Consecutive Angles of the Pythagorean Angle Lattice: Skeletons, an Exact Gap Formula, the Cubic Law and the Critical Constant

Consecutive angles of the Pythagorean Angle Lattice are governed by odd–odd skeletons with an exact floor formula for the gap determinant, the cubic law 4/(k(k+1)(k+2)), and the critical constant C_PAL = −6.2004356943…. 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_W = Arg(W²) of squares of Gaussian integers W = a + ib, and W = 2 + i gives the (3,4,5) triangle. The acute lattice angles split into the angles θ_W opposite the even leg, carried by the vectors (a,b) with a > b > 0 coprime and a + b odd, and their complements, carried by the coprime odd–odd vectors. We order the first family by angle, with hypotenuse at most R², and study consecutive angles. Every gap between consecutive angles has an odd–odd skeleton t = (u + v)/det(u,v), which is the only complementary angle of hypotenuse at most R²/2 that can lie inside the gap, and the gap determinant is given exactly by D = ⌊x − ω_t⌋ + ⌊x + ω_t⌋, where x = √(R² ‖t‖² − 1)/‖t‖² and ω_t is the fractional part of Re γ_t(i), the real part of the orbit point of i under a matrix of SL₂(ℤ) with bottom row t; one inequality decides which skeletons carry a gap. As R grows, gaps are born with determinant one (or two, for the skeletons (p,1)) and split by the rule D ↦ (1, D + 1); the sum of the determinants is exactly 2N(R) − n(R), the maximal determinant is attained on the skeleton (3,1) of the (3,4,5) triangle with an explicit formula, and the second moment obeys an exact history identity. Using the Weil–Estermann bound for Kloosterman sums we prove that the pairs (x, ω_t) are asymptotically distributed by x⁻³ dx dω on {x ≥ max(ω, 1 − ω)}; consequently the determinants follow the cubic law 4/(k(k + 1)(k + 2)), the elliptic-region analogue of Haynes’ theorem for Farey fractions with odd denominators, with a power saving uniformly for k ≤ R^κ, κ < 1/3, and birth ratio and age of a gap are independent. The critical second moment is 4 log R + C_PAL + O_ε(R^{−1/4+ε}) with C_PAL = 4γ + (8/3) log 2 + 2 log π − 8 log Γ(1/4) + 48 log A − 2π − 8 = −6.2004356943…, which equals the sum of the finite parts of the subcritical and supercritical moment families; the supercritical constants are values of a zeta function of the complementary angles, 2^{−s/2−1} [ζ(s/2) β(s/2)/((1 + 2^{−s/2}) ζ(s)) − 1]. Finally the angular excess R² Δ − D is strictly positive at every cutoff, has the unconditional mean π²/2 − 2, and has an explicit limit law with tail (1/B)[1 − 2/(1 + s) + 2 log 2/(1 + s)] for Y ≥ 8, B = Y + 1, s = √(1 − 8/B), and an extreme profile carried by determinant-one gaps at the cutoff circle.