The Nexohedron: A Seventh Genesis Solid and the Golden Chain φ = kβ · ρ in the AP3L(5, c) Family

This work introduces the Nexohedron, a new convex polyhedron identified as the member AP3L (5, ρ) of the three-ring prismatoid family, where ρ = φ / kβ. Its defining identity φ = kβ · ρ completes the AP3L (5) quadruplet, placing the Nexohedron as the missing geometric link between the Cosmohedron and the Trinhedron. The solid has V = 15, E = 35, F = 22, χ = 2, and D₅h symmetry, with three distinct edge classes and a characteristic set of long-span edges arising from its position below the triangulation threshold. All metric and topological properties are derived analytically and verified computationally. Comparisons with existing catalogs and known polyhedral families indicate that the Nexohedron is most likely new. Its dual structure, combinatorial regime, and near-coincidence eₗat ≈ √2 are also documented, situating it within the broader UMCR polyhedral program.