Emergent Fermionic Valleys from a Quasi-Periodic Internal Vacuum Texture

Title: Emergent Fermionic Valleys from a Quasi-Periodic Internal Vacuum TextureDescription / AbstractThis work explores a geometric framework addressing the flavor replication and mass hierarchy of fermions in the Standard Model. The approach proposes that flavor multiplicity emerges from a quasi-periodic internal texture of the vacuum, inspired by moiré interference patterns known from condensed matter systems. 1, 3 By defining this structure in an internal coordinate space, the model preserves macroscopic Lorentz invariance. 1, 4Within this framework, a triangular internal potential with C₃ symmetry naturally organizes the low-energy spectrum into three inequivalent valleys, providing a geometric origin for the three fermion families. 1, 2 We show that effective fermion masses arise from overlap integrals between localized valley modes and a vacuum-order field, offering a structural explanation for the observed mass hierarchy and family mixing. Numerical analysis of the potential's Hessian matrix confirms the stability of these valleys as true local minima. 1 This model suggests that the complexity of the fermion sector may reflect an underlying geometric texture of the vacuum itself. KeywordsMoire Vacuum; Lattice Field Theory; Flavor Problem; Fermion Generations; Mass Hierarchy; Beyond Standard Model; Hyperdiamond Lattice