OpenAI says an unreleased AI model has solved the Navier-Stokes existence and smoothness problem. The mathematical question has remained unresolved for nearly 90 years and is one of the Clay Mathematics Institute’s Millennium Prize Problems.
The company said about 10,000 AI agents worked on the problem and reached the result in 88 hours. However, the claim has not yet received broad independent acceptance. It has also drawn attention after NYU mathematics professor Tristan Buckmaster raised questions about OpenAI’s research timeline and possible model training data.
The Navier-Stokes equations describe how fluids such as water and air move. They have uses in aircraft design, weather forecasting, and studies of blood flow. The unresolved mathematical question asks whether smooth three-dimensional fluid motion can develop a singularity, where speed becomes unbounded within finite time.
OpenAI said its system found a construction in which an initially smooth fluid develops such a singularity. The solution uses a smooth external force and establishes statements ‘C’ and ‘D’ in the Clay Institute’s official formulation. Those statements allow researchers to resolve the problem by showing that smooth solutions can break down under specified conditions.
The company began training its new internal model on August 28. After hearing rumors on September 1 that two Millennium Prize problems may have been resolved, OpenAI directed groups of agents toward the remaining problems.
For Navier-Stokes, roughly 10,000 concurrent agents took part. They exchanged about 2.7 million messages and generated approximately 130 billion output tokens. OpenAI said the agents reached the result on September 5, around 88 hours after the project began. GPT-6 Astra then spent another 17 hours on Lean formalization and verification.
Buckmaster and mathematician Levent Alpöge had separately worked on closely related fluid equations for most of the previous year. They used several AI systems, including OpenAI’s Codex and Anthropic’s Claude. Their work built on research by Diego Córdoba and Luis Martínez-Zoroa involving fluid blowups with external forcing.
Buckmaster said he and Alpöge obtained smooth-forcing blowup results for the Boussinesq and Euler equations on August 15. They completed Lean verification on August 22. Their published work did not establish OpenAI’s full Navier-Stokes result.
In his statement, Buckmaster said he contacted an OpenAI mathematician on September 3 after hearing that information about their progress may have reached the company.
Buckmaster later asked whether OpenAI’s internal model had been trained on Codex sessions containing their drafts. He wrote, "I asked again, about training, and I did not get an answer.”
He also said that after telling OpenAI he would publicly describe their discussions, he received the response: ‘Why would you ruin your career?’ Buckmaster stressed that he did not know whether OpenAI had used his research data and said he was not making a direct accusation about how the company obtained its proof.
OpenAI said neither its researchers nor its AI agents saw Buckmaster and Alpöge’s work before the researchers released it publicly. The company also said no specific user data was accessed to solve the Navier-Stokes problem.
However, OpenAI added that it could not completely rule out whether de-identified data derived from their use of its products had helped improve its models. It said its proofs differ from the researchers’ work and noted that the Euler results address different versions of the problem.
The Clay Mathematics Institute continues to describe Navier-Stokes as a Millennium Prize Problem carrying a USD 1 million award. OpenAI said it does not plan to claim the prize and released both its mathematical paper and a Lean-formalized proof for review.