Four days ago OpenAI announced that its models had made progress on one of the seven Millennium Prize problems. Twelve hours before that announcement, two mathematicians had posted a closely related result of their own, and within hours one of them said OpenAI had pressured him over the credit. This briefing sets out what was claimed, what the problem actually requires, what is disputed and by whom, and what a reader who is not a mathematician can take from it. It is written on 12 September with the dispute live and no proof public, and it will be updated rather than rewritten.
What OpenAI says it did
The post went up on 8 September under the title "On the Navier, Stokes Millennium Prize Problem". The work is attributed not to a released product but to an unreleased internal model described only as significantly more capable than GPT-6 Astra, the model OpenAI had shipped five days earlier. The construction, as reported from that post, is a fluid starting at rest under a smooth external force in which a small vortex spirals inward and stretches, spinning faster while its total energy stays finite, until the mathematics predicts infinite speed at a finite time.[1]
- ~10,000
- Agents run on the Navier, Stokes problem, with about 100 more on a related Euler problem [1]
- 88 hours
- Time to the claimed solution, plus 17 further hours of formal verification in Lean [1]
- ~130bn
- Tokens consumed on this problem alone, across 2.7 million messages [1]
OpenAI says explicitly that it is not claiming the $1m prize. That is the right call for a reason worth stating: the Clay Institute's own rules require that a proposed solution be published in a qualifying outlet, that at least two years pass after publication, and that it receive general acceptance in the global mathematical community.[2] As of today there is no preprint, no arXiv number, and no public Lean repository for either OpenAI's work or Buckmaster and Alpöge's. One headline put the position accurately: OpenAI says it solved Navier, Stokes, and nobody has seen the proof.
What the Clay problem actually requires
This is where most of the commentary has gone wrong, including an earlier version of this briefing. Charles Fefferman's official problem statement offers four alternative targets, and two of them permit an applied force. Statement (D) reads: "Breakdown of Navier, Stokes Solutions on R³/Z³. Take ν > 0 and n = 3. Then there exist a smooth, divergence-free vector field u°(x) on R³ and a smooth f(x,t) on R³×[0,∞), satisfying (8), (9), for which there exist no solutions (p,u) of (1),(2),(3),(10),(11) on R³×[0,∞)."[3] The f in that sentence is the force. Proving a forced blow-up is therefore a legitimate way to settle the problem, but only if the force obeys the decay condition Fefferman specifies, a bound of the form |∂ᵅ∂ᵗf(x,t)| ≤ C(1+|x|+t)⁻ᴷ for every choice of exponents.
That bar has caught people before. Diego Córdoba and Luis Martínez-Zoroa had already produced a singularity using what Quanta described as awkward, ungainly forcing functions, which satisfied the singularity conditions but failed the forcing requirement.[4] Buckmaster's own view of that earlier work is not grudging: "Luis Martínez-Zoroa deserves a Fields Medal."[4] Fefferman himself, asked about the announcement, said he was thrilled the problem was solved, and explained why the periodic setting matters: without a boundary, it is the fluid doing the crazy stuff rather than the walls.[4] So the question is not whether forcing disqualifies a proof. The question is whether this particular force meets this particular bound, and nobody outside OpenAI can answer that yet.
The other paper
Tristan Buckmaster of NYU and Levent Alpöge, who works at Anthropic, posted their own result at around midnight on 7 September, roughly twelve hours before OpenAI's announcement. It is a finite-time blow-up for the three-dimensional incompressible Euler equations with forcing, from axisymmetric initial velocity with swirl supported in a fixed torus, along with forced blow-up results for the two-dimensional inviscid Boussinesq system and for incompressible porous media. All are claimed to be Lean-verified. They also claim, unverified, a proof of a somewhat easier version of Navier, Stokes.[5][4]
Their use of AI is instructive precisely because it is unglamorous. Claude identified key elements of the proof and reproduced arguments; OpenAI's Codex drafted the main text and handled the bookkeeping of constants; later models audited and iterated. Buckmaster's verdict on the first draft is the most quotable sentence in the episode: it "was the most horrendous I have ever read".[4] The released write-up is described as closer to model output under human direction than to a human-written paper.[5] This is what AI-assisted mathematics currently looks like from the inside: not a machine producing a theorem, but a machine producing a great deal of material that a mathematician then has to make true.
