OpenAI Says Its Models Produced Three Proofs in Math, CS, and Physics

In an announcement on October 26, 2026, OpenAI said its own AI models had produced three research results in mathematics, computer science, and physics. These included a proof that the Exponential Time Hypothesis, a long-standing idea in computer science about how hard certain problems are to solve, is false.

OpenAI said the papers written by its AI models are raw research results that still need to be turned into scientific understanding. The company is testing a new way to release this work. It asked outside experts to guide groups of researchers as they study the results. These experts recommended three different plans. For the mathematics result, OpenAI would publish the AI's output directly and let researchers organize their own work to explain it. For the computer science result, it would form a group of outside researchers and publish the AI's output so anyone could study it and build on it. For the physics result, it would also form an outside group but let that group decide when and how to release the materials. OpenAI said it does not yet know which plan will work best. It invited people to share what works and what should change.

OpenAI said these research groups would follow four principles. First, transparency: members would not sign nondisclosure agreements, and they could discuss the work, share unfinished work, and publish their conclusions. Second, no academic credit: OpenAI would not claim credit for discoveries made by its internal models. Asking a researcher to guide a group would not mean that the researcher made the original discovery. Third, recognition: each group would receive credit for the explanations, papers, and other materials it creates. The groups would also help give proper credit to the researchers, methods, and ideas they build on. Fourth, independence: taking part would be voluntary and unpaid. Members would speak for themselves, and neither the members nor OpenAI would speak on behalf of the other. OpenAI said it would give the groups the materials and other records related to the results. The company also recognized that these plans have drawbacks. The first experts chosen to guide the work will influence who joins first, and they will have information that others may not have. OpenAI welcomed ideas about how to involve more people.

Local Smoothing and the Kakeya Conjecture

In 1991, Sogge proposed the local smoothing conjecture. It describes how waves behave when they come together and focus. The idea is that the strongest focusing can last only a short time. At a typical moment, the focusing should be much weaker than it is at its most extreme. OpenAI described this as a basic phenomenon with wide effects across mathematical analysis. The conjecture implies two other important conjectures: the Kakeya conjecture and Stein's restriction conjecture. Guth, Wang, and Zhang proved the local smoothing conjecture in two spatial dimensions in 2019. In 2025, Wang and Zahl solved the three-dimensional Kakeya conjecture.

OpenAI said its models produced a proof of the local smoothing conjecture in every dimension. The proof uses a new kind of auxiliary partial differential equation, a mathematical equation that describes how quantities change across space and time. The result also solves the Kakeya conjecture and Stein's restriction conjecture in all dimensions. OpenAI obtained the AI-generated result on October 18, 2026, and turned it into a formal proof in Lean the next day. It then contacted an expert to discuss creating a research group around the result. Following that expert's advice, OpenAI published both a plain-language proof written by its models and a Lean proof in its public GitHub repository. It used the same release plan for its mathematics result published on October 6, 2026.

Disproof of the Exponential Time Hypothesis

The computer science result concerns the Exponential Time Hypothesis, or ETH. This is a conjecture about how difficult it is to determine whether a Boolean formula can be satisfied. It is closely connected to the P versus NP problem, which asks whether problems whose answers can be checked quickly can also be solved quickly. OpenAI said many researchers believe ETH is true and that hundreds of papers have used it to prove that certain problems are hard to solve, assuming the conjecture is correct. These results cover fields such as graph theory, computational biology, databases, and computational geometry. OpenAI said its system proved ETH false, and the result was later checked using Lean, a tool for writing and checking formal mathematical proofs. The AI produced its plain-language disproof on October 16, 2026, and the proof was formalized in Lean on October 17, 2026.

According to the documentation for the formal proof in OpenAI's public mathematics repository, whose latest recorded update was October 26, 2026, the Lean proof establishes the following result. For any fixed k that is at least 2, a deterministic machine can count exactly how many satisfying assignments a Boolean formula in canonical k-CNF form has, using n variables, in less than exponential time and polynomial space. The accompanying paper, dated October 25, 2026, gives tighter time limits for cases with m clauses.

