On September 8 local time, OpenAI announced a major breakthrough: an internal large language model, not yet publicly released and with significantly stronger mathematical capabilities than GPT-6 Astra, completed the proof for the existence and smoothness of solutions to the Navier–Stokes equations. This was achieved by coordinating approximately 10,000 AI agents, taking only 88 hours.
This is one of the seven Millennium Prize Problems proposed by the Clay Mathematics Institute in 2000, carrying a prize of $1 million, and has puzzled the mathematical community for nearly 90 years.
The Navier–Stokes equations are fundamental equations describing fluid motion, widely used in weather forecasting, aircraft design, and fluid simulation. The core question of this problem is: if the initial state of a fluid is sufficiently smooth and its total kinetic energy remains finite, will the fluid velocity tend toward infinity within a finite time, thereby creating a singularity?
OpenAI's proof conclusion is that such blow-up singularities can exist.
The entire workflow consisted of two steps: 10,000 AI agents explored in parallel to obtain the core proof strategy within 88 hours; subsequently, another 17 hours were spent using the Lean formal proof system to complete rigorous verification, resulting in a publicly released 165-page paper and formalized code.
The computational cost for this task was enormous. Solving the problem consumed approximately 130 billion tokens, with the total project consumption nearing 300 billion tokens. The project cost amounted to millions of dollars.
Of the seven historical Millennium Prize Problems, only the Poincaré conjecture has been proven by human mathematicians. If this achievement passes strict peer review by the global mathematical community, it will be the second Millennium Problem to be solved.
However, many leading mathematicians have raised questions, holding different views on the proof's logic and attribution of credit. Some scholars have also raised copyright disputes regarding prior related research. Currently, the Clay Mathematics Institute has not officially recognized the proof as valid, nor has it awarded the Millennium Prize.
It is worth mentioning that GPT-6 Astra, the most powerful model ever released by OpenAI, recently solved five long-standing Erdős mathematical problems that had remained unsolved by humans. These include the 1931 dissociation set conjecture, which Erdős proposed at age 18 and called his "first serious problem," as well as the renowned Erdős–Sós conjecture in extremal graph theory. These problems have stumped countless top mathematicians for decades.