The Bare Geometric Angles at the Planck Scale: Derivation from Gauge Coupling Ratios

The Higgs modulation directions in the internal 3D coupling space determine the Yukawa coupling ratios between generations. These directions are parameterized by four angles: the azimuthal angles φ_u, φ_d and the polar angles θ_u, θ_d. In the standard treatment, the effective low-energy values of these angles are calibrated to observed mass ratios. The bare values at the Planck scale, before renormalization group evolution, have not been derived. What this paper derives: We derive the bare geometric angles at the Planck scale from the gauge coupling ratios. The derivation uses only the primitives of the canvas model and the closed-form gauge couplings g₁² : g₂² : g₃² = 1 : 2/3 : 2/π with g₃ = 5π/32. The results are: φ_u = 73.6°, φ_d^bare = 163.6°, θ_u^bare = θ_d^bare = arctan(√(2/3)) ≈ 39.2° The azimuthal angles are fixed by the conjugation geometry: φ_u is determined exactly by the chirality and angle primitives (P5, P7), and φ_d^bare = φ_u + 90°. The polar angles are equal at the Planck scale because the conjugation Φ → iσ₂Φ* affects only the azimuthal direction in the (1,2) plane, not the polar angle from axis 3. The bare polar angle is determined by the ratio of SU(2) to U(1) gauge couplings at M_P: tan θ_H^bare = g₂/g₁ = √(2/3). This gives θ_H^bare ≈ 39.2°. Why this matters: The effective low-energy angles (θ_u^eff ≈ 80.8°, φ_d^eff ≈ 39.85°, θ_d^eff ≈ 89.00°) differ from the bare values due to renormalization group evolution. The RG evolution is driven primarily by the large top Yukawa coupling, which affects the up-type and down-type sectors differently. The effective angles are predictions of the theory, obtained by evolving the bare angles from M_P to the electroweak scale. No experimental mass ratios are used to determine the bare angles. The effective angles, when used in the mass formula, reproduce the observed quark mass spectrum. The geometric parameters of the Higgs sector are fully derived from the primitives. No free parameters remain. Keywords: Higgs modulation directions, Yukawa couplings, gauge coupling ratios, canvas model, renormalization group evolution, conjugation geometry, fermion masses