IT之家 October 27 news: OpenAI has announced that its AI models have achieved three major breakthroughs in mathematics, computer science, and physics, demonstrating AI's ability to independently discover and prove complex scientific theories, rather than merely answering questions or assisting research.
In mathematics, the AI proved the local smoothing conjecture, which applies to all dimensions, and thereby solvedthe local smoothing conjecture and the Kakeya conjecture。
In computer science, the AI refutedthe long widely accepted exponential time hypothesis, meaning faster algorithms may be found in the future.
In physics, the AI proposeda rigorous mathematical existence proof for quantum chromodynamics and Yang–Mills theory, touching on one of the world's seven Millennium Prize Problems.
OpenAI stated that the mathematics and computer science results have been formally verified with the Lean mathematical verification tool and the proofs made public; the physics result is being reviewed by experts including Nobel laureates in physics. The results still require further independent confirmation from the academic community.
IT之家 presents the gist of OpenAI's official statement as follows:
Local smoothing conjecture and Kakeya conjecture
The new result in mathematics comes from the field of harmonic analysis.
In 1991, mathematician Christopher Sogge proposed the local smoothing conjecture, attempting to explain the behavior of waves when they focus. According to the conjecture, even if a wave focuses extremely intensely, it can only persist 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 occur in mathematics: a phenomenon can reach a very extreme degree in the worst case, but after averaging, what is actually observed is much milder.
The phenomenon described by the local smoothing conjecture has fundamental significance, so it touches on many problems in mathematical analysis. Among them, the Kakeya conjecture and the Stein restriction conjecture can both be derived from the local smoothing conjecture.
In 2019, Guth, Wang Hong (Hong Wang), and Zhang Ruixiang proved the local smoothing conjecture in two-dimensional space. In 2025, Wang Hong and Zahl proved the three-dimensional Kakeya conjecture.
OpenAI's model has now given a proof of the local smoothing conjecture in all dimensions, thereby also solving 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 formalized proof the next day. Subsequently, OpenAI contacted external experts to discuss how to establish a collaboration mechanism around the result.
On the experts' advice, OpenAI decided to publish the AI-generated proof directly, in the same way as it released its research results on October 6. Currently, both the natural-language proof written by the model and the Lean formalized proof have been uploaded to a public GitHub repository.
Exponential time hypothesis refuted
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 and is closely related to the famous P and NP problems.
For a long time, academia hasgenerally believed ETH to be true. Hundreds of research papers have used this hypothesis as a premise to derive conclusions about computational hardness in fields such as graph theory, computational biology, databases, and computational geometry.
Recently, however, OpenAI's systemsproduced a proof that ETH does not hold, and completed a formal verification via Lean.
With ETH overturned, researchers need to re-examine many computational tasksand determine how much speedup can still be obtained. The new result also helps guide research into more efficient algorithms. However, finding algorithms that can realize these improvements, and translating the 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 verification, study, and interpretation of this result: Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of the Massachusetts Institute of Technology.
All three experts have long-standing research records in this field:
Impagliazzo and Paturi, together with Francis Zane, proposed ETH, and over the past 30 years have published numerous influential papers built around this hypothesis.
Williams previously 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 demands from the collaborating team: first, to make all raw data of the research results public so that other researchers can independently verify them; second, for OpenAI to fund an academic conference discussing the overall impact of the new result on the field of computer science.
The collaborating team also invited the academic community to jointly organize a public review of the ETH proof and to discuss the questions raised by this AI research breakthrough, including how to verify results, how to open up research materials, how to attribute academic contributions, and how scientific research should be conducted in the future.
On October 16, OpenAI generated the natural-language proof that overturns ETH, and on October 17 it completed the Lean formalized proof. OpenAI then began establishing the aforementioned external collaboration mechanism.
The model-generated natural language proofand Lean proofare now available in OpenAI's public GitHub repository.
The Existence of Quantum Chromodynamics and a Unified Completion of the Standard Model
On the physics side, OpenAI presented a research result on the existence of quantum field theory.
This result proves, in a rigorous mathematical sense, that certaininteracting quantum field theories containing gauge fields and fermions do indeed exist in four-dimensional space, including quantum chromodynamics (QCD), as well as a unified completion theory of the Standard Model of particle physics.
The proof adopts the research approach proposed by Balaban in constructive quantum field theory. In addition, the research resolved a subproblem, namely proving the existence of pure Yang-Mills theory. This problem is part of the Quantum Yang-Mills Theory Millennium Prize Problem 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.
Since the Standard Model itself cannot be directly extended to arbitrarily short distance scales, OpenAI's existence proof targetsthe simplest grand unified theory containing the Standard Model's gauge fields and fermions。
. Quantum chromodynamics studies how particles such as quarks and gluons are connected through the strong interaction. Pure Yang-Mills theory studies a narrower scope, involving only the self-interactions of gluons that mediate the strong interaction, not including matter particles such as quarks.
OpenAI invited David Gross to lead the research, with the other team members including 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 the research on asymptotic freedom of Yang-Mills theory laid the foundation for quantum chromodynamics to become the recognized theory of the strong interaction, and earned him the 2004 Nobel Prize in Physics. Asymptotic freedom is also the key mechanism for rigorously proving that these theories exist mathematically.
Arthur Jaffe: He jointly formulated the Quantum Yang-Mills Theory Millennium Prize Problem with Witten, was one of the proposers of the idea to establish the Millennium Prize Problems, and co-founded 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 apply gauge field/string theory duality to the study of confinement, an important property of quantum chromodynamics. He currently leads the Simons Foundation's collaboration on confinement and QCD strings.
Nikita Nekrasov: He achieved groundbreaking results in exact solution methods for generalized forms of Yang-Mills theory, and received the Dannie Heineman Prize for Mathematical Physics in 2023.
References