Emergence of Fermion Mass Hierarchy from the Weighted Cayley Graph of a Z3-Triality Vacuum Lattice

We derive the fermion mass hierarchy from the weighted Cayley graph of a 44-vector $\mathbb{Z}_3$-triality vacuum lattice, with zero free parameters. Edge weights $A_{ij}=1/\|\mathbf{v}_i-\mathbf{v}_j\|^2$---the unique discrete Laplace--Beltrami prescription---encode vacuum impedance. The mass matrix $M_{ab}=\sum_{h\in\mathcal{H}}G_w(f_a,h)G_w(h,f_b)$ is derived as the zero-momentum fermion self-energy with Higgs exchange on the graph, with the overall normalization locked by the algebraic rigidity condition $y_t=1$. The absolute mass scale emerges from a 12-level nested chain compressing $M_{\rm Pl}$ to $v_{\rm geo}\approx 185$~GeV (RG-evolved to the experimental $v_{\rm EW}=246$~GeV). Shell assignments are uniquely determined by $\mathrm{SU}(3)$ coupling range, $\mathrm{SU}(2)_L$ doublet structure, and color-singlet/hypercharge selection rules---not by fitting. The bare lattice predictions are: $m_t:m_c:m_u=1:0.0162:1.3{\times}10^{-5}$, $m_b:m_s:m_d=1:0.0145:8.9{\times}10^{-5}$, $m_\tau:m_\mu:m_e=1:0.0144:9.9{\times}10^{-7}$. After quantitative 1-loop SM renormalization group evolution from $M_{\rm Pl}$ to $M_Z$, all nine mass ratios converge to within $12\%$--$40\%$ of experiment, with the five-order-of-magnitude hierarchy correctly reproduced. The framework simultaneously determines the absolute electroweak scale ($246$~GeV) and the top quark mass ($174$~GeV) from $M_{\rm Pl}$ as the sole dimensionful input.