left entirely to itself. But Diego Córdoba and Luis Martínez-Zoroa spent years showing that a gentle push, shaped precisely enough, could drive the fluid to break down.
Buckmaster’s contention is that this route is obscure enough that nobody arrives at it in a few days from the bare problem statement.
He and Alpöge used a push of this kind to prove breakdown for three sets of equations, ending with three-dimensional incompressible Euler, which is Navier–Stokes with the friction taken out.
OpenAI says its agents proved breakdown for Euler with no outside push at all, a stronger result, before going on to the equations that include friction, with a push.
Terence Tao, the UCLA mathematician and Fields medallist whose 2014 paper on blowup for an averaged version of the equations is among the better-known attacks on the problem, wrote on his blog on September 7 that Buckmaster and Alpöge do not quite reach those goals yet, but have made enough of a breakthrough that completing them in the near future looks very feasible. He was assessing their papers, which he had discussed with Buckmaster by phone. OpenAI had not yet published at the time.
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Rumour and announcements
Buckmaster dates his and Alpöge’s central results to August 15 and their machine verification to August 22, after about a year of work using models from several companies, including OpenAI’s Codex, paid for out of his own research funds. They intended to release a polished paper alongside the formal certificate.
On September 1, by OpenAI’s account, the company heard a rumour that two Millennium Prize problems had been resolved and set its newest internal model, in training since August 28, across all of them.
On September 3, with the rumour spreading and Alpöge tipped off that word of their progress had reached OpenAI, Buckmaster emailed a prominent mathematician at the company — pre-emptively, to supply the facts so the rumour could be corrected. The reply the same day asked for details to avoid competing and offered him compute. He asked to speak the following week. He was pressed to meet on September 4, declined, and on September 6, was asked whether he could meet at any point that day. Two calls followed with Bubeck on them and Alpöge on neither. The three papers and the statement went up on the morning of September 8, one of them, in Buckmaster’s description, something that “can only be described as AI slop”. OpenAI published its Navier–Stokes claim later in the day.
Competing claims
Buckmaster says he was told on the calls that an internal model had produced a proof of roughly 100 pages, that he was shown a prompt and told the model had simply been handed the problem statement, and that Bubeck had told Alpöge very little human input was involved.
He says a different picture emerged during the call, as colleagues sent Bubeck corrections over OpenAI’s internal chat. A whole team had been working on the problem, this was one of several attempts, the agents had been given easier problems first, the prompt he had been shown was itself written by another OpenAI model, and the computing power used had been vast.
OpenAI’s published post describes that process in detail, though not the initial account Buckmaster says he was given. The post also says OpenAI began the work on September 1, the day the rumour reached the company.
Two arrangements were then put to him, Buckmaster says. Under the first, the pair would post their Euler result and OpenAI would post its Navier–Stokes result the next day. Under the second, Buckmaster alone would write a paper presenting the Navier–Stokes result, acknowledging that an internal OpenAI model had resolved it. He says Bubeck twice asserted that he wanted Alpöge removed from authorship, and said in substance that it would all be simple were it not so annoying that Alpöge works at Anthropic.
Buckmaster declined both.
He says he was told that going public would ruin his career and that when he asked why, the reply was: “If you don’t want me to be nice, then I don’t have to be nice.” He does not say who said this to him.
OpenAI describes the same outreach as an offer of concurrent release and a joint announcement recognising the pair’s priority, with sight of all its prompts and later of the proof itself. It makes no mention of authorship, and none of the fact that Buckmaster had written to the company first, five days earlier.
Bubeck has denied the allegations about his conduct without addressing the individual proposals. Buckmaster is careful in the statement about the limits of what he knows, saying that he has not seen the proof and cannot say what the model did. “I am not accusing anyone of anything,” he writes in the statement, posted on his page at the Courant Institute on Tuesday alongside the three papers.
OpenAI’s denial and the training question
Buckmaster asked on the calls whether the model had been trained on, or had access to, the Codex sessions holding every draft of the project. He says he was told the model does not look up user data, asked again about training, and got no answer.
OpenAI’s position is that neither its researchers nor its agents saw the work before publication and that no specific user data was accessed, while adding that it cannot rule out that de-identified data derived from their use of its products helped improve its models.
On the two sides’ dates, the training run OpenAI describes started on August 28, by which point every draft the pair had produced over the preceding year, including the finished result of August 15 and its verification on August 22, was in Codex.
Buckmaster hardened his position after OpenAI published that caveat on Tuesday. Having written that morning that he accused nobody, he wrote on Mastodon, the social platform where he had posted the papers, that the training window OpenAI describes opens after the date his own result was obtained. “Is it ethical to use customer’s data to try to scoop their customer?” Buckmaster wrote.
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No peer review yet – for either claim
Lean is a piece of software known as a proof assistant. A mathematician rewrites an argument in a formal language the computer can read, and the software then checks every step, confirming that nothing has been skipped or assumed. Mathematicians have begun using it on results too long or too intricate to check by eye.
The check works in one direction only. A mathematician writes the claim at the top of the file and the argument beneath it, and the software confirms that the argument establishes that claim. The software never asks whether the claim at the top is the famous problem, so a file can pass every check and still prove something weaker than the prize requires.
So, the real check on this prize claim will be for the conditions Fefferman laid down, particularly the conditions on the force. As of Tuesday, nobody outside OpenAI has run that check on its proof. The Buckmaster–Alpöge preprints are on his university page rather than arXiv and are unrefereed too. The Clay Mathematics Institute still lists Navier–Stokes as unsolved, and OpenAI has said it does not intend to claim the prize.
Buckmaster has put the weight of this on the mathematics rather than on either lab. His statement gives the credit for the underlying idea to Córdoba and Martínez-Zoroa, adding that “I believe Luis Martínez-Zoroa deserves a Fields Medal”, and calls what has happened in the past month “a Deep Blue-Kasparov moment” for a discipline with no settled answer yet on how it trains students, assigns credit or referees work of this kind. Writing again after OpenAI’s announcement, he said the far bigger story was “the sheer magnitude of what frontier models can now do”, and added: “I hope the labs can see this and set the petty posturing aside.”
For now, nobody has produced a blow-up for Navier–Stokes without an external force, which is the version the Clay Institute is actually asking about. Buckmaster is holding back a fourth result, for a version of the equations in which the friction is weakened rather than removed, until its verification finishes. He says it points towards the unforced case. Tao thinks the remaining distance can be covered soon.
