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.

One 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.

OpenAI 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.

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