Autonomous proof generation

OpenAI announced that an artificial intelligence model solved the Navier-Stokes equations, one of seven Millennium Prize Problems established by the Clay Mathematics Institute. The computation required 88 hours of processing by more than 10,000 AI agents working simultaneously across coordinated groups. The system generated over 130 billion tokens and exchanged 2.7 million messages, followed by 17 hours dedicated to formalization and verification. Mark Chen, head of research at OpenAI, stated that the computing power used during the run cost millions of dollars. OpenAI described the underlying model as possessing capabilities higher than its recently released GPT-6 Astra system.

“It is a significant step in AI research. It is the promise that it will become possible to answer other questions among the most complex.”

Reasoning run88

Verification17

Reasoning run

88 hours

Verification

17 hours

Data from the article text

Fluid dynamics and the Millennium Prize

The Navier-Stokes equations describe how viscous fluids such as water and air move over time. French engineer Henri Navier established the initial mathematical framework in 1822, and British physicist George Gabriel Stokes completed the definitive version in 1845. The equations serve practical applications in aviation wing design, weather forecasting, tornado modeling, and blood circulation analysis. In 2000, the Clay Mathematics Institute designated Navier-Stokes as a Millennium Prize Problem with a $1 million award for a verified proof. Until this announcement, the Poincare conjecture, solved by Russian mathematician Grigori Perelman in 2002, was the only solved problem on the list. OpenAI stated that its proof demonstrates how a vortex can narrow and elongate, causing fluid velocity to become theoretically infinite in finite time while maintaining limited total energy.

Academic credit and Spanish mathematicians

The announcement prompted scrutiny from independent researchers regarding proper attribution of prior work. Spanish mathematicians Luis Martinez of CUNEF and Diego Cordoba of the Institute of Mathematical Sciences developed foundational methods on fluid evolution that OpenAI incorporated into its process. OpenAI initially omitted references to their published work before updating its release to cite both mathematicians following public criticism. Martinez stated that the Navier-Stokes problem had resisted mathematical solutions for generations and that their method was essential for the AI system to reach a conclusion. The proposed proof remains subject to formal peer review and verification by the global mathematical community before any official prize confirmation.

Corporate competition and Anthropic rivalry

The project developed directly from industry competition between OpenAI and AI rival Anthropic. OpenAI chief executive Sam Altman acknowledged that the company began the effort after hearing rumors that Anthropic models had solved a Millennium Prize problem. New York University mathematician Tristan Buckmaster and Anthropic researcher Levent Alpoge had spent August working on the related Euler equations using internal Anthropic models. OpenAI researcher Sebastien Bubeck published an account on 8 September following private calls with Buckmaster, denying that he attempted to strip Alpoge of credit while apologizing for remarks made during tense discussions.

“When they are two of the biggest companies in the world, they need to get over it. I am just a bystander in these petty games.”