Exact Minimal Polynomial and Algebraic Degree of the Optimal 10-Point Packing Distance in a Unit Square

We establish the exact algebraic characterization of the optimal dispersion distance d₁₀ for packing 10 points in a unit square [0,1]². Starting from the 12-contact rigidity graph of the numerically optimal configuration, we perform a complete symbolic elimination via Sylvester resultants, removing all 11 intermediate coordinate variables in eight steps. The result is an irreducible degree-18 integer polynomial P₁₈(d) = 1180129·d¹⁸ − 11436428·d¹⁷ + ⋯ + 819200 = 0 whose smallest positive real root equals d₁₀. We certify irreducibility over F₁₇ via a finite-field reduction certificate, construct a rational isolating interval, audit the sign branches introduced by squaring, and verify that all 33 non-contact point-pair distances exceed d₁₀ with minimum clearance > 0.05007. Our exact value matches Specht’s authoritative benchmark [1] to within 10⁻²⁵.