The Millennium Problem is solved. The authorship isn't.

At noon on September 8th, OpenAI published a 166-page document announcing it had solved one of the seven millennium problems: existence and smoothness of the Navier-Stokes equations, the mathematical model describing how fluids move, with a million-dollar bounty the Clay Mathematics Institute had left standing since 2000. Behind it sat a system of roughly 10,000 agents running in parallel for 88 hours, 2.7 million messages exchanged between them, 130 billion output tokens, and a cost the company itself put at “millions of dollars” — outside estimates range from $10 million to$23 million, depending on who’s counting and at what rate. All of it, in theory, to win a $1 million prize that OpenAI has already said it won’t claim.

That gap between what it cost and what the prize is worth should be the first alarm bell about what was actually being purchased. Twelve hours before the announcement, NYU mathematician Tristan Buckmaster had posted a statement accusing OpenAI of learning about his unpublished work and arriving at its solution by following the same path. A week later, 25 Fields medalists had signed a joint letter warning that this kind of announcement is doing real damage to mathematics as a discipline. The equation, in the end, turned out to be the easy part.

The Headline Solved a Different Problem

The Navier-Stokes problem, as Charles Fefferman formalized it for the Clay Institute, actually poses four distinct statements — (A), (B), (C), and (D) — and only two of them, the ones without any external force acting on the fluid, are what the mathematical community has spent 25 years treating as the real question. What OpenAI says it proved are options (C) and (D): that, with an external forcing term added, smooth solutions exist that become singular in finite time. That’s a genuine result if it holds up, and it fits squarely inside the program Diego Córdoba and Luis Martínez-Zoroa had already been advancing before a single OpenAI agent existed. But it isn’t the version of the problem — unforced, the one that has actually kept the field up at night since 1934 — that most headlines treated as settled.

And even that narrower version isn’t verified as of today. The Clay Institute’s rules require publication in a peer-reviewed journal and two years without serious objections before the prize is even considered; neither arXiv nor a company’s own website counts. On that timeline, formal validation wouldn’t arrive before 2028 or 2029, if it arrives at all. The institute itself, through its director Martin Bridson, called the announcement “exciting” while noting that any real evaluation “will be careful and slow.” That’s the correct response to the most mathematically significant claim a private AI company has ever put out — and it says something that it needed to be said out loud at all.

An Eight-Day Race Against the Clock

The timeline explains this controversy better than the equations do.

On August 15th, Buckmaster and researcher Levent Alpöge — who works at Anthropic — used that same Córdoba–Martínez-Zoroa method, with help from several language models including Codex, to prove that the Euler equations (the simpler, viscosity-free cousin of Navier-Stokes) do blow up in finite time. Weeks later, by Buckmaster’s account, someone tipped him off that word of his unpublished work had reached OpenAI. On September 1st, OpenAI launched its all-out attempt at the six remaining open millennium problems, aiming explicitly at Navier-Stokes. On September 2nd and 3rd, the two mathematicians reached out to OpenAI on their own to flag their ongoing research. On September 6th, OpenAI’s Sébastien Bubeck wrote to tell them the company already had a complete proof, and offered to “share everything, including the human prompts.” That same afternoon there was a tense meeting between Buckmaster and Bubeck — Alpöge didn’t attend — in which OpenAI floated two ways out: let Buckmaster publish his Euler results first, or rewrite OpenAI’s own proof to include him as an author, with no clarity on whether Alpöge would be part of that deal. Buckmaster turned down both. At one point Bubeck asked him why he’d risk his career going to the press with “baseless accusations”; Buckmaster read that as a threat, and Bubeck later apologized for the choice of words.

At 11:48 pm on September 7th, Buckmaster published his own statement laying out the Euler results, twelve hours before OpenAI announced Navier-Stokes at noon the next day. That is not the timeline of an independent discovery that happens to land at the same moment. It’s the timeline of two parties who each knew about the other and raced to get there first.

”We Can’t Fully Rule It Out”

The technical core of the accusation is whether OpenAI, in some form, saw Buckmaster and Alpöge’s work before arriving at it independently. Buckmaster had fed drafts and notes into Codex, OpenAI’s AI coding tool, and publicly wondered whether any of that content could have reached, in some roundabout way, the system that solved the problem.

OpenAI’s first answer, on September 8th, went like this: “no specific user data was accessed in order to solve this problem,” but “while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.” It’s a sentence engineered to be true and reassuring at once, and it manages the first at the expense of the second: it says nobody looked at the files, while leaving the door open to the model having trained, at some point, on a trace of that work. Mark Chen, OpenAI’s chief research officer, hardened the line afterward: “no people or AI systems searched through user data to solve this problem.” By September 9th the company was calling it “categorically impossible” that Buckmaster’s prompts could have influenced the system; by the 10th, after looking into it further, that had softened to the prompts “could not have influenced the system in any way, including through training.” Going from “impossible” to “could not have influenced it, including through training” in 24 hours isn’t a clarification. It’s a retreat to a line of defense that can no longer be checked from the outside.

Bubeck offered another piece that doesn’t quite fit either: he said the team behind the proof included “no one who could meaningfully contribute to the mathematical content,” meant to underline that the credit belonged almost entirely to the model. It’s a line built to sell the system’s autonomy, and it makes it that much harder to explain how a team with no mathematical expertise of its own landed on the exact Córdoba–Martínez-Zoroa program that Buckmaster and Alpöge had spent months developing in private.

The Bibliography That Grew Quietly

This is the part that made the fewest headlines and says the most about how OpenAI handles a mistake once it’s been pointed out.

