OpenAI Posts Three More 'Major' Results in Mathematics, Computer Science and Physics, Pending Expert Review

OpenAI announced on October 26, 2026 three further results that it calls "major" discoveries, in mathematics, computer science and physics. Each is at the stage of expert scrutiny rather than settled acceptance. The company's announcement places each result under the guidance of outside researchers. It came less than a month after OpenAI released hundreds of manuscripts that it said contained solutions to long-standing mathematics questions.

The mathematics result is titled "Local smoothing for the Euclidean wave equation." It uses an auxiliary partial differential equation, a supporting equation describing how a quantity changes across space and time, to establish Sogge's local smoothing for waves in every spatial dimension of at least two. According to The Verge, the same proof also yields the Kakeya maximal and dimension conjectures and Stein's spherical restriction estimates throughout the interior of the conjectured range. OpenAI describes the work as harmonic analysis, the branch of mathematics that studies how functions break down into waves of different frequencies.

The computer science paper claims a deterministic algorithm, one that makes no random choices, which exactly counts satisfying assignments of fixed-width CNF formulas (a standard format for writing logic problems) in subexponential time and polynomial space. Subexponential means the running time grows more slowly than an exponential function. The paper's 3-SAT case is claimed to refute the Exponential Time Hypothesis, the conjecture that 3-SAT requires exponential time in the number of variables. OpenAI places this work in complexity theory.

An independent working group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine and explain the proof. Impagliazzo and Paturi, both of UC San Diego, were part of the group that originally proposed the Exponential Time Hypothesis in 1999. Williams is an MIT professor. The computer science group has released the AI output write-up and a Lean proof on GitHub. Lean is a proof assistant that checks each step of a formal argument mechanically. The group has also published an open letter asking the community for input on how to examine the claim.

The physics claim is described as a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions. The families named include Quantum Chromodynamics and a unified completion of the Standard Model. David Gross will lead a group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov to shepherd the physics work.

OpenAI said external experts it calls "shepherds" are guiding the computer science and physics results. For the mathematics result, the company said it was advised to release the AI output so that researchers could work through their own explanations. The two routes therefore differ. The mathematics output is public with explanation left to the community, while the other two results are being guided by named groups.

The announcement sits within a broader run of activity. OpenAI launched GPT-Rosalind for life sciences research in April 2026, according to Reuters. In July, it said it would give 100,000 academic researchers free access to its most advanced AI models. In early October, it published new results on open problems in mathematics from an internal frontier model and shared Lean proof formalizations. In late September, Reuters, citing the Wall Street Journal, reported that OpenAI had scrapped the planned October debut of GPT-6.1 Astra after internal safety testing.

The broader context here is verification. A claim in harmonic analysis or complexity theory is assessed by experts working through the argument line by line, and a Lean formalization gives reviewers a machine-checkable artifact to test. Checking that a formal statement matches the intended informal claim remains a human task, however, and that gap is where many disputes over formal results have started. Physics claims of this kind follow a different path, because an existence result for four-dimensional gauge theories with fermions would be judged by a community that has worked on these theories for decades.

In my view, the most useful aspect of this announcement is the open process rather than the headline. The GitHub materials and the open letter invite outside checking at a scale that a single lab's assertion cannot supply. If the proofs hold, that process will show it. If they do not, the same process will show that too, and it will do so in public.