 Elliptic existence of a Witten spinor

On complete smooth asymptotically flat spin initial data without inner boundary, satisfying the Einstein constraints and the dominant energy condition, the Witten spinor equation admits a spinor approaching each prescribed asymptotic constant, under the standard weighted decay assumptions. The integrated Witten-Nester identity gives a coercive estimate and excludes a decaying homogeneous kernel. Weighted elliptic solvability then supplies the desired correction to a spinor extended from infinity. Positivity of the boundary integral is a conditional identity until this analytic existence step is supplied; see the rigorous analytic proof.