Static zero-energy parallel-spinor rigidity ($E=0\Rightarrow {\nabla }_{\mu }ϵ=0$)

The vacuum Witten-Nester identity makes a zero-energy Witten spinor parallel under the Sen spinor connection on a complete asymptotically flat slice. In an orthogonal static slicing, extrinsic curvature vanishes, so this spinor is intrinsically parallel. Contracting its spin-curvature integrability condition gives zero spatial Ricci tensor; in three dimensions this implies flatness. The static vacuum equation $D_i{D}_{j}N=N{R}_{ij}(h)$ then gives a parallel lapse gradient, which asymptotic normalization forces to vanish. The time-independent spinor consequently extends to a spacetime parallel spinor, with Minkowski spacetime the complete regular asymptotically flat example.