The Pythagorean Angle Lattice and the Gaps between Rational Points on the Unit Circle: Determinants, Correlations and Runs

The gaps of the index-two subgroup U₀ of rational points on the unit circle are governed by the missing coset: every gap is controlled by an odd–odd vector, yielding an explicit five-mark law, hard edge 2/π², determinant law 4/(k(k+1)(k+2)), and a non-Markov colour process whose runs are arithmetic progressions. 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 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. The points ei θ_W are the points in the upper half-plane of the index-two subgroup U₀ of U consisting of the points with even second numerator, and the complementary coset i U₀ is the second cusp of the theta group. Korolev and Ustinov determined the gap density of the whole group U ordered by denominator. We give a complete statistical theory of the gaps of U₀, that is, of the angles of the primitive Pythagorean triangles with hypotenuse at most R², and we show that it is governed throughout by the missing coset. Every gap is controlled by a unique primitive vector with both coordinates odd. A 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), and the determinant has the law 4/(k(k+1)(k+2)). Along the sequence of gaps, two consecutive determinants exceeding one form one of the pairs (2,2), (2,3), (3,2), 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), and monochromatic runs are arithmetic progressions with length spectrum R/(4|z|) and phases governed by square roots of −1. 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, the higher moments are values of Epstein zeta functions, and at finite height the largest gaps and the largest determinant are determined exactly.