A Rigorous Holographic Bound on AdS Scale Separation

We give an elementary proof of the following property of unitary, interacting four-dimensional $\mathcal{N}=2$ superconformal field theories: at large central charge $c$, there exist at least $\sqrt{c}$ single-trace, scalar superconformal primary operators with dimensions $\Delta \lesssim \sqrt{c}$ (suppressing multiplicative logarithmic corrections). This follows from a stronger, more refined bound on the spectral density in terms of the asymptotic growth rate of the central charge. The proof employs known results on the structure of Coulomb branch operators. Interpreted holographically, this bounds the possible degree of scale separation in semiclassical AdS$_5$ half-maximal supergravity. In particular, the bulk must contain an infinite tower of charged scalar states of energies parametrically below the large black hole threshold, $E_{\rm BH} \sim c$. We address the extreme case of AdS$_5$ pure supergravity, ruling it out as the asymptotic limit of certain sequences in theory space, though the general question remains open.