On September 8th local time, OpenAI announced that its internal, undisclosed AI model (which has performance far exceeding the recently released GPT-6 Astra) has successfully solved the "Navier-Stokes existence and smoothness problem," one of the seven "Millennium Prize Problems" proposed by the Clay Mathematics Institute in 2000, which has puzzled the mathematical community for about 90 years.
Previously, Anthropic had announced last month that its internal research model, Claude, made significant progress in solving the Riemann Hypothesis, raising the long-term lower bound of the zero point ratio on the critical line of the Riemann zeta function from 41.6% to 67.2%.
Agent Collaboration Generates Proof, Lean Formal Verification Completed in 17 Hours
OpenAI stated that the team had around 10,000 AI agents collaborate to generate the proof within approximately 88 hours. The proof shows that a initially smooth stationary fluid might develop a "singularity," where velocity increases infinitely, within a finite time. The proof achieved this singularity by designing a "vortex" that rotates and stretches inward, maintaining finite energy under physical laws; the proof is based on the premise of smooth external forces, corresponding to the "C" and "D" scenarios in the official problem statement.
OpenAI provided an analysis of the proof and formal verification using the Lean programming language, which was completed by GPT-6 Astra in approximately 17 hours. OpenAI researcher Sébastien Bubeck called this an "amazing peak" of the trajectory seen over the past twelve months.
Controversy Over Priority and Tao's Concerns
The announcement quickly sparked controversy. On the previous day, New York University mathematics professor Tristan Buckmaster and Anthropic researcher Levant Alpoge had published related research on fluid equations. Buckmaster questioned whether OpenAI had only turned its resources to its ongoing "externally driven" approach after learning about their progress and worried that his research data stored in OpenAI Codex might have been used for training or reference, but he did not receive a clear response. Bubeck denied these accusations, emphasizing that neither researchers nor AI agents accessed specific user data, but OpenAI also admitted that "it cannot be ruled out" that anonymized data from its products may have helped improve the model.
Mathematician Terence Tao compared such problems to a "lighthouse" that attracts scientists' efforts, while warning that if AI can solve the most difficult problems without deep human involvement, it may weaken human understanding of mathematics. It is estimated that the computational cost could reach between $15 million and $22.5 million (approximately 101 million to 151 million yuan). OpenAI stated that it does not intend to claim the $1 million (approximately 6.728 million yuan) prize from the Clay Mathematics Institute.
