OpenAI reported this week that one of its AI models produced a proof for the Navier-Stokes existence and smoothness problem, a long-standing unsolved question in mathematics. If verified by external mathematicians, the claim would mark a major step forward for AI, showing it can contribute to real scientific discovery, not just generate convincing text.
The paper is now entering a peer review process with mathematicians outside the company. The problem is one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute in 2000. Solving one carries a $1 million prize, and to date, only one of the seven, the Poincaré conjecture, has been solved previously.
The Navier-Stokes equations describe the motion of fluids. They are fundamental to weather forecasting, aerodynamics, and numerous other fields of fluid mechanics. The Millennium Problem specifically asks whether the equations always yield smooth, physically meaningful solutions in three dimensions, or whether, under certain conditions, they can develop a singularity-a point where the mathematical model breaks down and the equations cease to produce valid results.
OpenAI stated that its proof identifies such a breakdown. The company emphasized that this finding doesn’t imply that water or air will behave in physically impossible ways. Instead, the result concerns the limits of the Navier-Stokes equations as a mathematical framework, not a prediction of real-world physical violation.
OpenAI didn’t use one large model for this. Instead, several groups of AI agents worked on different reasoning paths at the same time, while researchers moved back and forth between them, sharing useful insights. Dan Roberts, one of the researchers involved, likened it to “a bumblebee cross-pollinating across different groups and delivering different bits of information.”
Crucially, the AI agents worked within Lean, a formal proof language. Lean is a tool that checks whether each logical step in an argument follows the system’s rules. While it doesn’t judge the importance of a result, it can rigorously establish whether the steps of a proof are valid under a defined set of assumptions. This is particularly useful for AI-generated mathematics, where models are known to produce convincing but flawed reasoning in ordinary prose.
The project is part of a wider effort in the AI industry to train models on tasks with clear pass-fail signals. A mathematical proof can be tested, and a computer program can be run, which provides reinforcement-learning systems with a concrete way to learn through trial and error. This method is much harder to apply in fields where correctness is subjective, such as creative writing or ethical decision-making.
The Navier-Stokes claim is more consequential than previous AI-for-math announcements due to the problem’s prominence and difficulty. Over the past year, OpenAI and other companies have reported progress on open problems, such as an Erdős problem reportedly solved by OpenAI and Harmonic in January. However, some mathematicians noted that result resembled earlier human work, questioning whether a genuinely new approach was created.
OpenAI said it turned its attention to the Navier-Stokes problem after learning that other researchers were pursuing related work, adding that it “did not see any of their work through any means.” The announcement came shortly after Tristan Buckmaster, a mathematics professor at New York University, said he had been investigating related questions with a mathematician who works at Anthropic, an OpenAI competitor.
The computing resources required for this project were significant. According to OpenAI, the process involved as many as 10,000 agents running simultaneously, likely costing millions of dollars in compute. Research scientist Noam Brown acknowledged it was “a very expensive process,” but said he expects costs to decrease as the company’s systems and infrastructure improve.
For many in the mathematics world, the bigger question isn’t just whether this proof turns out to be correct. Terence Tao, a UCLA professor and one of the most respected mathematicians today, has warned that AI solving major problems with little human involvement could end up weakening the field. He previously told The New York Times, “The effort needed to solve problems is often very instructive.”
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