OpenAI announces three AI-generated results in mathematics, computer science, and physics

On October 26 local time, OpenAI announced three new discoveries in the fields of mathematics, computer science, and physics, involving harmonic analysis, complexity theory, and quantum field theory, respectively. However, this time OpenAI heeded external expert advice on releasing the results and adopted a different release strategy. All three results currently await independent verification by the academic community.

The mathematics result involves harmonic analysis and is related to the local smoothing conjecture and the Kakeya conjecture. The local smoothing conjecture, proposed in 1991, concerns the behavior of waves when they focus; the conjecture holds that the most extreme focusing must necessarily be brief. The amount of focusing at a typical random moment is always far smaller than the amount of focusing that could occur in the worst case. This pattern is very common in mathematics: what can happen in the worst case is more extreme than what is observed on average. Because of the foundational nature of this conjectured phenomenon, it has broad influence in mathematical analysis; for example, it implies the Kakeya conjecture and the Stein restriction conjecture.

OpenAI stated that its model produced a proof of the local smoothing conjecture in all dimensions, and therefore also resolved the Kakeya conjecture and the Stein restriction conjecture in all dimensions. The new proof is based on a new auxiliary partial differential equation; the AI output was obtained on October 18, and Lean formalization was completed the next day. On expert advice, OpenAI will make the AI-generated results public on GitHub.

In computer science, the Exponential Time Hypothesis (ETH) is a long-standing conjecture concerning the computational complexity of the Boolean formula satisfiability problem. OpenAI stated that the academic community generally believes ETH to be true, that hundreds of papers rely on it, and that it is used to derive conditional computational hardness results in areas such as graph theory, computational biology, databases, and computational geometry.

"Surprisingly, our system recently proved that ETH is false, and the result was subsequently verified in Lean," OpenAI stated. On October 16, its model found a natural-language disproof of ETH, and the next day it was formalized in Lean. The result has been made public on GitHub. The refutation of ETH raises anew the question of how much these kinds of computational tasks can still be sped up, and this result may help in the search for better algorithms. Finding such algorithms and turning theoretical progress into practical performance gains will still require a great deal of follow-up research. "The field of computational complexity still has a large number of open problems and conjectures, and for a long time scientists have been unable to complete their proofs or disproofs. We hope that collaboration between humans and AI can bring new understanding to this field and clarify the inherent limits of what efficient computation can do."

A team led by Russell Impagliazzo and Ramamohan Paturi, professors at the University of California, San Diego, and Ryan Williams, a professor at MIT, will check and explain this result. Russell and Paturi, together with computational complexity theorist Francis Zane, were the first to propose ETH.

In an open letter, Russell, Paturi, and Ryan stated that their role is not to endorse. "Our involvement should not be regarded as certification of this claim. A result of such magnitude requires careful, independent review, and it requires enabling others to reconstruct, question, and build on it."

The physics collaboration project concerns the existence of quantum chromodynamics and the completion of the Standard Model. The result rigorously proves, mathematically, a four-dimensional interacting quantum field theory containing gauge fields and fermions, including quantum chromodynamics (QCD) and a unified, complete version of the Standard Model of particle physics. As a subproblem, the work simultaneously proves the existence of pure Yang–Mills theory, a key component of the Millennium Prize Problem "Yang–Mills existence and mass gap."

OpenAI stated that its model generated a natural-language proof of Yang–Mills existence on September 9, generated a proof of QCD on September 18, and completed formal verification on October 16. The project's external expert team is led by David Gross, winner of the 2004 Nobel Prize in Physics; on the advice of the external expert team, OpenAI has decided not to release the AI-generated artifacts for the time being.

(Original title: OpenAI announces three new discoveries in mathematics, computer science, and physics, including the Kakeya conjecture and Yang–Mills theory)