  We further investigate center vortex percolation and Coulomb gauge remnant
symmetry breaking in the SU(2) gauge-Higgs model. We show that string breaking
is visible in Polyakov line correlators on the center projected lattice, that
our usual numerical tests successfully relate P-vortices to center vortices,
and that vortex removal removes the linear potential, as in the pure gauge
theory. This data suggests that global center symmetry is not essential to the
vortex confinement mechanism. But we also find that the line of vortex
percolation-depercolation transitions, and the line of remnant symmetry
breaking transitions, do not coincide in the SU(2)-Higgs phase diagram. This
non-uniqueness of transition lines associated with non-local order parameters
favors a straightforward interpretation of the Fradkin-Shenker theorem, namely:
there is no unambiguous distinction, in the SU(2) gauge-Higgs models, between a
"confining" phase and a Higgs phase.
