Three proofs from OpenAI and the “shepherds” its own panel warns against

Less than a month after releasing hundreds of manuscripts^?preprint: A paper published before peer review — available at once, but not yet checked by reviewers. containing solutions to open mathematical questions, OpenAI announced three more results on 26 October 2026 — one each in mathematics, computer science and theoretical physics. The strongest claim is the computer-science one: an algorithm said to refute the Exponential Time Hypothesis, widely believed to be true since 1999. What changed, though, is not only the content but the release mode.

Key takeaways

• Three results: local smoothing for the Euclidean wave equation, a refutation of the Exponential Time Hypothesis (ETH), and an existence proof for certain interacting quantum field theories
• The computer-science result is being examined by three experts: Russell Impagliazzo and Ramamohan Paturi of UC San Diego, and Ryan Williams of MIT
• The computer-science materials — a write-up and a Lean proof — went public on GitHub immediately, at the verifying group's own request
• The physics result has not been released in full, and the "shepherd" group is to be led by David Gross
• A day earlier the AGMAI panel of mathematical advisers recommended that AI companies not designate certain people to receive output first

Let us picture a situation

A model generates a 40-page proof of a theorem the field has worked on for decades. Nobody knows whether it holds — settling that takes months of work by a handful of people who know the area.

The question that must be answered before publication is not mathematical but procedural: does the result go to everyone at once, or first to a narrow group that will say whether it makes sense at all.

But here is the problem: whoever receives the result first also gains an advantage in publishing the work that follows from it.

That procedure is what the argument around this announcement is about.

What exactly was announced?

Mathematics

The first result is in mathematics. An OpenAI model produced a proof of the local smoothing conjecture for waves in every spatial dimension no smaller than two, using an auxiliary partial differential equation. The same proof, per the company, yields the Kakeya maximal and dimension conjectures plus Stein's spherical restriction estimates throughout the interior of the conjectured range.

Set that against the field's own calendar: a few months earlier Hong Wang of NYU received a Fields Medal for resolving the Kakeya conjecture in three dimensions. The new claim covers all dimensions.

Theoretical computer science

The second result is theoretical computer science and is the most concrete. The paper gives a deterministic algorithm that exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space. The 3-SAT case refutes the Exponential Time Hypothesis.

Theoretical physics

The third result, in physics, has not been released in full. OpenAI calls it a mathematically rigorous existence proof of certain interacting quantum field theories with gauge fields and fermions in four dimensions, including Quantum Chromodynamics and a unified completion of the Standard Model.

What is the "shepherd" mode?

What is new is not the result but how it was let out. For computer science and physics, OpenAI named external expert groups it calls "shepherds", to guide the understanding and verification of each result.

In computer science that is three people: Russell Impagliazzo and Ramamohan Paturi of UC San Diego — two co-proposers of the original 1999 hypothesis — and Ryan Williams of MIT. In physics the group is led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov named as members.

For mathematics the procedure came out differently. The advisers recommended releasing the raw model output immediately and letting researchers work through their own explanations — as in the 6 October release.

This is a simplified picture of how the three results were released, not a description of a formal procedure endorsed by the research community.

The computer-science group asked for immediate publication, so the write-up and the Lean proof landed on GitHub the same day. The group also published an open letter asking how such a claim should be examined.

What do these names mean?

The announcement rests on several terms that are empty strings without a translation.

• Exponential Time Hypothesis (ETH) — the assumption that 3-SAT (can logical variables be set to satisfy all three-variable clauses?) cannot be solved much faster than exponential time. An assumed speed limit many complexity results rest on.
• Subexponential time — a rate growing faster than polynomial but slower than exponential. The difference between "infeasible" and "expensive".
• Lean — a language in which a proof is written so that a computer checks every step itself. The equivalent of unit tests for mathematics.
• Kakeya conjecture — a question about the minimal "size" of a set containing a line segment in every direction. A geometric curiosity on which harmonic analysis results depend.
• Shepherd — a company-designated external expert leading the understanding and verification of a result before the wider community gets to it.

The first five entries on this list describe what the results say. The last describes the procedure — and that is what the argument is about.

Why not simply show the proof to experts before publishing?

The standard in the exact sciences is a preprint plus review in which reviewers gain no priority privilege. Naming "shepherds" changes that balance, because it merges two functions: judging correctness, and access to the result ahead of everyone else.

The AGMAI recommendation cited above targets exactly this point. It does not dispute experts' right to judge a proof — it disputes tying that judgement to the privilege of earlier access.

Instead the panel recommended publishing on scholarly repositories with Lean-formalised proofs, information on how the problems were chosen, and funding for those working to understand the results. The company itself concedes its approach may have problems of its own.

A second issue concerns credit. If a model generates a proof and a person spends a year checking, fixing and explaining it, what counts as the scientific contribution is not obvious. The verifying group's open letter asks that directly.

Is it live yet?

These are announced claims under verification, not confirmed results. The computer-science materials — write-up and Lean proof — have been public on GitHub since 26 October 2026. The mathematics result was released as raw model output. The physics result is not published in full.

None of the three has passed independent community verification. For computer science it is to be led by the named group of three, with no date given.

What expectations should govern early access to AI-generated results, disclosure of conflicts of interest, and the use of unpublished material for subsequent research?

From the open letter by the group verifying the Exponential Time Hypothesis result.

Why does it matter?

The argument is no longer about whether a model can prove a theorem. It is about who reads it first and whose name stays attached. That moves the conflict from the technical layer to the institutional one, where a commercial lab sets the norms the research community must live with.

What comes next?

• The Impagliazzo, Paturi and Williams group is to examine and explain the proof refuting ETH — the materials are public, so parallel community verification can start at once.
• David Gross's group is to take on the physics result, which is not published in full — how much gets released is the first point to settle.
• The AGMAI panel recommended publication on scholarly repositories with Lean proofs plus funding for verification work — whether OpenAI adopts those recommendations is the nearest test of its relationship with the research community.

Sources

• The Verge — OpenAI announces three big science discoveries and a new way to work with academics
• OpenAI — Working with academic communities to build scientific understanding