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S_{i+1}$ with $S = 1$ on a ring of L sites. Prove inf_L (E_1(L) - E_0(L)) > 0 (numerically the gap is approximately $0.4105 J$), together with a unique infinite-volume ground state with exponentially decaying correlations. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of a uniform positive gap, possibly computer-assisted.","status_note":"Tasaki (Physical Review Letters, 2025; arXiv 2407.17041) proved that the ground state is in a nontrivial symmetry-protected phase if the gap exists; the gap itself is unproven.","title":"Haldane gap in the spin-1 Heisenberg 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