OpenAI announced that it has found a solution to one of the most complex and ancient problems in mathematics: the existence and smoothness problem of Navier-Stokes, which is part of the seven Millennium Prize Problems established by the Clay Mathematics Institute.
According to the company, an internal artificial intelligence (AI) system generated an analytical proof demonstrating that the Navier-Stokes equations can exhibit a singularity at a specific time. OpenAI also made a formalization of this proof available in the Lean language, which is used for mathematical verification of the result.
This announcement marks a notable progress in the application of AI to mathematics. The Navier-Stokes equations are crucial for modeling fluid motion, being applied in fields such as weather forecasting, aircraft design, and blood flow study.
The question that remained unanswered for decades was whether these equations maintained a smooth solution in three dimensions or if, under certain conditions, they could collapse and generate a singularity, a moment when fluid velocity would increase infinitely in a finite period.
The problem was included in the seven Millennium Prize Problems in 2000, with a prize of US$1 million (equivalent to R$5 million) for anyone who presented a correct solution.
How the solution was achieved
To reach this result, OpenAI employed a system composed of coordinated AI agents. It is estimated that about ten thousand agents worked simultaneously on solving Navier-Stokes, receiving variations of the problem and being distributed into groups to explore different methods.
The process began after OpenAI researchers heard rumors on September 1st that two millennium problems might have been solved. The company then decided to test its new internal model on the remaining challenges.
The system was also used to investigate a similar question related to Euler's equations, which do not contain the viscosity term present in Navier-Stokes. About 100 agents dedicated approximately 50 hours to solving this second problem.
After obtaining this initial result, OpenAI directed its efforts toward Navier-Stokes. The agent groups began exchanging findings, using Codex to consolidate the most relevant information among the various teams.
The agents finally reached the Navier-Stokes solution on September 5th, about 88 hours after the start of the work. The formalization and validation phase of the proof in Lean consumed another 17 hours.
During all tests conducted in the project, the agents sent 4.9 million messages and consumed approximately 300 billion tokens. Specifically in the Navier-Stokes work, 2.7 million messages and about 130 billion tokens were recorded.
The computational cost was also substantial. According to supporting material, the operation demanded processing power estimated in millions of dollars due to the vast amount of resources required to operate so many AI systems simultaneously.
Although it considers the finding as a resolution of the Navier-Stokes problem, OpenAI stated that it does not intend to claim the US$1 million prize offered by the Clay Mathematics Institute.
The company released the work as a demonstration of the advancement of its AI models, publishing both the description of the solution and its formalization in Lean, allowing other mathematicians to examine the result.
Scientific Dispute Involved
However, the announcement did not end the debate about the achievement. In addition to the mathematical analysis of the proof, a controversy arose involving researchers working on related topics and the possibility that ideas developed by them had reached OpenAI.
OpenAI's announcement occurred amidst a scientific dispute involving researchers linked to the company itself and Anthropic. Tristan Buckmaster, a mathematics professor at New York University (USA), and Levent Alpöge, a researcher at Anthropic, were working on issues related to fluid dynamics.
Both researchers had presented results obtained with 'significant help' from language models, including systems from Anthropic and OpenAI. Their work focused on Euler's equations and presented a phenomenon called 'blow-up', or singularity. Part of these results had also been formalized in Lean.
The situation took on the contours of a dispute over credit when Buckmaster alleged that rumors about the method developed by his team might have been transmitted to OpenAI. He maintained that an internal team at the company had used its own model to advance in the complete Navier-Stokes problem.
There was also discussion about authorship. Buckmaster stated that OpenAI offered him exclusive authorship on a paper acknowledging that the internal model had solved the problem, but without including Alpöge, who works for a competing company.
OpenAI denied having accessed the researchers' work before its public disclosure. The company assured that its researchers and agents did not view the material from Buckmaster and Alpöge by any means before publication.
In its statement, OpenAI added that, upon contacting the two researchers after completing its own work, it discovered that they had solved a distinct version of the problem: the case of Euler's equations with external force. The company recognized the researchers' priority in this result.
This advancement also reignites the debate about the role of AI in generating scientific knowledge. AI systems are being trained to solve mathematical problems through techniques such as reinforcement learning. By repeatedly interacting with problems whose answers can be objectively verified, the models learn which strategies are effective and which lead to errors.
The Lean language enhances this process by enabling mathematical proofs to be converted into code and submitted to formal verification.
OpenAI's result is particularly relevant because the Navier-Stokes problem was among the most difficult in modern mathematics. For decades, mathematicians sought to determine whether the equations could develop a singularity or remain always smooth.
Thus, the feat transcends a mere demonstration of computational capability; it brings AI closer to a function that was long considered exclusively human: discovering new approaches to fundamental mathematical questions.
Simultaneously, the dispute between OpenAI and researchers linked to Anthropic raises a question that may become increasingly crucial as AI actively participates in scientific research: who holds the authorship of a discovery when humans and machines collaborate—and how to protect unpublished ideas while they are being developed with the aid of large AI laboratories?