  Supersymmetric localization has lead to remarkable progress in computing
quantum corrections to BPS black hole entropy. The program has been successful
especially for computing perturbative corrections to the Bekenstein-Hawking
area formula. In this work, we consider non-perturbative corrections related to
polar states in the Rademacher expansion, which describes the entropy in the
microcanonical ensemble. We propose that these non-perturbative effects can be
identified with a new family of saddles in the localization of the quantum
entropy path integral. We argue that these saddles, which are euclidean
$AdS_2\times S^1\times S^2$ geometries, arise after turning on singular fluxes
in M-theory on a Calabi-Yau. They cease to exist after a certain amount of
flux, resulting in a finite number of geometries; the bound on that number is
in precise agreement with the stringy exclusion principle. Localization of
supergravity on these backgrounds gives rise to a finite tail of Bessel
functions in agreement with the Rademacher expansion. As a check of our
proposal, we test our results against well-known microscopic formulas for
one-eighth and one-quarter BPS black holes in $\mathcal{N}=8$ and
$\mathcal{N}=4$ string theory respectively, finding agreement. Our method
breaks down precisely when mock-modular effects are expected in the entropy of
one-quarter BPS dyons and we comment upon this. Furthermore, we mention
possible applications of these results, including an exact formula for the
entropy of four dimensional $\mathcal{N}=2$ black holes.