The first version of the 166-page document — the one announced to the world on September 8th — cited sixteen references and included none of the joint work by Buckmaster and Alpöge, none of Chen and Hou’s, none of Elgindi’s: precisely the names whose research program most resembled the one OpenAI said it had used. Hours later, with no changelog and no announcement, a second version appeared with twenty-two references, this time crediting Córdoba and Martínez-Zoroa along with related work by Zheng. Mathematician Rafael Cao-Labora, who had flagged the gap, acknowledged the addition while insisting relevant work was still missing. And there’s one name neither version ever cited, at any point: Levent Alpöge.

The fact that the bibliography changed without notice doesn’t, on its own, prove bad faith — papers get corrected, and a fast correction beats no correction at all. But a company that markets its result as the achievement of “very little human input,” and that responds to the first public sign it had left out the very people whose method it used by quietly editing the document instead of issuing a correction notice, is choosing to look like it was never wrong over saying it fixed something. It’s the same logic, applied to a bibliography, that showed up when Anthropic promised watermarks “for the audience” without ever making clear who had actually asked for them: a gesture that patches the surface of the problem for whoever’s watching, without ever being announced as the fix that it is.

Tao Applauds, Then 25 Fields Medalists Sign a Warning

The mathematical community’s response came in two waves, and the second one matters more than the first.

Terence Tao, a Fields medalist himself, first praised Buckmaster and Alpöge’s Euler result as “a remarkable achievement” and said he saw “nothing in principle” stopping the method from extending all the way to Navier-Stokes. But days later he published, alongside 24 other Fields medalists — Peter Scholze and 2026 laureate Yu Deng among them — a joint statement titled “A Severe Misalignment of AI in Mathematics.” The letter doesn’t deny that the models have improved. It denies that solving famous problems is the right way to measure that improvement, warning that turning historic open questions into benchmark trophies “harms the science of mathematics and the mathematical community.” Tao put it in terms that should sting any lab that needs a headline every quarter: there’s a real opportunity cost in taking a problem that spent decades generating new questions and training generations of mathematicians, and reducing it to a viral post advertising one model’s progress, instead of letting it keep doing what it was doing before.

Everyone else’s reaction points the same way. Diego Córdoba, whose research program underlies the disputed proof, was blunt: “if their work hadn’t existed, the AI wouldn’t have solved the problem,” making clear the original idea was still a human one. Zsuzsanna Dancso, of the University of Sydney, framed it as a matter of basic scientific ethics: cite your sources, credit your collaborators. Peter Sarnak praised Buckmaster for “behaving the way mathematicians behave” — an implicit contrast with how the company had behaved. Michael Harris warned of a broader risk: that these announcements convince the public human mathematicians are already obsolete, something he called “extremely damaging” to the field. None of them are disputing whether the math checks out. All of them are disputing who gets to decide what gets told, and how.

On Hacker News, Nobody Argued About Lean

The technical crowd, for its part, was skeptical for a different reason: that a proof compiling in Lean, the formal verification assistant OpenAI used to check the result, doesn’t guarantee it says anything mathematically interesting. A badly stated theorem, or a proof that exploits a gap in the verifier itself, can compile just as cleanly as a real one; “verification in Lean is empirical evidence, but mathematical truth rests on expert consensus,” as one of the most-discussed threads put it. There’s a track record of AI “proving” false theorems and vacuous statements, and that history weighs more than the fact that the code compiles without warnings.

The other point of friction was process, not content. Publishing a result of this magnitude on the company’s own website, without peer review, before there’s even a real path to claiming the prize until 2028 or 2029, read like an announcement built for the press rather than for the mathematical community. Someone reached for the obvious comparison: Grigori Perelman, who proved the Poincaré conjecture in 2002 and turned down both the Fields Medal and the Clay Institute’s million dollars on principle. Stacking that against a company that spent somewhere between $10 million and$23 million in compute to “win” a million-dollar prize it has already said it won’t collect doesn’t leave OpenAI looking good under either reading: either the money was never the point and the headline was, or the result isn’t solid enough to risk claiming formally.

Gary Marcus, as summarized by John Gruber on Daring Fireball, went further and framed the whole episode as part of what he called a “desperate financial agenda” heading into a possible IPO, against a backdrop of slowing scaling laws and growing investor skepticism about whether frontier models will ever turn a profit. Gruber’s own takeaway, after reading it, was the most uncomfortable line in the entire saga for anyone who actually uses these tools to work: don’t use Codex or ChatGPT unless you’re fine with OpenAI helping itself to your work whenever it finds it useful.

The Million Nobody Is Going to Collect

Sam Altman has insisted OpenAI acted “in good faith” and that both sides’ methods “appear to be different,” which in his view is enough to call the two efforts independent. He may well be right on that technical point, and still be missing what actually matters: in a week when 25 Fields medalists sign a joint letter, a mathematician says he felt pressured on a call, a bibliography changes without notice, and the company itself announces it won’t claim the prize it said it had just won, the open question isn’t whether the equations hold up. It’s whether the story of the people presenting them can be trusted.

This blog has already made the point that when danger is the product, it’s the lab itself that decides what to disclose about its own research, with no outside party auditing it before it hits the press. The same mechanism just played out here with the sign flipped: instead of a threat, an achievement; instead of a system card, a 166-page paper. The pattern is identical — announce first, let scrutiny arrive later, if it arrives at all — and so is the outcome: by the time anyone can check it carefully, the headline has already done its job. Whether the millennium problem is actually solved will be known by around 2029. The question of who it was owed to already has an answer, and it didn’t take a week to get one.