$E_4={\rm Riem}^2 - 4{\rm Ric}^2 + R^2$ encode dependence on the background geometry[^1]. The coefficients $c,a$ depend on the matter content of the theory and they have been studied in great detail; e.g. using perturbation theory in small curvature around flat space. Their values for fields with simple couplings to the background have long been established and are summarized in table \[table:centralcharges\] (later). These well known coefficients are such that, for the field content of ${\cal N}=8$ supergravity, their sum does not vanish. This fact establishes a divergence that is present already at one loop.

However, there are equally well established perturbative nonrenormalization theorems based on the helicity supertraces over the on-shell spectrum $$\label{eq:supertrace}
\sum (-)^{2h} h^n=0~,$$ for $n<{\cal N}=8$. These sum rules imply powerful cancellations for perturbative amplitudes in asymptotically flat space and related supertrace formulae are influential in particle phenomenology because they survive spontaneous breaking of supersymmetry. For us the important point is that the helicity sum rules establish one-loop nonrenormalization in ${\cal N}=8$ AdS$_4$ supergravity (gauged ${\cal N}=8$ supergravity) [@Gibbons:1984dg; @Inami:1984vp; @Nicolai:1983me]. These cancellations even generalize to all massive levels obtained from Kaluza-Klein compactification of ${\cal N}=1$ supergravity in $11$ dimensions.

We will argue that despite appearances 