I will derive the analytic structure of the S-matrix for classical wave scattering on a Schwarzschild black hole in 4-dimensions. The argument relies on the analytic continuation of the gravitational potential, with the singularity behind the horizon playing a crucial role. We observe the emergence of Stokes phenomena leading to a surprising branch cut in the upper half plane, seemingly in tension with causality. I will discuss how this apparent paradox is solved and mention some bootstrap applications of these results.