Towards a Geometric Framework for the Higgs Scale and Fermion Generations

The Lagrangian of the Standard Model contains two fundamental input dependencies: the negative mass-squared term of the Higgs potential must be introduced by hand to trigger electroweak symmetry breaking, and the Yukawa coupling constants of fermions can only be fitted phenomenologically from experimental data. Starting from a new geometric picture---that the particle states and interactions observed at low energies are projections of internal higher-dimensional geometry onto a discrete boundary---this paper works backwards to determine the minimal internal structure required to produce the Standard Model gauge group SU(3) × SU(2) × U(1). This structure consists of a six-dimensional C³/Z₃ quotient space singularity (color symmetry) and a two-dimensional sphere S² with spinor structure (electroweak symmetry and chirality), with an external three-dimensional cubic lattice Λ³ serving as the projection boundary without dynamical degrees of freedom, yielding a total spacetime dimension of exactly ten. We present two phenomenological consequences of this framework. First, through the geometric projection of the S² ground-state energy, the electroweak scale satisfies the scaling relation v ~ (R/L)², which gives v ≈ 246\,GeV under natural hierarchy assumptions. Second, the compactness of S² restricts the angular momentum quantum number to l ≤ 3, providing a geometric origin for the finiteness of three fermion generations, and predicts a neutrino mass distribution m_ν ∈ [0, 0.31]\,eV (mean ~ 0.028\,eV), compatible with current cosmological constraints, while also predicting a superheavy neutrino with mass around 0.3\,eV.