OpenAI says its AI disproved the Exponential Time Hypothesis and proved Kakeya

What this is about

On October 26, 2026, OpenAI announced three new research results that its internal AI models reportedly produced: one in mathematics (harmonic analysis), one in theoretical computer science (complexity theory) and one in physics (quantum field theory). Less than a month earlier, the company had already released hundreds of manuscripts with solutions to long-open math questions (the October 6, 2026 release). What's new this time is mainly the process: OpenAI asked outside experts to act as "shepherds," organizing groups that understand and explain the results. OpenAI itself regards the AI manuscripts as "raw research data" that only becomes scientific understanding through human work.

An important caveat up front: these are the vendor's claims. Independent peer review is still pending. Only Lean formalizations (machine-checkable proofs) exist for several parts.

Three results, three publication strategies

The experts recommended a different route in each case:

• Mathematics: The AI output is published directly; researchers organize the explanatory work themselves.
• Computer science: An outside group explains the result, and the AI output is also publicly available.
• Physics: An outside group is being formed and decides on publishing the materials. It has decided to withhold the AI artifacts for now and write up the result itself.

OpenAI admits it doesn't yet know which approach is most useful and asks for feedback. It also concedes that the choice of the first "shepherds" determines who is involved first and creates an information gap.

Four principles of collaboration

1. Transparency: The collaborations are public. There are no nondisclosure agreements, and participants are free to talk about them, share interim results and publish conclusions.
2. No academic credit: OpenAI claims no scientific credit for its models' discoveries. Leading a group doesn't automatically mean having made the discovery.
3. Recognition: The groups should receive credit for understanding, texts and materials, and should make sure the prior work they draw on is properly acknowledged.
4. Independence: Participation is voluntary and unpaid. Members don't speak for OpenAI, and vice versa.

Mathematics: local smoothing and Kakeya

In 1991, Sogge formulated the local smoothing conjecture. It describes how waves behave when they bunch together (focus): the most extreme focusing can only be short-lived. At a random moment, the focusing is much weaker than at the worst moment, a pattern often seen in mathematics (the worst case is more extreme than the average). The conjecture has far-reaching consequences. It implies the Kakeya conjecture (a question about the minimal "size" of sets in which a line segment can be rotated in every direction) and Stein's restriction conjecture.

Until now: in 2019, Guth, Wang and Zhang proved the case of two spatial dimensions; in 2025, Wang and Zahl proved the Kakeya conjecture in three dimensions. According to OpenAI, the models have now proved the local smoothing conjecture in every dimension, and with it Kakeya and Stein in all dimensions. The proof relies on a novel auxiliary differential equation. The Verge adds that this covers dimensions two and up, and that Stein's spherical restriction estimates follow in the interior of the conjectured range.

OpenAI's timeline: AI output on October 18, 2026, Lean formalization the next day, then contact with an expert. On the expert's advice, the natural-language proof description and the Lean proof were published in a public GitHub repository.

Computer science: the Exponential Time Hypothesis is supposedly false

The Exponential Time Hypothesis (ETH) says, roughly, that the satisfiability of Boolean formulas (SAT) can't be solved much faster than in exponential time. It is closely tied to the P vs. NP question and widely believed. Hundreds of papers use it to derive conditional hardness results in graph theory, computational biology, databases and geometry.

OpenAI says the system has disproved the ETH. According to The Verge, the work provides a deterministic algorithm that counts the satisfying assignments of fixed-width CNF formulas exactly, in subexponential time and polynomial space; the 3-SAT case disproves the ETH. The Lean documentation (as reported by Unite.AI) states: for fixed k ≥ 2, a deterministic machine counts the satisfying assignments of a k-CNF formula with n variables in subexponential time and polynomial space; sharper time bounds in terms of the number of clauses m appear in a manuscript dated October 25, 2026.

The AI found the natural-language proof on October 16, and the Lean formalization followed on October 17. In practice, this doesn't mean faster software right away: OpenAI stresses that the disproof opens questions about how much faster these tasks could become, and that practical improvements would take a lot more research.

The review group consists of Russell Impagliazzo and Ramamohan Paturi (both UC San Diego), who proposed the ETH in 1999 together with Francis Zane, and Ryan Williams (MIT), who has refuted strong versions of the ETH. At the group's request, all raw data will be made public, and OpenAI is funding a conference on the implications. The group invites the community to an open review and raises questions about early access and conflicts of interest, about credit for discovery versus verification and explanation, and about supporting students and researchers.

Physics: existence of Yang-Mills theory and QCD

The third result is said to be a mathematically rigorous existence proof for certain interacting quantum field theories with gauge fields and fermions in four dimensions. These include quantum chromodynamics (QCD, the strong force between quarks and gluons) and a unified extension of the Standard Model. The work builds on Balaban's approach to constructive quantum field theory. As a sub-problem, it contains the existence of pure Yang-Mills theory, which is part of the Clay Mathematics Institute's Millennium Prize Problem. The proof concerns the simplest "grand unified theory," which contains the gauge fields and fermions of the Standard Model, since the Standard Model itself can't be extrapolated to arbitrarily short distances.

The models produced the Yang-Mills proof on September 9 and the QCD proof on September 18, 2026. Because of their length and complexity, OpenAI waited for the Lean formalization, which was completed for Yang-Mills on October 16.

The group is led by David Gross (2004 Nobel Prize for asymptotic freedom), with Michael Douglas, Arthur Jaffe (a co-formulator of the Millennium Problem), Igor Klebanov and Nikita Nekrasov. The materials have not been released so far; requests are to go to a dedicated email address.

Funding and criticism of the process

OpenAI also points to its $5 million contribution to the Simons Foundation for workshops and programs on understanding AI-generated results. The Verge notes that AGMAI, the advisory board OpenAI uses, advised companies a day earlier not to adopt the practice of giving certain people the results first, citing unequal access, "borrowed reputation" and biased selection. Instead, labs should publish on scientific repositories, with Lean proofs, details on how problems were chosen, and funding for those doing the work of understanding.

Our take

If the results hold up, they'd be breakthroughs: Kakeya in all dimensions, the end of a central assumption in complexity theory, and progress on a Millennium Problem. Lean formalizations are a strong indicator, but they don't replace expert review, for instance to check whether the formal statements capture exactly what was meant. For the physics result, nothing public is available to check yet. Open concerns include the choice of reviewers, the information gap, and how credit for AI discoveries and human verification work will be divided. Every claim so far comes from OpenAI. The coming weeks of open review will show what holds up.