In prior work [13], we introduced a procedure for deriving minimum formulas in first-order logic (FOL) for the reachable states of a restricted class of multi-sorted distributed protocol specifications: protocols with sorts representing unbounded sets of symmetric (indistinguishable) elements. This paper provides a deeper analysis of this idea that yields additional insights about the oft-cited observation that the behavior of such protocols can be inferred from analyzing relatively small finite instances whereby a protocol’s behavior becomes invariant, i.e. saturates [24], beyond certain cutoff sizes of its sorts. The paper discusses several issues in previous work [13] and provides more succinct FOL formulas of the reachable states for a collection of common protocols.