  In this paper, we develop a general study of contributions at infinity of
Bochner-Weitzenb\"ock-type formulas on asymptotically flat manifolds, inspired
by Witten's proof of the positive mass theorem. As an application, we show that
similar proofs can be obtained in a much more general setting as any choice of
an irreducible natural bundle and a very large choice of first-order operators
may lead to a positive mass theorem along the same lines if the necessary
curvature conditions are satisfied.
