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The coarse Novikov conjecture for bounded-geometry spaces: is coarse assembly rationally injective?

For a uniformly discrete metric space XX of bounded geometry, coarse K-homology KX∗(X)=lim→⁡rK∗lf(Pr(X))KX_*(X)=\varinjlim_r K_*^{lf}(P_r(X)) is the limit over Rips complexes, and the coarse assembly map μX:KX∗(X)→K∗(C∗(X))\mu_X:KX_*(X)\to K_*(C^*(X)) sends a class to its index in the K-theory of the Roe algebra. Roe formulated the coarse index theory and its Novikov-type injectivity question, and Higson and Roe (1995) stated the coarse Baum-Connes conjecture that μX\mu_X is an isomorphism. Surjectivity was known to fail for some expanders (Higson-Lafforgue-Skandalis), while injectivity was proved under geometric hypotheses such as coarse embeddability in Hilbert space (Yu). The coarse Novikov conjecture asks: is μX⊗Q\mu_X\otimes\mathbb Q injective for every bounded-geometry space XX?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Higher index theory; coarse geometry; operator K-theory
Posed by
John Roe, Coarse cohomology and index theory on complete Riemannian manifolds (Memoirs AMS 497, 1993); Nigel Higson and John Roe, On the coarse Baum-Connes conjecture (1995), Conjecture 6.6
Year posed
1993
Years open
33y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims a uniformly discrete bounded-geometry space XX, a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order class α∈KX1(X)\alpha\in KX_1(X) whose image under ordinary coarse assembly into K1K_1 of the reduced locally compact Roe algebra vanishes, so μX\mu_X fails to be injective integrally and rationally. The companion (5 October 2026) claims the same kind of class is killed already by maximal coarse assembly, which implies the reduced statement. Earlier kernel examples (Yu; Dranishnikov-Ferry-Weinberger) lacked bounded geometry or died under coarsening. It does NOT give a counterexample to the Novikov conjecture or to Baum-Connes for groups, nor a space with property A or coarsely embeddable into Hilbert space.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. Both manuscripts are credited to OpenAI with no human author named. The family has two manuscripts: the reduced counterexample (23 September 2026), taken as principal here, and a 5 October 2026 strengthening to maximal Roe algebras. The reduced result has a Lean comparator challenge in the release.

Verification

No independent mathematician has checked this yet. Theorem 1.1 of both manuscripts was read against the coarse Novikov conjecture. A Lean challenge lean/ComparatorChallenges/CoarseAssembly.json (solution module OAI.Topology.CoarseAssembly, present at the pinned commit) exists but is not in the formalization catalogue; found through lean/docs/307.md. Its statement was read here and does not settle the tier: the 6600-line challenge defines its own graphs, Roe algebra, K-theory and an assembly map from a bespoke analytic index family, and that family (separatedIndexFamily) is defined to be the zero map whenever its scale-wise indices are not compatible. In that case the asserted vanishing μ(α)=0\mu(\alpha)=0 and non-injectivity hold trivially, so the formal statement need not express the headline. Under the strict tier rule the entry stays Unreviewed. Not rebuilt here.

Sources

Changelog1 change

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