On October 26, 2026, OpenAI announced three scientific results from an unreleased internal model: a proof of the local smoothing conjecture in every dimension, a subexponential-time algorithm that it says refutes the Exponential Time Hypothesis (ETH), and existence results for four-dimensional quantum field theories, including Yang-Mills theory. As of October 28, none had been independently confirmed. This post focuses on the verification question: what Lean formalization does and does not establish, who is reviewing each claim, and what the public can actually inspect. For the larger catalogue these results sit alongside, see our earlier coverage of OpenAI's 722 math manuscripts and the Navier-Stokes claim.
OpenAIThe three OpenAI claims, side by side
According to The Next Web, OpenAI says the model found its ETH disproof on October 16 and had it checked in Lean the next day. Crypto Briefing reports the local smoothing proof on October 18, formalized in Lean on October 19. A report from Kingy AI lists the Yang-Mills work as starting September 9, with Lean completion on October 16. All dates are company-reported.
Where each claim sits on the verification ladder
Status as reported by outlets on Oct 28, 2026. Green is reported done, amber is partial, grey is not reported.
The ladder shows the pattern: the two mathematics and computer science results have public write-ups and Lean files, while the physics material has neither.
Why the Exponential Time Hypothesis matters
ETH, proposed by Russell Impagliazzo and Ramamohan Paturi, says that satisfiability problems cannot be solved in subexponential time. TNW describes it as widely believed and as underpinning hardness results in graph theory, computational biology and databases. Much of fine-grained complexity depends on a close relative, the Strong Exponential Time Hypothesis (SETH): for example, Backurs and Indyk showed that if SETH holds, edit distance has no strongly subquadratic algorithm.
According to ccleaks, OpenAI's preprint describes a deterministic algorithm that exactly counts satisfying assignments of k-CNF formulas in polynomial space, in roughly 2^(n/(log n)^(1/2+o(1))) time. Our own reading, not a reported conclusion: an algorithm that fast for every k would also sit uncomfortably with SETH, which would put many of those conditional lower bounds in question. That is an inference for experts to confirm or reject.
Kingy AI notes that the manuscript and the Lean scope state slightly different exponent bounds. Both are subexponential, so the headline conclusion would hold under either if the formalization checks out. The three computer scientists leading the review, Impagliazzo, Paturi and Ryan Williams, said in an open letter that their involvement "should not be taken as certification of the claim," per TNW. They report signing no NDAs and being unpaid.
If the claim is correct, it will change an important part of theoretical computer science.
Impagliazzo, Paturi and Williams, via The Next Web
Local smoothing, Kakeya and Stein: the mathematics claim
Christopher Sogge posed the local smoothing conjecture in 1991. Per Crypto Briefing, the two-dimensional case was settled in 2019 and the three-dimensional Kakeya conjecture in 2025, and OpenAI says its proof covers every dimension and thereby settles the Kakeya and Stein restriction conjectures in all dimensions. Kingy AI credits Larry Guth, Hong Wang and Ruixiang Zhang for the two-dimensional result, and Hong Wang and Joshua Zahl for the 2025 three-dimensional Kakeya paper.
Crypto Briefing says OpenAI credits a newly introduced auxiliary partial differential equation as the key ingredient. ccleaks adds that the local smoothing Lean file lists six declarations. No outside mathematician confirming the proof had been quoted in the coverage we reviewed.
What Lean does and does not guarantee
Lean checks that a formal statement follows from its axioms and imports. That is strong, but it has edges that matter here.
- Statement fidelity. The proof is only as meaningful as the formal statement. Experts must confirm the Lean theorem faithfully captures Sogge's conjecture or the ETH, with no simplification. Crypto Briefing makes this same point.
- Scope. ccleaks reports the final ETH Lean statement covers exact counting and SAT on one fixed run, and does not say that the paper's approximation, sampling or Max-k-SAT claims are formalized.
- Axioms. The ETH comparator configuration reportedly allows only the standard axioms (propext, Quot.sound, Classical.choice), which is a reassuring constraint.
- Independent checking. Reporting so far restates OpenAI's statement that the files compile. Kingy AI explicitly says it did not rebuild the proofs.
- Physics. Yang-Mills has no public Lean file to inspect at all. The existence result also does not address the mass gap, which the Millennium problem requires, per Kingy AI and MindStudio.
What history says about verification timelines
Announcement-to-consensus gaps in this field are long and uneven. In October 2025, an OpenAI executive's post suggested GPT-5 had solved ten open Erdos problems. Thomas Bloom, who runs the Erdos problems site, said the model had instead found existing literature, and the posts were deleted, per Yahoo News and The Decoder coverage. OpenAI's own October 2026 catalogue saw withdrawals: ccleaks reports that on October 7 it withdrew three manuscripts after a sign error and revised 14 others. Formal verification reduces that risk, but it moves the risk to the statement, which is why reviewers rather than compilers are the bottleneck.
Who verifies what: claim, status, reviewer, public material
| Claim | Status (Oct 28) | Named reviewers | Public material |
|---|---|---|---|
| ETH refuted | Under open review, not complete | Impagliazzo, Paturi, Williams | Manuscript and Lean files on GitHub |
| Local smoothing, all dimensions | No independent review reported | None named in coverage | Manuscript and Lean files on GitHub |
| Yang-Mills and QCD existence | Private write-up in progress | Physics group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov, Nikita Nekrasov | Nothing released |
On October 26, ccleaks reports, the Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study asked OpenAI to release the natural-language and Lean outputs for the physics proofs without delay, saying understanding "should develop in the open, with equal access." That characterization of the request comes from a single secondary source, and we could not reach the group's own statement.
What to watch next
- Whether Williams and colleagues publish a verdict, and whether OpenAI funds the community conference both sides reportedly agreed to.
- Whether independent analysts confirm that the Lean statements match the original conjectures.
- Whether the physics outputs are released, and whether a mass gap claim ever follows.
For teams using AI in research workflows, the practical lesson is to treat model output as a draft and keep the checking step independent of the model that produced it. Working across several models, as in Metir, makes that separation easier to maintain.
Sources:
- OpenAI: Building scientific understanding with academic communities (primary announcement; not retrievable by our tools)
- The Next Web: OpenAI says ETH is false, with Kakeya and Yang-Mills claims
- Crypto Briefing: OpenAI proves local smoothing conjecture
- MindStudio: OpenAI discoveries in math, physics and computer science
- Kingy AI: OpenAI scientific discoveries
- ccleaks: OpenAI AI proofs on ETH, Kakeya and Yang-Mills
- Abboud, Hansen, Williams, Williams: Simulating Branching Programs with Edit Distance and Friends
- The Decoder: OpenAI researcher announced a GPT-5 math breakthrough that never happened
- TechCrunch: OpenAI's embarrassing math
Image credits
- Hero: Fuld Hall, Institute for Advanced Study, Princeton, looking north, photo by Zeete (2023), via Wikimedia Commons, CC BY-SA 4.0. The photo is illustrative and does not depict the announcement.
- In-body: Institute Pond, Institute for Advanced Study, Princeton, photo by Zeete (2023), via Wikimedia Commons, CC BY-SA 4.0.