  We analyze a set of contributions to the superconformal index of 4d
$\mathcal{N} = 4$ $SU(N)$ super Yang-Mills using the Bethe Ansatz approach.
These contributions dominate at the large $N$ limit, where their leading order
in $N$ reproduces various supersymmetric Euclidean black hole saddles in the
dual theory, and they also dominate for finite $N$ in high temperature
Cardy-like limits. We compute the $O(N^0)$ terms, including those exponentially
suppressed in the Cardy limit, and show that there are no $1/N$ corrections
beyond them. Under certain assumptions, it implies that the gravitational
perturbative series around these black hole saddles is 1-loop exact.
