OpenAI Reports AI Results in Math, Computer Science and Physics, Hands Them to Outside Experts
OpenAI says its models have produced three new results in harmonic analysis, complexity theory and quantum field theory, and it is handing them to external expert groups to verify and explain. According to OpenAI, the claims include a proof of the local smoothing conjecture in every dimension and a refutation of the Exponential Time Hypothesis.
- OpenAI announced AI-produced results in harmonic analysis, complexity theory and quantum field theory.
- It claims a proof of the local smoothing conjecture in every dimension, implying the Kakeya conjecture in all dimensions.
- It claims the Exponential Time Hypothesis is false, verified in Lean, with a group led by Impagliazzo, Paturi and Williams.
- The physics group led by David Gross is withholding AI artifacts to write up the work itself.
- Collaborators are unpaid and independent, and OpenAI says it seeks no academic credit.
A new release approach: shepherds for raw research data
OpenAI describes the manuscripts as "a form of raw research data" and says turning them into scientific understanding requires the academic community. For this release it asked external experts to act as shepherds and organize groups to understand the results.
The shepherds advised three different release strategies:
- Mathematics: release the AI output and leave researchers free to organize their own efforts to explain it.
- Computer science: form an external group of collaborators and make the AI output public.
- Physics: form an external group and defer to it on the release of materials.
OpenAI says it does not yet know which approach will prove most useful and welcomes feedback. It also acknowledges a tradeoff: choosing initial shepherds shapes who gets involved first and creates an asymmetry of information.
The collaborations will follow four principles, according to OpenAI. They are transparent, so participants will not be asked to sign nondisclosure agreements. OpenAI says it does not seek academic credit for discoveries produced by its internal models. The groups receive recognition for the understanding and writeups they create. Participation is voluntary and unpaid, and members do not speak for OpenAI, nor does OpenAI speak for them.
Local smoothing and the Kakeya conjecture
The mathematics result concerns the local smoothing conjecture, proposed by Sogge in 1991. It states that the most extreme focusing of waves must be short-lived, so focusing at a typical time is much smaller than at the worst-case time. The conjecture implies the Kakeya conjecture and Stein's restriction conjecture.
Guth, Wang, and Zhang proved the conjecture in two spatial dimensions in 2019, and Wang and Zahl proved the Kakeya conjecture in three dimensions in 2025. OpenAI says its models produced a proof of the local smoothing conjecture in every dimension, which would also resolve the Kakeya and Stein restriction conjectures in all dimensions. The proof is based on a novel auxiliary partial differential equation.
According to OpenAI, the AI output was obtained on October 18th, 2026 and formalized in Lean the next day. Following an expert's recommendation, the company released both the natural language writeup and the Lean proof in its public GitHub repository.
The Exponential Time Hypothesis claimed false
The computer science result concerns the Exponential Time Hypothesis (ETH), a longstanding conjecture about the computational complexity of Boolean formula satisfiability, which is closely related to the P vs. NP problem. OpenAI notes that the ETH has been widely believed and used in hundreds of papers to derive conditional hardness results in fields such as graph theory, computational biology, databases and computational geometry.
OpenAI says its system proved the ETH false and that the result was verified in Lean. The company states that the refutation reopens questions about how much faster these tasks could become, but that finding better algorithms and making practical improvements will require significant further research.
A group led by Russell Impagliazzo (UC San Diego), Ramamohan Paturi (UC San Diego) and Ryan Williams (MIT) will verify and shepherd the work. OpenAI agreed to the group's requests to make all raw data public and to fund a conference on the implications. The AI disproof was found on October 16th, 2026 and formalized in Lean on October 17th, 2026. The natural language proof and Lean formalization are now public on GitHub.
Quantum chromodynamics and the Standard Model
The physics result is described as a mathematically rigorous existence proof of certain interacting quantum field theories with gauge fields and fermions in four dimensions. These include Quantum Chromodynamics (QCD) and a unified completion of the Standard Model. The work builds on Balaban's approach in constructive quantum field theory, and as a subproblem includes a proof of the existence of pure Yang-Mills theory, part of a Clay Mathematics Institute Millennium Prize problem.
David Gross will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. According to OpenAI, the group has decided to hold off on releasing the AI-produced artifacts and to write up the work itself.
What it means
All three claims come from OpenAI and are now in the hands of outside experts for verification. The differing release choices, from full public release to a delayed write-up, will serve as a test of how academic communities want to handle AI-generated results. OpenAI invites feedback on broadening participation, and the ETH group has called for open examination of questions about verification, access, credit and the future of research.
