The Pythagorean Angle Lattice

A single primitive Pythagorean triple unlocks a dense lattice of angles that generates metallic means, rational logarithms, zeta connections, and unifies seven mathematical languages—from Euclidean geometry to analytic number theory. We study the Pythagorean angle lattice, a countable set of real numbers obtained by applying the cotangent function to rational divisions of finite sums and differences of acute angles of primitive Pythagorean triples. The lattice is dense in R⁺. For every positive real constant T > 0, there exists a sequence of primitive Pythagorean triples such that cot((α₀ + ϕ(uₖ, vₖ))/D) → T. We extend the lattice to all trigonometric and hyperbolic functions. The Even-D Collapse theorem establishes that for D = 2D′, the field of cot(arg(W²)/D) equals that of cot(arg(W)/D′). The hyperbolic lattice H = e^G is dense in R⁺. The Bridge Identity connects the lattice counting function to the Riemann zeta function. Rational logarithms ln((a + c)/b) are shown to arise from primitive Pythagorean triples. A Galois group upper bound is given. The framework unifies seven mathematical languages in the integer right triangle.