OpenAI announced that it used a network of roughly 10,000 autonomous AI agents to tackle significant portions of the Navier-Stokes existence and smoothness problem, completing the effort in 88 hours. The system exchanged nearly 3 million messages and generated about 130 billion output tokens during the run, representing a computational effort estimated at roughly $10 million based on public pricing for OpenAI's highest-tier models.

The Navier-Stokes equations describe how fluids move and are central to understanding phenomena like turbulence. A rigorous proof of the problem's core questions has eluded mathematicians for about 90 years and is one of seven Millennium Prize Problems identified by the Clay Mathematics Institute, each carrying a $1 million award.

OpenAI said its work resolves two of the four statements needed for a complete proof under the Millennium Prize framework. The company stated it does not intend to seek the $1 million prize, framing the effort instead as a demonstration of its AI models' capabilities. However, the proposed solution has not yet undergone independent verification or been formally accepted by the Clay Mathematics Institute.

What Sparked the Controversy Over OpenAI's Timing?

The announcement has prompted significant questions about when OpenAI began its work. Tristan Buckmaster, a mathematics professor at New York University, publicly raised concerns that OpenAI may have learned from his concurrent research on the same problem.

Buckmaster explained that he and Anthropic mathematician Levent Alpöge had been working on the Navier-Stokes problem using OpenAI's Codex tool. He stated that he learned on September 3 that "information about our progress had been passed to OpenAI." Buckmaster said OpenAI did not begin working on the equations until "after information about our work had reached OpenAI".

Buckmaster

"What I was told, when, and what was proposed to me...because the alternative is to let a sequence of announcements say something I know to be false," explained Buckmaster, describing his decision to raise the issue publicly.

Tristan Buckmaster, Mathematics Professor at New York University

OpenAI responded by describing Buckmaster and Alpöge's concurrent work as "remarkable" and denied having reviewed their private work or user data before it was publicly released. The company stated it had not seen "any of their work through any means until they released it publicly".

How Did OpenAI Address the Data Sharing Concern?

While OpenAI denied direct access to Buckmaster and Alpöge's research, the company acknowledged a potential indirect pathway. OpenAI stated that while unlikely, it "cannot rule out that de-identified data derived from their usage of our products helped improve our models." The company emphasized, however, that "our proofs differ significantly and even the precise results proved are different".

OpenAI also noted that it had "heard rumours that two Millennium Prize problems had been resolved" on September 1, before its Navier-Stokes results were released. The company described the internal model behind the project as "significantly more capable" than its previous releases and called the findings a "milestone".

Steps to Understanding AI's Role in Mathematical Research

• Agent-Based Approach: OpenAI deployed approximately 10,000 autonomous AI agents working in parallel, each tackling separate approaches to the problem and exchanging results to build toward a solution.
• Computational Scale: The effort generated roughly 130 billion output tokens and exchanged nearly 3 million messages, demonstrating the massive computational resources required for cutting-edge mathematical work.
• Verification Gap: The proposed solution has not yet undergone independent peer review or been formally accepted by the Clay Mathematics Institute, which requires publication and acceptance by the mathematical community before awarding the Millennium Prize.

The scale of OpenAI's effort reflects a growing trend of using AI systems for mathematical research, where multiple agents can work on separate approaches simultaneously and consolidate their findings. This represents a shift in how complex mathematical problems are being tackled, moving from individual researchers to coordinated networks of AI systems.

The lack of independent verification means OpenAI's findings do not yet constitute an accepted solution to the Millennium Prize problem. The Clay Mathematics Institute requires a proposed solution to meet specific criteria, including publication and acceptance by the mathematical community, before a Millennium Prize can be awarded.