  Let $\mathcal{E}$ be an asymptotically Euclidean end in an otherwise
arbitrary complete and connected Riemannian spin manifold $(M,g)$. We show that
if $\mathcal{E}$ has negative ADM-mass, then there exists a constant $R > 0$,
depending only on $\mathcal{E}$, such that $M$ must become incomplete or have a
point of negative scalar curvature in the $R$-neighborhood around $\mathcal{E}$
in $M$. This gives a quantitative answer to Schoen and Yau's question on the
positive mass theorem with arbitrary ends for spin manifolds. Similar results
have recently been obtained by Lesourd, Unger and Yau without the spin
condition in dimensions $\leq 7$ assuming Schwarzschild asymptotics on the end
$\mathcal{E}$. We also derive explicit quantitative distance estimates in case
the scalar curvature is uniformly positive in some region of the chosen end
$\mathcal{E}$. Here we obtain refined constants reminiscent of Gromov's metric
inequalities with scalar curvature.
