SS-9: Conditional Derivation of Simplicial Alpha-Polytope Connectivity from CPP Lattice Geometry

Abstract: SS-9 advances OPEN-SS-24, the question of why the alpha-cluster contact graph used in SS-7 is simplicial. Under refined-C1 alpha rigidity, contact-face relations, the K_3 collective mode, and four new paper-level hypotheses (C5 ground-state minimization, C6 cluster surface-realization, C7 contact-graph planarity, C8 FvdW centroid-realizability), the paper proves a conditional theorem: the ground-state contact graph at N_alpha ∈ {4,5,6,7,8,9,10,12} is the 1-skeleton of a simplicial convex 3-polytope with 3 N_alpha - 6 edges and every face triangular, realized as the unique Freudenthal-van der Waerden convex deltahedron with vertices at alpha centroids and uniform edge length. The proof routes through three lemmas via Euler's formula and Steinitz's theorem, promoting C4 from a B-tier hypothesis to a derived statement at the C5+C6+C7+C8 inheritance tier. Unconditional promotion is pending closure of four sub-problems (OPEN-SS-29/30/33/37); the deltahedra-gap range N_alpha ∈ {11,13,14} is registered as OPEN-SS-31.