Open case for a closed universe

We establish a new no-go theorem for cosmology: spatially flat ($k=0$) and open ($k=-1$) Friedmann--Robertson--Walker (FRW) non-static spacetimes cannot be simultaneously nonsingular, geodesically complete, and consistent with the averaged null energy condition (ANEC). Equivalently, any dynamic flat or open universe that is complete must violate the ANEC. By contrast, closed universes ($k=+1$) uniquely admit nonsingular, geodesically complete, ANEC-consistent solutions, with global de Sitter space as the canonical realization that saturates the ANEC. Furthermore, we analytically demonstrate that positive spatial curvature naturally mimics the phenomenology of phantom dark energy ($w<-1$), biasing flat-model reconstructions of $w(z)$ at the $\sim 1\%$ level. These results augment the classical singularity theorems, establish a new classification of eternal cosmologies, and motivate renewed scrutiny of spatial curvature in both theory and observation.