OpenAI's Navier-Stokes proof differs fundamentally from the claimed Spanish approach

OpenAI announced a 166-page solution to the Navier-Stokes equations, a celebrated million-dollar problem, claiming it built on a program created by Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa. The company’s internal large language model generated the proof using an "agent swarm" of roughly 10,000 copies that collectively processed about 88 hours and 130 billion tokens to produce a blow-up—a mathematically infinite fluid speed that cannot occur physically.

A public dispute emerged when Tristan Buckmaster of NYU, who had been collaborating with Levent Alpöge on a similar line of inquiry, alleged OpenAI had appropriated their progress and even threatened him. However, mathematicians such as Gonzalo Cao-Laborra and Camillo De Lellis now say the OpenAI proof follows a fundamentally different approach, despite sharing a loophole in the Clay Institute’s problem framing. The proof was encoded in Lean for formal verification, and other research groups are employing additional LLMs to distill a more understandable version. Nonetheless, the black-box nature of the AI means the exact reasoning may never be fully revealed.