Geometric Origin of Neutrino Masses and the PMNS Mixing Matrix in the Zhang Effective Metric

The Standard Model fails to explain the extremely small masses of neutrinos and the peculiar structure of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix, which features large mixing angles in stark contrast to the quark CKM matrix. In this paper, we derive the neutrino mass spectrum and the PMNS matrix within the rotational-order geometric unification framework based on the Zhang Effective Metric. Neutrino masses arise from the topological torsion of the internal S³ fibre, exponentially suppressed by the geometric warp factor via internal wavefunction overlap. The large mixing angles are shown to be the direct geometric consequence of the tri-axial rotational order n̂ᵃ: the parallel-transport SO(3) holonomy generates the three mixing angles, while a relative topological winding phase on the third flavour sector supplies the physical, observable CP-violating Dirac phase. This framework provides a purely geometric origin for neutrino phenomenology without invoking discrete flavor symmetries or heavy right-handed neutrinos.