  We obtain holographic constraints on the microscopic degeneracies of black
holes by computing the exact macroscopic quantum entropy using localization,
including the effects of string worldsheet instantons in the supergravity
effective action. For $\frac14$-BPS black holes in type II string theory on $K3
\times T^{2}$, the constraints can be explicitly checked against expressions
for the microscopic BPS counting functions that are known in terms of certain
mock modular forms. We find that the effect of including the infinite sum over
instantons in the holomorphic prepotential of the supergravity leads to a sum
over Bessel functions with successively sub-leading arguments as in the
Rademacher expansion of Jacobi forms -- but begins to disagree with such a
structure near an order where the mock modular nature becomes relevant. This
leads to a systematic method to recover the polar terms of the microscopic
degeneracies from the degeneracy of instantons (the Gromov-Witten invariants).
We check explicitly that our formula agrees with the known microscopic answer
for the first seven values of the magnetic charge invariant.
