Path-dependent phase coupling induced by a dimensionally extended geometric background in neutrino oscillations

A minimal, unitary extension of the standard three-flavor oscillation scheme is formulated in which a projected geometric background, associated with a dimensionally extended configuration space, introduces a path-dependent Hermitian term without adding a fourth mass eigenstate. The effective Hamiltonian is written as H_eff = H_3 ν + V_MSW + λ_DQ (E/E₀)^n G(x) Q, where G(x) is a projected dimensionless scalar, Q is a traceless Hermitian generator in flavor space, and λ_DQ fixes the coupling energy scale. In the diagonal mass limit, the additional contribution shifts the interference phase according to δΦ_ij = λ_DQ Δq_ij (E/E₀)^n I_G, with I_G = ∫G dℓ. A null test is proposed that compares samples whose standard phases and matter effects are matched, but with different integrated geometric exposures. The main degenerate limits are identified, this term is explicitly distinguished from conventional Berry/Pancharatnam geometric phases, and low-dimensionality benchmarks are defined to avoid overly flexible parameterization. The model exactly recovers the standard theory when λ_DQ = 0. The goal is not to retrospectively explain a specific anomaly, but to provide a predetermined functional signal that can be subjected to global analysis with neutrinos from accelerators, reactors, and the atmosphere. Keywords: neutrino oscillations; DQ-12; extra dimensions; effective Hamiltonian; path-dependent phase; null proof; physics beyond the Standard Model; falsifiability.