OpenAI Models Generate Proofs in Mathematics, Computing and Quantum Physics

Harmonic analysis and wave concentration

On Oct. 18, 2026, artificial intelligence models at OpenAI generated a mathematical proof for the local smoothing conjecture, an open problem in harmonic analysis formulated by Christopher Sogge in 1991, as reported by Interesting Engineering. The conjecture describes the behavior of concentrating waves and states that extreme spatial focusing cannot sustain itself over extended durations. The lab reported that its system proved the conjecture across all spatial dimensions, while also resolving both the Kakeya conjecture and Stein's restriction conjecture.

Prior work in this domain had resolved restricted dimensional cases, including a 2019 proof in two spatial dimensions by Larry Guth, Hong Wang, and Ruixiang Zhang, followed by Wang and Zahl's 2025 proof of the three-dimensional Kakeya conjecture. To allow immediate scientific inspection of this theoretical innovation, OpenAI converted the natural-language proof into Lean, an interactive theorem prover that enables automated verification, on Oct. 19, 2026, publishing the full materials on GitHub according to Interesting Engineering.

Challenging the Exponential Time Hypothesis

Beyond pure mathematics, OpenAI stated that its models produced a disproof of the Exponential Time Hypothesis (ETH) on Oct. 16, 2026, formalizing the result in Lean the following day, according to Interesting Engineering. ETH is a central hypothesis in computational complexity theory that sets theoretical bounds on algorithms solving Boolean satisfiability problems. The conjecture serves as an axiomatic foundation for hundreds of published papers in computational geometry, graph theory, databases, and computational biology.

While refuting ETH does not instantly yield faster practical algorithms for real-world software, it challenges core assumptions used to prove conditional lower bounds across computer science. To assess the mathematical validity of the disproof, researchers Russell Impagliazzo and Ramamohan Paturi from the University of California, San Diego, alongside Ryan Williams of the Massachusetts Institute of Technology, will lead an independent review panel supported by an OpenAI-funded conference, as detailed by Interesting Engineering.

Quantum field theory and independent academic verification

The third claimed milestone targets mathematical physics and the rigorous foundations of quantum field theory. The models reportedly generated existence proofs for interacting four-dimensional gauge fields and fermions, encompassing quantum chromodynamics (QCD) and pure Yang-Mills theory, as reported by Interesting Engineering. Mathematical existence and the associated mass gap problem in Yang-Mills theory constitute one of the unresolved Millennium Prize problems designated by the Clay Mathematics Institute.

Unlike the mathematics and complexity results published directly on repository platforms, the physics materials will undergo an independent analysis led by Nobel laureate David Gross, alongside researchers Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. OpenAI confirmed that academic evaluation teams will operate with complete editorial autonomy, supported by a $5 million fund previously allocated to the Simons Foundation to coordinate evaluation workshops and academic seminars, as confirmed by Interesting Engineering.

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