OpenAI Releases Three Landmark Proofs in a Row: ETH Disproved, Yang-Mills Existence Has a Solution

On October 26, OpenAI released three new discoveries at once, falling respectively in mathematics, computer science, and physics. All three proofs were formally verified with Lean—every reasoning step was machine-checked, not merely asserted by the model.

Mathematics first. The model proved that Sogge's local smoothing conjecture holds in all dimensions. This conjecture studies the behavior of waves when they focus: the most extreme focusing cannot persist, and focusing at typical times is far smaller than the worst case. It is a foundational result that directly implies the Kakeya conjecture and the Stein restriction conjecture. Previously, humans had only settled two-dimensional local smoothing and three-dimensional Kakeya; this time AI covered all dimensions. The proof used a new auxiliary partial differential equation. The AI output was obtained on October 18, and Lean formalization was completed the next day.

The computer science side is even more explosive. The model proved that the Exponential Time Hypothesis (ETH) is false. ETH is a widely believed conjecture in computational complexity, roughly saying that Boolean satisfiability problems such as 3-SAT cannot be solved in subexponential time. It has been used as the foundation for conditional hardness results in hundreds of papers, involving graph theory, computational biology, databases, computational geometry, and more. Now that foundation has been pulled out, and many conclusions need to be re-examined. The counterexample given in the paper is a deterministic algorithm that can exactly count satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space.

The people responsible for verifying and interpreting this result are Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams. Interestingly, Impagliazzo and Paturi are precisely the original proposers of ETH. Now they are leading the check of the AI-provided disproof, which closes the loop. Both the original natural-language proof and the Lean formalization have been made public on GitHub.

The physics result may be the most profound. The model gave an existence proof for an interacting quantum field theory in four dimensions with gauge fields and fermions, including quantum chromodynamics (QCD) and a unified complete version of the Standard Model. As a subproblem, it also incidentally proved the existence of pure Yang-Mills theory—one of the Clay Mathematics Institute's Millennium Prize Problems. Note that it proves existence, not constructs a solvable model. Physicists have always taken it for granted that QCD is self-consistent, but there was no rigorous mathematical proof. Now AI has crossed this step.

A group led by David Gross is responsible for interpreting this work. Members include Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. David Gross won the Nobel Prize for establishing asymptotic freedom in 1973; Jaffe is one of the formulators of the Millennium Problems; Douglas has followed this problem for twenty years. The team decided not to release the AI-generated raw materials for the time being, but to digest them themselves first and then write them up.

The release method is also worth mentioning. OpenAI did not follow the old routine of simply throwing out papers; instead, it found external experts to serve as “shepherds” and asked them to organize teams to understand the results. The three results used three different strategies: for mathematics, the AI output was made public directly and researchers were left to organize themselves; for computer science, all materials were made public and an external group led the interpretation; for physics, release was delayed, letting the external group write it up themselves first. OpenAI said it does not know which method is most effective and welcomes feedback.

A few observations:

First, Lean formalization makes the accusation of “AI hallucination” untenable. In a machine-checked proof, every step cannot escape scrutiny. But as some netizens have pointed out, the real focus of review has become whether the formalized statement is consistent with the theorem humans care about. Deviations in the translation process are what mathematicians need to watch next.

Second, if these results pass independent verification, AI will no longer be merely an auxiliary tool. It has directly produced new knowledge from theorem proving. Although it is still impossible to predict the speed of actual algorithmic improvements, at least it proves that AI can push the boundaries of theory.

Third, OpenAI simultaneously announced a $5 million donation to the Simons Foundation to support related workshops and projects. It seems they understand that release is only the beginning, and getting academia to truly digest these results is a longer road.

References:

• OpenAI official announcement
• GitHub mathematics repository