The spacetime Penrose inequality: invariant ADM mass bounds the minimal area enclosing a trapped boundary, for general asymptotically flat initial data

Theorem 1.3: for smooth one-ended initial-data exteriors in every dimension $n\ge3$ with integrable constraint densities, dominant energy condition, weakly future trapped boundary ($\theta^+\le0$, nothing on $\theta^-$), the stated decay, $E>|P|$ and $A_{\min}>0$: $m=(E^2-|P|^2)^{1/2}\ge\frac12(A_{\min}/\omega_{n-1})^{(n-2)/(n-1)}$, sharp; outermostness and outer area minimization are not assumed. Corollary 1.4 derives $E>|P|$; Theorem 1.5 ($n=3,4$) needs only two metric derivatives of decay. Companions: rigidity (equality only for Schwarzschild-Tangherlini slices, under horizon hypotheses), a conformal-flow Riemannian inequality in all dimensions, the boundary deformations, and a Bondi-mass version for CKS hyperboloidal data. Not shown: evolution or cosmic censorship. Priority: Allen-Bryden-Kazaras-Khuri (2025) proved the bound with a non-sharp constant.

Theorem 1.3: for smooth one-ended initial-data exteriors in every dimension $n\ge3$ with integrable constraint densities, dominant energy condition, weakly future trapped boundary ($\theta^+\le0$, nothing on $\theta^-$), the stated decay, $E>|P|$ and $A_{\min}>0$: $m=(E^2-|P|^2)^{1/2}\ge\frac12(A_{\min}/\omega_{n-1})^{(n-2)/(n-1)}$, sharp; outermostness and outer area minimization are not assumed. Corollary 1.4 derives $E>|P|$; Theorem 1.5 ($n=3,4$) needs only two metric derivatives of decay. Companions: rigidity (equality only for Schwarzschild-Tangherlini slices, under horizon hypotheses), a conformal-flow Riemannian inequality in all dimensions, the boundary deformations, and a Bondi-mass version for CKS hyperboloidal data. Not shown: evolution or cosmic censorship. Priority: Allen-Bryden-Kazaras-Khuri (2025) proved the bound with a non-sharp constant.