OpenAI’s progress on another million-dollar problem is finally public. Experts appointed by the company to vet the work say they’ve now verified its artificial-intelligence-generated proof of the “existence” of the quantum Yang-Mills theory—a vital foundation for the Standard Model of particle physics.
Like many AI proofs, this one is messy—too lengthy and complicated for human minds to immediately grasp. Seeking to avoid another “slop drop”—a term used for when OpenAI unleashed a flood of more than 700 AI-generated proofs on a single day in October—the company enlisted the eminent American theoretical physicist David Gross to lead a committee to supervise the Yang-Mills proof’s release.
“We update the progress of our committee in shepherding the LLM [large language model] produced result,” the group said in a statement on November 9 previewing its findings. The team’s work confirms that OpenAI’s result “proves the existence of pure Yang-Mills theory according to the problem statements which are standard in physics and mathematics.”
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On the night of November 10, Gross and the four colleagues he selected for the committee posted a document explaining how the proof works, alongside their affirmation of the proof’s formal verification in Lean, a programming language that checks mathematical work. This “shepherding” approach, however, contradicts recommendations for transparency from an OpenAI-affiliated advisory group and has riled some mathematicians who have seen it as an act of needless gatekeeping.
“I’m happy the shepherds are releasing the Lean proof, finally, so we can see what the result is. And I hope, in the future, the AI labs will share their results widely with the community,” says Scott Armstrong, a mathematician at New York University. “This review can be done more efficiently and quickly in the open.”
OpenAI’s blockbuster proof addresses the “Yang-Mills problem,” which is one of five remaining unsolved “Millennium Prize Problems,” a set of seven problems codified by the Clay Mathematics Institute in 2000. Successfully solving a Millennium Prize Problem comes with a $1-million prize. But solving Yang-Mills would carry a bigger reward by tidying up the mathematical sloppiness of modern particle physics.
By many measures, the Standard Model is the most thoroughly tested and successful scientific theory ever devised, predicting precisely what happens when physicists smash subatomic particles together at facilities such as CERN’s Large Hadron Collider near Geneva. But no one has managed to boil down the Standard Model’s tenets to simple logic-based statements called axioms and therefore prove that it’s mathematically sound.
The Yang-Mills Millennium Prize Problem has two parts. The first is to prove that this theory underpinning the Standard Model “exists,” meaning that it can be constructed through valid mathematical steps. The second is more quantitative (and far more difficult), and it involves showing that an essential physical quantity called a “mass gap” emerges from the Yang-Mills equations.
In late October OpenAI announced that its model had solved the first part of the problem—the “existence” problem. The company’s claim cut ahead of another mathematician’s public, ongoing collaboration with an employee of OpenAI’s rival Anthropic to prove the same result. The “mass gap” remains unresolved.
The shepherds’ document is too cursory to provide a deep understanding of how the LLM’s proof works. Indeed, Gross and his colleagues admit they still lack that themselves. “We used OpenAI’s Astra [model] to assist with the digestion of the supplied Lean code,” they wrote in the their assessment. “We do not claim to fully understand the proof yet.” But with the proof and its Lean code now released to the world for deeper scrutiny, experts in the community can use their own agents to reverse-engineer and begin to parse it.
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