The coarse Novikov conjecture
From OpenAI Aftermath
The coarse Novikov conjecture predicts that the coarse assembly map of a uniformly discrete metric space of bounded geometry, from its coarse K-homology to the K-theory of its Roe algebra, is rationally injective; it is the injectivity half of the coarse Baum–Connes conjecture of Higson and Roe. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), listed together as its result 307, claim counterexamples: a manuscript dated 23 September 2026 for ordinary assembly into the reduced Roe algebra, and a manuscript dated 5 October 2026 for the stronger maximal assembly map. The collection includes a Lean formalization of the reduced statement. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- Roe developed coarse index theory and its Novikov-type formulation, and Higson and Roe clarified the role of bounded geometry and of coarsening K-homology over Rips complexes, formulating the coarse Baum–Connes conjecture.
- Yu proved the coarse assembly isomorphism for bounded-geometry spaces of finite asymptotic dimension and for those that coarsely embed into Hilbert space; Skandalis, Tu and Yu placed the latter in a groupoid framework, and further injectivity results cover coarse embeddings into uniformly convex Banach spaces and fibred coarse embeddings. Willett and Yu proved injectivity for coarse unions of graphs with girth tending to infinity.
- Yu constructed a kernel class for a space without bounded geometry, and Dranishnikov, Ferry and Weinberger gave integral noninjectivity examples for a map from locally finite K-homology whose classes die under coarsening. Higson, Lafforgue and Skandalis constructed injectivity and surjectivity obstructions for groupoid assembly maps; their Hausdorff groupoid example combining a Bott class with a Kazhdan projection is described as a close methodological antecedent.
The original results
- Claim, reduced assembly. Theorem 1.1 of A counterexample to the coarse Novikov conjecture states that there is a uniformly discrete bounded-geometry metric space, a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order class in its degree-one coarse K-homology whose image under ordinary coarse assembly into the K-theory of the reduced Roe algebra vanishes, so that rational injectivity fails.
- Claim, maximal assembly. Theorem 1.1 of Failure of rational injectivity for maximal coarse assembly states the same failure for the maximal coarse assembly map, with a different construction. The manuscript notes that this strengthens the reduced result, since the maximal index maps to the reduced index, and that the stronger conclusion requires a proof in the universal norm.
- Method, as described: in the September manuscript, finite congruence quotients of SL3 over the integers are placed above the vertices of triangular-lattice discs, trivial on the boundary cycles and growing with depth, with twisted matchings between neighbouring fibres. The twisting keeps short loops from winding around the centre, which detects every multiple of the boundary class at each Rips scale, while property (T) uniformly approximates the projections onto constant vectors in the fibres, so that an averaging isometry carries the filled discs into the Roe algebra and the boundary class has zero index. The October manuscript uses a construction built on three-dimensional tori.
- Formalization, as described by the repository. The scope note, titled after the maximal result, links only the September manuscript and states that the formalization constructs a coarse disjoint union of bounded-degree finite graphs with an infinite-order class of degree-one coarse K-homology whose image under ordinary coarse assembly into the reduced Roe algebra vanishes, and which stays nonzero after tensoring with the rationals. The Comparator statement is
CoarseAssembly.lean. - Sources (openai/math repository, commit
7ef5c0c): A counterexample to the coarse Novikov conjecture, dated 23 September 2026; Failure of rational injectivity for maximal coarse assembly, dated 5 October 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | Coarse assembly: persistent topology and vanishing indices, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the claim that an infinite-order class can have zero coarse index for a separated union of finite graphs of uniformly bounded degree, the September manuscript treating the reduced Roe algebra and the October manuscript giving a different construction for the maximal algebra, both separating a topological persistence argument from an averaging argument that destroys the index. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates this family "partial" and records a sandboxed Comparator pass for CoarseAssembly. Its note states that the formal statement concerns the reduced Roe algebra, the September companion claim, while the headline failure for the maximal assembly map is strictly stronger. It also notes that the formal assembly map is built with fallback definitions that could make it zero from the statement alone, and that the solution proves the genuine branches are taken. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | C. Toledo, OpenAI's math release: what it claims beyond what it checks | Review (scope of the formal statements) | Finding G4 states that the headline concerns the maximal coarse assembly map while the Lean statement concerns the reduced map on the September construction, and that the scope note links only the September manuscript under the maximal title; it lists the maximal result among the headlines carried by no machine-checked statement. Its kernel-run table records that the run for CoarseAssembly ended without a result. |
Self-published report by the author of cert-machine; findings dated 10 October 2026 at repository commit fd4aeeb.
|
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_513 and rank_466 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Both reviews report no gap. The review of the September manuscript states that it read the whole paper and checked each step of the proof chain, including the generation of the congruence tower, the trace separation and the small-loop lemma. The review of the October manuscript states that it read all six sections in full and checked, among other steps, the central-extension identity numerically and the passage to divisibility in K-theory, and that the transfer and sliding arguments carry over to the reduced setting. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-18 | AI Proved What?, episode 307 Lost in assembly?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claims: a coarse disjoint union of finite connected graphs of uniformly bounded degree with an infinite-order class that the maximal coarse assembly map sends to zero, which also breaks the reduced map, and a companion construction for the reduced map directly, disproving the coarse Novikov conjecture. The episode recalls Yu's theorem for spaces that coarsely embed into Hilbert space, the surjectivity failures of Higson and of Higson, Lafforgue and Skandalis for expanders, and Willett and Yu's remark that a bounded-geometry counterexample would be very interesting. It states that nothing is claimed about groups, that the classical Novikov conjecture is untouched, and that there is no conflict with Yu's theorem, because these graphs contain expanders. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, two papers, 40 pages, dated September 23 and October 5, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: in part: the reduced companion, not the maximal result. The page states that neither paper has been revised, that the site did not build the formalization, whose challenge statement builds Roe algebras, K-theory and the assembly map in about 6,600 lines, that the maximal result is not formalized, that the proof shows the fallback zero map is never used, and that it found no specialist comment. |
Notes
- The background summarizes the manuscripts' own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
