OpenAI announced on October 26, 2026, that its models had produced results in mathematics, computer science and physics. The claims range from a proof of a mathematical conjecture to an alleged refutation of a complexity hypothesis and existence proofs for quantum field theories. OpenAI also described different ways the work would be examined and released.
NeoTeo previously covered other AI-assisted mathematics claims, including OpenAI’s Navier–Stokes work and Anthropic’s formalization of Fermat’s Last Theorem. Those are separate developments from the three claims discussed here.
Three claims, different routes for examination
OpenAI’s October 26 announcement describes what its models produced and how the work was handled in each field. The mathematics and computer-science materials were made public in natural-language and Lean forms. Lean is a proof assistant, software that checks whether formalized steps follow from stated definitions, assumptions and rules. The physics group chose to hold back the AI-generated materials while preparing its own exposition.
| Field | What OpenAI says its model produced | Dates and examination path described by OpenAI |
| Mathematics | A proof of the local-smoothing conjecture in any dimension, which OpenAI says implies the Kakeya and Stein restriction conjectures in all dimensions. | OpenAI dates the natural-language result to October 18, 2026, and its Lean formalization to October 19. The mathematics materials were made public for examination. |
| Computer science | A refutation of the Exponential Time Hypothesis (ETH), a conjecture in computational complexity. | OpenAI dates the result to October 16, 2026, and its Lean formalization to October 17. The company said an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams would verify and study the work. |
| Physics | Existence proofs for certain interacting four-dimensional quantum field theories, including quantum chromodynamics (QCD), as well as pure Yang–Mills theory. | OpenAI dates the natural-language Yang–Mills result to September 9, 2026, and the QCD result to September 18. It says the Yang–Mills existence proof was formalized in Lean on October 16. The physics group chose to prepare its own exposition before releasing the AI-generated materials. |
OpenAI’s announcement described examination and publication arrangements, not academic acceptance of all three claims. The different paths also affect what readers can inspect: the mathematics and ETH materials were public, while the physics group withheld the AI-generated versions.
The mathematics claim starts with local smoothing
OpenAI says its model produced a proof of the local-smoothing conjecture for any dimension. The company says that result implies the Kakeya and Stein restriction conjectures in all dimensions. In other words, the primary claim OpenAI describes is the local-smoothing result; Kakeya and Stein restriction are stated consequences.
OpenAI places the claim alongside earlier mathematical milestones: local smoothing in two spatial dimensions was proved in 2019, and the Kakeya conjecture in three dimensions was proved in 2025. The company-reported all-dimensions result is broader in scope than those examples.
What the ETH claim does and does not say
OpenAI says its model refuted ETH, then formalized the result in Lean. That is a claim about a hypothesis in computational complexity, not a statement that P = NP has been proved. OpenAI said further research would be needed to find algorithms and turn theoretical results into practical improvements.
The external group OpenAI named to study the ETH work was led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT. The company described the group’s role as verification and study, rather than reporting a completed academic verdict.
The physics results concern mathematical existence
OpenAI describes proofs that certain interacting four-dimensional quantum field theories exist, including theories with gauge fields and fermions. Its examples include QCD and a unified completion of the Standard Model. The announcement also describes a pure Yang–Mills existence proof, a subproblem related to the Clay Mathematics Institute’s Yang–Mills problem.
An existence proof is a mathematical claim about whether a theory satisfying specified conditions exists. It is not an announcement of a newly observed physical phenomenon. OpenAI’s stated scope does not include a proof of the Yang–Mills mass gap, the component associated with the full Millennium problem.
OpenAI said the physics group chose to delay publication of the AI-generated materials and prepare its own exposition. That left the physics work on a different publication path from the mathematics and ETH claims, whose materials were released for examination.
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