OpenAI claims three major discoveries in math, CS and physics

Less than a month after releasing hundreds of manuscripts it said contained solutions to long-standing mathematical questions, OpenAI has announced three new "major" discoveries, this time across mathematics, computer science and physics. The claims come from OpenAI itself and have not yet been independently verified.

Unlike the earlier release, independent experts are involved in the process this time. OpenAI calls them "shepherds." In computer science and physics, these experts will guide how the results are examined. In mathematics, on the experts' recommendation, the model's output was published directly, leaving researchers to develop their own explanations of it.

Local smoothing and the Kakeya claims in mathematics

The mathematics result concerns the local smoothing problem for the Euclidean wave equation. According to OpenAI, the proof uses an auxiliary partial differential equation to establish Sogge's local smoothing conjecture in all spatial dimensions two and above. The company says the same output also covers the Kakeya maximal and dimension conjectures, as well as Stein's global restriction estimates, throughout the interior of the predicted range.

OpenAI says it published this output directly as model output, as it did in the October 6 release. The three-dimensional case of the Kakeya conjecture was settled a few months ago in work that earned NYU professor Hong Wang a Fields Medal; OpenAI now claims its model has resolved the conjecture in all dimensions.

A claimed refutation of the Exponential Time Hypothesis

The computer science result claims that the Exponential Time Hypothesis (ETH), widely believed to be true, is false. The paper describes a deterministic, subexponential-time, polynomial-space algorithm that exactly counts the satisfying assignments of constant-width CNF formulas. According to the paper, the 3-SAT case refutes the hypothesis.

An independent group made up of Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will review the result and explain it. Impagliazzo and Paturi were on the team that proposed the hypothesis in 1999, while Williams is a professor at MIT. At the group's request, the AI output and a Lean proof were made public on GitHub today. An open letter was also published asking the community for input on how such claims should be examined.

The physics result and criticism of the shepherd approach

The full physics result has not yet been published. OpenAI describes the work as a mathematically rigorous existence proof for certain interacting quantum field theories in four dimensions that include gauge fields and fermions, among them quantum chromodynamics (QCD) and a unified completion of the Standard Model. The work is said to be guided by a group chaired by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.

The company acknowledges that this approach may have problems of its own. AGMAI, a panel of mathematics advisers OpenAI convened, yesterday recommended that AI companies not adopt the practice of giving outputs first to selected individuals, citing unequal access, borrowed prestige and biased selection. It suggests instead that outputs be posted in academic repositories, accompanied by Lean-formalized proofs, information on how problems were chosen, and funding for people working to understand the results.

The open letter, for its part, puts several questions to the community: what rules should govern early access and conflicts of interest, how the labor of checking, correcting, simplifying and explaining should be recognized alongside the discoveries themselves, and how students and researchers should be supported as methods change.