  We consider the problem of the doped Ising antiferromagnet on the frustrated
triangular lattice in the limit where the hole kinetic energy is much smaller
than the Ising exchange. For a {\it single} hole we prove a ``frustrated
Nagaoka theorem'' showing that the ground state is magnetized and breaks
translational symmetry, in contrast to the parent insulating state that is
unmagnetized and spatially homogenous. The extension of this physics to finite
dopings depends on the strength of a density-density coupling that is
inevitably present - we find either phase separation of the holes, or a
superconducting state that is {\it also} magnetized and breaks translational
symmetry in a feat of spatial self-organization. Finally, we derive an
effective interaction between dilute holes at temperatures in excess of the
hopping and find an oscillatory, long-ranged form reflective of the
correlations in the underlying classical magnet which presages the breaking of
translational symmetry at zero temperature.
