\vspace{-1.1em}A four-point counterexample to the verbose pullback triangle inequality

The fixed-degree pullback matching distance of verbose Vietoris--Rips barcodes fails the triangle inequality. In degree one, three four-point ultrametric spaces, with nonzero distances only $1$ and $2$, have pairwise distances $0,0,1$. More generally, in every degree $k\ge1$ and over every field, the same phenomenon occurs on three spaces of equal cardinality $k+3$. Every barcode involved is nonempty. A rank-counting argument proves the positive lower bound over all finite common pullbacks, not merely an enumerated set. The common cardinality $k+3$ is minimal. The degree-one zero-distance pair used here is already in Mémoli--Zhou, Example 5.10; the third space and the uniform lower bound complete a counterexample.