OpenAI claims AI proofs of Yang-Mills and a false ETH

OpenAI said on October 26 that its internal models disproved the Exponential Time Hypothesis and proved the local smoothing conjecture in every dimension. It put Lean formalizations of both on GitHub, so anyone can check those two proofs today. The third result, an existence proof for quantum chromodynamics that includes pure Yang–Mills theory, stays unpublished while a group led by David Gross writes it up.

In OpenAI’s post, the company calls the manuscripts “a form of raw research data” and describes three release routes, each chosen with outside experts it calls shepherds: the math output goes out as is, the computer science result goes to an external group while the AI output stays public, and the physics materials stay with the physicists. Participants sign no nondisclosure agreements and work unpaid, and OpenAI says it seeks no academic credit for the discoveries. The company also names the cost of its own setup: picking the first shepherds “creates asymmetry of information”.

What the ETH paper actually claims

The manuscript defines the Exponential Time Hypothesis as the claim that deciding 3-SAT on n variables takes 2^Ω(n) time in the worst case. OpenAI says hundreds of papers in graph theory, computational biology, databases and computational geometry have used it to derive conditional hardness results. The paper claims more than the post does. Its Theorem 1.1 “refutes ETH, SETH, Gap-ETH, and NSETH” along with their bounded-error randomized versions, and an appendix derives a circuit lower bound from a 2013 theorem by Ryan Williams, one of the three people now reviewing the claim. Those wider claims are in the PDF and were not formalized in Lean.

The paper’s sharpest bound is 2 to the power n·log log n/√log n, times a polynomial. OpenAI itself says that finding faster algorithms and turning the theory into practical improvements “will require significant further research”.

Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Williams of MIT lead the review. At their request OpenAI agreed to make all raw data public and to fund a conference on the result. Their own open letter is cautious: “Our involvement should not be taken as certification of the claim.” They say the effort should not confer ownership “on a company or a selected group of researchers, including ourselves”, and they promise a findings repository, a blog and a feedback form. On Bluesky, Williams wrote: “We do not have access to their internal models. We only got the PDF.”

That is a different setup from May. When OpenAI announced that a model had disproved the Erdős unit distance conjecture, it published reactions from mathematicians it had given early access, including Tim Gowers, and Scientific American noted that OpenAI had solicited that commentary. This time the named reviewers say outright that they are not vouching for the claim.

Local smoothing, released without a group

Sogge’s 1991 conjecture says that when waves focus, the most extreme focusing is short-lived, and the conjecture implies the Kakeya conjecture. Guth, Wang and Zhang proved it in two spatial dimensions in 2019, and Wang and Zahl proved Kakeya in three dimensions in 2025. OpenAI says its proof for every dimension rests on “a novel auxiliary partial differential equation”. The company dates the AI output to October 18 and the Lean formalization to the next day. On an expert’s advice, it released the writeup and the Lean proof without forming a working group.

The writeup sits in the same repository where, on October 7, a single sign error forced OpenAI to withdraw three papers, so its natural-language manuscripts have needed correction before. The Lean proof is what can be checked independently of that record.

The physics result stays private

OpenAI says its models produced a natural-language existence proof for Yang–Mills theory on September 9 and one for QCD on September 18. It then held the release until the Lean formalization of Yang–Mills existence was done on October 16. The proof covers certain four-dimensional quantum field theories with gauge fields and fermions, including QCD and the simplest grand unified theory containing the Standard Model’s fields, and builds on Balaban’s approach. Pure Yang–Mills existence is part of a Clay Millennium Prize problem. Gross, who shared the 2004 Nobel Prize for asymptotic freedom, leads Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. Jaffe formulated that Millennium problem with Witten. The group has chosen to write the result up itself before releasing any AI-produced artifacts, and wants all questions sent to YANGMILLS4D@gmail.com.