The RFW Orthotope: A Sandbox Model for Discretized Fermion Masses and Charges

The Standard Model (SM) accurately describes fermionic interactions but lacks a fundamental explanation for the free parameters in fermion mass spectra and the origin of fractional electric charges. Within the Higgs–Yukawa framework, this work establishes a minimal geometric mapping model based on the self-similarity of a three-strand RFW-Orthotope, enabling a unified, parameter-free description of fermion mass and charge spectra. In the charge sector, differential differences in the rotational moment of inertia of the three constituent strings reproduce the fractional charges of all nine charged fermions, including six quarks and three charged leptons, as well as their antiparticles, with weak hypercharges consistently derived from standard electroweak relations. In the mass sector, the Yukawa coupling is identified as a geometric scaling factor $y_f=2^{-k/3}$, which yields a discrete mass sampling formula. The experimental masses of the nine fermions are precisely mapped to integer quantum numbers $k$. Distinct from conventional parametric fitting, this mapping is purely geometric and completely free of tunable parameters. Number-theoretic analysis shows that the $\mathbb{F}_7$ algebraic structure emerging from the rotational inertia of the RFW-Orthotope generates three disjoint coset orbits, where each fermion generation of the Standard Model occupies a separate orbit. This discrete geometric correspondence provides a natural interpretation for the three-generation structure of fermions. Dynamically, the discrete selection of the quantum number $k$ is conjectured to arise from intrinsic geometric boundary conditions of the RFW-Orthotope, rather than continuous dynamical mechanisms. Using experimentally measured neutrino mass-squared differences as empirical constraints, this model predicts the masses and corresponding quantum numbers of the three neutrino flavors ($k=125,\ 4.988\times 10^{-2}\ \mathrm{eV}$; $k=132,\ 9.897\times 10^{-3}\ \mathrm{eV}$; $k=135,\ 4.948\times 10^{-3}\ \mathrm{eV}$), offering conditionally falsifiable predictions for future experimental tests. Fully consistent with the well-established Higgs mechanism and quantum electrodynamics of the Standard Model, this study heuristically investigates the discrete geometric origin of Yukawa couplings and fractional charges. Constructed on three phenomenological postulates, the proposed RFW-Orthotope model provides a refined geometric perspective for understanding the intrinsic spectral architecture of fundamental fermions.