On September 8 (local time), OpenAI announced that an unreleased internal model generated a proof demonstrating the solution to the "Existence and Smoothness of the Navier-Stokes equations," one of the Millennium Prize Problems. This proof indicates that a three-dimensional incompressible fluid starting from a smooth state can develop a "singularity" where the velocity diverges to infinity within a finite time. Along with an explanation of the proof, OpenAI also released the formalization created using the proof assistant "Lean."

According to OpenAI's announcement, the proof was generated by an internal model more powerful than GPT-6 Astra, utilizing a method that ran approximately 10,000 agents in parallel. The company stated that this effort began after they heard rumors regarding a solution to the problem on September 1, reaching a conclusion on September 5 after approximately 88 hours of computation. Furthermore, the company stated that it will not claim the Millennium Prize award money.

Meanwhile, prior to this announcement, Tristan Buckmaster, a mathematician at New York University, released a statement asserting that there were issues with the process of OpenAI's effort. Buckmaster pointed out that OpenAI deployed computational resources using a similar approach immediately after learning about his research results. Sebastian Bubeck, OpenAI's head of mathematical research, countered this by stating that the claims are "contrary to fact and inflammatory," sparking a dispute over the validity of the results and the history of the announcement.

Source: