The Omni-Metallic Framework, Pythagorean Angle Lattices, and Universal Approximation

This work unifies primitive Pythagorean triples, the Omni-Metallic family, and the Pythagorean angle lattice into a single geometric-algebraic framework that proves constructive universal approximation of any positive real number and derives the constants π, e, and the golden ratio directly from integer right triangles. We present a comprehensive unified framework connecting primitive Pythagorean triples, metallic means, and fundamental constants through the Pythagorean angle lattice. The Omni-Metallic family ℧m,r(n, N), defined as the unique positive real root of the polynomial equation x^m − 2n xᵐ⁻¹ − N^r = 0 (m ≥ 2, n, N, r ∈ ℕ), provides the algebraic foundation. The framework rests on three master identities and proves that the Pythagorean angle lattice is dense in ℝ⁺ and constructively universal. We derive fundamental constants (π, e, golden ratio) from the lattice and establish a universal Diophantine approximation framework with accelerated convergence using the Omni-Metallic family.