Depth Arithmetic of the Pythagorean Angle Lattice: The Divisibility Index, Depth Genealogies and Depth Spectra

Depth in the Pythagorean Angle Lattice is governed by the divisibility index J(W) = gcd(g, 2s): rational depths are the divisors of 2J, power-orbit genealogies obey explicit limiting laws, and every spectrum is a truncated arithmetic progression on a single Gaussian power ray. Let U be the group of rational points of the unit circle, μ₄ = {±1, ±i} and 𝒜 = {Arg z : z ∈ U \ μ₄}. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}, with depth-D layers PAL_D; its angles are the arguments θ_W = Arg(W²) of the squares of primitive Gaussian integers. This paper studies depth: when the depth-D coordinate cot(θ/D) of a lattice angle is rational, at which depths a given number of the lattice occurs, and how these depths behave under products, powers and counting. The lifted angle group 𝒢 = {ψ ∈ ℝ : e^{iψ} ∈ U} is free on π/2 and the angles of the Gaussian primes, and every question about depth reduces to divisibility in 𝒢. We introduce the divisibility index J(W) = gcd(g, 2s), where g is the gcd of the Gaussian-prime exponents of W and s ∈ (1/2)ℤ is its phase, and prove: the rational numbers q with q θ_W ∈ 𝒢 form the group J(W)^{-1} ℤ; the rational depths of θ_W are the divisors of 2J; the exact odd depths, the selected and principal Gaussian roots, the value spectra and the depth strata are all governed by J. Along the power orbit of a lattice angle the index becomes the depth genealogy J_n = gcd(gn, 2(ns − c_n)) with c_n the nearest integer to n θ_W/π; we determine its limiting law P_{g,ε} for every g and each of the two source sectors ε (for g = 1, ℙ(J = j) = 4/(π² j²) or 12/(π² j²) according to the parity of j), its moments and tail 8/(π² Y), its growth exponent 1 − 1/μ, and prove that it determines the angle, the exponent gcd and the phase class; the folded powers carry rotation and parity cocycles, and identifying a generator of height at most B from it needs at least of order log log B adaptive queries. The set of depths at which a number of the lattice occurs is a truncated arithmetic progression with an explicit common difference, and the ratio of exponent gcd to depth is the same for all its realizations. We also give the genealogy and hypotenuse power law of rational divisions, intersection numbers, probe lattices and continued-fraction renormalization of folded powers, sharp minimum dyadic heights with the complete list {1, 5, 13, 17, 29, 37, 65, 85, 1073} of two-step height gcds, a crossing law for products, an exact exponent law for Gaussian-power cancellation with Dickman, mod-Poisson and joint four-way limits, counting laws with their depth zeta function, and a classification of the moduli that detect the phase.