We study quantum corrections in the gravitational path integral around nearly 1/16-BPS black holes in asymptotically AdS5 × S5 space, dual to heavy states in 4D \( \mathcal{N} \) = 4 super Yang-Mills. The analysis provides a gravitational explanation of why 1/16-BPS black holes exhibit an exact degeneracy at large N and why all such states have the same charges, confirming the belief that the superconformal index precisely counts the entropy of extremal black holes. We show the presence of a gap of order N−2 between the 1/16-BPS black holes and the lightest near-BPS black holes within the same charge sector. This is the first example of such a gap for black hole states within the context of AdS5 holography. We also derive the spectrum of near-BPS states that lie above this gap. Our computation relies on finding the correct version of the \( \mathcal{N} \) = 2 super-Schwarzian theory which captures the breaking of the SU(1, 1|1) symmetry when the black hole has finite temperature and non-zero chemical potential. Finally, we comment on possible stringy and non-perturbative corrections that can affect the black hole spectrum.
