  We propose a formula for computing the (moduli-dependent) contribution of
multi-centered solutions to the total BPS index in terms of the
(moduli-independent) indices associated to single-centered solutions. The main
tool in our analysis is the computation of the refined index Tr(-y)^{2J_3} of
configurational degrees of freedom of multi-centered BPS black hole solutions
in N=2 supergravity by localization methods. When the charges carried by the
centers do not allow for scaling solutions (i.e. solutions where a subset of
the centers can come arbitrarily close to each other), the phase space of
classical BPS solutions is compact and the refined index localizes to a finite
set of isolated fixed points under rotations, corresponding to collinear
solutions. When the charges allow for scaling solutions, the phase space is
non-compact but appears to admit a compactification with finite volume and
additional non-isolated fixed points. We give a prescription for determining
the contributions of these fixed submanifolds by means of a `minimal
modification hypothesis', which we prove in the special case of dipole halo
configurations.
