OpenAI announces three mathematics and physics results, inviting external experts to lead interpretation

The biggest difference from previous practice lies in the divergence of release strategies: for the computer science and physics results, OpenAI invited external experts to lead teams in interpreting the results; whereas for the mathematics problem, the experts recommended simply making the AI output public and letting researchers organize the explanatory work themselves. In its official blog, OpenAI framed the three manuscripts as a kind of "raw research data," and admitted that "we do not yet know which of these three approaches will prove most useful."

Among the three results, the mathematics result concerns harmonic analysis. In 1991, Sogge proposed the "local smoothing conjecture," which discusses the behavior of waves when they focus—the most extreme focusing must necessarily be brief, and the focusing intensity at random moments is far lower than in the worst case. Because this conjecture is extremely fundamental, it simultaneously implies the Kakeya conjecture and the Stein restriction conjecture. In 2019, Guth, Wang, and Zhang proved the two-dimensional case, and in 2025, Wang and Zahl proved the three-dimensional Kakeya conjecture; OpenAI's model produced a proof of the local smoothing conjecture in arbitrary dimensions, thereby resolving the Kakeya conjecture and the Stein restriction conjecture in all dimensions. The new proof is based on an entirely new auxiliary partial differential equation; the model output was obtained on October 18, 2026, and Lean formalization was completed the next day.

On the computer science side, the result points to the refutation of the "Exponential Time Hypothesis" (ETH). ETH is a long-standing conjecture about the computational complexity of Boolean formula satisfiability, closely related to the P vs. NP problem, and has been used in a large number of papers in the past to derive conditional hardness results in areas such as graph theory, computational biology, databases, and computational geometry. OpenAI claims that its system proved ETH false and completed verification in Lean: the paper presents a deterministic algorithm that can exactly count the number of satisfying assignments of fixed-width CNF formulas in subexponential time while using only polynomial space, and the 3-SAT case constitutes a refutation of ETH. This work was examined and interpreted by an independent working group led by Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams; Impagliazzo and Paturi were precisely the original proposers of ETH (together with Francis Zane). At the working group's request, the relevant materials were made public immediately; OpenAI also committed funding to support a conference to discuss the significance of this result for the entire field. The natural-language-level counterexample was produced on October 16, and formalization was completed on October 17.

The physics result provides a rigorous existence proof for certain interacting quantum field theories in four dimensions that include gauge fields and fermions, covering quantum chromodynamics (QCD) as well as a unified completion of the Standard Model. This work builds on Balaban's methods in constructive quantum field theory, and as a subproblem it also includes an existence proof for pure Yang–Mills theory—the latter being part of the Clay Mathematics Institute's Millennium Prize Problem on "Quantum Yang–Mills Theory." The Standard Model cannot be extrapolated to arbitrarily short distances, whereas this existence proof concerns the simplest grand unified theory containing the Standard Model's gauge fields and fermions. This work was led by David Gross, winner of the 2004 Nobel Prize in Physics, with members including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov; Jaffe, together with Witten, once proposed the Yang–Mills Millennium Problem and helped found the Clay Mathematics Institute, and Douglas has followed this problem for twenty years. The working group requested that all related correspondence be sent uniformly to [email protected]. The model's natural-language proof gave the Yang–Mills existence result on September 9 and QCD on September 18; because of the extremely high length and complexity, release was postponed until Lean formalization was completed on October 16; the group decided not to make the AI-generated artifacts public for the time being, but instead to write the paper themselves.

OpenAI set four principles for these external collaborations: transparency—participants are not required to sign confidentiality agreements and may freely discuss, share intermediate products, and publish conclusions; no claim to academic credit—it does not seek academic credit for discoveries produced by its internal models, and inviting researchers to participate does not mean that they made the original discoveries; recognition of contributions—the working groups receive recognition for the understanding they established, the manuscripts they wrote, and other materials, and are responsible for appropriately citing the researchers, techniques, and ideas drawn upon; independence—participation is voluntary and unpaid, members do not speak on behalf of OpenAI, and OpenAI does not speak on their behalf.

However, this model itself is also controversial. OpenAI acknowledges that choosing the initial "shepherds" will inevitably determine who participates first and create information asymmetry, and therefore welcomes outside opinions on how to broaden participation. A more direct criticism came from OpenAI's own AGMAI mathematics advisory group: just the day before (October 25), the group explicitly stated that it "recommends that AI companies not adopt" the practice of designating specific groups to receive priority access to outputs, citing unequal access, borrowing others' reputations, and bias in selection.

In terms of supporting scientific research, OpenAI mentioned the previously announced $5 million investment in the Simons Foundation; these funds are managed and disbursed by the foundation to support workshops, conferences, and special programs that help researchers understand and continue working on the basis of AI-generated results.