Ordered kinematic endpoints for 5-body cascade decays

We present expressions for the kinematic endpoints of 5-body cascade decay chains proceeding through all possible combinations of 2-body and 3-body decays, with one stable invisible particle in the final decay stage. when an invariant mass can be formed in multiple ways by choosing different final state particles from a common vertex, we introduce techniques for finding the sub-leading endpoints for all indistinguishable versions of the invariant mass. in contrast to short decay chains, where sub-leading endpoints are linearly related to the leading endpoints, we find that in 5-body decays, they provide additional independent constraints on the mass spectrum.