In the history of mathematics, there have been two moments that felt like the world was about to end for the people doing the job: Deep Blue beating Kasparov in 1997, and last Tuesday, when OpenAI announced that an internal model had produced a solution to the Navier-Stokes Millennium Prize Problem in 88 hours.
The difference is that Kasparov lost one game. What OpenAI’s announcement threatened wasn’t a single result — it was the entire idea of how mathematical progress gets made, claimed, and credited. And the story that’s actually unfolding has more drama in it than any of the press releases.
The claim
The Navier-Stokes equations describe how fluids move — air, water, blood in your veins. They underpin aircraft design, weather forecasting, and a fair bit of engineering. The Millennium Prize version of the problem, set by the Clay Mathematics Institute, asks whether solutions to the equations can break down, or “blow up”, in finite time. It has been open for around 90 years in its modern form, and it’s one of seven problems each carrying a $1 million prize. Only one — the Poincaré conjecture, claimed by Grigori Perelman — has been solved since the list was published in 2000.
On 8 September, OpenAI said a new internal model, “significantly more capable” than its released GPT-6 Astra, had produced a proof of finite-time blowup. The setup was almost absurdly baroque: roughly 10,000 AI agents working on the problem simultaneously, passing messages back and forth, running for 88 hours. Mark Chen, OpenAI’s chief research officer, told reporters the compute bill ran “emphatically in the millions of dollars” — about 1,000 times what the company spent on its earlier mathematical results.
The trigger was almost farcical. OpenAI says it started training the new model in late August, and on 1 September “heard rumours that two Millennium Prize problems had been resolved” — so it pointed the system at all six remaining open problems and let it run. Navier-Stokes, it says, showed “unexpected promise”.
OpenAI also says it won’t claim the million dollars.
The dispute
Here’s where it stops being a press release and becomes a story. On the same day — hours before OpenAI’s announcement — Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpöge, who works at OpenAI’s rival Anthropic, published three related results of their own: finite-time blowup with smooth forcing for incompressible porous media, for the Boussinesq equations, and for the 3D incompressible Euler equations. They believe they also have a blowup result for hypo-dissipative Navier-Stokes, but they’re not releasing it yet — the Lean verification isn’t finished.
Buckmaster’s statement, published on his NYU page, is extraordinary reading. He and Alpöge had been working on the problem as a purely personal collaboration — no institutional agreements, no employer involvement, Buckmaster paying for the AI tools out of his own research funds. They used Claude, and OpenAI’s Codex, including GPT-5.6 Sol. And on 3 September, with a rumour circulating that “Anthropic had resolved a major open problem”, Buckmaster emailed a prominent mathematician at OpenAI to say, in effect: we’re about to publish, and we’d rather you had the facts.
What happened next, in Buckmaster’s account, is where the temperature drops. On 6 September he was put on a call with OpenAI’s Sebastien Bubeck and told that an internal model had produced a proof of finite-time blowup for forced Navier-Stokes. Buckmaster’s words: “When I heard ‘forced’, it was a bright red flag.” The forced route is precisely the path two other mathematicians — Diego Córdoba and Luis Martínez-Zoroa, whose years-old programme of work he and Alpöge had built on — had opened, and the one his pair had quietly chosen to attack. “Almost nobody else I know of was working on it.”
He was shown a prompt and told the model had simply been given the problem statement, with “very little human input”. Over the course of the call, that turned out not to be true: an entire team had been working on it, easier problems first, the prompt itself had been written by prompting Codex, and “an insane amount of compute” had been used.
Then came the two offers. Either Buckmaster and Alpöge would post their Euler result and OpenAI would post its Navier-Stokes result the next day — or Buckmaster alone would write a paper presenting the Navier-Stokes result, acknowledging the internal OpenAI model had solved it. Bubeck twice said he wanted Levent removed from authorship. The reason, in Buckmaster’s words: “it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic.”
Buckmaster declined both, and said he’d go public if OpenAI released its result in the way proposed. The reply, as quoted in the statement: “Why would you ruin your career?” Then: “If you don’t want me to be nice, then I don’t have to be nice.”
What OpenAI says
OpenAI’s response was measured. It called the “concurrent work” of Buckmaster and Alpöge “remarkable”. Bubeck denied at a press briefing that the company had used the pair’s work or accessed material shared with OpenAI’s servers, and said the two proofs “differ significantly and even the precise results proved are different”. There was one telling concession: “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”
Buckmaster, for his part, is careful to say what he is not claiming. He has not seen OpenAI’s proof. He doesn’t know what their model did or how, or whether his and Alpöge’s data was used. “I am not accusing anyone of anything. I am stating what I was told, when, and what was proposed to me,” he writes, “because the alternative is to let a sequence of announcements say something I know to be false.”
It’s a remarkable line from a man describing what is, by his own framing, “a Deep Blue-Kasparov moment”. And it’s worth pausing on: the most credible version of the OpenAI story is that an unreleased model, given enormous compute and a team’s worth of human direction, found a route to a famous problem in days. The most credible version of Buckmaster’s story is that the same model was pointed at that problem after OpenAI heard rumours about the very people working on it with OpenAI’s own tools. Both can be true. That’s what makes it uncomfortable.
The real story
There’s a broader pattern here, and it’s been building all year. In May, an OpenAI model disproved the Erdős unit-distance conjecture — an 80-year-old problem first posed in the 1940s — discovered while the model was being evaluated. In early August, OpenAI’s Astra generated machine-checkable proofs for ten decades-old problems for roughly $2,000. And last month, OpenAI released a new AI agent after admitting one of them had gone rogue.
The mathematics community is now watching the same sequence of events play out at the summit of the discipline: AI systems producing results on problems that define careers, with verification lagging years behind. The Clay Institute’s rules are almost comically slow by 2026 standards — a solution must be published in a peer-reviewed journal and survive two years before the institute even convenes a committee to consider it. Martin Bridson, the Clay’s president, told AFP that “the process of evaluation is deliberately unhurried”.
So the actual state of play, as of this week: OpenAI claims a solution that nobody has independently verified. Two mathematicians claim a closely related body of work that was, by their account, already in motion before OpenAI’s model was pointed at the problem. The prize money is, by the rules, at least two years away from anyone. And the question that matters — not who gets the million, but what counts as a mathematical result when the thing doing the proving is a model you can’t fully inspect — has just become everyone’s problem.
Buckmaster said he’d much rather be talking about the mathematics, about Córdoba and Martínez-Zoroa’s ideas, and about what all of this means for how the field trains students and assigns credit. Fair enough. But for the moment, the story that’s actually happening is the one where the proof is 100 pages long, nobody outside OpenAI has seen it, and the two humans closest to the problem are in a shouting match over who got there first.
Deep Blue beat Kasparov in six games. This one is going to take a lot longer to settle — and it’s already produced the first genuinely awkward moment in the history of AI and mathematics.
Sources: BBC, The Guardian, The Journal/AFP, OpenAI, Buckmaster’s statement
