A cluster carries its own coherence, or it is not a finding

k-means returns k clusters whether or not k themes exist in the data. The inverse crosswalk's first run reported 24 of them as "themes"; reading the rows showed roughly half were mush — one labelled "school" held building violations and lobbyist registrations.

Every cluster now carries the mean cosine of its members to its own centroid, and below 0.62 it is published as diffuse rather than read as a theme. Cluster labels also require a term to cover a fifth of the members, not merely to be distinctive: lift alone labelled a 144-member cluster "inch, sea, rise" off a handful of coastal datasets.

The rule generalises past clustering: any method that always returns an answer has to be made to report how much of an answer it actually found.