there is no contradiction, but rather a topological distinction encoded in the boundary conditions. The basis for the sum rules (\[eq:supertrace\]) is Lorentzian AdS$_4$ which, after Euclidean continuation, gives rise to $S^1\times S^2$ boundary conditions with the $S^1$ corresponding to Euclidean time. In this geometry the Euler characteristic $$\label{eq:gaussbonnet}
\chi =\frac{ 1}{ 32\pi^2} \int E_4 + ~{\rm bndy} ~,$$ vanishes. This is significant because the divergences uncovered by the curvature expansion are proportional to $\chi$ and so they are not captured by AdS$_4$ with $S^1\times S^2$ boundary conditions. On the other hand, we will easily reproduce them from Euclidean AdS$_4$ with $S^3$ boundary conditions since this geometry has Euler invariant $\chi=1$.

One might wonder if these divergences have any physical significance. We argue in the affirmative by computing a finite and nonvanishing one-loop correction to the cosmological constant in maximal AdS$_4$ supergravity. In this computation it is manifest that the helicity supertrace relations (\[eq:supertrace\]) are violated in spacetime with $S^3$ boundary conditions. Interestingly, the violation is rather mild so all power law corrections in fact cancel. Thus the cosmological constant acquires just logarithmic running. This feature is intriguing since it might offer a mechanism that could describe dark energy without 