Rédiger un PREreview

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.