The Arithmetic of the Pythagorean Angle Lattice: Growth, Hardy Sums, Norm Shells and Depth Layers

The arithmetic of the Pythagorean Angle Lattice is governed by Pick’s theorem, Hardy sums, a single square root of −1 controlling every odd–odd vector, and exact depth-layer degree laws. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}, where 𝒜 is the set of arguments of the rational points of the unit circle other than ±1, ±i. Its classical layer is carried by the sources, the primitive Gaussian integers W = a + bi with a, b > 0 and odd norm, through the angles θ_W = 2 arg W, and its depth-D layer consists of the numbers cot(θ_W/D). We study the arithmetic of this lattice. When the sources of squared norm at most N are ordered by slope, every gap (u, v) carries a unique odd–odd mediator h, the only sources that can later enter the gap are u + h and v + h, and every entry obeys an area-one law, which is Pick’s theorem in the ideal (1 + i)ℤ[i]. The determinants are conserved, ∑ det = 2M − 1; each mediator is born at an explicit norm, with activation law 2√t − 1; and the mediators form a parity-selected Stern–Brocot tree whose depth parity is χ₄(h₁h₂) and is equidistributed with a power saving. The running sum of det − 2 is Hardy’s sum 8s(q, 2p) + 8s(2q, p) − 20s(q, p). A single square root of −1 modulo the norm of an odd–odd vector governs its gap determinant, its birth, its fans and its position in its norm shell. On the norm shells we prove two-level determinant splittings, fluctuation laws with an explicit variance floor and an exact transport identity, and, for three primes, we show that the cubic Walsh coefficient of the roots is, up to sign and a factor 1/4, the indicator of a folded triangle in the local roots. In the deeper layers the degree of cot(θ_W/p) is 1, p − 1 or p, the sources with classical deep coordinates are explicit D-th powers and are counted exactly, and two Gaussian D-th powers are never unimodular neighbours.