A Mathematician, an AI, and a Million-Dollar Problem

A mathematician, an artificial intelligence, and a million-dollar problem. Inside: a war between companies, a threatened career, a "did they steal it or not" fog, and a deep anxiety about the future of mathematics. Its name: Navier–Stokes. Its story exploded today.

The problem: the secret of water and air

The Navier–Stokes equations govern how water and air move. Written in the 19th century, still unsolved. The core question: from a smooth start, does the solution always stay smooth, or does it "blow up" in finite time? The Clay Mathematics Institute put a $1 million prize on it; one of the millennium problems.

Here "blow-up" means the collapse of the solution's mathematical regularity. Charles Fefferman wrote Clay's official statement, and there are four options: A and B for the unforced case, C and D for exhibiting collapse under a smooth external force. So a "forced blow-up" can also win the prize, provided it meets the conditions. Keep that nuance in mind; it becomes critical below.

The triumph: in a month, with AI

NYU's Tristan Buckmaster and Anthropic's Levent Alpöge took exactly that narrow technical path. For a year, with help from Claude and Codex (GPT-5.6 Sol and more recently Astra), they advanced a research program opened by Diego Córdoba and Luis Martínez-Zoroa.

On August 15, 2026 they obtained blow-up results for Boussinesq and Euler; on August 22 they machine-verified them in the Lean proof assistant. The result: "finite-time blow-up" for 3D incompressible Euler, Boussinesq, and porous-medium (IPM) equations, under a smooth external force. Fields Medalist Terence Tao called it "a remarkable achievement" and added that the method would "very plausibly" extend to Navier–Stokes.

But by Buckmaster's own admission, the first LLM-produced proof was "the most horrifying thing he had read," and the Euler write-up could only be described as "AI slop." Correctness was there; readability was not.

Up to here, a lovely AI–mathematics triumph story. Then things went off the rails.

The leak

Thursday, September 3, 2026. A rumor — "Anthropic has solved a major open problem" — started circulating in math circles. Tips reaching Alpöge suggested something else: word of the work had leaked to OpenAI. Buckmaster emailed a leading mathematician at OpenAI directly: "This is a personal collaboration, not a corporate one. I pay for the tools out of my own research funds, including a large bill to OpenAI."

The reply was polite and came the same day: "Sharing information helps to avoid competing; we are always excited to see mathematicians advance with our models. OpenAI stands ready to provide compute if needed."

The phone call

Sunday, September 6, 12:45. A same-day meeting request came from Buckmaster. OpenAI researcher Sébastien Bubeck — author of the "Sparks of AGI" paper, former Microsoft VP — joined the call. That afternoon, the three of them (Buckmaster, the OpenAI mathematician, and Bubeck) met twice. Alpöge was not in the meetings.

And they told Buckmaster this: OpenAI's internal model had produced a roughly 100-page proof for forced Navier–Stokes.

In Buckmaster's words, the exact sentence: "Existence of forced blowup in R3 and T3, with the forcing function is smooth option c and d in Fefferman."

That exact narrow path: Fefferman's options C and D. The route Buckmaster and Alpöge had quietly pursued for a year, which almost nobody works on. When Buckmaster heard the word "forced," he saw a "bright red flag." Because nobody randomly lands on that narrow technical path in a few days by handing a model nothing but the problem statement.

The confessions

Then the confessions started coming — again, according to Buckmaster's account. At first it was said that "very little human intervention" had been used. But as corrections and details flowed to Bubeck from the OpenAI team's own internal chat during the call, the real picture emerged: a team had been working on the problem for a long time, this was one of several attempted paths, they had started from the unforced problem, they had first pointed the model at easier warm-up problems like Euler, even the prompt shown to Buckmaster had been written with Codex, and "crazy" amounts of compute had been used.

Buckmaster asked the most critical question: when was the first prompt sent? A straight answer took a long time to come. It was finally admitted: within "the last few days" after their work had reached OpenAI.

The data question

Throughout the project, Buckmaster and Alpöge had been using OpenAI's Codex tool. All their draft proofs, working notes, and materials sat in Codex sessions. Buckmaster asked whether OpenAI's internal model had accessed this user session data, or been trained on it.

To the first question he got the answer "the model doesn't look at user data." But the second, more specific question — whether it had been trained on the data — went unanswered. Buckmaster states this plainly: he is not claiming the data was used, only emphasizing that the question was left hanging. Still, it touches a broad industry anxiety about whether OpenAI uses user data in model training.

