Effective Mass-Squared Operators from Flavor Holonomy on a Compact Euclidean Four-Torus

We propose a unified hydro-topological framework in which neutrino flavor oscillations emergewithout introducing fundamental Dirac or Majorana mass parameters (mbare = 0). The underlyingmanifold is taken to be a flat Euclidean four-torus containing three extended spatial directions andone highly compactified spatial cycle. Effective Minkowski space-time arises only as a low-energyhydrodynamic description of excitations confined to a three-dimensional minimal-surface domainwall separating alternating vacuum domains. The three active neutrino flavors correspond to discreteincompressible flow modes of a quasi-liquid propagating tangentially along the wall.Localized geometric frustration of the wall induces a non-trivial flavor holonomy around thecompact cycle. The associated flavor connection modifies the Kaluza–Klein momentum spectrumand generates an effective non-diagonal mass-squared operator in the emergent low-energy theory.Oscillation observables are therefore controlled by geometric phase differences rather than byfundamental rest masses.