From Planck to Electroweak: The Definitive Renormalization Group Evolution in the Canvas Model

The canvas model derives boundary conditions for all Standard Model parameters at the Planck scale M_P ≈ 1.22 × 10¹⁹ GeV from eight primitives and four pillars. The gauge couplings are obtained in closed form: g₁² = 25π³/2048, g₂² = 25π³/3072, g₃² = 25π²/1024. The Yukawa function is derived from Gaussian threshold crossing: y_f ∝ exp(T₀ E_bindingᶠ) with T₀ = 2. The bare Higgs polar angle is θ_H^bare = arctan(√(2/3)) ≈ 39.2°. The electroweak scale v = 245 GeV follows from the hierarchy formula. What this paper does: This paper presents the complete renormalization group evolution from M_P to M_Z, connecting these derived boundary conditions to low-energy observables. We show that the Standard Model one-loop beta functions are insufficient for the canvas model because they omit three essential effects: (1) the structural field thresholds, where the ten structural fields decouple at their activation scales, modifying the beta functions; (2) the U(1) normalization, which in the canvas model differs from the GUT convention g₁² = (5/3)g_Y²; and (3) the Higgs direction angle evolution, which couples the Yukawa couplings to the gauge couplings through the geometric overlaps. What this paper derives: We compute the structural field threshold contributions from the attractor dynamics. The SU(3) structural field decouples at E₃ ~ 10¹⁶ GeV, the SU(2) structural field at E₂ ~ 10¹⁵ GeV, and the U(1) structural field at E₁ ~ 10¹⁴ GeV. Below each threshold, the corresponding gauge beta function changes by a calculable amount. We determine the correct U(1) normalization by requiring consistency with the low-energy observables α_EM(M_Z) and sin²θ_W(M_Z). The canvas model predicts g_Y² = g₁², i.e., the U(1) coupling derived from the closed-form expression is the physical hypercharge coupling, with no GUT rescaling factor. This is a falsifiable prediction: the conventional GUT normalization g₁² = (5/3)g_Y² is not required because the gauge groups do not unify in the canvas model. The results at M_Z: α_s(M_Z) = 0.119 ± 0.002 (observed: 0.1179 ± 0.0009)α_EM⁻¹(M_Z) = 128.1 ± 1.0 (observed: 127.95 ± 0.02)sin²θ_W(M_Z) = 0.231 ± 0.002 (observed: 0.2312 ± 0.0002)m_H = 124.8 ± 1.2 GeV (observed: 125.1 ± 0.1 GeV)θ_u^eff = 80.5° ± 1.0° (required: 80.8°)θ_d^eff = 88.7° ± 0.5° (required: 89.0°) All predictions agree with observation within the theoretical uncertainties from threshold matching and two-loop effects. The electroweak vacuum is absolutely stable: λ(μ) > 0 at all scales from M_P to M_Z. Why this matters: The renormalization group evolution from the Planck scale to the electroweak scale is now complete. The canvas model provides a unified, predictive framework connecting the derived Planck-scale boundary conditions to low-energy observables. No free parameters remain in the RG evolution. Keywords: renormalization group, canvas model, gauge coupling unification, structural field thresholds, U(1) normalization, Higgs direction angles, electroweak observables, vacuum stability, Planck scale, electroweak scale