Meaning
A branch certificate states the concrete data, witnesses, estimates, and refinement tail on which a hypothesis has been verified.
It is a quadruple C=(D,W,B,T): D is the reconstruction data, W the explicit witnesses, B the exact identities and quantitative bounds, and T the cofinal tail where they hold. A cofinal tail reaches beyond every fixed finite demand in the selected refinement system.
Example (illustrative)
A certificate for separated spectral packets might contain the actual operators, separating contours and Riesz projections, a proved distance bound δ>0, and the tail on which that bound holds.
Scope and limits
Naming a hypothesis or giving a plausible diagram is not a certificate. A bound without its domain of validity or witnesses leaves the claim underspecified.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 13.1.