Satoshi Pupil

Less than a month after releasing hundreds of solutions to long-standing mathematical problems, OpenAI has announced three new breakthroughs achieved by its artificial intelligence in the fields of mathematics, computer science, and physics. These include the proof of mathematical conjectures, the disproof of the exponential time hypothesis, and discoveries related to quantum field theory. The company has already engaged external experts to verify these results, interpret them, and assess their implications for science.
A distinct publication approach was adopted for each study. Some materials have already been made publicly available, while groups of scientists have been assembled to verify others.
Proving Conjectures Across All Dimensions
The first result concerns harmonic analysis—a branch of mathematics that, among other things, studies the behavior of waves. In 1991, mathematician Christopher Sogge formulated the "local smoothing conjecture," which describes patterns in wave focusing and predicts that the most extreme instances of such focusing must be short-lived.
The concept suggests that, in the worst-case scenario, a wave can focus much more intensely than it would at a typical moment in time. This conjecture carries broader implications for mathematical analysis; specifically, it is linked to the Kakeya conjecture and the Stein restriction conjecture. Prior to this, the local smoothing conjecture had been proven for two spatial dimensions in 2019, and the Kakeya conjecture for three dimensions in 2025. According to a statement from OpenAI, its models have proven the local smoothing conjecture, the Kakeya conjecture, and the Stein restriction conjecture for all dimensions. The new proof relies on an auxiliary partial differential equation.
The company achieved this result on October 18, 2026, and the following day formalized the proof using Lean—a system for verifying mathematical proofs. OpenAI has published a description of the result and the formalized proof in a public GitHub repository.
AI Refutes the Exponential Time Hypothesis
A second result concerns the Exponential Time Hypothesis (ETH), which describes the expected complexity of solving the Boolean satisfiability problem. This problem involves determining whether there exists an assignment of variable values ​​that makes a given logical formula true.
ETH is widely used in theoretical computer science to evaluate the complexity of algorithms. It is also linked to the P versus NP question—one of the most famous unsolved problems in theoretical computer science, which concerns the relationship between problems whose solutions can be verified quickly and problems that can be solved quickly.
OpenAI claims that its system has refuted the ETH, and the result was subsequently verified using Lean. The hypothesis has been used in hundreds of scientific papers, including research in graph theory, computational biology, and database systems.
If the refutation is confirmed by independent researchers, it could reshape our understanding of the complexity of many computational problems and aid in the search for faster algorithms. At the same time, the practical application of this result will require further research. The verification process will be coordinated by experts in computational complexity theory: Russell Impagliazzo and Ramamohan Paturi from the University of California, San Diego, as well as Ryan Williams from the Massachusetts Institute of Technology.
At the request of this group, OpenAI has released raw data and proofs, allowing scientists to independently verify the claim. The company has also agreed to fund a conference dedicated to discussing the result and its potential implications.
Proofs for Quantum Field Theory
The third result concerns quantum field theory—the mathematical foundation of modern theoretical physics. OpenAI states that its models have generated proofs for the existence of certain interacting quantum field theories in four dimensions, specifically those related to quantum chromodynamics (QCD) and the Standard Model of particle physics.
Quantum chromodynamics describes the strong interaction that binds quarks and gluons together. The work also covers pure Yang-Mills theory, the existence of which is linked to one of the Clay Mathematics Institute’s Millennium Prize Problems—a challenge with a one-million-dollar reward for its solution.
The results will be verified by a group led by physicist David Gross, with the participation of Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. The team members have extensive research experience in mathematical physics and Yang-Mills theory.
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OpenAI’s models generated a proof for the existence of Yang-Mills theory in natural language on September 9, 2026, followed by a proof for QCD on September 18. The company delayed the release of these findings to complete their formalization; the proof for the existence of Yang-Mills theory was verified using Lean on October 16.
Unlike the other two research tracks, the physics group decided not to release the AI-generated materials directly, opting instead to prepare their own presentation of the results. Their conclusions are expected to help assess whether these findings can contribute to making quantum field theory more mathematically rigorous.
OpenAI will fund further research
The company stated that it would support scientists who verify, explain, and build upon results generated by AI. This new collaboration program complements OpenAI’s previously announced $5 million contribution to the Simons Foundation.
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The Simons Foundation will independently manage the funding, directing it toward seminars, conferences, and specialized programs for researchers. The goal is to provide scientists with the resources to independently examine, explain, and further develop these findings.
OpenAI notes that it will continue to gather feedback from the academic community regarding how AI-generated results are disclosed. The company also acknowledges that involving a limited number of initial curators could influence who gains early access to the research, potentially creating disparities in access to information.
Therefore, a key objective moving forward is to broaden the participation of scientists and find ways to make the process of verifying and interpreting results more open. The significance of these findings for mathematics, computer science, and physics will depend on the extent to which researchers can independently verify the claims.
Last month, OpenAI established a Mathematics and AI Advisory Group at the Institute for Advanced Study (IAS). A group of nine scientists operates autonomously and without funding from OpenAI to independently assess the academic significance of discoveries, develop recommendations for the safe release of findings, and determine the impact of AI systems on the development of fundamental mathematics.