Meaning
The represented destination family lists the effective destination sectors actually carried by the nonzero marked supports.
At a finite window, each nonzero atom of the limiting marking algebra selects a subspace and its effective destination realization, taken modulo zero-cost internal equivalence. Compatible finite-window families then form the cofinal destination family.
Example (illustrative)
If three nonzero marked atoms support destinations s₁, s₂, and s₃, those classes belong to the represented family. A formal label whose projection is zero does not add a destination.
Scope and limits
A recorded family is not automatically complete. Exhaustion and fiberwise effectivity are additional hypotheses in the destination-completeness theorem.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 6.2.