On October 26, 2026, OpenAI announced three new research results in mathematics, computer science and physics: an OpenAI math proof of the local smoothing conjecture, a refutation of the Exponential Time Hypothesis (ETH), and an existence proof for certain quantum field theories. According to OpenAI, all three came from its own AI models. The key caveat is that independent experts have not yet finished checking them.
This article sorts out what OpenAI claims, what is already public, and what remains open.
Key Takeaways
- OpenAI reports results in three fields: harmonic analysis, complexity theory and quantum field theory.
- For the math and computer science results, write-ups and Lean formalizations are available in OpenAI’s public GitHub repository. The physics materials are being held back for now.
- External working groups are meant to examine and explain the results. Independent confirmation is still pending.
- OpenAI says it does not seek academic credit for discoveries produced by its models.
The OpenAI math proof: local smoothing and Kakeya
The mathematics result concerns the local smoothing conjecture, proposed by Sogge in 1991. It describes how waves focus: the most extreme focusing must be short-lived, and on average it is much weaker than in the worst case. The conjecture implies the Kakeya conjecture and Stein’s restriction conjecture.
According to OpenAI’s announcement, the conjecture had been proved in two spatial dimensions (Guth, Wang and Zhang, 2019), and the Kakeya conjecture in three dimensions (Wang and Zahl, 2025). OpenAI’s models are now said to have produced a proof for every dimension, based on a novel auxiliary partial differential equation. The company says the AI output was obtained on October 18 and formalized in Lean the next day. A natural-language write-up and the Lean proof are in the public GitHub repository.
The Exponential Time Hypothesis is said to be false
The second result is especially striking for computer scientists. The ETH holds that 3-SAT cannot be solved in subexponential time. Hundreds of papers use it as the basis for conditional hardness results in graph theory, databases, geometry and other fields. OpenAI says its system disproved the ETH and that the result was then verified in Lean.
As The Verge describes it, the paper gives a deterministic algorithm that exactly counts satisfying assignments of fixed-width CNF formulas in subexponential time and polynomial space. The 3-SAT case then refutes the ETH. An independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the proof. Impagliazzo and Paturi are among the original proposers of the hypothesis.
OpenAI itself stresses that practical algorithms do not follow automatically. Turning the theoretical advance into practical improvements will require significant further research. If you build SAT solvers or verification tools, nothing needs to change today.
Existence proof for quantum field theories
The third result is the most ambitious. OpenAI describes a mathematically rigorous existence proof of certain interacting quantum field theories in four dimensions, including quantum chromodynamics (QCD). As a subproblem, the work is said to include a proof of the existence of pure Yang-Mills theory, which is part of one of the Clay Mathematics Institute’s Millennium Prize problems.
A group led by Nobel laureate David Gross is to shepherd the result. That group has decided to hold off on releasing the AI-produced artifacts and to write up the work itself. The public therefore has no basis for an assessment yet. Anyone already calling a Millennium problem solved is getting ahead of expert review.
A new process: external “shepherds”
The handling of the results is also new. OpenAI calls the AI manuscripts “raw research data” that experts must turn into scientific understanding. External experts act as “shepherds” and organize the review. Participants sign no nondisclosure agreements, work unpaid and do not speak for OpenAI. The company also points to a $5 million contribution to the Simons Foundation, which will fund workshops and conferences to help researchers understand AI-produced results.
The approach is contested. The Verge reports that a panel of math advisers assembled around OpenAI recommended the day before that AI companies not adopt the practice of designating certain people to receive output first, citing unequal access, borrowed reputation and biased selection. The panel instead suggested releasing results on scholarly repositories with Lean-formalized proofs and information on how problems were chosen.
Timeline according to OpenAI
These details come from OpenAI’s own publication and cannot be independently checked:
- September 9 and 18, 2026: The models produce a proof text for Yang-Mills existence, then for QCD. Release is postponed until the Lean formalization is complete.
- October 16, 2026: The system finds the ETH refutation. The same day, OpenAI finishes formalizing the Yang-Mills existence result.
- October 17, 2026: The ETH refutation is formalized in Lean.
- October 18 and 19, 2026: The local smoothing output arrives and is formalized.
- October 26, 2026: The three results are announced together with the external working groups.
The short gap between AI output, formalization and release stands out. Human review now has to catch up quickly while the materials are discussed in public. According to OpenAI, it had already published a batch of manuscripts on October 6 and chose the same strategy for the math results then.
What it means for developers and readers
- A claim is not a confirmation: Lean formalization raises confidence in a proof’s correctness, but it does not remove the question of whether the formalized statement says exactly what is claimed.
- Check, don’t just believe: The math and computer science materials are public, which is an improvement over press releases alone.
- Open questions: How should the work of checkers and explainers be credited, and who gets early access? OpenAI puts these questions up for discussion itself.
Similar patterns appear elsewhere in the industry. Verifiability also matters for Claude’s solution of the NSA elliptic curve puzzle and for Microsoft’s MAI-DxO. For how governments respond to the risks of powerful AI models, see our article on the UK AI Biosecurity Lab.
Conclusion
If they hold up, the OpenAI math proof and the two other results would be remarkable advances. As of today, none of them is confirmed. The reports of the independent working groups over the coming weeks will decide. We will update you as soon as reliable assessments are available.
