The announcement arrived on October 26 as a blog post with an unusual shape. OpenAI did not only present three results; it presented three different plans for what happens to each of them next. One is fully public, one is public and under examination, and one is held back while its examiners write it up.

The three claims at a glance#

Every row is a claim by OpenAI about its own models, not an accepted finding.

FieldClaimWhat is public
MathematicsLocal smoothing in every dimensionProof and Lean formalization on GitHub
Computer scienceThe Exponential Time Hypothesis is falseProof and Lean formalization on GitHub; under external examination
PhysicsExistence proof for certain four-dimensional quantum field theoriesNothing yet; the examining group writes its account first

The computer science claim#

The Exponential Time Hypothesis, or ETH, is a conjecture about a basic question: how hard is it to determine whether a collection of true-or-false conditions can all be satisfied at once? The hypothesis says the task needs exponential time, and it has been widely believed to be true. Hundreds of papers have used it to derive conditional hardness results across graph theory, computational biology, databases and geometry.

OpenAI says its system found a natural-language disproof on October 16, 2026, and formalized it in Lean, a proof-checking tool, the next day. Both documents are now public on GitHub. The examination is led by UC San Diego's Russell Impagliazzo and Ramamohan Paturi — two of the hypothesis's original proposers — with MIT's Ryan Williams. At the group's request, OpenAI will make all raw data public and fund a conference on the result's implications. OpenAI cautions that turning this into better algorithms will need significant further research. If it holds, conditional results need re-examination; if not, it becomes a cautionary tale about machine-generated proofs.

Abstract illustration of waves focusing into sharp geometric patterns
AI-generated editorial illustration

The mathematics claim#

The local smoothing conjecture, proposed by Christopher Sogge in 1991, concerns how intensely waves can focus. In plain terms, it says the most extreme focusing must be short-lived. Settling it would ripple through mathematical analysis — including, OpenAI says, resolving the Kakeya conjecture about needle-like shapes and Stein's restriction conjecture.

Progress had been partial: Guth, Wang and Zhang proved local smoothing in two spatial dimensions in 2019, and Wang and Zahl proved the Kakeya conjecture in three dimensions in 2025. OpenAI says its models produced a proof in every dimension, built on a novel auxiliary partial differential equation. The output arrived on October 18 and was formalized in Lean the next day; the company published both documents on GitHub. It is the most openly available of the three claims.

The physics claim#

The physics claim is the least checkable. OpenAI describes a mathematically rigorous existence proof for certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including quantum chromodynamics (QCD) and a unified completion of the Standard Model. Built on Balaban's constructive quantum field theory approach, it includes a pure Yang-Mills existence proof as a subproblem. The proof covers the simplest grand unified theory containing the Standard Model's fields.

OpenAI says its models produced the Yang-Mills proof on September 9 and the QCD proof on September 18, postponing release until the Yang-Mills Lean formalization finished on October 16. The examining group, led by Nobel laureate David Gross with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, has chosen to hold off on releasing the AI-produced artifacts and write the work up itself. Outsiders therefore cannot audit the physics output yet. This is an existence proof about mathematical foundations, not an experimental discovery.

Abstract illustration of a glowing lattice of particles and force lines
AI-generated editorial illustration

A new way to publish#

OpenAI frames its manuscripts as raw research data, and says the scientific community's job is to convert that data into understanding. To that end it asked outside experts to act as “shepherds,” organizing groups to examine each result. The collaborations follow four stated principles: transparency (no nondisclosure agreements, participants free to publish their conclusions); no academic credit for OpenAI; recognition for the groups' explanations and write-ups; and independence (participation voluntary and unpaid, conclusions the groups' own).

The context: a company still working out how to disclose machine-made results. On October 6 it released hundreds of AI-produced mathematics manuscripts on GitHub. On October 25, its mathematics advisory panel recommended against giving designated people first access to outputs, The Verge reported — at odds with a shepherd model that does exactly that, at first. OpenAI concedes the tension: choosing initial shepherds shapes who gets involved first. One caveat: Lean checks a proof's logical steps, not whether the formal statement matches the headline claim. OpenAI has committed $5 million to the Simons Foundation to support the checking work.

What to watch#

  • The ETH examination. Whether Impagliazzo, Paturi and Williams's group confirms the disproof or finds a gap.
  • The physics write-up. When the Gross group's account appears, and whether the withheld AI artifacts follow it into the open.
  • The Lean proofs. Whether independent readers find the formalizations match the headline claims.
  • The publishing model. Whether other labs copy the shepherd approach.

Until those questions resolve, the honest summary is narrow: OpenAI says its models produced three remarkable proofs, two of which anyone can now read. The proofs are not yet knowledge. They are homework, assigned to everyone.

Sources: OpenAI announcement, Gizmodo, AI Affairs, PANews.