Spacetime Positive Mass Theorem with Boundary for Multiple Time Dimensions

We prove a positive mass theorem for one-ended asymptotically flat spin initial data with multiple time directions and a smooth compact boundary. Assuming the trace-norm dominant energy condition and pairwise commutativity of the second fundamental forms, a fixed auxiliary time direction on each boundary component determines a local chirality condition. We identify the surviving mixed boundary term and obtain the inequality $E\ge \mathrm{\parallel }\mathcal{P}{\mathrm{\parallel }}_{\mathrm{t}\mathrm{r}}$ under the boundary condition $H+{\mathrm{tr}}_{\mathrm{\Sigma }}{k}^T+\mathrm{\parallel }{\mathcal{Q}}_T{\mathrm{\parallel }}_{\mathrm{t}\mathrm{r}}\le 0$. The proof constructs a Dirac–Witten harmonic spinor in an affine energy space and evaluates the mass identity by approximation. We also establish the pointwise sharpness of this boundary condition for the fixed projector and give two-time data with a nonzero mixed boundary term.