This document formalizes the Hyper-Matrix framework as a research program investigating the distri-bution of prime numbers as spectral signatures of a π-centered periodic manifold [0, 2π]. We establishthe Invariant System Radius (rbase) from transcendental growth convergence and propose a candidatespectral operator derived from the n-th Order Radical Engine (Rn). We derive the stability of thecritical line ℜ(s) = 0.5 as the mandatory geometric solution for Asymptotic Boundary Alignment re-quired to prevent structural divergence at the manifold’s topological singularities (π/2, 3π/2). Finally,we define the Reverse Intercept Law as a falsifiable numerical law for zeta-zero locations.