which proves that all connected diagrams generate sizeconsistent and size-extensive contributions. For disconnected diagrams, there are two cases, because the two Fermion lines could either be on the same or on different fragments. We note that the existence of diagrams of the former type is only possible if there is some component with an odd number of indices. Nonetheless, in these types of diagrams there is obviously no walk that starts at one unlinked index and ends at the other, which means that Eq. B2 no longer applies. Therefore, terms generated from disconnected diagrams with unlinked indices on different fragments evaluate to size-inconsistent terms. If both unlinked indices are on the same fragment, we can factor the expression into each connected contribution. Since the factorised prefactor to the expression scales with system size, this causes an incorrect total scaling, resulting in an overall size-consistent, but not size-extensive and also not unique term.