The Pythagorean Angle Lattice: Galois Classification for All Projection Branches

A unified Galois classification of all projection branches of the Pythagorean Angle Lattice, delivering explicit holomorph extensions of the Gaussian rationals from the geometry of integer right triangles : We study the D-th projection branch of the Pythagorean Angle Lattice. For each primitive Gaussian integer W = u + vi ∈ ℤ[i]prim, the lattice gives a polynomial f_W^{(D)}(y) of degree D, defined by: f_W^{(D)}(y) = P_D(y) − γ Q_D(y) = 0, where P_D, Q_D are the multiple-angle polynomials for cotangent and γ = cot(arg(W²)) = (u² − v²)/(2uv). We prove the following classification: 1. For D = 2, f_W^{(2)} splits completely over ℚ, so Gal = {1}. 2. For even D ≥ 4, f_W^{(D)} is always reducible over ℚ(i) with the explicit factor: P_{D/2}(y) − cot(arg W) Q_{D/2}(y). 3. For odd prime D, f_W^{(D)} is irreducible if and only if W is not a D-th power in ℤ[i]. In this case: Gal(K_W^{(D)}/ℚ(i)) ≅ ℤ/D ⋊ (ℤ/D)×. 4. For odd composite D, the classification is open. The exceptional set of D-th powers has natural density zero for all D ≥ 2. For D = 3, we recover the dihedral representation ρ₂ = Ind_{A₃}^{S₃}(ψ) with fixed field ℚ(i, √3) = ℚ(i, ζ₃). This provides an explicit family of holomorph extensions of ℚ(i) for odd prime D, arising from the geometry of integer right triangles.