A Lattice of Cotangents from Angles of Primitive Pythagorean Triples: Galois Groups of Kummer-Cyclotomic Type, Density in R+, and the Omni-Metallic Zeta Function

Let A be the set of acute angles of primitive Pythagorean triples. This paper studies the set of real numbers obtained by applying the cotangent function to rational divisions of finite sums and differences of elements of A. This set, denoted P, is shown to have two fundamental properties. First, for Z ∈ ℤ[i] with square norm and D ∈ ℕ, the number x = cot(arg(Z)/D) is an algebraic number. Its Galois group over ℚ is a subgroup of ((ℤ/Dℤ)× ⋊ C₂) ⋉ C_D (with simplifications when 4 | D). Even D cases collapse to quadratic fields (recovering classical metallic means), while genuinely new higher-degree content appears for odd D. Second, P is dense in ℝ⁺. Every positive real number can be approximated arbitrarily closely by elements of P. The construction uses the Brahmagupta-Fibonacci identity (multiplication in Gaussian integers) and the cotangent half-angle formula. The algebraic structure connects to the four-parameter Omni-Metallic family.