A 6D periodic lattice projected to 3D produces three things simultaneously: a quasicrystal, a spectral gap, and a triadic frustration. The spectral gap and the frustration feed independently into a single β-function whose RG flow — from (ξ, 𝒟) = (0, 3) to (1, 2) — determines the organizational hierarchy of physical systems, from elementary particles through black holes to cosmic expansion. The core theory develops this chain. Supporting results provide detailed computations and observational tests. Applications extend the constraint geometry into engineering, biology, and social systems.