The Pythagorean Angle Lattice: Angle Hierarchy and Spectral Structure

Abstract

Every primitive Pythagorean triple generates an infinite angle-halving hierarchy that recovers the metallic means precisely when c−b equals 1, 2 or 8, and naturally produces a convergent spectral zeta function. We study the Pythagorean Angle Lattice, a countable set of real numbers defined by cot(arg(Z)/D) where Z = W² or iW² and W belongs to a restricted set of primitive Gaussian integers. For every primitive Pythagorean triple (a,b,c) with d = c−b we construct a generating right triangle with legs 1 and a/d. Angle halving is realized by the map T(x) = x + √(x²+1) and produces an infinite hierarchy of right triangles. Angle doubling recovers the original triple. When d ∈ {1,2,8} the hierarchy collapses exactly to the metallic means; otherwise it yields general quadratic irrationals. The halving direction yields a discrete spectrum E_k = T^k(a/d), spectral bands, and a convergent spectral zeta function ζ_L(s). The framework unifies Pythagorean triples, metallic means, quadratic irrationals and a simple spectral theory inside a single geometric construction.