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

The Standard Model (SM) provides a precise description of fermionic interactions yet cannot fundamentally explain the free parameters of fermion mass spectra and the physical origin of fractional electric charges. Within the Higgs–Yukawa framework, this study develops a minimal geometric mapping model grounded on the self-similarity of a three-dimensional RFW orthonormal body, which achieves a unified parameter-free characterization of fermion mass and charge spectra. In the charge sector, differential variations in the rotational moment of inertia of three constituent strings reproduce the fractional charges of all nine charged fermions, namely six quarks and three charged leptons, as well as their antiparticles, while the corresponding weak hypercharges are derived via standard electroweak formalism. In the mass sector, the Yukawa coupling is defined as a geometric scaling factor $y_f=2^{-k/3}$, yielding a discrete mass sampling formula that maps the experimentally measured masses of nine fermions to a set of integer quantum numbers $k$. Unlike conventional parametric fitting, the present mapping involves no adjustable parameters and realizes a pure geometric sampling of fermion mass values. Number-theoretic analysis reveals that the $\\mathbb{F}_7$ algebraic structure inherited from the RFW rotational inertia forms three mutually disjoint coset orbits, with each generation of SM fermions occupying an individual orbit. This geometric correspondence offers a natural discrete interpretation of the three-generation hierarchy of fermions. Dynamically, the specific selection of quantum number $k$ is speculated to be governed by intrinsic geometric boundary conditions of the RFW structure, rather than conventional continuous dynamical effects. Constrained by experimental neutrino mass-squared difference data, this model preliminarily predicts the masses and quantum numbers of the three neutrino flavors: $k=125( 4.988\\times 10^{-2}\\ \\mathrm{eV})$; $k=132( 9.897\\times 10^{-3}\\ \\mathrm{eV})$; $k=135( 4.948\\times 10^{-3}\\ \\mathrm{eV})$, delivering conditionally falsifiable predictions for future experimental validation. Strictly compatible with the well-established Higgs mechanism and quantum electrodynamics of the Standard Model, this work heuristically explores the discrete geometric origin of Yukawa couplings and fractional charges. Built upon three phenomenological postulates, the proposed model presents a viable alternative geometric perspective for investigating the intrinsic spectral structure of fundamental fermions.