OpenAI Says AI Models Disprove ETH and Resolve Major Math and Physics Problems

In mathematics, OpenAI said the model proved the local smoothing conjecture proposed by Christopher Sogge in 1991. The conjecture concerns how waves focus: extreme focusing can occur briefly, but average behavior over time is less concentrated. OpenAI said its proof introduces an auxiliary partial differential equation and works in every dimension. Because local smoothing implies the Kakeya conjecture and the Stein restriction conjecture, OpenAI said the result resolves the Kakeya conjecture in all dimensions. In early 2025, Wang Hong and Joshua Zahl proved the three-dimensional Kakeya conjecture; the new claim extends to all dimensions through a stronger statement. In 2019, Larry Guth, Wang Hong and Zhang Ruixun proved the two-dimensional local smoothing conjecture, according to the report.

OpenAI said the model generated the natural-language proof on Oct. 18 and completed a Lean formalization the next day. Both documents were posted to a public GitHub repository.

In computer science, OpenAI said the model disproved the exponential time hypothesis, or ETH, an assumption advanced around 1999 by Russell Impagliazzo, Ramamohan Paturi and Francis Zane. ETH holds that 3-SAT, a Boolean satisfiability problem, cannot be solved in subexponential time in the number of variables, written as 2^{o(n)}; it has underpinned hundreds of conditional lower bounds in graph theory, computational biology, databases and other areas. OpenAI said the model generated a natural-language disproof on Oct. 16 and finished Lean formalization the following day.

OpenAI said the external team will be led by Impagliazzo, Paturi and Ryan Williams of MIT. The team asked OpenAI to release all raw data and to fund an academic conference on the implications for computational complexity theory; OpenAI said it accepted. If the result holds, some problems thought to require exponential time may have faster algorithms, and conditional conclusions built on ETH would need review.

In theoretical physics, OpenAI said its model constructed rigorous mathematical existence proofs for certain interacting quantum field theories in four dimensions involving gauge fields and fermions. The work covers quantum chromodynamics, a unified completion of the Standard Model and pure Yang-Mills theory, OpenAI said. The Yang-Mills existence and mass gap problem is one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute in 2000; OpenAI's claim addresses existence, not the mass gap, and extends to QCD and a related unified theory.

OpenAI said the model generated the Yang-Mills existence proof on Sept. 9, a QCD proof on Sept. 18 and completed Lean formalization of the Yang-Mills existence proof on Oct. 16. A five-physicist team led by David Gross, a 2004 Nobel laureate in physics, will study the work. Other members are Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. Jaffe helped formulate the Yang-Mills Millennium Prize problem and co-founded the Clay institute. The team decided not to release the raw AI-generated materials for now and will organize, review and write up the research.

The announcement came after OpenAI published 722 AI-generated mathematical manuscripts on Oct. 6, covering 372 unsolved problems. Some proofs were later found to contain errors; OpenAI retracted three manuscripts and revised 14 others. In the new release, OpenAI said it would treat AI-generated proofs as raw research data and invite external scientists to verify, explain and organize them. It said it would not seek academic authorship for the findings and would support the research through a previously announced $5 million grant to the Simons Foundation.