  Motivated by the surface of topological insulators, the Dirac anomaly's
discontinuous dependence on sign of the mass, $m/|m|$, is investigated on
closed topologies when mass terms are weak or only partially cover the surface.
It is found that, unlike the massive Dirac theory on an infinite plane, there
is a smoothly decreasing current when the mass region is not infinite; also, a
massive finite region fails to exhibit a Hall current edge--exerting an
extended proximity effect, which can, however, be uniformly small--and
oppositely orientated Hall phases are fully quantized while accompanied by
diffuse chiral modes. Examples are computed using Dirac energy eigenstates on a
flat torus (genus one topology) and closed cap cylinder (genus zero topology)
for various mass-term geometries. Finally, from the resulting the properties of
the surface spectra, a potential application for a flux-charge qubit is
presented.
