<p>In Type II Calabi-Yau string compactifications, S-duality predicts that suitable generating series of BPS indices counting microstates of D4-D2-D0 black holes are in general mock modular forms of higher depth. The non-holomorphic contributions needed to cancel the anomaly under modular transformations involve certain indefinite theta series with kernels constructed from generalized error functions. Physically, these contributions are expected to arise from a spectral asymmetry in the continuum of scattering states of <i>n</i> BPS dyons with mutually non-local charges. For <i>n</i> = 2, the (standard, depth one) error function completion was derived long ago by explicitly computing the bosonic and fermionic density of states in the two-body supersymmetric quantum mechanics. Here we derive the general non-holomorphic completion for an arbitrary number of centers by evaluating the refined Witten index of the supersymmetric quantum mechanics using localization. In a nutshell, the index reduces to an integral over <i>ℝ</i>^{3<i>n</i> − 3} (the relative location of the centers), and splits into an integral over the 2<i>n</i> – 2 dimensional phase space of BPS ground states times an integral over <i>n</i> – 1 transverse directions, which ultimately produces the expected generalized error functions.</p>