Mathematicians are finally making sense of the biggest event in decades for their field\u2014and it\u2019s stranger than anyone thought.<\/p>\n
In early September OpenAI posted an inscrutable 166-page solution to the Navier-Stokes problem<\/a>, one of math\u2019s famed \u201cmillion-dollar problems<\/a>.\u201d Initial accounts\u2014including from the company itself\u2014suggested that the proof, produced by an internal large language model (LLM), followed an approach devised by two Spanish mathematicians, Diego C\u00f3rdoba and Luis Mart\u00ednez-Zoroa.<\/p>\n On supporting science journalism<\/p>\n If you’re enjoying this article, consider supporting our award-winning journalism by subscribing<\/a>. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<\/p>\n The model\u2019s purported reliance on this \u201cSpanish program\u201d sparked a very public conflict<\/a> 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 Alp\u00f6ge, an employee of OpenAI\u2019s archrival, Anthropic. The pair cried foul after OpenAI\u2014whose public models Buckmaster and Alp\u00f6ge had been using\u2014announced a Navier-Stokes proof that seemed to build upon and then leapfrog over their hard-won progress. But the resemblance between the two groups\u2019 methods now seems merely skin-deep.<\/p>\n \u201cWhen you go to the details, it is a different approach,\u201d 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\u2019s increasingly clear that the company\u2019s proof does not follow the Spanish program.<\/p>\n The Navier-Stokes equations are the math behind every fluid. OpenAI solved the million-dollar problem by uncovering a flaw in the equations. The company\u2019s artificial intelligence concocted a situation in which the equations declared that the speed of a fluid\u2019s flow should be infinite\u2014a so-called blowup that can\u2019t happen in real life.<\/p>\n OpenAI solved the problem with an \u201cagent swarm,\u201d some 10,000 copies of its new, math-adept LLM, which spent about 88 hours of computing time and 130 billion tokens collectively constructing a blowup. The company\u2019s math team threw all its resources at the problem after learning of significant progress made by Buckmaster and Alp\u00f6ge. Buckmaster has alleged that OpenAI then attempted to bribe and threaten him to prevent him going public with insinuations of malfeasance, which he ultimately did.<\/p>\n Even top experts have found OpenAI\u2019s proof of blowup to be completely impenetrable. But the agents coded their arguments into Lean, a programming language used to automatically verify that the logical steps toward a conclusion in a given sequence are all valid. Several prominent research groups are now using both the Lean code and the paper, with the help of other LLMs, to get a simplified, bird\u2019s-eye view of the math involved.<\/p>\n