Meaning
A fiberwise local algebra is the algebra of admitted observables supported in one region of one physical realization.
It is generated by the supported thick Wilson observables, electric and magnetic insertions, and seam-compatible operators, with products and adjoints defined on a common invariant domain. “Fiberwise” fixes the realization; “local” fixes its admitted region.
Example (illustrative)
If two admitted operators A and B preserve the common domain in region O, their products and adjoints belong to the generated algebra there. Moving to a larger admitted region requires the specified inclusion map.
Scope and limits
The algebraic definition does not assert a C∗ or von Neumann closure, a continuum net, or transport to a different realization. Those structures need their own certificates.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 8.2.