Geometric Configuration of the Standard Model Mass Spectrum: A Precision Validation Study via Dimensionless Pure π Formulas

The fermion and boson masses in the Standard Model of particle physics have long been treated as independent free parameters. This paper proposes a Geometric Phenomenological Framework based on the topology of the unit 3-sphere (S³), demonstrating that these masses obey a geometric progression governed by integer powers of π. Statement of Theory and Verification Status: This work does not claim to offer a microscopic alternative to Quantum Field Theory at this stage but presents a zero-free-parameter formula-numerical validation study. By normalizing the electron mass to a dimensionless unit (m_e ≡ 1), we derive a series of pure geometric mass formulas and perform precision comparisons with the latest CODATA/PDG experimental data: Proton-Electron Mass Ratio: The deviation between theory and experiment is only +5.25 ppb (5.25 × 10⁻⁹), and this deviation carries systematic physical significance. Weak Mixing Angle (cos θ_W): Locked directly by the geometric lattice integer ratio 514/583, with a precision of 0.02%. Pion-Proton Ratio: Precision reaches 7.5 ppm. These universal high-precision results, spanning strong, weak, and electromagnetic interactions, strongly suggest that mass is not a random parameter but a projection of the intrinsic geometric structure of the vacuum. This work responds to the ultimate question posed by Pauli and Feynman—"Why α ≈ 1/137?"—and provides a geometric answer: α⁻¹ = 4π³ + π² + π. While numerical precision is not a final proof, it constitutes undeniable Strong Evidence demanding serious scientific scrutiny.