The Pythagorean Angle Lattice: Density and Galois Structure of Cotangent Projections

Let A denote the set of acute angles of primitive Pythagorean triples. We study the set P = {cot(Θ/D) : Θ ∈ G, D ∈ ℕ}, where G is the smallest set containing A and closed under finite sums, finite differences, and division by positive integers. We prove two results. First, for any Z ∈ ℤ[i] with square norm and any D ∈ ℕ, the number x = cot(arg(Z)/D) is algebraic. Its Galois group over ℚ is a subgroup of ((ℤ/Dℤ)× ⋊ C₂) ⋉ C_D. Second, P is dense in ℝ⁺. The construction is explicit and relies on the Gaussian integer correspondence and the density of rational points on the unit circle. This framework naturally generalizes the known connection between primitive Pythagorean triples and metallic means.