  The microstates of supersymmetric black holes in asymptotically flat
four-dimensional spacetime are expected to be bosonic due to the spherical
symmetry of their horizons. This implies that the index counting the difference
between bosonic and fermionic black hole microstates must be positive, as
conjectured by Sen in arXiv:1008.4209. We show that the conjecture is satisfied
by the index which counts 1/4-BPS states in N = 4 string theory forming single
centered black holes, called immortal dyons. The proof relies on the relation
between the indexed degeneracies and the coefficients of a family of mock
modular forms, thus showing that black holes predict the sign of the Fourier
coefficients of mock modular forms.
