Maximum two-distance sets in Euclidean spaces of dimension nine, ten, eleven and twelve

A two-distance set is a finite set of points in ℝᵈ whose pairwise distances take only two values. Let g(d) be the largest cardinality of a two-distance set in ℝᵈ. The values g(d) for d ≤ 8 have been known since Lisoněk's work of 1997; the case d = 9, where 45 ≤ g(9) ≤ 55, has remained open since then. We prove that g(9) = 45 and that, up to similarity, there are exactly two 45-point two-distance sets in ℝ⁹: the Johnson set J(10, 2) and Lisoněk's 45-point set in ℝ⁸. We also prove g(10) = 55, g(11) = 66, g(12) = 78, each attained only by J(d + 1, 2), and that Lisoněk's set is the unique 45-point two-distance set in ℝ⁸. The proofs combine a structural reduction with an exhaustive, symmetry-reduced computer search: the Larman–Rogers–Seidel theorem forces the distance ratio √2, after which the short differences of the configuration generate an ADE root lattice, and every configuration with connected short-distance graph is a clique in a finite graph built from the vectors of norm 2 and 4 of a root lattice of rank ≤ d. Disconnected configurations are handled by a separate geometric lemma. The search reproduces every previously known value g(d), d ≤ 8, and the new negative results were cross-checked with an independent SAT-based implementation. The authors are listed in alphabetical order and contributed equally.