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Prof, OpenAI spar as headway made in Millennium Problem
OpenAI was the outside pressure, according to Tristan Buckmaster of New York University’s Courant Institute, who published first
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When water seeps through sand, air rises over a warm pavement, or tea swirls in a stirred cup, one family of equations explains the behaviour of all three. Those equations underpin weather forecasts, aircraft wing design and simulations of blood moving through an artery. But for 90 years, mathematicians have been unable to prove that the equations never break down. Nobody has ruled out that a flow starting perfectly calm and smooth could wind itself into ever smaller, ever faster swirls until the equations say a speck of the fluid is moving infinitely fast.
On Tuesday, two breakthroughs were announced within hours of each other: First, from an American mathematician who published under protest, apologising for the state of his papers, which, he said, outside pressure had forced him to release early. The second came from OpenAI, which said about 10,000 of its AI agents, working with one another, had gone further in 88 hours.
OpenAI was the outside pressure, according to Tristan Buckmaster of New York University’s Courant Institute, who published first. In a four-page statement he said OpenAI turned to the problem only after hearing about his work, pressed him to accept publication arrangements he refused, and twice asked him to drop his co-author because that co-author, the number theorist Levent Alpöge, works at Anthropic, a competitor.
Sébastien Bubeck, the OpenAI researcher named in the statement, called the allegations “false and inflammatory”. “Anyone who knows me knows that academic standards are of the highest importance to me,” he wrote, promising a fuller response.
A Millennium Prize challenge
The Navier–Stokes equations, written down in the nineteenth century by Claude-Louis Navier and George Gabriel Stokes, treat a fluid as one unbroken substance rather than a swarm of molecules, are the foundation of the models used to forecast weather, design aircraft wings and simulate blood moving through an artery.
Mathematician Jean Leray found the limit of what can be proved in 1934. He showed the equations follow a calm, smooth fluid faithfully for at least a short stretch of time, and that a blurrier description, one tracking the fluid patch by patch rather than point by point, survives for all time. Nobody has shown that the sharper description lasts. It either keeps making sense forever, or reaches a moment at which the equation predicts part of the fluid is moving infinitely fast, which mathematicians call a finite-time blow-up.
In 2000, the Clay Mathematics Institute made proving that the equations never break down one of seven Millennium Prize problems. Millennium Prize, in total seven unsolved mathematical challenges at the time, carry a prize of $1 million. Only one has been solved.
What each side proved
Clay Institute had laid down what would count as an answer, outlining four possible approaches. Two allow the fluid to be pushed from outside by a force written into the equations, provided that push is gentle, smooth everywhere and fades far from the action.
Most mathematicians however ignore that approach — of using force to test how the equation will hold — and study a fluid left to itself. Diego Córdoba and Luis Martínez-Zoroa spent years showing that a gentle push, shaped precisely enough, could drive a fluid to break down.
Buckmaster and Alpöge used a push of that 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, wrote on September 7 that the pair fall short of those goals but are close enough that finishing looks feasible. He was assessing their papers; OpenAI had not published at the time.
What happened between OpenAI and the academics
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.
Buckmaster emailed a prominent mathematician at the company on September 3 to correct the rumour. The reply that day asked for details to avoid competing and offered him compute. He asked to speak the following week, 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.
He was told on the calls, he says, that an internal model produced a proof of roughly 100 pages, and was shown a prompt and told the model had simply been handed the problem statement. Colleagues then sent Bubeck corrections over OpenAI’s internal chat: a whole team had been working on it, this was one of several attempts, the prompt he had been shown was itself written by another OpenAI model, and the computing power used had been vast.
In other words, between the start of the call and the end of it, according to Buckmaster, OpenAI’s account of how the proof was produced changed from a single model working unaided to a team, several attempts and a large amount of compute.
OpenAI then put two arrangements to him, he 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 – the OpenAI scientist -- 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. It makes no mention of authorship, or of Buckmaster having written first, five days earlier. Bubeck has denied the allegations about his conduct without addressing the individual proposals.
The three papers went up on the morning of September 8, one of them, in Buckmaster’s description, something that “can only be described as AI slop”.
A fight about credit
Buckmaster and Alpöge worked on the problem for about a year using models from several companies, including OpenAI’s Codex, paid for from his own research funds. Every draft of the project sat in those Codex sessions, including the central result of August 15 and its machine verification on August 22.
OpenAI’s training run began on August 28. The push route had taken Córdoba and Martínez-Zoroa years, and Buckmaster argues it is too specialised for anyone, or anything, to have reached in a few days from a bare problem statement. He asked on the calls whether the model had been trained on, or had access to, those sessions. He says he was told the model does not look up user data, asked again about training, and got no answer.
OpenAI says neither its researchers nor its agents saw the work before publication and that no specific user data was accessed, but it added that it cannot rule out that de-identified data from customers’ use of its products helped improve its models.
Buckmaster hardened his position after that caveat. Having written on Tuesday morning that “I am not accusing anyone of anything”, he said later on Mastodon that the training window opens after the date his own result was obtained. “Is it ethical to use customer’s data to try to scoop their customer?” he wrote.
For now, whether the Millennium Prize has been cracked will need peer review.
The real check will be for the conditions Clay Institute laid down, particularly those on the force.
Buckmaster, on his part, 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 the past month “a Deep Blue-Kasparov moment”, referring to a 1997 chess match that was is widely regarded as the first time a supercomputer defeated a world champion. The far bigger story, he wrote after OpenAI’s announcement, was “the sheer magnitude of what frontier models can now do”, and: “I hope the labs can see this and set the petty posturing aside.”
