OpenAI says its models have produced mathematical proofs that challenge established theories in harmonic analysis and computer science. The work also includes a claimed existence proof for quantum field theories, including quantum chromodynamics (QCD). Researchers will now examine the results, verify their implications and explain their significance to the wider scientific community.The company has adopted different release strategies for the three projects. It has published the mathematical and computing proofs, while an external physics group will prepare its own account of the research.Math results challenge established theoriesOne proof addresses the local smoothing conjecture, a problem in harmonic analysis proposed by Christopher Sogge in 1991. It describes how waves behave when they concentrate, suggesting that extreme focusing cannot persist for long periods.OpenAI says its models proved the conjecture across every spatial dimension. The result also resolves the Kakeya conjecture and Stein’s restriction conjecture, according to the company.Previous research established important milestones in this area. In 2019, Larry Guth, Hong Wang and Ruixiang Zhang proved the local smoothing conjecture in two spatial dimensions. Wang and Zahl subsequently proved the three-dimensional Kakeya conjecture in 2025.OpenAI generated its new proof on Oct. 18, 2026, and formalized it in Lean the following day. Lean is a proof assistant that helps mathematicians verify arguments with computer-based checks. The company has published both the natural-language proof and its formal counterpart on GitHub.Computing theory faces fresh scrutinyOpenAI also says its models disproved the Exponential Time Hypothesis (ETH), a longstanding conjecture in computational complexity. The hypothesis concerns the difficulty of solving Boolean satisfiability problems and underpins conditional hardness results across numerous research fields.Researchers have used ETH in hundreds of papers covering graph theory, computational biology, databases, and computational geometry. If the new result withstands scrutiny, it could force researchers to reassess assumptions behind those studies.The models produced the disproof on Oct. 16, 2026, before formalizing it in Lean the following day. However, refuting ETH would not automatically deliver faster algorithms for existing computing problems. Researchers must still determine what the result means for practical computation.Russell Impagliazzo and Ramamohan Paturi of UC San Diego will lead the review alongside Ryan Williams of MIT. OpenAI has agreed to fund a conference examining the claim and released its proof materials for independent evaluation.Quantum field theory proof awaits reviewThe physics work concerns the mathematical foundations of quantum field theory. OpenAI says its models produced existence proofs for interacting theories involving gauge fields and fermions in four dimensions. The results include QCD, which describes the strong force governing interactions between quarks and gluons.The work also addresses pure Yang-Mills theory, a key component of a Millennium Prize problem from the Clay Mathematics Institute. That problem concerns the mathematical existence of quantum Yang-Mills theory and the presence of a mass gap.Physicist David Gross will lead an external group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. The team plans to develop its own explanation before releasing the model-generated physics materials.OpenAI says all three collaborations will operate independently, with participants free to publish their findings. The company has also pledged support through a previously announced $5 million contribution to the Simons Foundation. That funding will help researchers organize workshops and conferences to examine the work and assess its broader implications.