正文 · AI 翻译
IT Home reported on October 27 that OpenAI announced three major breakthroughs by its AI models in mathematics, computer science and physics, demonstrating AI's ability to independently discover and prove complex scientific theories, rather than merely answer questions or assist research.
In mathematics, AI proved the local smoothing conjecture in all dimensions, thereby resolving the local smoothing conjecture and the Kakeya conjecture.
In computer science, AI overturned the long and widely accepted Exponential Time Hypothesis, suggesting that faster algorithms may be found in the future.
In physics, AI proposed a rigorous mathematical proof of the existence of quantum chromodynamics and Yang–Mills theory, involving part of one of the world's 7 Millennium Prize Problems.
OpenAI said the mathematics and computer science results have been formally verified using the Lean proof assistant, and the proofs have been made public; the physics result is under review by experts, including a Nobel laureate in physics. The results still require further independent confirmation from the academic community.
IT Home provides the gist of OpenAI's official statement below:
The local smoothing conjecture and the Kakeya conjecture
The new mathematics result comes from harmonic analysis.
In 1991, mathematician Christopher Sogge proposed the local smoothing conjecture to explain how waves behave when they focus. According to the conjecture, even extremely intense focusing of waves can last only a short time. If one selects a representative moment at random, the degree of focusing observed will always be far below the most extreme case.
Similar phenomena often arise in mathematics: something can be extreme in the worst case, but after averaging, what is actually observed is much less severe.
The phenomenon described by the local smoothing conjecture is fundamental, so the conjecture bears on many problems in mathematical analysis. Both the Kakeya conjecture and the Stein restriction conjecture can be derived from it.
In 2019, Guth, Hong Wang and Ruixiang Zhang proved the local smoothing conjecture in 2 dimensions. In 2025, Hong Wang and Zahl also proved the 3-dimensional Kakeya conjecture.
OpenAI's model has now produced 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 a formal proof in Lean the next day. OpenAI then contacted external experts to discuss how to establish a framework for collaboration around the result.
On the experts' advice, OpenAI decided to release the AI-generated proof directly, using the same approach it took when publishing research results on October 6. Both the model's proof in natural language and the formal proof in Lean have now 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 longstanding conjecture in computational complexity theory concerning the computational difficulty of the Boolean satisfiability problem. It is closely related to the well-known P versus NP problem.
For a long time, researchers generally believed ETH to be true. Hundreds of papers have used it as a premise to derive conclusions about computational difficulty in fields including graph theory, computational biology, databases and computational geometry.
Recently, however, OpenAI's system produced a proof that ETH is false and formally verified it in Lean.
With ETH overturned, researchers need to reconsider how much faster many computational tasks might be performed. The new result may also help guide research into more efficient algorithms. However, finding algorithms that deliver these improvements and turning the theoretical result into practical applications will still require substantial further research.
OpenAI invited 3 experts in algorithm design and computational complexity theory to lead the verification, study and interpretation of the result: Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of the Massachusetts Institute of Technology.
All 3 experts have long records of research in the field:
Impagliazzo and Paturi proposed ETH together with Francis Zane and have published several influential papers on the hypothesis over the past 30 years.
Williams has proved that certain stronger versions of ETH are false, used ETH to derive conclusions about computational difficulty, and established connections between algorithms and lower bounds in computational complexity.
OpenAI accepted 2 requests from the collaborating team: first, to release all the raw data behind the research so other researchers can verify it independently; and second, to fund an academic conference to discuss the new result's overall impact on computer science.
The collaborating team also invited the academic community to jointly organize a public review of the ETH proof and discuss issues raised by AI research breakthroughs, including how to verify results, how to make research materials available, 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 formal proof in Lean on October 17. It then began establishing the external collaboration framework described above.
The model-generated natural-language proof and Lean proof have now been published in OpenAI's public GitHub repository.
The existence of quantum chromodynamics and a unified completion of the Standard Model
In physics, OpenAI proposed a research result concerning the existence of quantum field theories.
In a rigorous mathematical sense, the result proves the existence in 4 dimensions of certain interacting quantum field theories containing gauge fields and fermions. These include quantum chromodynamics (QCD) and a unified completion of the Standard Model of particle physics.
The proof uses a research method developed by Balaban in constructive quantum field theory. The research also resolves a subproblem by proving the existence of pure Yang–Mills theory. That problem is part of the Clay Mathematics Institute's Yang–Mills quantum theory Millennium Prize Problem.
The Standard Model of particle physics describes interactions among fundamental subatomic particles. It covers 3 of nature's 4 fundamental interactions, excluding gravity.
Because the Standard Model itself cannot be directly extended to arbitrarily short distance scales, OpenAI's existence proof concerns the simplest grand unified theory containing the Standard Model's gauge fields and fermions.
Quantum chromodynamics studies how particles such as quarks and gluons interact through the strong interaction. Pure Yang–Mills theory has a narrower scope: it deals only with self-interactions among gluons, which carry the strong interaction, and excludes matter particles such as quarks.
OpenAI invited David Gross to lead the research. Other members of the team include Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.
All 5 scholars have made significant contributions to quantum field theory and mathematical physics:
David Gross: In 1973, his work with others 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 a key mechanism for rigorously proving that these theories exist mathematically.
Arthur Jaffe: He and Witten jointly formulated the Yang–Mills quantum theory Millennium Prize Problem. He was also among those who proposed the idea of the Millennium Prize Problems and helped found the Clay Mathematics Institute.
Michael Douglas: He has helped advance research on this problem over the past 20 years, reported on progress in 2004, and provided a comprehensive review of the latest results earlier this year.
Igor Klebanov: He was among the first 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.
Nikita Nekrasov: He has made pioneering contributions to methods for finding exact solutions to generalized forms of Yang–Mills theory and received the 2023 Dannie Heineman Prize for Mathematical Physics.
References
Official announcement: Working with academic communities to build scientific understanding
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