Covering Laws for Powers of the Pythagorean Angle Lattice: Pell Twins and a Ternary Invariant

Sharp covering laws for powers of the Pythagorean Angle Lattice are governed by Pell twins and a ternary invariant of the symmetric Gaussian cubic form, yielding the cubic law ρ₃(Q) ≍ Q-3/4 and a general lattice-number conjecture. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}, where the set 𝒜 of Pythagorean angles consists of the arguments of the non-torsion elements of the group 𝒰 of rational points of the unit circle. Its D-th power is 𝒰^D; a point eiα lies in 𝒰^D exactly when a branch cot((α + 2πj)/(2D)) of the depth-2D value cot(α/(2D)) is rational or infinite. Let 𝒰_Q be the points of 𝒰 of height at most Q and ρ_D(Q) the covering radius of 𝒰_Q^D = {w^D : w ∈ 𝒰_Q}. We prove the sharp cubic covering law ρ_3(Q) ≍ Q-3/4, with (0.513 + o(1)) Q-3/4 ≤ ρ_3(Q) ≤ (3√6 + o(1)) Q-3/4 and lim sup ρ_3(Q) Q3/4 ≥ 1.916. The cubic image is governed by the directions of three rotated copies of ℤ² inside ℤ[eπi/6]: Pell twins in two of the copies create the holes, and the integrality of the symmetric Gaussian cubic form 𝒲(z₀, z₁, z₂) = ¯z₀ z₁ z₂ + z₀ ¯z₁ z₂ + z₀ z₁ ¯z₂, through a ternary invariant equal to −|𝒲|², forbids larger ones. For all powers, ρ_D(Q) ≍ Q-1/2 exactly when D ∈ {1, 2, 4}, ρ_D(Q) ≪ Q-3/4 when 3 | D, and ρ_D(Q) ≤ (2D + o(1)) Q-2/3 otherwise. We also show that two branches of the cubic image cover only at scale Q-2/3, that the cubic image has minimal spacing ≍ Q⁻² and lies at distance ≍ Q⁻¹ from the units and from every fixed point whose three inverse slopes are badly approximable, and we determine the local spacing at the units and in every coset of 𝒰^D for D ∈ {1, 2, 3, 4, 6, 8, 12}. We propose a lattice-number law ρ_D(Q) = Q-L/(L+1)+o(1), L = D / gcd(D, 4), consistent with all proved cases and with numerical evidence; for D = 8 we show that the matching lower bound is equivalent to an explicit arithmetic criterion on √2-twins, and we give floating-point numerical evidence, not a certified bound, that ρ_8(Q) ≥ 0.56 Q-2/3 for 5 · 10⁵ ≤ Q ≤ 5 · 10³⁷.