E₈ Symmetry and Spectral Geometry in Quantized Spacetime: A Geometric Origin of Fermion Mass Hierarchies and Koide’s Relation

The Standard Model introduces fermion masses through Yukawa couplings, yet it provides no underlying principle governing their values or generation structure. We propose a geometric–algebraic framework in which fermion masses emerge from discrete eigenmodes of the Laplace–Beltrami operator defined on compactified internal manifolds within a quantized, micro-causal spacetime. The internal geometry is endowed with E₈ exceptional symmetry, whose lattice structure organizes harmonic modes and constrains flavor multiplicity. Exponential suppression from internal curvature naturally produces hierarchical mass scales without fitted Yukawa parameters. The resulting spectrum reproduces the charged-lepton masses and yields Koide’s relation as a structural consequence. Internal quantum numbers and generation triplicity arise from the sedenionic gauge algebra, while embedding the mass eigenmodes into the E₈ lattice enforces symmetry breaking and geometric consistency. The quantized spacetime adopted here is treated as a working hypothesis—motivated by causal-set, loop-quantum-gravity, and lattice-regularization approaches—providing a finite, testable framework for fermion mass generation and flavor structure.