The Pythagorean Angle Lattice and the Calkin–Wilf Tree: Exact Dimension, a Lagrange Theorem and the Lyapunov Constant

The stationary law of the PAL forest is exact dimensional of dimension log 3/(2λ) ≈ 0.85548, with Lyapunov constant λ ≈ 0.64210 rigorously enclosed, and satisfies a Lagrange theorem for quadratic irrationals. 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. The primitive W = a + bi with a, b > 0 and odd norm, the sources, carry the lattice angles θ_W = 2 arg W ∈ (0, π) and the depth-D coordinates cot(θ_W/D). Contracting the Calkin–Wilf tree to its vertices of odd norm arranges the sources in a forest of two ternary trees with state-dependent branches, and the uniform laws of its generations converge to a unique stationary law ν_*. We prove that the stationary slope laws and the induced laws μ_D of the depth-D lattice coordinates are exact dimensional, with dim_H μ_D = log 3 / (2λ), where λ is the Lyapunov exponent of the branch cocycle, and we prove the one-moment formula λ = (2/3) 𝔼 log(1 + X) for the stationary slope X. Jensen’s inequality gives λ > (2/3) log(1 + √2), so that the dimension is below 1 and the laws are singular, and a compact transfer operator with an explicit cylinder enclosure, evaluated in exact rational and correctly rounded decimal arithmetic, gives 0.6421026587 ≤ λ ≤ 0.6421028427, so that dim_H μ_D = 0.8554801… for every depth D. We also prove a Lagrange theorem: the state distribution functions of ν_* are rational at an irrational slope exactly when the slope is a quadratic irrational. Further results include a dilation and a reciprocity symmetry, an independent geometric renewal decomposition, exact values at √2, 1 + √3 and the golden ratio, the description of ν_* as a reversible Markov measure on Stern–Brocot words, an exact binomial law for the Calkin–Wilf length of the forest paths, and a Jacobsthal count of the sources by Calkin–Wilf depth.