  We report on a breakdown of both monopole dominance and positivity in
abelian-projected lattice Yang-Mills theory. The breakdown is associated with
observables involving two units of the abelian charge. We find that the
projected lattice has at most a global $Z_2$ symmetry in the confined phase,
rather than the global U(1) symmetry that might be expected in a dual
superconductor or monopole Coulomb gas picture. Implications for monopole and
center vortex theories of confinement are discussed.
