OpenAI Claims AI Breakthroughs Across Mathematics, Complexity Theory, and Physics

IT Home, October 27 — OpenAI has announced three major breakthroughs by its AI models in the fields of mathematics, computer science, and physics. These achievements demonstrate the AI’s ability to independently discover and prove complex scientific theories, rather than merely answering questions or assisting in research.

In mathematics, the AI has proven the local smoothing conjecture for all dimensions, thereby resolving both the local smoothing conjecture and the Kakeya conjecture.

In computer science, the AI has overturned the long-accepted Exponential Time Hypothesis (ETH), implying that faster algorithms may be found in the future.

In physics, the AI has proposed a rigorous mathematical existence proof for quantum chromodynamics and Yang–Mills theory, addressing one of the seven Millennium Prize Problems in mathematics.

OpenAI stated that the achievements in mathematics and computer science have been formally verified using the Lean mathematical verification tool, and the proofs have been made public. The physics achievement is currently under review by experts, including Nobel laureates in physics. These findings still require further independent confirmation from the academic community.

IT Home provides a summary of the official statement from OpenAI:

The Local Smoothing and Kakeya Conjectures

The new mathematical result comes from the field of harmonic analysis. In 1991, mathematician Christopher Sogge proposed the local smoothing conjecture to explain the behavior of waves during focusing. According to the conjecture, even when waves experience extremely intense focusing, it only lasts for a very short duration. If a representative moment is selected at random, the observed degree of focusing will always be far lower than the most extreme case.

Similar phenomena often appear in mathematics: a phenomenon may reach extreme levels in the worst case, but when averaged out, the actual observed outcome is much milder.

The phenomenon described by the local smoothing conjecture is fundamental, and thus it touches upon numerous problems in mathematical analysis. Both the Kakeya conjecture and the Stein restriction conjecture can be derived from the local smoothing conjecture.

In 2019, Guth, Wang, and Zhang proved the local smoothing conjecture in two dimensions. In 2025, Wang and Zar proved the three-dimensional Kakeya conjecture.

OpenAI’s model has now provided a proof for the local smoothing conjecture in all dimensions, thereby also resolving the Kakeya and Stein restriction conjectures for all dimensions. The key to the proof is a newly constructed auxiliary partial differential equation.

OpenAI obtained the AI-generated proof on October 18, 2026, and completed the Lean formal verification the following day. Subsequently, OpenAI contacted external experts to discuss how to establish a collaborative mechanism around these results.

Following expert advice, OpenAI decided to directly release the AI-generated proof, adopting the same approach as the research results released on October 6. Currently, both the natural language proof written by the model and the Lean formal proof have been uploaded to a public GitHub repository.

The Exponential Time Hypothesis Overturned

The computer science discovery concerns the Exponential Time Hypothesis (ETH). ETH is a long-standing conjecture in computational complexity theory that studies the computational difficulty of the Boolean satisfiability problem, which is closely related to the famous P vs. NP problem.

For a long time, the academic community has generally accepted that ETH is true. Hundreds of research papers have used this hypothesis as a premise to derive computational difficulty conclusions in fields such as graph theory, computational biology, databases, and computational geometry.

Recently, however, OpenAI's system provided a proof that ETH is false, and this has been formally verified via Lean.

With ETH overturned, researchers need to re-examine how much of a speedup is actually possible for many computational tasks. The new findings also help guide the research of more efficient algorithms. However, finding the algorithms capable of achieving these improvements and translating these theoretical results into practical applications will still require extensive follow-up research.

OpenAI invited three experts in algorithm design and computational complexity theory to lead the verification, research, and interpretation of this result: Russell Impagliazzo and Ramamohan Paturi from the University of California, San Diego, and Ryan Williams from the Massachusetts Institute of Technology.

The three experts have long-standing research backgrounds in this field:

• Impagliazzo and Paturi, along with Francis Zane, proposed ETH and have published numerous influential papers on the hypothesis over the past 30 years.
• Williams has previously proven that certain strengthened versions of ETH are false; he has also derived computational difficulty conclusions based on ETH and established links between algorithms and computational complexity lower bounds.

OpenAI accepted two requirements proposed by the collaboration team: first, to publicly release all raw data of the research so that other researchers can independently verify it; second, for OpenAI to provide funding to host an academic conference to discuss the overall impact of the new results on the computer science field.

The collaboration team has also invited the academic community to organize a public review of the ETH proof and to initiate discussions on the issues raised by this AI research breakthrough, including how to verify results, how to open research materials, how to recognize academic contributions, and how scientific research should be conducted in the future.

OpenAI generated the natural language proof overturning ETH on October 16 and completed the Lean formal proof on October 17. Subsequently, OpenAI began the work of establishing the aforementioned external cooperation mechanism.

The model-generated natural language proof and the Lean proof are now available on OpenAI’s public GitHub repository.

Existence of Quantum Chromodynamics and the Complete Unification of the Standard Model

In physics, OpenAI has proposed research regarding the existence of quantum field theories.

This result proves, in a rigorous mathematical sense, that certain interacting quantum field theories in four dimensions containing gauge fields and fermions do exist, including quantum chromodynamics (QCD) and a unified completion theory of the Standard Model of particle physics.

The proof employs the research methodology proposed by Balaban in constructive quantum field theory. Additionally, the research solved a sub-problem, namely the existence of pure Yang–Mills theory. This problem is part of the Millennium Prize Problem for Quantum Yang–Mills Theory established by the Clay Mathematics Institute.

The Standard Model of particle physics describes the interactions between subatomic fundamental particles and covers three of the four fundamental interactions of nature, excluding gravity.

Because the Standard Model itself cannot be directly generalized to arbitrarily short distance scales, OpenAI’s existence proof targets the simplest Grand Unified Theory that includes the gauge fields and fermions of the Standard Model.

Quantum chromodynamics studies how particles like quarks and gluons interact through the strong force. Pure Yang–Mills theory has a narrower scope, focusing only on the self-interaction of gluons that mediate the strong force, excluding matter particles like quarks.

OpenAI invited David Gross to lead the research, with other team members including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov.

The five scholars have all made significant contributions to the fields of quantum field theory and mathematical physics:

• David Gross: In 1973, his work on the asymptotic freedom of Yang–Mills theory laid the foundation for quantum chromodynamics to become the accepted theory of the strong interaction, earning him the 2004 Nobel Prize in Physics. Asymptotic freedom is also a key mechanism for rigorously proving the mathematical existence of these theories.
• Arthur Jaffe: Together with Witten, he formulated the Millennium Prize Problem for Quantum Yang–Mills Theory. He was one of the proposers of the Millennium Prize Problem concept and helped found the Clay Mathematics Institute.
• Michael Douglas: Over the past 20 years, he has been involved in advancing research on this problem, reporting on progress in 2004 and conducting a comprehensive review of the latest findings earlier this year.
• Igor Klebanov: He was an early adopter of using the duality between gauge fields and string theory to study confinement, an important property of quantum chromodynamics. He currently leads the Simons Foundation’s collaboration on Confinement and QCD Strings.
• Nikita Nekrasov: He has made groundbreaking achievements in precise solution methods for generalized forms of Yang–Mills theory and received the Dannie Heineman Prize for Mathematical Physics in 2023.

Reference Official Announcement: Working with academic communities to build scientific understanding