Real Parts of Gaussian Powers on the Pythagorean Angle Lattice: Seed Counts, Value Spectra and Primitive Divisors

Real parts of first-lobe Gaussian powers on the Pythagorean Angle Lattice obey Beta seed-count laws, have spectra controlled by primitive divisors of Lucas sequences, and almost every integer occurs only at depth two (or not at all). 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}; its angles are the arguments θ_z = Arg(z²) of squares of primitive Gaussian integers z, and the (3,4,5) triangle corresponds to z = 2 + i. For a seed z = p + iq (p > q > 0 coprime of opposite parity) and an exponent D with D θ_z < π, the power z^D is again a generator of a lattice angle, namely D θ_z, and F_D(p,q) = Re z^D is the numerator of its coordinate in the layer PAL₂. We study the integers F_D(p,q), calling D the depth. At every depth D ≥ 3 the number of seeds with F_D ≤ K is (1/(π² D)) B(1/2, 1/2 − 1/D) K^{2/D} + O_D(K^{1/D} log 2K): continued-fraction bases and Roth’s theorem control the cusp, and at the three depths D = 3,4,6, where a Pell unit fixes the cusp, an explicit tiling gives the same error with effective constants. Depth two is solved completely, including the second-order term of the seed count; every value has at most τ(n) representations at a fixed depth; and for D = 2ʳ m with r ≥ 2 and m odd, ν₂(F_D − Nm(z)^{D/2}) = 2r + 2ν₂(pq) − 1. The value spectrum Spec(n) is the set of depths at which n occurs. Using primitive divisors of Lucas sequences we prove that the values of one seed at distinct depths never coincide except along the Pell family (2 + √3)^j at depths 1 and 3; that the even part of Spec(n) is the union of the divisor ideals of the endpoint indices of the factor-pair generators of n; and, with Ljunggren’s equation X² + 1 = 2Y⁴, that the spectrum of an odd prime power n consists of 2, of 3 exactly when (n² − 1)/3 is a square, and of 4 exactly when n ≡ 1 (mod 4) and (n² + 1)/2 is a square. Every depth D ≠ 3 in Spec(n) satisfies D ≤ (P(n) + 1)/2; odd depths D ≥ 5 satisfy D ≤ (P(n) + 1)/4 and τ(D) ≤ ω(n); and the integers occurring at some depth D ≥ 3 have natural density zero, so that almost every odd n has Spec(n) = {2} and almost every even n has empty spectrum. Further results are cyclotomic factorizations, depth ceilings, 2-adic necessary conditions for values 2ᵃ rᵇ, the complete family 8rᵇ, and an effective procedure, through S-unit equations, that lists all first-lobe values of depth at least three supported on a given finite set of primes.