OpenAI claims Yang-Mills proof, beating Anthropic mathematicians to it

What's going on

According to Scientific American, OpenAI has used a proprietary internal large language model (LLM) to present a proof of one part of the Yang-Mills problem, which mathematicians have been working on for decades. The problem is one of the Clay Mathematical Institute's Millennium Prize Problems. The proof overtakes the work of two people who were tackling the same question with Anthropic's Claude: Levent Alpöge (an Anthropic employee) and Ethan Sussman (Northwestern University). According to the source, the story is still developing.

What the Yang-Mills problem is

The Standard Model of particle physics explains all known forces with 17 interacting particles. They are described by "quantum fields," mathematical objects that permeate space and time and determine when and where particles appear. Physicists use this math to reliably predict experiments at the Large Hadron Collider. Mathematicians worry, however, that the theory has gaps.

The Yang-Mills problem is meant to put it on firm mathematical footing and has two parts. You have to prove that the theory is consistent, meaning it "exists," and that it has a "mass gap" greater than zero, which has important physical consequences. In the 1980s, Tadeusz Balaban sketched a rough path to the first part, running to thousands of pages and incomplete. According to Roland Bauerschmidt (NYU), experts didn't seriously doubt it could be completed with enough effort. Nobody did it.

How the human-AI teams got there

This summer, LLMs moved to the front of research mathematics, mainly thanks to their stamina for grind work. Sussman saw them as the right tool for Balaban's program, since the obstacle was "the amount of work," not a lack of ideas. Around the same time, Alpöge was getting results with Claude, including a reliably generated proof of about 100 pages on six-dimensional spheres.

On September 8, Sussman wrote to Alpöge, who invited him to collaborate. Alpöge used Anthropic's internal version of Claude, with many agent copies working together, to generate thousands of pages of mathematics. He checked them with further agents and condensed them into more readable summaries. Sussman: "The vast majority, if not all, of the errors in the version published on GitHub are mine." The agents didn't just fill gaps. They also found some new approaches.

On October 6, the two announced their progress in a public GitHub repository, because rumors were circulating that OpenAI would publish many results. On Sunday evening they updated it. They believed they had solved the three-dimensional version of the existence problem. That wouldn't meet the Clay criteria (four-dimensional), but according to Sussman the core ideas are probably the same. He estimated they were "maybe only a few weeks" away.

OpenAI's claim and the dispute

On Monday evening, OpenAI announced it had proved the "existence" part of the Clay problem. The proof isn't published. OpenAI handed it to a group of experts who are supposed to "shepherd" it into the literature. That sparked protest, including from the advisory panel of mathematicians that OpenAI itself convened after earlier criticism.

According to OpenAI's blog post, the model found the result on September 9 but kept it secret until it had been formalized in Lean. Lean is a programming language used to check proofs for correctness by machine. That reportedly took well over a month, versus 17 hours for the Navier-Stokes proof. Tristan Buckmaster (NYU) disagrees: "There is no way the Lean verification of Yang-Mills took a month." It would take a few days at most. His dispute with OpenAI over Navier-Stokes is largely unresolved.

Alpöge voiced outrage on X: OpenAI didn't reach out to collaborate, as is often suggested in public. Sussman says: "I won't say I'm not disappointed."

Reactions

Michael R. Douglas (Harvard, theoretical physicist and AI proponent) calls it a "major advance." Mathematicians have been trying for more than 50 years to make quantum field theory rigorous, and real-world theories like quantum chromodynamics seemed out of reach. He hopes for rapid further progress.

Our take

If the proof holds up, it would be a big step for mathematical physics and a strong argument for AI as a research tool. For now, though, it isn't publicly available and so hasn't been independently checked. According to Scientific American, the origin of mathematical ideas is also getting blurred, and the AI companies' race ahead of IPOs comes at the expense of scientific clarity and academic norms. Everyone involved says they want new norms, yet nobody is hitting the brakes. The second part (the mass gap) is also still open, as is the question of how credit and prize money would be divided.