We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
Our release process continues to evolve in response to feedback from the associated academic communities. In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach. We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies:
- For the computer science result, form an external group of collaborators to explain the result, while making the AI output publicly available for anyone to examine and build on.
- For the physics result, form an external group of collaborators and defer to them on the release of materials.
We do not yet know which of these three approaches will be most useful, and we welcome feedback on what works and what we can improve.
The external collaborations will follow four principles:
1. Transparency. We are transparent about the existence of our collaborations. Participants will not be asked to sign nondisclosure agreements and will be free to discuss the work, share intermediate products, and publish their conclusions.
2. No Academic Credit. We do not seek academic credit for the discoveries produced by our internal models. Inviting a researcher to help steward a collaboration also does not imply that they or their working group made the original discovery.
3. Recognition.This group should receive recognition for the understanding they develop, the writeups they author, and the other materials they create. The group will also help ensure that the work appropriately credits researchers, techniques, and ideas that it draws on.
4. Independence.Participation is voluntary and unpaid. Working-group members participate independently, and their conclusions are their own. They do not speak on behalf of OpenAI, and OpenAI does not speak on their behalf.
We will turn over materials and artifacts related to these results to the collaborations. For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
The result in mathematics concerns harmonic analysis. In 1991, Sogge proposed the local smoothing conjecture, which concerns the behavior of waves as they become focused. In particular, the conjecture states that the most extreme focusing must be short-lived. The amount of focusing at a typical random time is always much smaller than what can happen at the worst-case time. This is a familiar pattern throughout mathematics: what can happen in the worst case is more extreme than what is observed after averaging. Because of how fundamental this conjectured phenomenon is, it also has wide ranging implications throughout mathematical analysis. For instance, it implies the Kakeya conjecture and Stein’s restriction conjecture.
We obtained our AI output on October 18th, 2026, which was formalized in Lean the following day, and then we reached out to an expert to discuss forming a collaboration around this result. Following their recommendation, we decided to release the AI output publicly, following the same strategy for this result as our October 6th, 2026 release. We have released both an AI natural language output writeup(opens in a new window) produced by our models and aLean proof (opens in a new window)in our public GitHub repository.
The computer science collaboration will understand and explain a result concerning the Exponential Time Hypothesis (ETH), a longstanding conjecture about the computational complexity of Boolean formula satisfiability, a problem deeply related to the P vs. NP problem. The ETH has been widely believed to be true, and has been used in hundreds of papers to derive conditional computational hardness results across a wide variety of fields including graph theory, computational biology, databases, computational geometry, and more. Surprisingly, our system has recently proved that the ETH is false, a result that was subsequently verified in Lean. The refutation of the ETH reopens questions about how much faster these tasks could become, and may help guide the search for better algorithms. Finding such algorithms, and turning theoretical advances into practical improvements, will require significant further research.
This work will be verified and shepherded by a group led by Russell Impagliazzo (UC San Diego), Ramamohan Paturi (UC San Diego), and Ryan Williams (MIT). All three are experts in algorithm design and complexity theory.
Our AI found its natural language disproof for the ETH on October 16th, 2026 and then subsequently we formalized the result in Lean on October 17th, 2026, which led us to develop the collaboration described here. To that end, we have released both an AI natural language output writeup(opens in a new window) produced by our models and a Lean proof (opens in a new window)in our public GitHub repository.
The physics collaboration will understand and explain a result that gives a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics. Our work builds on Balaban’s approach(opens in a new window) in constructive quantum field theory. As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem(opens in a new window).
David Gross will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov that will shepherd this work.
Source: https://openai.com/index/building-scientific-understanding-with-academic-communities/