OpenAI published a proof on 8 September, produced by an internal model it calls significantly more capable than GPT‑6 Astra, that the three-dimensional Navier–Stokes equations can develop a singularity in finite time: a fluid that starts smooth and at rest, pushed by a smooth external force, ends with speeds growing without bound. OpenAI says this settles the Millennium Prize problem by establishing statements C and D of the official formulation, and released both a paper and a Lean formalisation.
A group of roughly 10,000 concurrent agents reached the proof on 5 September after about 88 hours; Lean verification took a further 17. The same system also proved blow-up for the unforced Euler equations.
The same week, Tristan Buckmaster (NYU) and Levent Alpöge (Anthropic) released AI-assisted, Lean-formalised preprints proving blow-up with smooth forcing for the 3D Euler, 2D Boussinesq and porous-medium equations. OpenAI says its effort began on 1 September after rumours of their work, recognises their priority on forced Euler, and says it saw none of it before release. Buckmaster disputes how OpenAI approached him.
The Clay Institute said on 11 September the problem has “apparently been settled” and that its review is “deliberately unhurried”. OpenAI says it will not claim the prize. Outside referees have not yet finished checking either proof.