Pythagorean Angle Lattice: Division Content, Oriented Kummer Structure, Dyadic Descent, and Arithmetic Height Spectrum

Abstract

Oriented Kummer structure and dyadic descent on the Pythagorean Angle Lattice produce a canonical division content whose exact-content height spectrum has poles at s = 1/C and maximal depth asymptotic to log X / (2 log log X) : The Pythagorean Angle Lattice (PAL) is an arithmetic hierarchy obtained by dividing the arguments associated with primitive Gaussian squares. This paper develops an intrinsic division theory for PAL angles in structural, explicit, and analytic forms. Passing from the rational norm-one torus to an oriented additive lift produces a free abelian angle group and a canonical division content. For a Pythagorean angle θ and D ≥ 1, D | cont(θ) ⇔ θ/D is the acute angle of a primitive Pythagorean triple, and then ht(θ) = ht(θ/D)^D. The discrepancy between oriented divisibility and power-divisibility on the rational torus is measured exactly by 0 → ℤ/(n/(n,4))ℤ → 𝒜/n𝒜 → T(ℚ)/T(ℚ)^n → 0. A Gaussian-coordinate formula expresses the full content as the gcd of the perfect-power exponent of the hypotenuse and an orientation–winding coordinate. This yields the first three dyadic criteria and, independently of Gaussian factorization, the elementary test 2 | cont(θ) ⇔ a² + (c − b)² is a square, whose iteration computes the full dyadic depth. The three integral metallic families c − b ∈ {1,2,8} are correspondingly constrained: none has content divisible by 4, the family c − b = 2 has content 1, and the family c − b = 1 has odd content. On the algebraic side, Dickson–Chebyshev maps realize PAL division and induce a real-cyclotomic conductor-decimation law. On the quantitative side, if N_C(X) counts exact-content-C angles of hypotenuse at most X, then N_C(X) = (1/(2π C)) X^{1/C} + O_C(X^{1/(2C)} log X). Inside the perfect-C-power layer the limiting content law is φ(C/d)/C, equivalently the law of gcd(C,U) for U uniform modulo C. Exact content C cannot occur below height cot^{2C}(π/(4C)), and the maximal content below height X is asymptotic to log X/(2 log log X). The same hierarchy appears analytically through distinguished poles s = 1/C of the exact-content height zeta functions; in addition, the PAL-primitive series has a secondary pole at s = 1/2 of residue −1/(8π). Thus finite division strata are topologically dense while becoming arithmetically sparse in a precisely quantified manner.