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 = vEW × (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 vEW = 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. Keywords: Standard Model mass hierarchy, A₂ lattice, exceptional Jordan algebra, hexagonalgeometry, E₈, particle mass derivation, 116 GeV scalar, falsifiable prediction.