OpenAI claims its AI models have led to three discoveries, in mathematics, computer science, and physics. This announcement comes after the publication, in early October, of 722 mathematical manuscripts generated by an internal model, posted on GitHub without independent verification. This time, the company is ceding academic credit to the researchers tasked with verifying its work.
The first publication sparked strong reactions in the scientific community. Mathematician Terence Tao said he was “deeply frustrated,” while Fields Medalist Hugo Duminil-Copin spoke of a “huge shock.” A group of mathematicians published an open letter, titled A Severe Misalignment of AI in Mathematics, calling the operation a “demonstration of power, not rigor.”
On October 26, OpenAI published a new press release, in which it states it is experimenting with “a new approach for this publication.” The company claims to have solved, in a few days of computation, three problems that have occupied research communities since the 1990s.
The Local Smoothing Conjecture
The first problem falls under harmonic analysis, which studies wave propagation. The question is as follows: when a wave concentrates at a point, can this intensity peak last? Formulated in 1991 by American mathematician Christopher Sogge, the conjecture answers negatively: a point peak is inevitable, but the wave smooths out on average over time.
For over thirty years, the question has remained open. The case of the second spatial dimension was solved in 2019 by Larry Guth, Hong Wang and Ruixiang Zhang, and Hong Wang and Joshua Zahl solved the Kakeya conjecture in three dimensions in 2025, a related problem. The general version, valid for all higher dimensions, remained unproven. OpenAI writes: “Our models produced a proof of the local smoothing conjecture in all dimensions and thus also solve the Kakeya conjecture and the Stein restriction conjecture in all dimensions. The new proof relies on a novel auxiliary partial differential equation.”
For now, OpenAI has not named a verification committee. However, the company has published a 59-page manuscript on GitHub, accompanied by a verification performed with Lean, a software that checks each logical step of a proof.
The Exponential Time Hypothesis
The second problem concerns the Exponential Time Hypothesis (ETH), formulated in 1999 by Russell Impagliazzo and Ramamohan Paturi. It postulates that certain problems are so complex that, as they grow, the computation time explodes exponentially, regardless of the algorithm used. For a problem with 100 variables, the number of operations would reach about 2^100, a number with about 30 digits, and a classical computer would have to run for a duration longer than the age of the universe.
According to OpenAI, its model found a refutation of the ETH “on October 16, 2026,” formalized in Lean the next day. The company states that the model constructed an algorithm that solves these problems in sub-exponential time. The computation remains very long, but it grows slower than what the conjecture deemed possible. Over 800 papers in theoretical computer science rely on this hypothesis. A committee including its authors, as well as Ryan Williams from MIT, is to examine these results, with no verdict to date. A few hours after OpenAI, Anthropic published its own refutation of the ETH, found independently by a model from the Claude family.
Yang-Mills Theory
The third result concerns quantum chromodynamics (QCD), which describes the strong interaction, and Yang-Mills theory, from which it originates. OpenAI claims to have produced a mathematically rigorous proof of existence for these two theories. This work touches on one of the seven Millennium Prize Problems, established in 2000 by the Clay Mathematics Institute, which offers one million dollars for each. The aspect the company says it addresses is proving that the theory exists in the mathematical sense. It does not claim to have solved the other aspect, which concerns the “mass gap.”
OpenAI writes that it completed its formalization of the existence of Yang-Mills theory on October 16, 2026. The committee tasked with examining it is chaired by David Gross, 2004 Nobel laureate, alongside Arthur Jaffe, co-author with Edward Witten of the official statement of the problem. Unlike the other two results, the physics committee will not disseminate the material generated by the model and will publish only its own written version of the proof. OpenAI also announced five million dollars, paid via the Simons Foundation, to fund workshops and conferences around its results. The company specifies that it claims “no scientific authorship of the discoveries produced” and that the participation of the evaluators is “voluntary and unpaid.”
