Pythagorean Angle Lattice: Division Content, Cyclotomic Decimation, and Arithmetic Reconstruction

A single intrinsic division content controls integer angle division on the Pythagorean Angle Lattice and is recovered equivalently from Gaussian coordinates, cyclotomic decimation, local ranks, and an analytic Euler product: Primitive Pythagorean triples determine rational points on the unit circle and hence rational norm-one Gaussian rotations. This paper develops an arithmetic theory of integer division for their oriented acute angles. The basic invariant is the division content L(∠), the largest integer by which an oriented rational Pythagorean angle is divisible inside the rational-angle group. We prove that the complete principal division spectrum is the divisor lattice of L(∠) and that every nonzero acute Pythagorean angle has a unique division-primitive core. Universal division polynomials and Gaussian power maps give an explicit degree-d description, including a three-state ramification law and fixed-degree counting exponent 2/d. For the forward arithmetic dynamics, if (b + ia)^n = b_n + ia_n and Δ_n = c^n − b_n, we prove the strong divisibility identity gcd(Δ_m, Δ_n) = Δ_gcd(m,n). A canonical Lehmer lift yields exact square factorizations of the cyclotomic gap layers: for n ≥ 3, the Möbius layer P_n = ∏d|n Δ_dμ(n/d) is an integral square. Combining this with classical primitive-divisor theory gives a precise exceptional classification on the Pythagorean locus. The inverse and forward theories are joined by the decimation map ρ_L(e) = e/(e,L). For U = V^L, cyclotomic layers and local prime ranks are transported by the same map. Its fibres are divisor intervals, it belongs to an adjoint triple on the divisibility lattice, and its kernel relation is a lattice congruence whose identity fibre is exactly Div(L). Finally, the collision multiplicities κ_L(n) = |ρ_L⁻¹(n)| have Dirichlet series K_L(s) = ζ(s) ∏p^a ∥ L ((a+1) − a p⁻ˢ), from which the complete prime-power factorization of L is reconstructed. Thus one intrinsic division invariant is recovered equivalently from oriented Gaussian coordinates, divisor genealogy, cyclotomic decimation, local rank contraction, and an analytic Euler modifier.