Offers and a threat

Two offers were put to Buckmaster. First: publish the Euler result, and OpenAI publishes its own Navier–Stokes result the next day. Second: after Euler is published, Buckmaster writes up the Navier–Stokes result alone and credits the internal OpenAI model.

Buckmaster rejected both. Allegedly, Bubeck twice suggested removing Alpöge from authorship. The reason: Alpöge working at Anthropic — OpenAI's direct rival — was an "annoying" complication. It was even reportedly said that if OpenAI published after them, they would say Buckmaster and Alpöge "deserved the Clay Prize" and were "the people closest to the problem."

An important nuance: this was not a demand to strip Alpöge's authorship from already published papers; it concerned a proposed future Navier–Stokes write-up.

Buckmaster said that if OpenAI published its results as proposed, he would share events publicly. Allegedly Bubeck's reply was: "Why ruin your career?" When Buckmaster asked "I'm an academic, how would sharing publicly ruin my career?", the answer came: "If you don't want me to be nice, I don't have to be nice."

Some time later Alpöge received a message from Bubeck directly, proposing a one-on-one chat: "I don't know if Tristan is fully rational right now." Alpöge declined and said conversations should go through him.

The other side

Bubeck, for his part, rejected all these accusations in a short post on X: "False and inflammatory accusations against me are currently circulating on social channels. To clarify, I entered the discussion following academic norms and I am disappointed it came to this. Anyone who knows me knows academic standards matter to me above all. More to say tomorrow."

But that short, general statement answered none of the specifics one by one: the Codex data question, the authorship pressure, the "ruin your career" remarks, the timeline of OpenAI's work. OpenAI has also yet to publicly present the alleged Navier–Stokes proof. The Clay Mathematics Institute still lists the problem as open; there is no accepted solution or prize announcement.

The bigger story

Buckmaster's own statement actually frames something wider than this conflict. To him this is a "Deep Blue–Kasparov moment": what matters is not the results themselves but "a mathematician and an LLM now being able to do all this work in a month." The implications for student training, citation culture, peer review, and "how we decide which subjects are worth a human lifetime" cannot be overstated.

And this anxiety is not Buckmaster's alone. In early August, OpenAI held a closed summit in San Francisco titled "The End of Mathematics" with some 40 mathematicians attending; Fields Medalist Jacob Tsimerman and Bubeck organized it together. Daniel Litt reportedly said there that we may be heading toward "a world with no high-quality mathematical research, with human mathematical expertise entirely gone." And Bryna Kra underlined the exact question Buckmaster lived: "AI can prove theorems, but can it explain how it proved them? A proof nobody understands doesn't become part of the literature."

Buckmaster's follow-up post on Mastodon is striking too: "Grateful to the math community for the support shown to me. But there is a much bigger story than my statement tells: the scale of what frontier models can now do and what it means for all of us. I hope the labs see it and set small calculations aside."

What we know, what we don't

Public and verifiable: Buckmaster and Alpöge published Lean-verified blow-up results for forced Euler, Boussinesq, and IPM (PDFs on an NYU server and an open GitHub Lean repo). Bubeck denied the accusations and promised more explanation. Clay still lists the problem as open; there is no public OpenAI proof.

One-sided claims (not independently verified): that OpenAI's internal model produced a ~100-page forced Navier–Stokes proof; that the work started after the tip-off; the authorship pressure; the "ruin your career" remarks; the unanswered Codex data question.

Unknowns: does the alleged proof really exist, is it valid? Does it meet all of Clay's assumptions? Were private drafts used? What is the full record of the meetings?

Let us end in Buckmaster's own words: "I want to be clear about what I am not claiming. I have not seen OpenAI's proof. I don't know what their models did or how they did it. I don't know whether our data was used. I am not accusing anyone of anything. I am only stating what I was told, when, and what was proposed."

And: "If an OpenAI model really has closed the gap to Navier–Stokes, that is remarkable, and it should be said loudly, by themselves, with the history kept complete."

This story may change tomorrow. Bubeck said "more to say tomorrow"; the detailed response may arrive September 9. Until then, this remains a story told by one side while the other has yet to fully answer — still developing.

Sources: Buckmaster's statement (NYU), the Euler/Boussinesq/IPM papers and Lean repo, Terence Tao's blog post, the Clay Mathematics Institute, Bubeck's response, Glitchwire, OfficeChai, Kingy AI.

PS. This piece is a narrativized account compiled from public sources, faithful to the facts. The alleged accusations are one-sided; Bubeck has denied them and promised a detailed response. The story is still developing.