Classical Shannon entropy and mutual information satisfy elegant axiomatic characterizations that make them essentially unique measures of pairwise statistical dependency. When one tries to generalize these notions to capture genuine higher-order dependencies among three or more random variables using the structure of partition lattices, the axiomatic foundations collapse. The partition lattice framework introduced in recent work produces a family of interaction measures indexed by subsets of variables, but it remains completely open whether there exists a minimal, independent set of natural axioms that forces these higher-order measures to be unique up to normalization, in the same way that Shannon's axioms pin down entropy.