Positive Mass Theorem with Arbitrary Ends and Noncompact Boundary

We prove a positive mass theorem for complete Riemannian manifolds with noncompact boundary, a distinguished asymptotically flat half-space end, and finitely many additional complete ends with no prescribed asymptotics. If $3\leq n\leq7$, $R_g\geq0$, and $H_{\partial M}\geq0$, then $ \mathfrak m(M,g,\mathcal E)\geq0$. Moreover, equality holds if and only if $(M,g)$ is isometric to the Euclidean half-space. The proof combines a density deformation near the distinguished end with doubling across the noncompact boundary, local smoothing, and a conformal correction. We also obtain the sharp Riemannian Penrose inequality when a compact outermost minimal hypersurface separates $\mathcal E$ from all the remaining ends; equality holds precisely when the exterior region is a Schwarzschild half-space exterior.