A counterexample to degree-zero universality of the interleaving distance

The universality conjecture for the interleaving distance asks whether every distance stable under uniform perturbations of sublevel functions is bounded by the interleaving distance on realizable persistence modules. Its formulation for each homology degree is false for ordinary zero-dimensional homology. For every field we construct a bounded, $0$-stable pseudometric on all four-parameter persistence modules and a pair realized on three-point discrete spaces for which this distance is $3$ and the interleaving distance is $1$. The algebraic pair is Bjerkevik's published free-module example. The additional argument recovers the upper supports of eventual sublevel components from the image of a module in its colimit and proves stability for arbitrary spaces and functions, with no finite-dimensionality assumption.