The Omni-Metallic Framework and the Pythagorean Angle Lattice: A Spectral Synthesis

We establish a synthesis between the Pythagorean angle lattice and the Omni-Metallic family ℧m,r(n,N), defined as the unique positive real root of x^m - 2n xᵐ⁻¹ - N^r = 0. The shared arithmetic kernel is the counting function r(k) = ⌊(k-1)/2⌋, with generating function F(x) = x^3 / ((1-x)(1-x^2)). The Mellin transform yields the Bridge Identity Z(s) = (1/2)ζ(s-1) - (1/2)(1+2⁻ˢ)ζ(s). We show that π, γ, and e admit alternative expressions in terms of the lattice’s spectral data. A product formula connecting the counting function to the Glaisher-Kinkelin constant A is established. The Generalized Integer Value Theorem is proven, the cubic case D=3 is resolved, and a conjectural spectral decomposition over odd D is presented.