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 Rₑff = 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 low energy) are both captured geometrically by the sign of the fractal perturbation. Keywords: running coupling constant, asymptotic freedom, fractal perturbation, Z₃ symmetry, Higgs mass scale, renormalization group, geometric derivation