OpenAI obtained a proof that Navier-Stokes flow reaches unbounded velocity under a specially chosen external force; mathematicians debate the method's origins

OpenAI obtained a proof that Navier-Stokes flow reaches unbounded velocity under a specially chosen external force; mathematicians debate the method's origins

On 22 October, Scientific American compared Tristan Buckmaster's lecture, his documents and OpenAI's response. The mathematics is being checked, while those involved and outside experts give differing accounts of the method's origins.

The Navier-Stokes equations describe the motion of a viscous fluid. The Clay Mathematics Institute's formulation of the problem allows a smooth external force. This force is chosen so that the model's flow velocity grows without bound in finite time while the force itself remains smooth. Diego Córdoba and Luis Martínez-Soroa proposed constructing the force in layers, with each new layer amplifying the previous one.

On 15 August, Buckmaster and Alpöge obtained a result for the Euler equations, which model a fluid without friction, with a smooth external force. On 22 August, they formalized the proof in Lean, a formal system that breaks an argument into steps and checks the logical connections between them. On 7 September, Terence Tao wrote that a version of their method would very likely extend to Navier-Stokes as well.

Following rumors that Anthropic had solved a Millennium Prize Problem, OpenAI assigned a new internal model to the open Millennium Prize Problems and the Euler equations on 1 September. Around a hundred agents, copies of the model running simultaneously, found a construction for the Euler equations without an external force. A swarm of roughly 10 thousand agents then received this proof as a starting point; by 5 September, it had produced a solution for Navier-Stokes with an external force.

Buckmaster describes this sequence as two branches of the same research program. In his view, the branch addressing the Euler equations without a force builds on the work of Córdoba and Martínez-Soroa. Fan Zheng, who worked with them, said:

“As far as I can tell, OpenAI's proof largely follows the framework of our program. But OpenAI manages to overcome several technical difficulties to remove the force.”

Buckmaster sees the route to Navier-Stokes as a more extensive development of that work, incorporating new ideas. Other experts who examined the proof see a different route: OpenAI uses the same external force permitted by the problem's conditions, but constructs it differently.

OpenAI states that Buckmaster's prompts in Codex and ChatGPT after 3 July had no influence on the model or its result. Outside observers can see the published texts, while the sequence of decisions made by the swarm remains internal to the company. The mathematician Camillo De Lellis described the limits of the dispute this way:

“To be honest, I do not see a factual resolution to this dispute. It is no longer a matter of clearly verifiable mathematical facts.”