  We show that the Bekenstein-Hawking entropy of large supersymmetric black
holes in AdS$_5\times S^5$ emerges from remarkable cancellations in the giant
graviton expansions recently proposed by Imamura, and Gaiotto and Lee,
independently. A similar cancellation mechanism is shown to happen in the exact
expansion in terms of free fermions recently put-forward by Murthy. These two
representations can be understood as sums over independent systems of giant
D3-branes and free fermions, respectively. At large charges, the free energy of
each independent system localizes to its asymptotic expansion near the leading
singularity. The sum over the independent systems maps their localized free
energy to the localized free energy of the superconformal index of $U(N)$
$\mathcal{N}=4$ SYM. This result constitutes a non-perturbative test of the
giant graviton expansion valid at any value of $N$. Moreover, in the
holographic scaling limit $N\to\infty$ at fixed ratio
$\frac{\text{Entropy}}{N^2}\,$, it recovers the 1/16 BPS black hole entropy by
a saddle-point approximation of the giant graviton expansion.
