OpenAI claims AI proofs on Kakeya, ETH and QCD

OpenAI said on Monday, October 26, 2026 that its internal models have produced proofs touching three long-standing problems: the Kakeya conjecture in harmonic analysis, the Exponential Time Hypothesis (ETH) in computer science, and the existence of quantum chromodynamics (QCD) in physics. None has been independently confirmed, and outside review is only starting. In its announcement, OpenAI also described a new arrangement in which outside researchers organize the checking, and it calls the manuscripts raw research data.

Three claims, three release plans

Experts OpenAI consulted advised a different path for each. For the math result, a claimed proof of Sogge's local smoothing conjecture for all dimensions that OpenAI says also settles the Kakeya and Stein restriction conjectures, it posted the model's write-up and a Lean formalization to its GitHub repository and left researchers to organize their own reviews, as it did with its October 6 batch. Hong Wang won a Fields Medal this year for the three-dimensional Kakeya case, The Verge notes.

According to OpenAI's repository notes, the paper gives a deterministic method for exactly counting the solutions to k-SAT formulas, using polynomial memory and beating every exponential running-time bound. Applied to 3-SAT, that would refute ETH. OpenAI says the proof was checked in Lean and notes that hundreds of papers rely on ETH for hardness results.

The physics result stays private. OpenAI describes an existence proof for interacting four-dimensional quantum field theories including QCD, covering pure Yang-Mills existence, part of a Clay Millennium Prize problem. A group led by Nobel laureate David Gross will decide when the materials come out, so the wider community cannot check it yet.

Who is checking, and what they say

The ETH review is led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, two of the hypothesis's original proposers, and Ryan Williams of MIT. In an open letter, they wrote that their involvement "should not be taken as certification of the claim." Williams posted that the group signed no NDAs, is unpaid, and secured OpenAI's agreement to release all raw ETH data and to fund a conference.

Williams and the letter both call it a claimed refutation. Caution is warranted: on October 7 OpenAI withdrew three manuscripts from its earlier batch over a sign error.

The Advisory Group on Mathematics and Artificial Intelligence (AGMAI), based at the Institute for Advanced Study, recommended on October 25 that labs stop handpicking researchers, citing unequal access and borrowed reputation. On October 26 it urged OpenAI to publish the Yang-Mills material immediately, arguing that the finished formalization removes any scientific reason for delay.

What we don't know yet

• Whether the ETH refutation and the all-dimension local smoothing proof survive independent review.
• When the physics materials will be released, and in what form.
• Which model produced the results and how much compute or how many attempts it took; OpenAI's post does not say.