OpenAI announced that its systems produced three new scientific results: a general proof in harmonic analysis (settling the local smoothing conjecture and, as consequences, the Kakeya and Stein restriction conjectures), a disproof of the Exponential Time Hypothesis (ETH) in computational complexity, and existence proofs for certain interacting four-dimensional quantum field theories including Yang–Mills theory and Quantum Chromodynamics (QCD). OpenAI engaged external expert "shepherds" to help organize verification and exposition, and each result is being handled with a distinct release strategy recommended by those experts.
Release approaches for the three results
The external advisors proposed three different approaches:
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For the mathematics result, the AI output was released publicly so researchers can independently organize to understand and expand on the claim. OpenAI published both the AI natural-language writeup and a Lean formalization on GitHub.
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For the computer science result, an external working group was formed to explain and shepherd the result while the AI output was also made public for independent inspection and further work.
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For the physics result, an external group of experts was formed and the group requested that the AI-produced artifacts be withheld while they write up and validate the material.
OpenAI notes it does not yet know which approach will be most effective and invites feedback to improve the process.
Principles governing the collaborations
The external collaborations follow four stated principles:
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Transparency: the collaborations are public; participants will not be asked to sign nondisclosure agreements and may share intermediate products and publish conclusions.
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No academic credit claimed by OpenAI: OpenAI does not seek academic authorship or credit for discoveries produced by its internal models; inviting researchers to steward work does not imply they made the original discovery.
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Recognition: the steward groups should receive credit for their explanations, writeups, and materials, and will help ensure proper attribution to prior researchers and techniques.
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Independence: participation is voluntary and unpaid; working-group members participate independently and do not speak on behalf of OpenAI, nor does OpenAI speak on their behalf.
What OpenAI has released and what remains private
OpenAI will hand over relevant materials and artefacts to the collaborations. For the computer science project, the group decided to make the natural-language proof and the Lean formalization public immediately, and OpenAI has published those artifacts to its public GitHub repository. For the physics project, the group requested that the AI-produced artifacts not be released immediately; they will work first on writing and validating the results themselves.
OpenAI acknowledges tradeoffs: shepherded collaborations help expert unpacking and contextualization but initial shepherd selection shapes who is involved first and creates information asymmetries. The company requests suggestions on broadening participation.
Mathematical result — harmonic analysis and local smoothing
The mathematics result concerns harmonic analysis and the local smoothing conjecture, originally proposed by Christopher D. Sogge in 1991. The conjecture addresses how waves can concentrate: the extreme focusing events must be short-lived, and typical (averaged) focusing is much weaker than the worst-case behavior. This type of distinction between worst-case and typical behavior appears frequently across analysis and has important implications: the local smoothing conjecture implies the Kakeya conjecture and Stein’s restriction conjecture.
Previously, Guth, Wang, and Zhang proved local smoothing in two spatial dimensions (2019), and Wang and Zahl proved the Kakeya conjecture in three dimensions (2025). OpenAI's models produced a proof of the local smoothing conjecture in every dimension, which therefore also resolves the Kakeya and Stein restriction conjectures in all dimensions. The new argument relies on a novel auxiliary partial differential equation.
OpenAI obtained the AI output on October 18, 2026; the Lean formalization was completed on October 19, 2026. Following the recommendation of an external expert, OpenAI released both the AI natural-language writeup and the Lean proof publicly, mirroring the approach taken in its October 6, 2026 release.
Computer science result — ETH refutation
The computer science result addresses the Exponential Time Hypothesis (ETH), a longstanding conjecture about the time complexity of Boolean formula satisfiability (SAT) closely related to the P vs NP question. ETH has been widely assumed and used across hundreds of papers to derive conditional hardness results in graph theory, computational biology, databases, computational geometry, and more.
OpenAI's system produced a disproof of ETH; that disproof was subsequently verified in Lean. If confirmed by the community, the refutation reopens questions about how much faster many of these tasks could become and may guide the search for improved algorithms. Translating theoretical improvements into practical gains will require substantial further research.
A verification and shepherding group is led by Russell Impagliazzo (University of California, San Diego), Ramamohan Paturi (University of California, San Diego), and Ryan Williams (Massachusetts Institute of Technology). Impagliazzo and Paturi were among the original proposers of ETH; Williams has previously refuted strong variants of ETH and has deep expertise connecting algorithms and lower bounds.
At the request of this group, all raw data related to the result will be made public to facilitate independent verification, and OpenAI has committed to funding a conference to assess the implications of the work for the field. The group invites the community to help organize an open examination of the ETH claim and to discuss broader issues that AI-generated breakthroughs raise regarding verification, access, credit, and the future of research.
OpenAI's models found the natural-language disproof on October 16, 2026, and the Lean formalization was completed on October 17, 2026. OpenAI has published both the AI natural-language writeup and the Lean proof in its public GitHub repository.
Physics result — Yang–Mills, QCD and interacting quantum field theories
The physics result provides a mathematically rigorous existence proof for certain interacting quantum field theories in four dimensions that include gauge fields and fermions, covering Quantum Chromodynamics (QCD) and a simple unified completion related to the Standard Model. The work builds on Balaban's constructive quantum field theory approach. As a subproblem, the work includes a proof of the existence of pure Yang–Mills theory, which is related to the Clay Mathematics Institute’s Quantum Yang–Mills Millennium Prize problem.
The Standard Model describes three of the four fundamental forces (excluding gravity); the existence proof targets the simplest grand-unified–type theory that contains the gauge fields and fermions of the Standard Model and can be extrapolated to arbitrarily short distances. QCD governs interactions of quarks and gluons through the strong force; pure Yang–Mills theory models glue self-interactions without matter fields.
David Gross will lead the shepherding group, joined by Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. Gross’s 1973 work on asymptotic freedom helped establish QCD and contributed to his share of the 2004 Nobel Prize in Physics. Arthur Jaffe co-formulated the Quantum Yang–Mills Millennium Prize problem and co-founded the Clay Mathematics Institute. Michael Douglas has been tracking progress on the problem for decades; Igor Klebanov and Nikita Nekrasov have made foundational contributions to gauge–string dualities and exact methods in generalized Yang–Mills contexts.
OpenAI's models produced a natural-language proof for Yang–Mills existence on September 9, 2026, and a proof for QCD on September 18, 2026. Due to the length and complexity of the work, publication was deferred until Lean formalization completed; the Yang–Mills formalization was finished on October 16, 2026, which prompted outreach to form the stewardship group. The physics group has requested that communications about the effort be sent to YANGMILLS4D@gmail.com.
The shepherds will review, validate, and write up the results before broader release as they see fit.
Support, next steps and significance
OpenAI says it is committed to supporting independent evaluation and subsequent work. The stewarding collaborations complement OpenAI's previously announced $5 million contribution to the Simons Foundation, which will be managed by the Simons Foundation to fund workshops, conferences, and programs to help researchers understand and build on AI-produced results.
These claims, if independently verified by the scholarly community, could have significant implications: the ETH refutation could reshape complexity-theoretic assumptions and algorithmic research; a general local smoothing proof would reshape parts of harmonic analysis; and rigorous constructions in four-dimensional quantum field theory could advance the mathematical foundations of theoretical physics.
OpenAI and the appointed shepherd groups are now focused on detailed verification, formal exposition, and community engagement. The organization requests feedback on disclosure practices and on ways to broaden and accelerate scientific scrutiny and participation.
