OpenAI said on Monday, October 26, 2026, that its models produced three major results in mathematics, computer science and physics, and that it is handing them to outside experts to verify. The claims include a proof of local smoothing for the wave equation that would also settle the Kakeya conjecture, and an algorithm that would refute the Exponential Time Hypothesis (ETH), a long-held assumption in complexity theory. None of the results has been confirmed by independent reviewers yet, so each should be read as a claim under examination.
The three results
According to The Verge’s account of the release, the first is a proof of Sogge’s local smoothing conjecture for waves in every spatial dimension of at least two. The model output also yields the Kakeya maximal and dimension conjectures and Stein’s spherical restriction estimates across the conjectured range. OpenAI released this output directly, as it did with its October 6 batch of math manuscripts.
The second is titled “The Exponential Time Hypothesis is False.” It describes a deterministic algorithm that exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space. Applied to 3-SAT, OpenAI says, this refutes ETH, which many theorists have treated as true since it was proposed in 1999.
The third, in physics, is described as a mathematically rigorous existence proof for certain interacting quantum field theories with gauge fields and fermions in four dimensions, including quantum chromodynamics (QCD) and a unified completion of the Standard Model. The Verge says this material has not been released in full.
A different release process
The Verge notes the release differs from October 6. For computer science and physics, OpenAI named outside “shepherds.” Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine and explain the ETH proof; Impagliazzo and Paturi were among those who originally proposed the hypothesis. David Gross will lead a group with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov for the physics work. The ETH group asked for the write-up and a Lean proof to be public immediately, and both are on GitHub.
For the math result, advisers recommended releasing the raw model output for researchers to work through on their own.
Disagreement over norms
The approach already has critics. The Verge reports that a panel of mathematics advisers convened by OpenAI said a day earlier that AI companies should not adopt the practice of giving selected people early access, citing unequal access, borrowed reputation and biased selection. The panel instead recommended publishing on scholarly repositories with Lean-formalized proofs, information on how problems were chosen and funding for people who work to understand the results.
OpenAI acknowledged the shepherd model could have its own issues. It also published an open letter asking the community how early access, conflicts of interest, credit for checking and explaining results, and support for students should be handled.
Why it matters
If the proofs hold, they would be among the most significant AI-produced results so far. A refutation of ETH would reshape assumptions behind many lower bounds in algorithms. But the history of large computer-generated claims counsels patience, and the checking process, not the announcement, is what counts.
What to watch next
- Statements from the ETH working group on whether the Lean proof and write-up check out.
- Mathematicians’ reactions to the local smoothing and Kakeya argument.
- Whether OpenAI publishes the full physics manuscript.
- Responses to the open letter and any shared norms for AI-generated results.
Key takeaways
- OpenAI announced three claimed discoveries on October 26, 2026.
- They cover local smoothing and Kakeya, the Exponential Time Hypothesis and QCD existence.
- Named external experts are verifying the computer science and physics results.
- OpenAI’s own math advisers warned against early access for chosen people.
- Nothing here is independently confirmed yet.
Frequently asked questions
What is the Exponential Time Hypothesis?
ETH is the conjecture, proposed in 1999, that 3-SAT cannot be solved in subexponential time. It underpins many results about how hard problems are.
Has anyone verified the proofs?
Not yet. OpenAI has named outside groups to examine them, and the ETH proof comes with a Lean formalization.
How does this relate to OpenAI’s October 6 release?
It follows that batch of AI-generated math manuscripts and tries a different process, with named outside experts for two of the three results.
