  The computation of multi-loop multi-leg scattering amplitudes plays a key
role to improve the precision of theoretical predictions for particle physics
at high-energy colliders. In this work, we focus on the mathematical properties
of the novel integrand-level representation of Feynman integrals, which is
based on the Loop-Tree Duality (LTD). We explore the behaviour of the
multi-loop iterated residues and explicitly show, by developing a general
formal proof for the first time, that contributions associated to displaced
poles are cancelled out. The remaining residues, called nested residues as
originally introduced in Ref. \cite{Verdugo:2020kzh}, encode the relevant
physical information and are naturally mapped onto physical configurations
associated to nondisjoint on-shell states. By going further on the mathematical
structure of the nested residues, we prove that unphysical singularities
vanish, and show how the final expressions can be written by using only causal
denominators. In this way, we provide a mathematical proof for the all-loop
formulae presented in Ref. \cite{Aguilera-Verdugo:2020kzc}.
