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We study the Hamiltonian index of $\frac{1}{16}$-BPS operators in 4d$\mathcal{N}=4$ U(N) super Yang-Mills (SYM) theory numerically for$N=2\,,\dots,10$. We show that the large-charge asymptotics agree with analyticresults in the Cardy-like limit, as consistent with the entropy ofsupersymmetric black hole in the dual AdS. The numerics also agree with thelarge-N analytic result, thus providing hints towards an exact formula for theindex. We then prove, using ideas from representation theory, that for valuesof charges (quantized in integer units) less than 2(N+1) the index agreesprecisely with the multi-graviton index, and then begins to deviate for largercharges. Thus the U(N) SYM index interpolates between multi-graviton values atsmall charge and black hole growth at large charges.

The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM