OpenAI claims AI proofs of local smoothing, a false ETH, and Yang-Mills, then hands them to shepherds

OpenAI did not wait for peer review to say the words Kakeya, ETH, and Yang-Mills in one post. It waited for shepherds.

On 26 October 2026 the company framed three AI-produced manuscripts as raw research data and asked named external experts how to release them. Math went public first. Complexity theory went public with a named verification group. Physics stayed closed while a Nobel-led group writes the account. The thesis worth tracking is the disclosure protocol itself: how raw AI mathematics enters the academy.

Three release strategies for three AI manuscripts

OpenAI's post "Working with academic communities to build scientific understanding" announces discoveries in harmonic analysis, complexity theory, and quantum field theory. Internally, the company treats the manuscripts as raw research data that still need scientific understanding. External shepherds advised three different strategies.

For mathematics, release the AI output and let researchers organise explanation themselves. For computer science, form an external collaborator group while publishing the AI output for anyone to examine. For physics, form an external group and defer to them on whether materials go public.

Four principles govern the collaborations: transparency without NDAs; OpenAI seeking no academic credit for model discoveries; recognition for the understanding and write-ups the groups produce; and unpaid, independent participation so neither side speaks for the other. The post admits the trade-off: shepherds accelerate unpacking, and they also choose who sees the material first.

Local smoothing in every dimension, ETH false in Lean, Yang-Mills existence held back

The mathematics claim concerns Sogge's 1991 local smoothing conjecture: the most extreme focusing of waves must be short-lived, so typical-time focusing is milder than the worst case. Local smoothing implies the Kakeya conjecture and Stein's restriction conjecture. Guth, Wang, and Zhang proved local smoothing in two spatial dimensions in 2019; Wang and Zahl proved Kakeya in three dimensions in 2025.

OpenAI says its models produced a proof of local smoothing in every dimension, and therefore Kakeya and Stein restriction in all dimensions, via a novel auxiliary partial differential equation. AI natural-language output arrived on 18 October 2026; Lean formalisation followed the next day. Both sit in the public openai/math repository, following the same release pattern as the company's 6 October 2026 maths dump.

The computer-science claim targets the Exponential Time Hypothesis. Impagliazzo, Paturi, and Zane proposed ETH; hundreds of papers use it for conditional hardness across graph theory, computational biology, databases, and geometry. OpenAI says its system proved ETH false, then verified the refutation in Lean.

A shepherd group led by Russell Impagliazzo (UC San Diego), Ramamohan Paturi (UC San Diego), and Ryan Williams (MIT) will explain the result. The group asked that all raw data be public and that OpenAI fund a conference on field implications. Natural-language disproof is dated 16 October 2026; Lean formalisation 17 October. Both are public on the same repository path.

The physics claim is a mathematically rigorous existence proof for certain interacting four-dimensional quantum field theories with gauge fields and fermions, including QCD and a unified completion of the Standard Model, building on Balaban's constructive QFT approach. As a subproblem it includes existence of pure Yang-Mills theory, part of the Clay Mathematics Institute Millennium Prize problem.

David Gross leads a group with Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. That group chose to hold AI-produced artefacts and write the work itself. OpenAI dates a natural-language Yang-Mills existence proof to 9 September 2026 and a QCD proof to 18 September; Lean formalisation of Yang-Mills existence finished 16 October. Communications for the group route through YANGMILLS4D@gmail.com.

Lean certificates, named proposers, and an asymmetric physics hold

Lean formalisation is OpenAI's internal verification bar for the public math and ETH packages. It is a machine-checkable certificate of a formal statement, not a substitute for community exposition, literature placement, or independent re-proof in the style journals expect. Naming Impagliazzo and Paturi as ETH shepherds is a fairness signal: the original proposers are in the room. Williams's prior work on strong ETH variants and algorithm-lower-bound links sits beside them. On physics, Jaffe co-formulated the Millennium problem with Witten; Gross's asymptotic-freedom work underpins why these theories can exist as rigorous objects. Holding the physics artefacts while publishing math and ETH is an intentional asymmetry the shepherds requested.

The collaborations sit next to OpenAI's recently announced $5 million contribution to the Simons Foundation for workshops and programmes on AI-produced results. Funding for independent examination is promised; dollar schedules and grant criteria beyond that announcement remain on the Simons and OpenAI pages, not fully restated here.

Peer acceptance, prize claims, and practical algorithms stay off the page

The post does not claim journal acceptance, Clay Prize eligibility, or that ETH's fall immediately yields faster practical SAT solvers. It says finding algorithms and turning theory into practice will need significant further research. Independent third-party reproduction of the Lean developments outside OpenAI's repository tooling is not evidenced on the announcement page. Whether the all-dimension local-smoothing argument survives specialist scrutiny in harmonic analysis, and whether the Yang-Mills write-up the Gross group produces matches the withheld AI artefacts, remain unsettled until those communities speak.

openai/math, Lean proofs, and a physics inbox

Public artefacts today: the 26 October OpenAI post; natural-language write-ups and Lean proofs for local smoothing / Kakeya and for the ETH refutation in the openai/math repository; the broader manuscript catalog OpenAI has been releasing through that repository. Physics materials are intentionally unpublished pending the working group. The shepherd email for Yang-Mills / QCD is public. Conference funding tied to the ETH collaboration is promised in principle; dates and programme chairs are not on the post.

Price the protocol, wait for the shepherds' write-ups

Labs watching disclosure norms should copy the dated three-track structure (public AI output; public output plus named explainers; deferred artefacts) rather than a single press-release template. Complexity theorists can begin from the public ETH Lean package and the Impagliazzo-Paturi-Williams invitation to organise open examination.

Harmonic analysts can start from the released local-smoothing Lean development without treating OpenAI's prose abstract as the final statement. Physics desks should route questions to the Gross group and treat Millennium-adjacent language as a subproblem claim under shepherd control. Buyers and policy staff should file this as a verification-governance story until peer write-ups land.

After the 6 October maths dump, shepherds become the interface

OpenAI already released a large maths manuscript collection on 6 October 2026. The 26 October post extends that pattern into complexity theory and constructive QFT while adding an explicit shepherd layer and four collaboration principles. The live record is the protocol: AI output entered three different academic intake valves on the same day. How those valves perform is now an empirical question for the shepherds and their fields.