The company said on Monday that its systems had produced mathematical arguments challenging problems that have resisted researchers for decades. Two results have been released with formal proofs checked using Lean software, while a group of physicists will oversee further examination of the third before its underlying materials are published.
The computer science claim concerns the Exponential Time Hypothesis, which holds that certain Boolean satisfiability problems cannot be solved in subexponential time. The assumption underpins hundreds of conditional results about the difficulty of computational tasks, including problems in graph theory, databases, computational biology and geometry.
OpenAI said its model generated a natural-language disproof on 16 October and that a formalisation was completed in Lean on 17 October. Lean checks whether a mathematical argument follows specified formal rules, although its use does not eliminate the need for specialists to examine definitions, assumptions and the significance of a result.
Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of the Massachusetts Institute of Technology will lead an external group examining the claim. Impagliazzo and Paturi helped formulate the original hypothesis alongside Francis Zane.
The group has requested publication of all raw material to facilitate scrutiny, and OpenAI has agreed to fund a conference exploring the implications. Its stated objective is to organise independent examination rather than simply endorse the company’s interpretation.
If upheld, the result would require researchers to reassess conclusions whose hardness guarantees depend on the hypothesis. It would not, by itself, settle the separate P versus NP question or demonstrate immediate improvements in commercial computing. Developing useful algorithms from the theoretical result would require additional work.
The mathematical announcement concerns the local smoothing conjecture, proposed by Christopher Sogge in 1991. It describes how extreme concentrations of waves diminish when their behaviour is averaged over time. OpenAI claims its system proved the conjecture across all dimensions covered by the problem, with consequences for the Kakeya conjecture and Stein’s restriction conjecture.
The company said the proof uses a new auxiliary partial differential equation. Its model produced the argument on 18 October, followed by Lean formalisation on 19 October. Both the natural-language account and formal proof have been made publicly accessible for examination by mathematicians.
Earlier research had established local smoothing in two spatial dimensions, while a separate breakthrough settled the three-dimensional Kakeya conjecture in 2025. OpenAI’s claimed extension would therefore reach beyond those established results, provided the argument withstands independent scrutiny.
The physics claim involves rigorous mathematical constructions of interacting quantum field theories in four dimensions. OpenAI says its systems established existence results involving quantum chromodynamics, which describes the strong nuclear interaction, and a unified completion of the Standard Model of particle physics.
The work also includes a claimed existence proof for pure Yang–Mills theory, a component of a Millennium Prize problem administered by the Clay Mathematics Institute. Whether the proposed construction meets the precise requirements of that problem remains a matter for expert assessment.
Nobel laureate David Gross will lead the physics review group, alongside Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. OpenAI said its systems generated the Yang–Mills argument on 9 September and the quantum chromodynamics argument on 18 September. Formalisation of the Yang–Mills result was completed on 16 October, according to the company.
Unlike the other two projects, the physics group’s AI-generated manuscripts are not being released immediately. The external researchers have opted to prepare their own explanations before deciding how to present the material.
OpenAI said its collaborators would participate voluntarily and without payment, would not be bound by nondisclosure agreements, and could publish independent conclusions. The company said it would not claim academic authorship for discoveries generated by its internal models.
The company also acknowledged that selecting initial experts creates unequal access to information. It has committed to supporting wider examination, alongside a previously announced $5 million contribution to the Simons Foundation for workshops, conferences and programmes examining AI-generated scientific work. The foundation will administer funding for activities designed to help researchers understand and build upon machine-generated results, with independent scientific participation remaining central to those efforts.
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