OpenAI Claims Three Major Scientific Breakthroughs: Local Smoothing Conjecture, ETH Refutation, and Yang-Mills Proof

• In mathematics, AI proved the local smoothing conjecture for all dimensions, thereby resolving the Local Smoothing Conjecture and the Kakeya Conjecture.
• In computer science, AI disproved the long-standing Exponential Time Hypothesis (ETH), implying that faster algorithms may be discovered in the future.
• In physics, AI provided a rigorous mathematical existence proof for Quantum Chromodynamics (QCD) and Yang–Mills theory, addressing one of the seven Millennium Prize Problems.

OpenAI stated that the results in mathematics and computer science have been formally verified using the Lean theorem prover, with proofs made publicly available; the physics results are currently under review by experts, including Nobel laureates in Physics. These findings still require further independent confirmation by the academic community.

IT Home has attached the gist of OpenAI's official statement below:

The Local Smoothing Conjecture and the Kakeya Conjecture

The new mathematical breakthrough comes from the field of harmonic analysis.

In 1991, mathematician Christopher Sogge proposed the Local Smoothing Conjecture to explain how waves behave when they focus. According to this conjecture, even if waves exhibit extremely strong focusing, such intensity can only persist for a very short duration. If one randomly selects a representative moment, the observed degree of focusing will always be significantly lower than the most extreme case.

Similar phenomena frequently occur in mathematics: a certain phenomenon may reach very extreme levels in the worst-case scenario, but after averaging, the actual observations are much milder.

The phenomenon described by the Local Smoothing Conjecture is fundamental, impacting numerous problems in mathematical analysis. Notably, 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 Zar proved the three-dimensional Kakeya Conjecture.

This time, OpenAI's model provided a proof of the Local Smoothing Conjecture for all dimensions, thereby also resolving the Kakeya Conjecture and the Stein Restriction Conjecture in all dimensions. A key element of the proof was a newly constructed auxiliary partial differential equation.

OpenAI obtained the AI-generated proof on October 18, 2026, and completed the formalized proof in Lean the following day. Subsequently, OpenAI contacted external experts to discuss establishing a collaboration mechanism around this achievement.

Based on expert recommendations, OpenAI decided to directly publish the AI-generated proof, adopting the same approach used when releasing research results on October 6. Currently, both the natural language proof written by the model and the formalized Lean proof have been uploaded to a public GitHub repository.

The Exponential Time Hypothesis Disproved

The discovery in computer science 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 (SAT), closely related to the famous P vs NP problem.

For a long time, the academic community widely assumed ETH to be true. Hundreds of research papers relied on 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 system provided a proof that ETH is false, which has been formally verified using Lean.

With ETH disproven, researchers must re-evaluate how much speedup is actually achievable for many computational tasks. This new result also helps guide research into more efficient algorithms. However, finding algorithms that realize these speedups and translating theoretical results into practical applications will still require significant 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 MIT.

All three experts have extensive research backgrounds in this field:

• Impagliazzo and Paturi, along with Francis Zane, originally proposed ETH and have published several influential papers on the hypothesis over the past 30 years.
• Williams previously proved that certain strengthened versions of ETH do not hold, derived computational hardness results based on ETH, and established connections between algorithms and lower bounds in computational complexity.

OpenAI accepted two requests from the collaborating team: first, to make all raw data of the research results public to facilitate independent verification by other researchers; second, for OpenAI to fund an academic conference to discuss the overall impact of the new results 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 issues arising from AI-driven research breakthroughs, 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.

OpenAI generated the natural language proof disproving ETH on October 16, and completed the formalized Lean proof on October 17. Subsequently, OpenAI began establishing the aforementioned external collaboration mechanisms.

The model-generated natural language proof and the Lean proof are now published in OpenAI's public GitHub repository.

Existence of QCD and Unified Completion of the Standard Model

In physics, OpenAI presented a research result concerning the existence of quantum field theories.

This result rigorously proves, in a mathematical sense, that certain interacting quantum field theories containing gauge fields and fermions indeed exist in four-dimensional space. These include Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics.

The proof employs the constructive quantum field theory methods proposed by Balaban. Additionally, the research resolved a sub-problem: proving the existence of pure Yang-Mills theory. This problem is part of the Yang-Mills Existence and Mass Gap Millennium Prize Problem established by the Clay Mathematics Institute.

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

Since the Standard Model itself cannot be directly extended to arbitrary short distance scales, OpenAI's existence proof targets the simplest Grand Unified Theory (GUT) containing the Standard Model's gauge fields and fermions.

Quantum Chromodynamics studies how particles such as quarks and gluons interact via the strong force. Pure Yang-Mills theory covers a narrower scope, dealing only with the self-interactions 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 made significant contributions in the fields of quantum field theory and mathematical physics:

• David Gross: In 1973, his work on asymptotic freedom in Yang-Mills theory laid the foundation for Quantum Chromodynamics (QCD) becoming the accepted theory of strong interactions, earning him the 2004 Nobel Prize in Physics. Asymptotic freedom is also a key mechanism in rigorously proving the mathematical existence of these theories.
• Arthur Jaffe: Together with Edward Witten, he formulated the Millennium Prize Problem concerning quantum Yang-Mills theory. He was also one of the proposers of the concept behind the Millennium Prize Problems and participated in founding the Clay Mathematics Institute.
• Michael Douglas: For the past 20 years, he has been actively involved in advancing research on this problem. He reported on progress in 2004 and provided a comprehensive review of the latest developments earlier this year.
• Igor Klebanov: He was among the first to apply the duality between gauge fields and string theory to study confinement, a crucial property of QCD. He currently leads the Simons Foundation’s collaboration project on confinement and QCD strings.
• Nikita Nekrasov: He achieved pioneering results in exact solution methods for 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