Geometric Derivation of the Running Coupling Constant via Fractal Perturbation of Z₃ Symmetry

In a prior working paper (Zenodo: 10.5281/zenodo.20059886), we derived the strong coupling constant αₛ = 2/3 − √3/π ≈ 0.1153 and the electromagnetic coupling α = αₛ/16 from the geometry of Z₃ rotational symmetry on the complex plane. These are fixed values valid at a specific reference scale. However, the measured value of αₛ varies with energy (running coupling constant), a phenomenon explained in QCD by the renormalization group equation. In this addendum, we show that the energy dependence of αₛ can be expressed geometrically as a fractal perturbation δ of the boundary of the Z₃ circles. The effective radius becomes R_eff = A(1+δ), where δ runs logarithmically with energy scale Q. A critical result emerges: the fractal perturbation δ vanishes exactly at Q ≈ 125 GeV — the Higgs boson mass scale. This means the pure Z₃ geometry is realized most cleanly at the Higgs scale, suggesting that the Higgs mass itself may be determined by the geometric structure of Z₃ symmetry. The running toward asymptotic freedom (δ < 0 at high energy) and toward confinement (δ > 0 at low energy) are both captured geometrically by the sign of the fractal perturbation.