Michael JohnsonOn October 26, 2026, OpenAI announced three results from an internal model in harmonic analysis, computational complexity theory, and quantum field theory. The central computer science claim is a refutation of the Exponential Time Hypothesis (ETH), which is used to derive conditional complexity estimates in many studies. The company published a manuscript and a Lean formalization; a group of external theorists is organizing further study of the result.
According to OpenAI’s October 26 announcement, the model formulated a refutation of ETH on October 16, and the result was formalized in Lean on October 17. The work describes a deterministic subexponential-time algorithm for exactly counting satisfying assignments of k-CNF formulas for fixed k. If the result withstands scientific scrutiny, it will affect conditional complexity estimates for problems in several fields.
What Lean actually verifies
The published note on the scope of the ETH formalization sets out the precise limits of the machine-checked result: for fixed k, Lean proves that a deterministic machine exactly counts the solutions of canonically encoded k-CNF formulas in subexponential time and with polynomial space usage. The October 25 manuscript gives a stronger time bound than the formalized statement.
Thus, the formal result concerns a specific problem formulation, encoding, and set of resource bounds. Lean checks the proof within a specified formal system; experts still need to assess how the claim relates to ETH, the argument’s presentation, and its scientific novelty. For now, the practical status of the work is a published refutation claim open for scrutiny, accompanied by a formalization and external review.
Who will study the work
OpenAI said the group studying and explaining the ETH result will be led by Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams. Impagliazzo and Paturi, together with Frances Zane, proposed the original hypothesis. In their letter to the theoretical computer science community, the researchers distinguish the group’s involvement from confirmation of the result: the claim requires careful study. OpenAI said the group requested open access to the source materials and funding for a conference to discuss the work’s implications.
The second result concerns the local smoothing conjecture for the wave equation. OpenAI says it has proved the conjecture in all dimensions and connects it to consequences for the Kakeya conjecture and the Stein Fourier restriction conjecture. The company published the result and a description of the scope of its Lean formalization. Mathematicians will need to assess the proof and its implications.
The physics result has been published differently so far
In physics, OpenAI claimed to have proved the existence of certain interacting quantum field theories in four dimensions, including quantum chromodynamics and pure Yang–Mills theory. An external group led by David Gross has decided not to publish the model-generated materials for now and to prepare its own account. As a result, readers cannot yet examine open proof materials for this claim in the same way they can for ETH and local smoothing.
The claim about the existence of Yang–Mills theory is related to the Clay Institute’s Yang–Mills and mass gap problem, but the two claims are not identical. The Clay problem requires proving both the existence of a quantum Yang–Mills theory and the existence of a mass gap; the institute notes that no proof of this property is currently known.
For computer scientists, the most concrete next step is to compare the ETH formalization with the accompanying manuscript and the conclusions of the independent group. OpenAI also notes that finding faster algorithms and turning theoretical improvements into practical ones will require further work. The main news on October 26 is three claims of scientific results, with different levels of access to the materials and different review processes; their place in science will depend on researchers’ further analysis.
