The Rundown:
OpenAI published a proof from an unreleased internal model, claiming to settle Navier-Stokes, one of the seven $1M Millennium Prize problems — with NYU's Tristan Buckmaster questioning whether OAI raced with help from his drafts in Codex.
The details:
OAI said it ran roughly 10,000 agents at once on a model “significantly more capable” than GPT-6 Astra, creating the proof in 88 hours.
The company estimated the compute cost at “millions of dollars,” with Sam Altman calling it “one of the most amazing moments for me in OpenAI history.”
Anthropic's Levent Alpöge and Buckmaster spent a year on a similar route, feeding drafts into Codex and posting partial results the night before OAI’s own.
Buckmaster released a statement saying OAI only started after hearing of their work, and never answered whether his Codex drafts trained the model.
OAI said it “did not see any of their work” and that “no specific user data was accessed,” but can’t rule out that usage data may have improved its models.
Why it matters:
The fight over credit has unfortunately overshadowed the biggest math breakthrough in an AI summer that has already changed the field. With an internal model already “significantly more capable” than the just publicly-released Astra, whatever ceiling you had in mind for what models can do this year probably needs to be raised.
Quote:The problem
The Navier–Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move. Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules. These equations are used for aircraft design, weather forecasting, and the study of blood flow.
A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behaviour of each particle individually.
https://openai.com/index/navier-stokes-solution/