The Genealogy of the Pythagorean Angle Lattice: Angular Census, Cubic Carries and Transfer Operators

The genealogy of the mediators of the Pythagorean Angle Lattice is sectorial, with exact angular census, a distinct Euclidean first-return tree, signed exponential-sum census, and a conditional cubic-carry law under equidistribution.

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 lattice angles θ_W = 2 arg W, and its gaps are mediated by the primitive vectors with both coordinates odd. We study the genealogy of the mediators and the structures attached to it. The genealogy of the mediators is sectorial: the descendants of an interior mediator are exactly the odd–odd vectors strictly between its two canonical parents, so the size of every subtree at height N is one more than the number of sources in a sector, and the total depth is an exact sum of sector lattice-point counts. We describe the Euclidean first-return tree of the odd–odd vectors, whose norm increments factor exactly, and show that it differs from the genealogy of the lattice. We characterise the realisability of unimodular chains as blocks of the lattice, write the signed census of the sources as the real part of an explicit exponential sum, and record the modulo-four state space with its shear pairing. For the cubic carry of three-prime norm shells we give the conditional fixed-pair law under a progression-restricted equidistribution hypothesis, a finite Fourier reconstruction, Walsh and character decompositions, and examples showing that no residue condition determines the carry. Finally we record the ingredients of a Dolgopyat estimate for the outer limit set of the Pythagorean forest and contraction estimates for the Stern 3-adic walk, with numerical evidence throughout.