OpenAI announces three breakthroughs by its model in mathematics, computer science, and physics

On October 27, OpenAI announced that its AI models have achieved three major breakthroughs in the fields of mathematics, computer science, and physics, demonstrating the ability of AI to independently discover and prove complex scientific theories, rather than merely answering questions or assisting research.

OpenAI stated that the mathematics and computer science results have been formally verified using the Lean mathematical verification tool, and the proofs have been made public; the physics result is being reviewed by experts including Nobel Prize laureates in physics. The results still require further independent confirmation by the academic community.

In 1991, mathematician Christopher Sogge proposed the local smoothing conjecture, an attempt to explain the behavior of waves when they focus. According to this conjecture, even if a wave focuses extremely strongly, it can only do so for a very short time. If a representative moment is chosen at random, the degree of focusing observed will always be far below the most extreme case.

Similar phenomena often appear in mathematics: a phenomenon may reach a very extreme degree in the worst case, but after averaging, the situation actually observed is much milder.

The phenomenon described by the local smoothing conjecture is of fundamental significance, and therefore relates to numerous problems in mathematical analysis. Among them, both the Kakeya conjecture and the Stein restriction conjecture can be derived from the local smoothing conjecture.

In 2019, Guth, Wang Hong, and Zhang Ruixiang proved the local smoothing conjecture in two-dimensional space. In 2025, Wang Hong and Zahl also proved the three-dimensional Kakeya conjecture.

OpenAI's model this time provided a proof of the local smoothing conjecture in all dimensions, thereby also resolving the Kakeya conjecture and the Stein restriction conjecture in 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 proof the next day. Subsequently, OpenAI contacted external experts to discuss how to establish a cooperation mechanism around this result.

Based on expert advice, OpenAI decided to directly publish the AI-generated proof, using the same approach as when it released research results on October 6. At present, both the natural language proof written by the model and the Lean formal proof have been uploaded to a public GitHub code repository.

The Exponential Time Hypothesis Is 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 formula satisfiability problem and is closely related to the famous P versus NP problem.

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

Recently, however, OpenAI's system provided a proof that ETH does not hold, and completed formal verification through Lean.

After ETH is overturned, researchers will need to re-examine how much speedup can still be obtained for many computational tasks. The new result also helps guide research into more efficient algorithms. However, finding algorithms that can achieve these improvements and translating theoretical results into practical applications will still require a great deal of follow-up research.

OpenAI invited three experts in algorithm design and computational complexity theory to lead the effort to verify, study, and explain this result: Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of MIT.

All three experts have long-standing research records in this field:

· Impagliazzo and Paturi, together with Francis Zane, jointly proposed ETH, and over the past 30 years have published multiple influential papers around this hypothesis.

· Williams has proved that certain strengthened versions of ETH do not hold, has also derived computational hardness conclusions based on ETH, and has established connections between algorithms and computational complexity lower bounds.

OpenAI accepted two requirements put forward by the collaboration team: first, to make public all raw data from the research results so that other researchers can independently verify them; second, for OpenAI to provide funding to hold an academic conference to discuss the overall impact of the new results on the field of computer science.

The collaboration team also invited the academic community to jointly organize a public review of the ETH proof and to discuss questions raised by AI research breakthroughs, 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. OpenAI then began establishing the aforementioned external cooperation mechanism.

The natural language proof and Lean proof generated by the model have now been published in OpenAI's public GitHub code repository.

The Existence of Quantum Chromodynamics and a Unified Completion of the Standard Model

In physics, OpenAI presented a research result on the existence of quantum field theory.

In a rigorous mathematical sense, this result proves that certain interacting quantum field theories in four-dimensional space that include gauge fields and fermions do indeed exist, including quantum chromodynamics (QCD), as well as a unified completion theory of the Standard Model of particle physics.

The proof adopts the research method proposed by Balaban in constructive quantum field theory. In addition, the research also solved a sub-problem, namely proving the existence of pure Yang-Mills theory. This problem is part of the Millennium Prize Problem on quantum Yang-Mills theory established by the Clay Mathematics Institute.

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

Because the Standard Model itself cannot be directly extended to arbitrarily short distance scales, OpenAI's existence proof targets the simplest grand unified theory that includes the Standard Model gauge fields and fermions.

Quantum chromodynamics studies how particles such as quarks and gluons are connected through the strong interaction. Pure Yang-Mills theory covers a narrower scope, involving only the self-interactions among the gluons that mediate the strong interaction, and does not include matter particles such as quarks.

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

All five scholars have made important contributions in the fields of quantum field theory and mathematical physics:

· David Gross: In 1973, his participation in research on asymptotic freedom in Yang-Mills theory laid the foundation for quantum chromodynamics to become the accepted theory of the strong interaction, and earned him the 2004 Nobel Prize in Physics. Asymptotic freedom is also the key mechanism for rigorously proving the mathematical existence of these theories.

· Arthur Jaffe: Together with Witten, he formulated the Millennium Prize Problem on quantum Yang-Mills theory, was one of the proposers of the idea of establishing the Millennium Prize Problems, and participated in founding the Clay Mathematics Institute.

· Michael Douglas: Over the past 20 years, he has been involved in advancing research on this problem, reported research progress in 2004, and comprehensively reviewed the latest results earlier this year.

· Igor Klebanov: He was among the earlier researchers to use 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 research project.

· Nikita Nekrasov: He achieved pioneering results in exact solution methods for generalized forms of Yang-Mills theory, and in 2023 received the Danny Heineman Prize in Mathematical Physics.