  As a consequence of axial current conservation, fermions cannot be localized
in the massless NJL$_2$ model. In the case of twisted kinks, this manifests
itself in a cancellation between valence fermion density and fermion density
induced in the Dirac sea. To attribute the correct fermion number to these
bound states requires an infrared regularization. Recently, this has been
achieved by introducing a bare fermion mass, at least in the non-relativistic
regime of small twist angles and fermion numbers. Here, we propose a simpler
regularization by a finite box which preserves integrability and can be applied
at any twist angle. A consistent and physically plausible assignment of fermion
number to all twisted kinks emerges.
