An Independent Symbolic Certification and Galois Group Proof for the 10-Point Square Packing Distance

Abstract We present a complete symbolic derivation and independent algebraic certification of the minimal polynomial of the optimal 10-point packing distance d₁₀ in a unit square [0,1]², together with a full determination of its Galois group. Starting from the corrected 12-contact rigidity graph (V1.1 corrects the erroneous edge (P₃,P₆) to the true contact edge (P₈,P₁₀)), 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) = 0 whose smallest positive real root equals d₁₀. P₁₈(d) = 1180129·d¹⁸ − 11436428·d¹⁷ + ⋯ + 819200 = 0 New in V1.4: We determine the Galois group Gal(P₁₈/ℚ) ≅ S₁₈ via a four-step certification: (i) transitivity from irreducibility; (ii) disc(P₁₈) is not a perfect square in ℚ; (iii) primitivity; (iv) Jordan’s theorem. As a corollary, d₁₀ is not expressible by radicals over ℚ.