Three results announced on 26 October 2026, one of them claimed twice, as reported by Scientific American.
On 26 October 2026, according to Scientific American's report of 27 October (Joseph Howlett and Peter Hall, marked there as a breaking story to be updated), OpenAI announced three large results: the Kakeya conjecture in every dimension, the existence half of the Yang–Mills problem, and a refutation of the Exponential Time Hypothesis (ETH). Within hours, Anthropic posted its own result disproving ETH. This chapter is the second ledger in this book's record of AI-lab mathematics claims; the first is "Claims Awaiting Verification". Its main source is the article text, supplied by the project's owner. The Anthropic ETH preprint's Hexagon page (hexagon:2610.00577v1) and the repository's About page have since been supplied as text by the project's owner; the reference list and provenance statement below are from that text, not from a summary. The paper's PDF was not read. No other paper, repository or lab announcement was opened, and OpenAI's own preprints were not.
This chapter was drafted with Claude, a model made by Anthropic. Anthropic is one of the parties in the ETH story below, and the preprint says its algorithm was found by one of Anthropic's Claude models. The claims are recorded as reported, the verifier checks arithmetic only, and the chapter takes no side between the labs.
| Result | Claimant | State |
|---|---|---|
| Kakeya conjecture, all dimensions (four dimensions were claimed in an earlier release) | OpenAI, internal model | Claimed. The article does not say whether a paper is public. |
| Yang–Mills, existence half only | OpenAI, internal model | Claimed. The paper has not been made public. The advisory group has called for its release. |
| Yang–Mills, mass gap half | — | Not claimed. Remains open on the article's account. |
| ETH refuted | OpenAI; then Anthropic, independently and within hours | Claimed twice. Anthropic's preprint is public (posted 27 Oct, unreviewed). OpenAI has asked Impagliazzo, Paturi and Williams to investigate and will sponsor a conference. No verdict reported yet. |
| Model, prompts, compute time, chain-of-thought | OpenAI | Not public, per the article. |
The article describes the conjecture in terms of volume: a needle turned to point in every direction, sweeping out as little space as possible. That is a popular simplification. In its standard form the Kakeya conjecture concerns dimension: a set in $\mathbb{R}^n$ containing a unit line segment in every direction should have Hausdorff and Minkowski dimension $n$. Volume is the wrong measure because sets of that kind with zero volume exist, as Besicovitch showed in the plane. The article's account of progress is that Hong Wang and Joshua Zahl proved the three-dimensional case last year, that Wang received a Fields Medal this summer, and that OpenAI's earlier release covered four dimensions. If the new claim holds, it settles a central problem in harmonic analysis. Whether it holds is the open question; the article reports no vetting.
The Clay Mathematics Institute's problem asks for two things, existence of a quantum Yang–Mills theory and a mass gap, and the article says OpenAI claims only the first. So even if the claim is right, the million-dollar problem as Clay states it would still be open. The path was sketched by Tadeusz Balaban in the late 1980s in long, dense papers that, according to Roland Bauerschmidt as quoted, he essentially completed but that nobody can now confidently certify. Ethan Sussman and Anthropic's Levent Alpöge had a public repository using language models to fill in the steps and believed they were close; the article says OpenAI's claim, with no public proof, appears to have beaten them to it. That is a priority dispute, and the priority claim carries no proof until the paper is released.
ETH says the 3-SAT problem needs time $2^{\delta n}$ for some fixed $\delta > 0$ as the number of variables $n$ grows. The article says OpenAI's algorithm is "just a tiny bit faster than exponential time". That phrase cannot be read literally. An algorithm running in $c^n$ for some $1 < c < 2$ is already faster than brute force, such algorithms are known, and they do not refute ETH, because $c^n$ is still exponential. Refuting ETH needs a running time of $2^{o(n)}$: for every $\delta > 0$ eventually below $2^{\delta n}$. The verifier script shows the difference numerically, with $\log_2(T)/n$ constant for $c^n$ and tending to zero for $2^{\sqrt{n}}$. What OpenAI proved is not in this chapter's sources. Anthropic's preprint, which is, says (per the page text) that a deterministic algorithm solves 3-SAT in $2^{o(n)}$ time, so it claims the full refutation and not just a better base, and that the algorithm also counts satisfying assignments (#3-SAT).
The page, titled "A $2^{o(n)}$ Time Algorithm for 3-SAT" (hexagon:2610.00577v1), is in the computational complexity category, contributed by Anthropic, version 1, 27 October 2026, licensed CC BY 4.0, and carries two labels: "Primarily AI-generated text" and "Human understanding: some parts". Its abstract says the authors refute ETH with a deterministic $2^{o(n)}$-time algorithm for 3-SAT that in fact solves #3-SAT, the counting version, and that the result was found by an internal Claude research model and formalized in Lean 4.
