Non-Abelian geometry and globally inequivalent symmetry reductions of the cylindrical Dirac doublet

Different exact cylindrical Dirac constructions are locally related within the same two-dimensional positive-energy sector, suggesting that they might be merely alternative choices of basis. We show that this equivalence can fail globally. Treating the cylindrical Dirac doublet as a rank-two bundle over momentum space, we identify transverse helicity, a mass-dressed transverse integral, and helicity as distinct symmetry-selected rank-one reductions, whose normalized restrictions organize the internal doublet through a Pauli algebra. The positive-energy Dirac $SU(2)$ connection Abelianizes exactly on fixed-azimuth meridians in the transverse-helicity basis, while its full three-dimensional curvature remains genuinely non-Abelian for nonzero mass. We derive the corresponding azimuthal Wilson-loop spectrum in closed form. The three reductions then display sharply different global structures: the transverse-helicity splitting terminates on the momentum axis, the mass-dressed splitting extends smoothly and is Chern trivial for $m>0$, whereas helicity defines line bundles with opposite unit Chern numbers. Thus a single Dirac doublet admits symmetry resolutions that are locally equivalent but globally inequivalent.