OpenAI claims three AI discoveries in math, computer science and physics

OpenAI said on 26 October 2026 that its models produced three new results: a proof of the local smoothing conjecture in harmonic analysis, a refutation of the Exponential Time Hypothesis in computer science, and an existence proof for quantum chromodynamics in physics [1]. None of the claims has been peer reviewed; outside expert groups will now examine them [1][3].

What does OpenAI claim its models discovered?

In mathematics, OpenAI says its models proved Sogge's 1991 local smoothing conjecture in every dimension, which would also resolve the Kakeya conjecture and Stein's restriction conjecture in all dimensions [1]. Earlier human work had proved local smoothing in two spatial dimensions in 2019 and the Kakeya conjecture in three dimensions in 2025 [1].

In physics, OpenAI describes a mathematically rigorous existence proof of certain four-dimensional quantum field theories, including quantum chromodynamics and a unified completion of the Standard Model [1]. As a subproblem, it says the work proves the existence of pure Yang-Mills theory, which is part of a Clay Mathematics Institute Millennium Prize problem [1]. This material has not been released [1][3].

Why is the Exponential Time Hypothesis claim so significant?

The Exponential Time Hypothesis, or ETH, holds that deciding 3-SAT on n variables needs exponential time in the worst case [2]. OpenAI says the ETH has been widely believed to be true and has been used in hundreds of papers to derive conditional hardness results [1]. OpenAI claims its models proved the Exponential Time Hypothesis false, a conjecture used in hundreds of papers.

The 67-page preprint, credited to OpenAI as author, describes a deterministic algorithm that counts the satisfying assignments of k-CNF formulas in subexponential time and polynomial space [2]. OpenAI says turning the result into practical faster algorithms will require significant further research [1].

Has anyone verified the proofs?

OpenAI released its math and computer science proofs as preprints on GitHub; none has been peer reviewed yet. OpenAI says both come with Lean formalizations, machine-checked proofs, also published in its repository [1][2]. Its own timeline says the ETH disproof was found on 16 October and formalized on 17 October 2026 [1].

A group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT will verify and shepherd the ETH result [1]. Impagliazzo and Paturi, with Francis Zane, originally proposed the hypothesis [1]. For physics, Nobel laureate David Gross will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov [1].

How is OpenAI changing the way it releases results?

OpenAI says it asked external experts to act as shepherds, and they advised three different release strategies: open release for math, an external group plus public files for computer science, and deferring to the group for physics [1]. OpenAI says it does not seek academic credit and that outside expert groups will examine the results.

Participation is voluntary and unpaid, and OpenAI says it agreed to fund a conference on the implications of the ETH result [1]. The Verge notes that OpenAI's own panel of math advisers had just recommended against giving selected people first access to AI output [3]. OpenAI concedes that choosing shepherds creates an asymmetry of information [1].

The release comes less than a month after OpenAI published hundreds of AI-written manuscripts on 6 October, according to The Verge [3]. OpenAI says the new collaborations complement its recently announced 5 million dollar contribution to the Simons Foundation, which funds workshops and programmes for researchers studying AI results [1].

Our read

Opinion: If the ETH refutation holds, it is the biggest claim yet from AI-driven mathematics, and putting the original proposers of the hypothesis in charge of checking it is the right call. But a Lean proof only certifies what was formalized, and extraordinary claims released by the company that profits from them deserve the slow, adversarial scrutiny that peer review exists for.

What we could not verify

• Whether any of the three claimed proofs is correct; all are AI-written preprints or unreleased material, and no peer-reviewed publication or independent verification was available at publication.
• Whether the Lean formalizations capture the full original conjectures; OpenAI says the ETH and local smoothing proofs were verified in Lean, but we could not check the formal statements ourselves.
• The physics proof for Yang-Mills theory and quantum chromodynamics, which has not been released; we rely solely on OpenAI's description.
• Which internal model or system produced the proofs; OpenAI does not name it.
• Any on-record reaction from the named shepherds, such as Russell Impagliazzo, Ryan Williams or David Gross; we found no public statement from them beyond what OpenAI reports.