On October 26, 2026, OpenAI said its internal models had produced three scientific results: a proof involving local smoothing, a refutation of the Exponential Time Hypothesis (ETH), and existence results in quantum field theory. The company announced a different publication and collaboration path for each field. OpenAI’s announcement describes the claims and those arrangements.
This was separate from OpenAI’s earlier reported plan to publish mathematical results on GitHub, covered in NeoTeo’s previous report.
What OpenAI says each result covers
OpenAI says the mathematics work proves the local smoothing conjecture in every dimension, with consequences for the Kakeya and Stein restriction conjectures in every dimension. The company says the proof uses a novel auxiliary partial differential equation. It dates the model output to October 18 and its formalization in Lean to October 19, 2026.
For computer science, OpenAI says its system refuted ETH, a conjecture about the computational complexity of Boolean formula satisfiability. The company says the result is related to P versus NP, but its announcement does not claim that P = NP. OpenAI dates the natural-language result to October 16, 2026, and its Lean formalization to October 17.
The physics claim concerns the existence of certain interacting four-dimensional quantum field theories with gauge fields and fermions. OpenAI says these include quantum chromodynamics (QCD) and a unified completion of the Standard Model, as well as pure Yang–Mills theory. The Yang–Mills claim concerns existence; it is not a claim to have solved the full Clay Mathematics Institute problem, which also requires a result about the mass gap.
OpenAI describes these as three separate workstreams. It does not say that one result produced the other two.
What Lean formalization and the ETH claim mean
Lean is a proof assistant used to encode formal mathematical arguments. OpenAI’s dates distinguish the natural-language results from the later formalizations: October 16 and 17 for ETH, and October 18 and 19 for local smoothing. Formalizing an argument is a distinct step in presenting the work, not a claim that the scientific result has been independently accepted.
A claimed refutation of ETH also does not mean computers will immediately run faster. OpenAI says finding algorithms that use the theoretical result, and turning them into practical improvements, will require significant further research.
How publication and collaboration differ by field
| Field | OpenAI’s stated claim | Announced publication and collaboration path |
| Mathematics | Local smoothing in every dimension, with claimed consequences for Kakeya and Stein restriction | OpenAI released the model output and Lean proof for examination by researchers. |
| Computer science | A refutation of ETH concerning Boolean formula satisfiability | OpenAI made the natural-language result and Lean formalization public. Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams were named to lead the external group examining the work. |
| Physics | Existence results for specified four-dimensional quantum field theories, including QCD, a unified Standard Model completion, and pure Yang–Mills | The physics group chose to withhold the AI-produced artifacts while preparing its own write-up. David Gross leads the group, whose members include Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. |
OpenAI says collaboration participation is voluntary and unpaid, and that participants work independently and may publish their conclusions. The company also says it does not seek academic credit for discoveries produced by its internal models.
