OpenAI has announced that an unreleased AI model successfully solved the Navier-Stokes existence and smoothness problem, a mathematical challenge that has stumped researchers for nearly nine decades and is listed among the Clay Mathematics Institute’s Millennium Prize Problems.
The company claims that approximately 10,000 AI agents collaborated to reach the result in just 88 hours. However, the announcement has not yet been met with widespread independent acceptance, and questions have been raised by NYU mathematics professor Tristan Buckmaster regarding the research timeline and potential use of training data.
Massive AI Agent Collaboration
The Navier-Stokes equations describe the motion of fluids like water and air, with applications ranging from aircraft design to weather forecasting and blood flow analysis. The core unresolved question is whether smooth three-dimensional fluid motion can develop a singularity, where speed becomes unbounded within finite time.
OpenAI says its system discovered a construction in which an initially smooth fluid develops such a singularity. The solution employs a smooth external force and establishes statements ‘C’ and ‘D’ from the Clay Institute’s official formulation, allowing researchers to resolve the problem by demonstrating that smooth solutions can break down under specified conditions.
Training for the new internal model began on August 28. After hearing rumors on September 1 that two Millennium Prize problems might have been solved, OpenAI directed groups of agents toward the remaining challenges. For Navier-Stokes, roughly 10,000 concurrent agents participated, exchanging about 2.7 million messages and generating around 130 billion output tokens. The agents reached their result on September 5, roughly 88 hours after the project commenced. GPT-6 Astra then spent an additional 17 hours on Lean formalization and verification.
Academic Scrutiny and Concerns
Buckmaster, along with mathematician Levent Alpöge, had been independently working on closely related fluid equations for most of the previous year. Their efforts utilized various AI systems, including OpenAI’s Codex and Anthropic’s Claude, building on prior research by Diego Córdoba and Luis Martínez-Zoroa.
Buckmaster claims he and Alpöge obtained smooth-forcing blowup results for the Boussinesq and Euler equations on August 15, with Lean verification completed on August 22. Their published work, however, 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 might have reached the company. He later asked whether OpenAI’s internal model had been trained on Codex sessions containing their drafts. “I asked again, about training, and I did not get an answer,” he wrote. Additionally, he noted that after telling OpenAI he would publicly describe their discussions, he received the response: ‘Why would you ruin your career?’ Buckmaster emphasized that he did not know whether OpenAI had used his research data and clarified that he is not making a direct accusation about how the company obtained its proof.
OpenAI’s Response
OpenAI has denied that its researchers or AI agents accessed Buckmaster and Alpöge’s work before it was publicly released. The company also stated that no specific user data was used to solve the Navier-Stokes problem. However, OpenAI acknowledged it could not completely rule out the possibility that de-identified data derived from their use of its products might have contributed to model improvements. The company maintains that its proofs differ from the researchers’ work and notes that the Euler results address different versions of the problem.
The Clay Mathematics Institute continues to list Navier-Stokes as a Millennium Prize Problem with a US$1 million award. OpenAI has stated it does not intend to claim the prize and has released both its mathematical paper and a Lean-formalized proof for public review.