The fixed-skeleton topology

Fix a finite labelled DAG and every finite local type. Let ${{\mathcal{X}}_{\Gamma }}^{\circ }$ be the circuits whose wire inequalities are strict:

Include the chosen finite list of identity waypoints in the vertex set. Interface types, vertex labels, and replacement slots are fixed.

For two assignments on this skeleton, define

If a fixed prepared state is retained as separate source data, add its trace distance; equivalently treat its preparation as a one-dimensional-input source channel.

This topology compares complete local channels, not only their effects on one prepared input. Equality in the completion is equality of the labelled local data. Equal terminal channels alone are not used to identify different skeleton-local experiments.