The Statistics of the Pythagorean Angle Lattice: Gaps, Colours, Correlations and Extremes

The gaps of the Pythagorean Angle Lattice obey an explicit five-mark law, with hard edge 2/π², determinant law 4/(k(k+1)(k+2)), and a non-Markov colour process whose monochromatic runs are arithmetic progressions controlled by square roots of −1. Let U be the group of rational points of the unit circle and 𝒜 = {Arg z : z ∈ U \ {±1, ±i}}. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}; its angles are the arguments Arg(W²) of primitive Gaussian integers W, and its classical layer is carried by the sources, the primitive W = a + bi with a, b > 0 and odd norm, through the angles θ_W = 2 arg W of the primitive Pythagorean triangles. We determine the statistics of the gaps between the lattice angles of norm at most R². Every gap is governed by a unique primitive vector with both coordinates odd. The five-mark law gives the joint limit law of the length, determinant, endpoint ratio, colour word and fluctuation of a gap as an explicit pushforward of Lebesgue measure; it rests on an equidistribution theorem for modular inverses on parity classes, proved with the Weil–Estermann bound. The normalized gaps have an explicit law with hard edge 2/π² and tail 16/(π⁴ T²), the determinant has the law 4/(k(k+1)(k+2)), and the second moments of the determinant and of the gap length are 4 log R + C_det and 32 log R + C_G with both constants in closed form. Along the sequence of gaps the colours form a stationary process that is not Markov of any finite order, in which L ≥ 3 consecutive sources share a colour with probability 1/(8L − 10); two consecutive determinants exceeding one form one of the pairs (2,2), (2,3), (3,2); and monochromatic runs are arithmetic progressions with length spectrum R/(4|z|) and phases governed by square roots of −1. At finite height the largest and the second largest gaps are determined exactly.