Meaning
A finite admissible region system specifies which regions and supported inclusions the realized construction actually permits.
Its regions include admitted tubes, collars, supported intersections, mediation regions, and finite unions. Inclusion must respect the carrier, seams, domains, and markings. Separation means separation of the full thick supports, including collars and any declared coupling regions.
Example (illustrative)
Two loops drawn far apart can still fail the separation test if their collars or a shared boundary-flux channel overlap. The region system records that additional support.
Scope and limits
Ordinary set inclusion or separation of thin cores does not establish an admissible inclusion or physical separation.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 8.1.