The dispute, in order
| Date | What happened |
|---|---|
| 1 Sep | OpenAI says it began the effort, "inspired by rumors that two Millennium Prize problems had been solved" |
| 3 Sep | Buckmaster, worried the rumour concerned his own unpublished work, emails a mathematician affiliated with OpenAI |
| 6 Sep | Two calls between Buckmaster, that mathematician and OpenAI researcher Sébastien Bubeck. Alpöge is not included |
| 7 Sep | Around midnight, Buckmaster and Alpöge post their Euler result |
| 8 Sep | OpenAI announces its Navier, Stokes claim. Buckmaster goes public with his account. Bubeck responds |
| 11 Sep | The Clay Institute and 25 Fields Medallists both publish statements |
Buckmaster's account of the 6 September calls is that he was offered two mutually exclusive proposals, both of which excluded Alpöge as a co-author: either he and Alpöge post the Euler result with OpenAI posting Navier, Stokes the next day and describing them as the closest humans to the problem, or Buckmaster alone write up the Navier, Stokes result while acknowledging an internal OpenAI model. He says Bubeck wanted Alpöge removed because it was "so annoying" that Alpöge works at Anthropic. When Buckmaster declined and said he would go public, he says he was asked "Why would you ruin your career?" and then told "If you don't want me to be nice, then I don't have to be nice."[6] His own framing of the allegation is careful: "I am not accusing anyone of anything."[6]
Bubeck's public response the same day was that "a series of false and inflammatory allegations against me are currently circulating on social channels", with a fuller reply promised.[7] OpenAI's institutional statement addressed the separate and more serious question of whether it had seen the unpublished work: "We (the researchers and the agents) did not see any of their work through any means until they released it publicly, in particular, no specific user data was accessed." It conceded it could not rule out that de-identified data derived from Buckmaster's use of Codex had helped improve its models generally, and said the proofs differ significantly.[8][9] That concession is the part a reader should sit with. It is not an admission of anything, and it is also not a denial of the thing people are actually worried about.
What the institutions said
The Clay Mathematics Institute published a short statement on 11 September: "The Navier-Stokes problem has apparently been settled," adding that "the process is deliberately unhurried, but we will provide updates."[10] On the same day NYU issued a release headlined that its mathematician had answered long-standing questions about the fundamental equations of fluid motion.[11] And on the same day again, 25 Fields Medallists, among them Avila, Deligne, Figalli, Hairer, Kontsevich, Scholze, Tao, Viazovska and Villani, signed a declaration posted on Tao's blog under the title "A Severe Misalignment of AI in Mathematics". Its argument is not that AI cannot do mathematics. It is that "solving problems is only a tool and proxy for achieving the primary goal", and that the "mass production at faster and faster pace of 'true/false' statements" risks destroying the fertile ground the discipline grows in.[12] Tao separately warned of a potentially tragic world in which nobody trusts the AI companies, if researchers' unpublished work is absorbed without consent.[13]
How to read it
Three rules serve a reader here. The first is the distinction Tao drew during the Erdős episode of October 2025, when OpenAI researchers said GPT-5 had solved ten open problems and the curator of those problems showed it had found existing papers: between a model that retrieves a result and one that produces it, and between a claim and a checked claim.[14] Verified AI contributions to mathematics since have been real and modest: a key idea in one Erdős proof published with its human authors in November, and two confirmed proofs out of the ten unpublished research problems posed by eleven mathematicians in the controlled First Proof trial.[15][16] The second rule is that the trustworthy results in this field are the ones whose output can be checked by something other than a model, which is why AlphaEvolve's algorithms are undisputed and this claim is not yet anything. A released Lean proof would settle the mathematics; it would not settle whether the problem it proves is the Clay problem, or whose idea it was. The third rule is about the source. OpenAI's materials are the only account of what was done, the figures in them are OpenAI's, and the field has reasons for caution: Epoch AI's audit of the FrontierMath benchmark in June found errors in roughly a third of its problems, and a Nature audit in March found fabricated citations spreading through the literature at roughly one paper in 280.[17][18]
None of that means the claim is wrong. It may be an important result, produced in a way that raises real questions about credit. It may be a forced-equation result whose force does not clear Fefferman's bound, presented as more. The community will know within months, and this briefing will say so when it does.