OpenAI announced a result on one of the best-known open questions in mathematics. An unreleased model produced a proof showing that a finite-time singularity can form in the Navier-Stokes existence and smoothness problem.
The day after the announcement, what academia was discussing was not the proof. The argument turned to how OpenAI picked this research direction in the first place.
Roughly ten thousand agents, 88 hours
According to the company, on 28 August it began training an internal model that performed markedly better at mathematics than its predecessors. On 1 September the team aimed that model at problems still open. The system first focused about 100 agents on an easier question in the Euler equations and reached a result in roughly 50 hours.
Resources were then shifted to Navier-Stokes and the agent count was raised to about 10,000. That single problem consumed 2.7 million messages and roughly 130 billion output tokens. Formal verification of the proof in Lean was completed in 17 hours using GPT-6 Astra. OpenAI research lead Mark Chen said the compute spent cost millions of dollars.
| Stage | Scale | Duration |
|---|---|---|
| Euler equations, trial run | ~100 agents | ~50 hours |
| Navier-Stokes proof | ~10,000 agents | 88 hours |
| Formal verification in Lean | GPT-6 Astra | 17 hours |
What the proof actually says
The Navier-Stokes equations describe how fluids such as water and air move. At the centre of the problem sits the question of whether a three-dimensional flow with smooth initial conditions always stays smooth. The Clay Mathematics Institute named it one of seven Millennium Prize problems in 2000 and attached a one million dollar award.
- The result shows that a singularity can develop in finite time when a smooth external force is applied to a fluid that starts at rest and smooth.
- OpenAI states that this satisfies options C and D of the problem as the institute defined it.
- That forced version is a different question from the force-free case, which is what the prize is actually attached to.
So there is a genuine mathematical result here, but it is not the branch that wins the Millennium award itself. That distinction went missing in most of the headlines.
Two mathematicians who spent a year on the same line
Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpöge, a mathematician working at Anthropic, published the results of roughly a year of research just before OpenAI's announcement. The work was not carried out on Anthropic's behalf; the pair drew on both Claude models and OpenAI Codex.
Buckmaster says the team's main tool was GPT-5.6 Sol, with Astra entering later for writing and proof checking. The critical advance in their work came on 15 August, and the Euler result was verified in Lean on 22 August. Both dates precede the intensive compute run OpenAI says it started on 1 September.
The objection is direction, not copying
Buckmaster is not making an accusation of copying. In his own statement he says he has not seen OpenAI's proof and does not know whether his data was used. His objection sits at a finer point: knowing which problem has ripened before you spin up tens of thousands of agents is not the same as ripening it.
He also stresses that the line builds on years of work by Diego Córdoba and Luis Martínez-Zoroa. Compute can accelerate a research line; finding that line is separate labour. There is still no settled convention for how AI companies credit that labour, and that is where the argument comes from.