Degree Rigidity in the Pythagorean Angle Lattice

The algebraic degree of every value in the Pythagorean Angle Lattice is completely determined by an explicit formula involving torsion phase, return depth and support conditions, yielding degree–return rigidity and sharp results on simultaneous divisions and Pell angle differences. The Pythagorean Angle Lattice consists of the cotangents cot(α/D) of the rational divisions of the angles α of primitive Pythagorean triangles, together with their oriented division closure. We determine the algebraic degree of every such value. Writing the Cayley transform ξ = (X + i)/(X − i) of a value X as ξ^D = ε A(mP), with a primitive Gaussian source P, a multiplicity m and a phase ε ∈ μ₄, we attach to X an explicit root of unity τ, its torsion phase, and prove the complete degree law [ℚ(X) : ℚ] = (1/2) φ(M) q 2^{r−h}, where M = lcm(4, ord τ), D/gcd(m, D) = 2^r q with q odd, and h ∈ {0, 1, 2} is decided by explicit support and congruence conditions. The degree never exceeds the minimal return depth, with equality exactly when τ has 2-power order; this contains the degree–return rigidity theorem for gcd(m, D) = 1. For the angles of Pythagorean triangles the torsion phase is read off from the content of the Gaussian source and a winding index, and we prove a quarter-turn dichotomy and a sector trichotomy: one triangle, together with its quarter-turned companion, yields three different degrees at the same denominator and the same return depth. We determine the intersection of the dyadic division fields of an acute angle with those of its complement and of the difference of the two acute angles, classify simultaneous rational divisions of integral affine angle expressions, prove sharp height bounds for dyadically close Pythagorean angles, describe the difference map as a tent map with exact height and division-depth laws, show that the quarter and eighth parts of the difference angle generate ℚ(√2) and ℚ(√2, √(c(2 + √2))), c the hypotenuse, and develop the source lattice, its realizations, finite configurations and the profinite division boundary. As an application we determine the rational divisions of differences of Pell angles: no such difference has a rational fourth part, and for coprime indices no rational Dth part with D > 2.