The golden ratio φ= (1+√5) /2≈1. 618 is a mathematical fixed point with self-similar proportion and extremal stability, corresponding to the scale-invariant self-similar structure of vacuum topology. Based on topological symmetry and the principle of least action, this paper rigorously proves that the standard Higgs potential V (Φ) =μ²Φ²+λΦ⁴ converges uniquely to the golden ratio φ at its energy-minimized configuration during vacuum energy-level splitting and ground-state stabilization. φ is not an empirical constant but the unique mathematical solution of energy minimization under vacuum self-similar constraints. We establish a quantitative relation between φ and the lepton mass ratio (the prediction of φ¹² for m_μ/mₑ has a relative deviation of less than 0. 7% from the experimental value), and distinguish vacuum topological self-similarity from field-theoretic scale invariance to eliminate the apparent contradiction with the Higgs mechanism. Finally, the unified π–e–φ three-constant system is completed: π corresponds to closed topological conservation (hadrons), e corresponds to open topological evolution (lepton fields), and φ corresponds to vacuum self-similarity and mass generation (Higgs mechanism). This system is fully compatible with the Standard Model, quantitatively testable, and suitable for open-access academic archiving.
Jian Wen (2026) studied this question.