The Pythagorean Angle Lattice: Arithmetic Structures and Spectral Geometry A Synthesis of the Omni-Metallic Framework, Non-Abelian Character Invariants, and the Geometry of Integer Triangles

Abstract

A self-contained arithmetic–geometric framework linking primitive Pythagorean triples to zeta values, modular geodesics, and a conjectural cancellation of non-abelian Artin characters. We present a complete, self-contained unification of the Pythagorean Angle Lattice L and the Omni-Metallic Framework. We prove that the discrete integer projection of the lattice maps exactly onto the arithmetic kernel r(k) = floor((k-1)/2), deriving its exact Mellin transform into the Bridge Identity Z(s) = (1/2)ζ(s-1) − (1/2)(1 + 2^(−s))ζ(s), which yields the Omni-Metallic Euler Constant γ_O = γ/2 − 5π²/48. We establish the S₃ Galois structure of the cubic projection branch, construct a many-to-one correspondence with closed geodesics on the modular surface via Pell matrices, and formulate a formal conjecture on the asymptotic vanishing of non-abelian character traces under the projective limit D → ∞.