Abstract
We disprove Kusner's 1983 conjecture that every equilateral set in ℓₚⁿ with 2 < 𝑝 < ∞ has at most 𝑛 + 1 points: there exist 58 points in ℝ⁵⁶ whose pairwise ℓ₅ distances are all equal, so the maximum equilateral-set size satisfies e(ℓ₅⁵⁶) ≥ 58 > 57. This is the first equilateral set of more than 𝑛 + 1 points in ℓₚⁿ for any finite 𝑝 ≥ 2 . The construction persists on an open interval of exponents around 5; since Ge, Xu and Zhou recently proved the conjecture for 2 ≤ 𝑝 ≤ 4 , the infimum of exponents at which it fails lies in [4,5) . The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.