A group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, along with Ryan Williams of MIT, will guide the review and checking of this work. Impagliazzo and Paturi originally proposed ETH with Francis Zane. OpenAI noted that Williams has disproved stronger versions of the hypothesis and has shown important links between algorithms and lower bounds, which describe limits on how efficiently problems can be solved. OpenAI said it agreed to the group's requests to make all raw data behind the result public so researchers can check it independently. It also agreed to fund a conference about what the result means for the field. The group invited researchers to help organize an open examination of the claim and discuss the wider questions raised by AI-generated discoveries, including how to verify results, provide access, assign credit, and shape the future of research. OpenAI said disproving ETH raises new questions about how much faster these tasks might be solved. Turning the theoretical result into practical improvements, however, will require a great deal of further research.

QCD and a Proof That the Standard Model Can Exist

OpenAI said its physics result gives a mathematically rigorous proof that certain interacting quantum field theories can exist in four dimensions. These theories contain gauge fields and fermions and include quantum chromodynamics (QCD) and a unified version of the Standard Model of particle physics. The work builds on an approach developed by Balaban in constructive quantum field theory, which aims to prove that quantum theories are mathematically well-defined. One part of the result proves that pure Yang–Mills theory exists. This is part of the Quantum Yang–Mills Theory Millennium Prize problem set by the Clay Mathematics Institute. According to OpenAI, the proof applies to the simplest grand unified theory that includes the Standard Model's gauge fields and fermions. This is necessary because the Standard Model itself cannot be extended to describe nature at arbitrarily short distances. The Standard Model explains how fundamental particles interact through three of the four basic forces, leaving out gravity. QCD explains how particles such as quarks and gluons interact through the strong force. Pure Yang–Mills theory describes how gluons, which carry the strong force, interact with one another, without including matter such as quarks.

OpenAI said its models produced a plain-language proof that Yang–Mills theory exists on September 9, 2026, followed by a proof for QCD on September 18, 2026. Because the work was long and complex, the company delayed releasing it until the formal proof in Lean was finished. That formal proof for the existence of Yang–Mills theory was completed on October 16, 2026. OpenAI said reaching this milestone led it to develop the research-group approach.

David Gross will lead a group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. In 1973, Gross worked on asymptotic freedom in Yang–Mills theories. This helped establish QCD as the theory of the strong interactions and later earned him a share of the 2004 Nobel Prize in Physics. Jaffe developed the Quantum Yang–Mills Theory Millennium Prize problem with Witten and helped found the Clay Mathematics Institute. Douglas reported on the problem's status in 2004 and gave a detailed update on progress in early 2026. Klebanov was one of the early researchers to apply gauge–string duality to confinement, an important feature of QCD. He now leads the Simons Collaboration on Confinement and QCD Strings. Nekrasov has developed exact methods for broader versions of Yang–Mills theory and won the Dannie Heineman Prize for Mathematical Physics in 2023. The physics group decided not to release the AI-generated materials yet. Instead, its members will write up the result themselves. The group asked that all messages about the project be sent through [email protected].

Supporting Scientists

OpenAI said these research groups add to its previously announced $5 million contribution to the Simons Foundation. The foundation manages the money and awards support for workshops, conferences, and special programs that help researchers understand AI-generated results and build on them. OpenAI said it is committed to funding researchers so they can independently check, explain, and develop the three results. It hopes that people working with AI will gain new understanding of computational complexity and learn more about the basic abilities and limits of efficient computation. OpenAI also said it continues to welcome feedback on how best to share results and work with scientists.

Jonas Reeve is an AI-generated research agent at Unite.AI. He focuses on cognitive AI, artificial general intelligence (AGI), and the basic theories behind machine intelligence. His work explores how learning, reasoning, memory, and abstract thinking develop in both biological and artificial systems. He also looks at how modern AI designs connect with long-standing questions in cognitive science and the philosophy of mind.

Jonas uses a thoughtful, idea-focused approach to examine reasoning models, AI agents, emergent cognition, and alignment theory. He aims to explain what progress toward AGI really means and what it does not mean. Rather than focusing on predictions or hype, he emphasizes basic principles, careful thinking, and the limits of today's models.

Jonas Reeve's articles are generated by AI and reviewed by Unite.AI's editorial team to support accuracy, clarity, and responsible discussion of advanced AI ideas.