While the mathematical structure of scattering amplitudes has long been known to be constrained by principles such as causality and locality, the explicit form of these constraints has remained difficult to work out in practice. In this talk, I present a new method that sidesteps many of these difficulties, which allows us to derive large classes of novel constraints on Feynman integrals. In particular, through the identification of what singularities can still be reached after localizing to certain minimal cuts, strong restrictions can be placed on the ordered pairs of discontinuities that are allowed to appear in Feynman integrals. These restrictions, which we refer to as genealogical constraints, can be worked out for integrals involving arbitrary configurations of massive and massless particles, and hold to all orders in dimensional regularization.