Mathematicians are finally making sense of the biggest event in decades for their field—and it’s stranger than anyone thought.
In early September OpenAI posted an inscrutable 166-page solution to the Navier-Stokes problem, one of math’s famed “million-dollar problems.” Initial accounts—including from the company itself—suggested that the proof, produced by an internal large language model (LLM), followed an approach devised by two Spanish mathematicians, Diego Córdoba and Luis Martínez-Zoroa.
The model’s purported reliance on this “Spanish program” sparked a very public conflict between the company and Tristan Buckmaster, a mathematician at New York University. Buckmaster had spent much of the past year using LLMs to carry out that program alongside Levent Alpöge, an employee of OpenAI’s archrival, Anthropic. The pair cried foul after OpenAI—whose public models Buckmaster and Alpöge had been using—announced a Navier-Stokes proof that seemed to build upon and then leapfrog over their hard-won progress. But the resemblance between the two groups’ methods now seems merely skin-deep.
“When you go to the details, it is a different approach,” says Gonzalo Cao-Labora, a mathematician at EPFL, the Swiss Federal Institute of Technology in Lausanne. Scientific American spoke with several groups of experts working through the OpenAI paper. All said that it’s increasingly clear that the company’s proof does not follow the Spanish program.
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