  We prove a Riemannian positive mass theorem for manifolds with a single
asymptotically flat end, but otherwise arbitrary other ends, which can be
incomplete and contain negative scalar curvature. The incompleteness and
negativity is compensated for by large positive scalar curvature on an annulus,
in a quantitative fashion. In the complete noncompact case with nonnegative
scalar curvature, we have no extra assumption and hence prove a long-standing
conjecture of Schoen and Yau.
