A Negative Answer to the Strong Geography Question of Alfieri and Binns

Alfieri and Binns say that an F[U]-module M satisfies the strong geography restriction if it has a direct summand F[U]/U^ℓ ⊕ F[U]/U^(ℓ−1) ⊕ ⋯ ⊕ F[U]/U, where ℓ is the least integer with U^ℓ M_red = 0. They showed that HF⁻(Y) satisfies it when Y is surgery on a knot in S³ or large surgery on a link, and asked whether it holds for every rational homology sphere Y. We show that the answer is no. For the Brieskorn sphere Y = Σ(30,47,83), with either orientation, the reduced Heegaard Floer homology is T_16 ⊕ T_14^10 ⊕ T_13^16 ⊕ ⋯ ⊕ T_1^92, where T_k = F[U]/U^k. So ℓ = 16, but F[U]/U^15 is not a direct summand of HF⁻(Y). The proof combines the Ozsváth–Szabó description of HF⁺ for plumbed manifolds, Némethi's reduction to the graded root of an explicit function τ, and an exact computer calculation. We give a certificate, the 1707 turning points of τ, from which the summand lengths can be rechecked by a short program. The manifold Y is an irreducible integral homology sphere, is not an L-space, and satisfies Lin's weaker restriction. It also shows that the word "large" cannot be removed from the link-surgery theorem of Alfieri and Binns. Computer searches with two independent programs show that among Seifert fibred integral homology spheres Σ(a_1,…,a_n), the counterexamples with the smallest product a_1⋯a_n are Σ(30,47,83) and three spheres with four singular fibres, all of product 117030. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14298580-008 (Oberwolfach Report 34/2024, A. Alfieri, "Is the geography of Heegaard Floer homology restricted or is the L-space conjecture false?", p. 1953; Alfieri–Binns, arXiv:2404.00490, Question 1.10).