The Pythagorean Angle Lattice and the Omni-Metallic Framework: Arithmetic Bridges and Spectral Relations

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 x^{m−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 the constants π 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 partially resolved, and a conjectural spectral decomposition over odd D is presented. The Crown Identity connects the Universal Metallic family to primitive Pythagorean triples.