A Symmetric Gaussian Cubic Form on the Pythagorean Angle Lattice: Unit Triples, a ℤ[√3] Composition Law and Pell Symmetries

Abstract

A symmetric Gaussian cubic form on the Pythagorean Angle Lattice admits a complete composition law in ℤ[√3], classifies all unit triples via Pell shells, and yields Pell symmetries with explicit spectral and arithmetic dynamics. The Pythagorean angle lattice is PAL = {cot(α/D) : α ∈ 𝒜, D ≥ 1}, where the set 𝒜 of Pythagorean angles consists of the arguments of the non-torsion elements of the group 𝒰 of rational points of the unit circle. Every point e^{iα} ∈ 𝒰 is z/¯z for a primitive Gaussian integer z = x + iy with x/y = cot(α/2), the depth-2 value of α in PAL. We prove that three points of 𝒰 of heights H₀, H₁, H₂ always satisfy |w₀ + w₁ + w₂| ≥ (H₀ H₁ H₂)^{-1/2}, and that this bound is governed by the symmetric Gaussian cubic form 𝒲(u, v, w) = ¯u v w + u ¯v w + u v ¯w. Its extremal solutions, the unit triples with 𝒲(u, v, w) a Gaussian unit, are the subject of the paper. We show that unit triples are exactly the solutions of a composition law in ℤ[√3]: with φ(x + iy) = x + y √3, one has 𝒲(u, v, w) = 1 if and only if d(u, v) φ(w) = φ(u) φ(v), where d(u, v) = 3 Re(¯u v)² − Im(¯u v)². Consequently φ(u) φ(v) φ(w) = d(u, v) d(v, w) d(w, u), the pairwise determinants multiply to values of x² − 3 y², and their prime factors ℓ > 3 split in ℚ(√3), while the ramified prime 3 divides at most one of them, and only to the first power. We classify the unit triples with a pairwise determinant 3 by the factorisations of the Pell shells m_k + 3 n_k i, prove that every pair with determinant 3 extends, classify the triples containing a unit, show that no nonconstant polynomial family exists and no one-variable rational family in which all three coordinates are unbounded, and prove that for each determinant there are effectively finitely many balanced triples. Finally we construct Pell symmetries of 𝒲 and determine their spectral geometry, determinant dynamics, trace algebra, ranks of apparition and escape behaviour.