A Geometric Derivation of the Standard Model

ABSTRACTWe present a zero-parameter geometric derivation of the Standard Model particle mass hierarchy. Byidentifying the vacuum at the Planck scale with the A₂ hexagonal root lattice constrained by theexceptional Jordan algebra J₃(��) particle masses emerge as topological necessities rather thanempirical inputs. The mass of any particle is given by M = v_EW × (n/27)^(1/d), where n is thesquared lattice norm of the particle's coordinate, d is its topological dimension (1 for vectors, 2 forscalars and fermions), and 27 = dim(J₃(��) is the state-space dimension of the vacuum. Theelectroweak scale v_EW = 246.22 GeV is the single external input, setting the physical scale; it isnot derived here but is under active investigation. All mass ratios are parameter-free. The frameworkreproduces the Higgs boson (0.095% error), Z boson (0.003%), top quark (1.1%), W boson (2.1%),and the observed 95 GeV scalar excess (0.66%), all pre-registered on viXra:2512.0067 (December26, 2025). The primary falsifiable prediction is a scalar resonance at 116.07 ± 0.05 GeV, testable atLHC Run 3 before July 2026.