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

The fermion mass hierarchy in the Standard Model spans six chief orders of magnitude and is conventionally explained by arbitrary Yukawa couplings. Here we explore a purely mathematical construction—a discrete vacuum geometry derived from a finite-dimensional 19D (12+4+3) $\mathbb{Z}_3$-graded Lie superalgebra with exact triality symmetry—and examine whether simple integer lattice vectors embedded in its extended $\mathbb{Z}^3$ lattice happen to produce mass ratios and other parameter values resembling those observed in nature. Using a geometric scaling $m \propto L^{-2}$ where $L$ is the Euclidean norm of selected lattice vectors, and anchoring to the top-quark mass (173 GeV), the framework yields the following curious numerical proximities: electron 0.49 MeV (4.6\% agreement), muon 118 MeV (12\%), qualitative up/down quark mass inversion $m_u < m_d$, exact Weinberg angle $\sin^2 \theta_W = 0.25$, a Higgs-related scale ratio of 0.727 (0.3\%), strong/weak coupling ratio $\approx 0.95$ (near equipartition), CKM CP phase $\approx 65.3^\circ$ (5\%), and neutrino mixing angles of exactly $45^\circ$ (maximal atmospheric) with $\cos^2 \theta_{12} = 1/3$ (exact tri-bimaximal solar angle). These alignments, along with geometric patterns resembling tri-bimaximal neutrino mixing, are presented as intriguing mathematical coincidences within an abstract algebraic framework and do not constitute evidence of physical relevance or predictive power. The construction offers a speculative geometric perspective that unifies gauge and flavor aspects in a single algebraic setting, extending previous work on the same structure, while emphasising that the observed numerical matches may reflect serendipity rather than deeper significance.