Four days ago OpenAI announced that its models had made progress on one of the seven Millennium Prize problems, and within hours the two mathematicians whose work it most resembles said that it resembled their work rather too closely. This briefing sets out what was claimed, what was proved before it, 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 it will be updated rather than rewritten.
The problem, precisely
The Navier–Stokes equations describe how fluids move. The Clay Mathematics Institute's prize problem, open since 2000 with $1m attached, asks whether smooth solutions to the three-dimensional equations always exist or whether a solution can develop a singularity, a point at which velocity becomes infinite, in finite time. The problem as stated concerns the equations without an external force, or with one that is smooth and bounded, and a proof of blow-up in the unforced case would be a resolution.[1] That last clause is where the argument sits.
What came before
On 18 September 2025 a team from Google DeepMind and mathematicians at NYU, Stanford, Brown and Princeton, among them Tristan Buckmaster and Javier Gómez-Serrano, published the first systematic discovery of unstable self-similar singularities in three equations related to Navier–Stokes, found by training physics-informed neural networks to near machine precision. The paper says plainly that the Navier–Stokes problem is not resolved; it is covered in our briefing on that work.[2] A follow-up in November pushed the precision further.[3] On 15 August 2026 Buckmaster and Levent Alpöge, a mathematician on Anthropic's technical staff, proved a blow-up result for the forced Euler equations, the frictionless relative of Navier–Stokes, with assistance from several labs' models; Terence Tao called it a remarkable achievement.[4][5]
What OpenAI claimed
- ≈10,000
- Agents OpenAI says ran concurrently on the problem, over 88 hours [6]
- 17 hours
- Further time OpenAI says was spent formalising the construction in Lean [6]
- 0
- Peer reviews, published papers or independent verifications of the result as of 12 September [7]
On 8 September OpenAI said an internal model, running as roughly ten thousand agents over 88 hours, had constructed a three-dimensional flow that starts at rest and develops unbounded velocity in finite time under a smooth applied force, with its kinetic energy remaining bounded, and that a further 17 hours had been spent formalising the construction in Lean.[6] Two things are contested. Mathematically, whether a forced blow-up of this kind answers the Clay problem as stated is disputed; the Nature news report of the announcement found mathematicians unconvinced that it does, and no paper, preprint or Lean artefact had been independently examined at the time of writing.[7] Ethically, Buckmaster has said publicly that OpenAI's approach mirrors work he and Alpöge shared in confidence and that the information reached OpenAI around 3 September; OpenAI has denied appropriation, pointed to methodological differences, and acknowledged that de-identified user interactions may have informed its models.[5][8] Tao publicly congratulated Buckmaster and Alpöge on their Euler result in the same week, a notably different tone from his handling of OpenAI's claim.[5]
| Result | Who | When | What is claimed | Status |
|---|---|---|---|---|
| Unstable singularities in porous-media, Boussinesq and bounded Euler equations | DeepMind, NYU, Stanford, Brown, Princeton | 18 Sep 2025 | New solution families; Navier–Stokes explicitly not resolved | Published preprint; not disputed |
| Finite-time blow-up for forced Euler | Buckmaster and Alpöge | 15 Aug 2026 | A proof, AI-assisted, for the frictionless equations | Preprint; welcomed by Tao |
| Finite-time blow-up for forced Navier–Stokes | OpenAI | 8 Sep 2026 | A construction with unbounded velocity and bounded energy | Unreviewed; scope and priority disputed |
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 the 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.[9] 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, a handful of problems by Tao's count since, and seven of ten unpublished research problems solved to publication standard in his controlled trial in May.[10][11] The second rule is that the verifiable 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 Lean proof would settle the mathematics, if one is released; 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 own materials are the only account of what was done, the figures in them are OpenAI's, and the Nature audit published in March found fabricated citations spreading through the literature at roughly one paper in 280, a reminder that the tools involved are not yet trustworthy narrators of their own work.[12]
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 of the kind Buckmaster and Alpöge had already produced for Euler, presented as more. The community will know within months, and this briefing will say so when it does.
