SS-7: Alpha-Cluster Regime and the 3N-6 Edge Formula for Medium-Mass Nuclei

Abstract: SS-7 extends the SS-5 cascade paradigm: for A ≥ 8 nuclei, alpha particles themselves act as rigid tetrahedral building blocks assembling into closed polytopes with quark-quark contact bonds across each shared face. With only two numerical inputs — the ⁴He binding and B_pair = 2.342 MeV — both inherited from SS-5, the binding energy of an N_alpha-alpha cluster nucleus comes out to B = N_alpha B_alpha + (3 N_alpha - 6) B_pair, where 3 N_alpha - 6 is the edge count of any simplicial triangulation of a convex polytope on N_alpha vertices via Euler's formula. Against AME 2020, this zero-fitted-parameter formula reproduces twelve strict-N=Z alpha-chain nuclei from ¹²C through ⁵⁶Ni within ±1.5% (RMS 0.80%). The N_alpha=2 case (⁸Be) gives zero edges; the single linear alpha-alpha contact must compete with Coulomb repulsion, yielding the observed 92 keV near-threshold unboundness. An adversarial stress test confirms the simplicial 3N-6 rule outperforms plausible lower-edge alternatives.