OpenAI announced AI-generated results in mathematics, computer science and physics on 26 October, alongside a release plan in which external academics help examine the work. The company has published the mathematical and computer science outputs for others to study, while the physics group has held back the AI-produced material and is writing its own account. OpenAI describes the manuscripts as raw research data that scientists must work through to develop an understanding of the results.
Key points
- OpenAI says its model proved a conjecture about waves in every dimension and, through it, resolved two other mathematical conjectures.
- Its claimed refutation of the Exponential Time Hypothesis has a public natural-language proof and a Lean formalisation.
- External groups will examine the computer science and physics results, but have chosen different approaches to releasing the AI output.
OpenAI’s three routes for academic review
The announcement follows OpenAI’s 6 October release of hundreds of manuscripts that it said addressed longstanding mathematical questions, The Verge reported. This time, OpenAI asked external experts to organise efforts to understand the results. On their advice, the company released the mathematics output directly, made the computer science material public alongside a working group’s examination, and left the timing of the physics material to another group.
OpenAI says members of the external groups will participate voluntarily and without pay. They will not be asked to sign nondisclosure agreements and can publish their own conclusions. The company says it will not seek academic credit for discoveries produced by its models, while the groups should receive recognition for the explanations and other work they produce.
That arrangement has an access problem OpenAI acknowledges: choosing the people who organise an examination gives them an earlier view of the work. On 25 October, its AGMAI panel of mathematics advisers recommended that AI companies “do not adopt” practices that designate particular people to receive output first, The Verge reported. The panel cited unequal access, borrowed reputation and biased selection, and recommended releases through scholarly repositories with formal proofs and information about how problems were chosen.
Local smoothing and the Kakeya conjecture
OpenAI says its models found a proof that settles the local smoothing conjecture in every dimension. Proposed by Sogge in 1991, the conjecture concerns how intensely waves can focus: an extreme concentration may occur at a particular instant, but should be short-lived. The company says its proof uses an auxiliary partial differential equation and also resolves the Kakeya conjecture and Stein’s restriction conjecture in all dimensions.
The scope matters because earlier work had addressed smaller parts of these problems. OpenAI says Guth, Wang and Zhang proved local smoothing in two spatial dimensions in 2019, while Wang and Zahl proved the Kakeya conjecture in three dimensions in 2025. The new all-dimensions result is OpenAI’s claim about its model output, rather than a conclusion attributed to the mathematicians who produced those earlier proofs.
OpenAI says it obtained the AI output on 18 October and formalised it in Lean the following day. A Lean formalisation expresses a proof as precise logical steps that software can check, much as a calculation must be written in a form a calculator can process. After consulting an expert about a possible collaboration, the company chose to publish both the model’s written argument and the Lean proof on GitHub for researchers to examine.
Impagliazzo’s group examines the ETH result
OpenAI also says its system disproved the Exponential Time Hypothesis, or ETH, a longstanding conjecture about the difficulty of determining whether a collection of true-or-false conditions can all be satisfied. The hypothesis has underpinned conditional results about computational difficulty in fields including databases and computational biology. According to the company, its AI found a natural-language disproof on 16 October, and the result was subsequently formalised in Lean on 17 October.
Checking whether a set of requirements can all be met could become faster if the theoretical result leads to better algorithms. OpenAI says finding such algorithms, and turning them into practical improvements, will require significant further research.
Russell Impagliazzo and Ramamohan Paturi of UC San Diego, together with Ryan Williams of MIT, will lead a group to examine and explain the ETH result. Impagliazzo and Paturi proposed the hypothesis with Francis Zane. At the group’s request, OpenAI has published the natural-language proof and Lean formalisation on GitHub and says it has agreed to make the raw result data public to support examination by others.
Gross’s group holds the physics output
In physics, OpenAI describes the result as a rigorous mathematical argument for the existence of certain interacting quantum field theories in four dimensions, including Quantum Chromodynamics and a completion of the Standard Model, The Verge reported. The account describes a result about the mathematical foundations of those theories. David Gross will lead the group examining it, alongside Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.
The physics group has chosen to hold back the AI-produced artifacts while its members write up the result themselves, OpenAI says. Unlike the mathematics and computer science proofs, the underlying physics output is therefore not part of the material the company has released for public examination.
