Meaning
An asymptotically central source-native marker is a branch-label operator whose commutators with the determining local observables tend to zero.
On the selected finite windows, the marker family must asymptotically commute with the compressed test observables and with itself in the source’s stated sense. Effectivity further requires norm convergence of its joint spectral projections to projections commuting with the limiting compressed physical algebra. Those projections are then central in the auxiliary algebra they generate with it.
Example (illustrative)
For a bounded operator a, let q=diag(0,1) commute with a and q_n=q+n⁻¹r for bounded r. Then ‖[q_n,a]‖≤2‖r‖‖a‖/n. This illustrates vanishing commutators; spectral-projection convergence and the branch conditions still need checking.
Scope and limits
Asymptotic commutation alone does not establish the full effective-marker conclusion. Centrality in an auxiliary marked algebra does not imply centrality inside the physical local algebra.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 5.20.