The Azimuthal Angle Evolution: Why φ_d Rotates from 163.6° to 39.85°

The down-type Higgs modulation direction in the canvas model is parameterized by two angles: the polar angle θ_d and the azimuthal angle φ_d. The bare values at the Planck scale are derived from the gauge structure: θ_d^bare = arctan(√(2/3)) ≈ 39.2° and φ_d^bare = 163.6°. The effective values at the electroweak scale are determined by the down-type quark mass ratios: θ_d^eff = 89.00° and φ_d^eff = 39.85°. What this paper does: The polar angle evolution has been derived in previous work: the gauge push drives θ_d from 39.2° toward 90°, and the bottom Yukawa is too small to reverse it. This explains the ~50° rotation. The azimuthal angle evolution has remained an open problem. φ_d rotates by approximately 123.75° (equivalent to 56.25° modulo 180°) over the same energy range. The CKM phase J ~ 10⁻⁵ is far too small to drive this rotation. So what does? What this paper derives: We show that φ_d is driven by the Yukawa couplings, not the CKM phase. The geometric overlaps for the down-type quarks depend on φ_d through g₁^(d) ∝ |cos φ_d + sin φ_d| and g₂^(d) ∝ |cos φ_d − sin φ_d|. The Yukawa couplings y_b and y_s therefore depend on φ_d, and their beta functions contain terms that drive φ_d evolution through the chain rule. The beta function for φ_d contains a factor of 1/sin(2φ_d). This factor is large near φ_d = 0° and φ_d = 90°, and minimal at φ_d = 45°. Starting from the bare value φ_d^bare = 163.6° (equivalent to 16.4° in the first quadrant), the evolution is initially rapid. As φ_d approaches 45°, the factor 1/sin(2φ_d) approaches its minimum of 1, and the evolution slows. The angle settles near 40°, close to the attractor at 45°. Why this matters: The mechanism is the same gauge push/Yukawa pull that drives the polar angle evolution, now applied to the azimuthal direction. The 1/sin(2φ_d) factor naturally explains both the large total rotation and the final value. No new physics is required. This completes the derivation of all four Higgs direction angles. The bare values are determined by the gauge structure at M_P. The effective values are determined by the renormalization group evolution, driven by the same gauge push/Yukawa pull mechanism that operates in the polar direction. No free parameters remain. Keywords: azimuthal angle, Higgs direction, Yukawa couplings, renormalization group evolution, canvas model, geometric overlaps, strange quark, attractor