  We examine the ultraviolet (UV)-sensitive part of the one-loop Casimir energy
which is induced when various higher-dimensional supergravities are
compactified to 4D on extra dimensions which are Ricci flat, but otherwise
arbitrary. We identify the leading dependence on the mass of very massive
higher-dimensional modes, as well as the UV divergent part of the contributions
of modes which are massless in the higher-dimensional sense (but which consist
of a KK tower of massive modes from the 4D perspective), and show how these are
constrained by higher-dimensional general covariance. Some of the implications
of co-dimension 2 branes are computed in the limit where their tension is small
compared with the extra-dimensional Planck scale (but not small compared with
the observed dark energy). Our results support the interpretation of
supersymmetric large extra dimensions (SLED) in 6 dimensions as a potential
solution to the cosmological constant problem (but do not yet completely clinch
the case).
