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

The Standard Model (SM) successfully describes interactions among elementary fermions, yet it lacks fundamental explanations for the free parameters in the fermion‑mass spectrum, the fractional charges of quarks, and the origin of three fermion generations. Within the Higgs‑Yukawa framework, this paper proposes a minimal geometric mapping model built upon the self‑similarity of the three‑dimensional RFW (Ren‑Fuwen) orthotope, aiming to characterize the mass and charge spectra of fermions. The model is solely determined by spatial dimension \\(N=3\\) and contains no continuous tunable parameters. On the charge‑spectrum side, differences in the moments of inertia of three orthogonally arranged strings map onto the fractional charges of the nine electrically charged fermions (six quarks and three charged leptons) together with their antiparticles. Weak hypercharge is subsequently computed via well‑established Standard‑Model electroweak relations. On the mass‑spectrum side, the Yukawa coupling is identified with the geometric scaling factor of the model, \\(y_f=2^{-k/3}\\). This yields a discretized mass‑sampling formula \\(E_k=(v/\\sqrt{2})\\,2^{-k/3}\\) without continuous free parameters. The experimentally measured masses of the nine fermions are mapped onto the integer set \\(k\\in\\{0,16,20,21,32,33,46,49,55\\}\\). It should be emphasized that this procedure is not a fit but a sampling mapping: no continuous free parameters, conceptually, one measures fermion masses using a fixed logarithmic ruler. From a number‑theoretic perspective, the moment‑of‑inertia structure of the RFW orthotope hints at an \\(\\mathbb{F}_7\\) algebraic structure, from which three disjoint coset orbits are constructed. When the mass quantum numbers \\(k\\) of the nine Standard‑Model fermions are projected onto these orbits, one observes that fermions of each generation occupy exactly one member from each orbit. This provides a discrete‑geometric correspondence for fermion‑generation structure. Dynamically, the trilinear‑coupling Lagrangian cannot single out particular values of \\(k\\). We therefore suggest that the selection of discrete \\(k\\) values may arise from intrinsic geometric boundary conditions of the RFW orthotope rather than from continuous dynamics. Constructed entirely within the Standard‑Model framework, the present model leaves the experimentally validated Higgs mechanism and electrodynamics unmodified. It offers only heuristic geometric explorations for the origins of Yukawa couplings and fractional charges, and presents conditionally falsifiable clues. The three sets of postulates are phenomenological inputs of the model and are not derived from first principles.