A panel of five physicists and mathematicians led by David Gross, who shared the 2004 Nobel Prize in Physics, published its first review of OpenAI's AI-made proof about Yang-Mills theory on Nov. 10. In the panel's judgment, the proof shows that the theory can be built as a quantum theory that meets a standard set of rules mathematical physicists require of one. Yang-Mills theory sits at the heart of how physicists explain the strong nuclear force, which is believed to keep quarks locked inside particles such as protons and neutrons. The proof covers the theory in its pure form, which describes only gluons, the particles that carry the force, and leaves out quarks. Adding quarks gives quantum chromodynamics, the full theory of the strong force. OpenAI announced on Oct. 26 that its AI had proved that pure Yang-Mills theory exists, and that quantum chromodynamics and a fuller version of the Standard Model, the theory of known particles and three of nature's four forces, exist too. The panel's review so far covers only the pure form.
Experiments and computer simulations have long backed the theory, but physicists calculate with it through approximations such as a grid, and no one had proved that those approximations add up to a well-defined theory. The proof starts from a version of the theory set on a grid of points inside a box of space and time. It shows that the results settle into a proper theory as the grid gets finer, in steps the proof picks, and that as the box grows to fill all of space, at least some chains of ever-larger boxes settle too, the panel says.
The version the panel checked is written in Lean, a programming language in which a computer can check every step. The panel rebuilt the code's 9,470 parts from scratch on an Apple Mac in about three hours with no errors. It found that the proof's main results rely on no unproved assumptions beyond the basic ones Lean allows. The panel notes it ran these checks without the usual safeguard that walls off the code being checked, so they do not rule out code that tampers with the check. A computer check shows only that a proof is right for the problem as it was typed into the computer. So the panel also read the definitions to see that they match the problem as physicists and mathematicians state it, a review it calls extensive but not exhaustive. After submitting the proof, OpenAI told the panel that the statement OpenAI had first given the computer to check left out some links to the grid version of the theory. The panel says that does not undermine its review, because it judged those links against a stricter statement of its own, which also passed the computer check.
The proof does not settle everything. It does not show a mass gap, which the Yang-Mills problem on the Clay Mathematics Institute's list of Millennium Prize Problems also requires. A mass gap means that, apart from empty space, every state of the theory carries at least some minimum energy, so its lightest particles cannot be massless. The institute says experiments and computer simulations suggest the gap exists, but no proof is known. Nor does the proof show that letting the box grow to fill all of space always gives the same theory. It finds at least one, and there could be others, though the panel says it believes this shortfall fits the general expectation that there is only one.
The panel says it used Astra, one of OpenAI's AI models, to help it read the code, and it does "not claim to fully understand the proof yet." It also raised concerns about the code's quality, including large amounts of code the main results never use and odd names such as "Cake" and "cookie". OpenAI asked Gross on Oct. 16 to take charge of the proof and decide how to release it, and Gross chose the other four members: Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. The panel says it has "no responsibility to OpenAI" and that, besides accepting OpenAI's submissions, its only tie to the company is its use of Astra. It published OpenAI's code with its review, along with instructions for rerunning its checks.
The release does not include OpenAI's 1,147-page written proofs, and the panel's check did not draw on them. The panel said in October it would publish them once satisfied with their quality and correctness. An independent group of mathematicians that advises AI companies on releasing math results had argued on Oct. 25 that labs should not hand results to mathematicians they choose. On Oct. 26 it called on OpenAI to publish, without delay, the written and computer versions of all three proofs it announced. The computer version of the pure Yang-Mills proof is now out; the others are not. A second OpenAI submission on pure Yang-Mills theory, dated Nov. 4, is still a work in progress for the panel. As of Nov. 15 the panel has not said what the submission adds or when it will finish.