Geometric Foundations of the Weak Interaction: Deriving the Fermi Constant and Mixing Angle from Mass-Charge Constraints

The Standard Model provides an exceptionally accurate description of electroweak phenomena, yet several of its defining parameters—most notably the weak mixing angle θ _W ​, the Fermi constant G _F ​, and the Higgs mass M _H ​—remain empirical inputs without established theoretical origin. In this work, we examine whether these quantities may arise from a geometric constraint imposed at a Quantum Turning Point (QTP), where mass and charge components of the electroweak vacuum obey a unified relation. We introduce a mass–charge identity, ( e / g ) ² + ( M _W / M _Z ) ² = 1, which consolidates standard electroweak relations into a single geometric constraint. Using the experimentally measured W and Z boson masses, this identity yields θ _W  ≈ 28.2°, in close agreement with precision determinations of the effective weak mixing angle. Interpreting g as the SU(2) _L weak-isospin coupling—and, within the QTP-weak framework, as an effective geometric charge—leads to a natural weak-coupling scale suggestive of a harmonic value α _g  ≈ 1/30. Substituting this coupling into standard electroweak expressions reproduces the magnitudes of the Fermi constant and the W-boson decay width within a few percent, indicating that these parameters may be largely determined by an underlying geometric scaling. A geometric projection of the vector-boson scale further yields M _H ≈ ( π /2)× M _W , consistent with the observed Higgs mass. Together, these results suggest that several electroweak parameters conventionally treated as independent may instead reflect a common geometric structure encoded at the Quantum Turning Point. This framework does not modify Standard Model dynamics but offers a compact and coherent geometric interpretation of its parameter values, motivating further exploration of the role of vacuum geometry in fundamental physics.