OpenAI claims a proof of Navier-Stokes blowup
The company published a proof that a singularity can form in finite time in the three-dimensional Navier-Stokes equations, together with a Lean formalisation, and said the system that produced it significantly exceeds GPT-6 Astra.
Why it matters
If it stands, a Millennium Prize Problem open for about ninety years was solved by machine, and the result arrived already formalised - so what is disputed is not correctness but priority and attribution.
The publication is dated 8 September 2026. OpenAI reports that an internal system produced both an analytic proof and a Lean formalisation showing that a singularity can form in finite time in an initially smooth fluid at rest: a smooth force is applied, the energy stays finite, and the velocity grows without bound. The company states that the result resolves the Millennium Prize Problem by establishing statements C and D in its official formulation. The solution is described as a vortex spiralling inward and stretching along its axis. OpenAI notes that it used an internal model that significantly exceeds GPT-6 Astra. According to Quanta Magazine, nearly 100 agents worked about 50 hours on the Euler regularity disproof, and roughly 10,000 agents worked 88 hours on Navier-Stokes, after which another model formalised the result in a further 17 hours; the agents exchanged almost five million messages. Twelve hours before OpenAI's announcement, Tristan Buckmaster of New York University and Levent Alpoge published results of their own; the parties give different versions of what happened. As of mid-September 2026 the proof had not cleared conventional peer review.