The provenance statement gives specifics. The core algorithm was found on 16 October by an internal Claude research model using 14 million output tokens and no human input; Anthropic researchers then worked with Claude to improve the presentation. It adds that an internal OpenAI model "independently and concurrently" claims a similar result, and the reference list cites an OpenAI preprint, "The Exponential Time Hypothesis is False", dated 26 October 2026. The path in that citation sits in a folder named for 25 October. That small mismatch is recorded here and not explained; the word "independently" is the authors' own and has not been checked.
The abstract says the result has been formalized in Lean 4, and the references list Lean 4 and mathlib. The page as pasted gives no link to a formalization and does not state its scope. That absence is partly explained by the host: the repository's About page says code such as Lean formalizations is not accepted as part of a submission, and that contributors may instead link to formalizations elsewhere. So the missing file is not itself a warning sign. But it means the claim "formalized in Lean 4" cannot be checked from this page, and the point from the Axiom chapter applies in full: the kernel certifies the statement as coded, and a person must confirm that statement is $2^{o(n)}$ for all of 3-SAT, deterministically, and not something weaker.
Most of the list is the expected literature: the original ETH paper of Impagliazzo, Paturi and Zane, the PPSZ and Schöning families of $k$-SAT algorithms, Hertli's and Scheder's refinements, the derandomization of Schöning's algorithm by Moser and Scheder, and the standard ETH-consequences surveys. The less expected cluster is algebraic: octonions and quasialgebra structure, Hurwitz's composition problem and the sum-of-squares work of Hrubš and coauthors, tensor networks and asymptotic tensor rank, vertex algebras, small-bias spaces, and a recent paper on the asymptotic rank conjecture and the set cover conjecture. A reference list is not an argument, and nothing here says how or whether these are used. It does indicate that the method, whatever it is, draws on algebra and not only on the usual local-search techniques.
One reference is checkable by arithmetic and supports the point above. Dantsin and coauthors (2002), cited on the list, give a deterministic $(2-2/(k+1))^n$ algorithm for $k$-SAT. For $k=3$ that is $1.5^n$, faster than $2^n$ and still exponential, so it does not refute ETH. The verifier checks it.
Hexagon's own About text says acceptance is a decision to disseminate and preserve a work, and is not a judgment about its correctness, quality or novelty. Moderation checks that the TeX compiles, that the work is a scholarly contribution, and that it is not abusive; it says incorrect proofs of famous conjectures will often be rejected without comment, and that AI systems cannot be contributors, so the contributor of record is an organization or person. Posting there therefore adds some friction against obvious junk and no evidence of correctness. Axiom's Unbalanced Group Theorem paper is on the same repository.
The article also says the result would not by itself resolve P versus NP, since ETH is a stronger assumption than $\mathrm{P}\neq\mathrm{NP}$ and refuting it does not show $\mathrm{P}=\mathrm{NP}$. Two groups claiming the same refutation independently is, if both are right, a form of corroboration. If one of them is wrong, it is a warning. The article reports neither.
The article spends as much space on process as on mathematics. The advisory group on mathematics and AI that OpenAI itself convened (AGMAI) issued a statement the day before the release about how experts should shepherd results into public view. The article says OpenAI's handling of two of the three results violated that statement, and that the model, prompts, compute time and reasoning traces are all withheld. The first ledger records the same tension from Science's reporting. A result that outside readers cannot reproduce on the same tool is something they can check only by reading, and reading is slow when the papers are enormous and the authors are not human.
First, the "Millennium" count. The article says Yang–Mills is one of seven problems set by Clay and one of five that remain open. Seven minus the one resolved problem (the Poincaré conjecture) is six. Five would imply that a second has been counted as resolved, perhaps the Navier–Stokes claim from September that this book has treated as disputed. The article does not say. The verifier prints the arithmetic and does not decide it.
Second, "just a few weeks after" the earlier release is 20 days from 6 to 26 October. That one is fine.
It does not know whether any of the three results is correct or what any of them says in detail. It does not know the state of OpenAI's, Anthropic's or the advisory group's own statements. It has not opened the other Scientific American pieces linked from the article, including one titled "The inside story behind OpenAI's million-dollar math proof". It has not read Balaban's papers, the Sussman–Alpöge repository, OpenAI's preprints, the Anthropic preprint's PDF, or any proof, and it has not seen any Lean file.
When a row changes state, change the row and its date and cite the new source. Do not add a source that was not opened.
Joseph Howlett and Peter Hall, "Dueling proofs from OpenAI and Anthropic astound scientists", Scientific American, 27 October 2026 (text supplied by the project's owner; not retrieved by this book).
Anthropic, "A 2^{o(n)} Time Algorithm for 3-SAT", hexagon:2610.00577v1, posted 27 October 2026 (hexagon:2610.00577v1; page text supplied by the project's owner).
Hexagon, "About Hexagon" (text supplied by the project's owner).