Discrete Vacuum Geometry Predicts the Hierarchical Mass Spectrum of Standard Model Fermions

The broad hierarchy of fermion masses in the Standard Model, spanning six orders of magnitude, is conventionally attributed to ad hoc Yukawa couplings. This work explores a possible geometric interpretation arising from a discrete $\mathbb{Z}_3$-graded vacuum structure, derived from a finite-dimensional (19D: $12+4+3$) Lie superalgebra with exact triality symmetry. Within this framework, the vacuum is organized into a two-layer lattice: a finite \textbf{Core Lattice} (44 vectors) that yields gauge unification with $\sin^2 \theta_W = 0.25$, and an infinite \textbf{Extended Lattice} ($\mathbb{Z}^3$) that may generate the fermion mass spectrum through a geometric seesaw-like relation $m \propto L^{-2}$. By associating specific integer lattice vectors with known fermions, we find that the resulting mass scales appear to align with those of the top quark, bottom quark, tau lepton, charm quark, muon, down quark, and electron. For instance, the electron mass is obtained within 4.6\% (0.488 MeV compared to the observed 0.511 MeV) across a $10^6$ range. Observed deviations for heavier quarks are qualitatively consistent with QCD renormalization effects, suggesting the lattice might correspond to bare parameters. These numerical coincidences, while intriguing, do not constitute a proof and may reflect mathematical serendipity. The approach offers a complementary geometric perspective that unifies forces and matter within a single algebraic setting, extending previous work on the Weinberg angle and other constants from the same structure.