  In the mirror fermion approach with Ginsparg-Wilson fermions, it has been
argued that the mirror fermions do not decouple: in the 345 model with Dirac-
and Majorana-Yukawa couplings to XY-spin field, the two-point vertex function
of the (external) gauge field in the mirror sector shows a singular non-local
behavior in the PMS phase. We re-examine why the attempt seems a "Mission:
Impossible" in the 345 model. We point out that the effective operators to
break the fermion number symmetries ('t Hooft operators plus others) in the
mirror sector do not have sufficiently strong couplings even in the limit of
large Majorana-Yukawa couplings. We observe also that the type of Majorana mass
term considered there is singular in the large limit due to the nature of the
chiral projection of the Ginsparg-Wilson fermions, but a slight modification
without such singularity is allowed by virtue of the very nature. We then
consider a simpler four-flavor axial gauge model, the 1$^4$(-1)$^4$ model, in
which the U(1)$_A$ gauge and Spin(6)(SU(4)) global symmetries prohibit the
bilinear terms, but allow the quartic terms to break all the other continuous
mirror-fermion symmetries. In the strong-coupling limit of the quartic
operators, the model is well-behaved and simplified. Through Monte-Carlo
simulations in the weak gauge coupling limit, we show a numerical evidence that
the two-point vertex function of the gauge field in the mirror sector shows a
regular local behavior, and we still argue that all you need is killing the
continuous mirror-fermion symmetries with would-be gauge anomalies non-matched.
Finally, by gauging a U(1) subgroup of the U(1)$_A$$\times$ Spin(6)(SU(4)) of
the previous model, we formulate the $2 1 (-1)^3$ chiral gauge model and argue
that the induced fermion measure term satisfies the required locality property
and provides a solution to the reconstruction theorem.
