Meaning
A coherent selected branch is a compatible family of physical choices—carriers, source and fusion sectors, comparison maps, associators, form domains, complete-ground lineage, and Hamiltonian-descent controls—on which the later analytic statements are actually defined.
The raw theory can offer many formally possible realizations and many possible passages between them. That abundance is not yet a physical branch. A selected branch records which realizations are being followed and why the operations used later in the proof remain compatible as the regulator changes.
The word coherent matters. It says that successive comparisons compose with the declared associator data and that the analytic domains, ground sectors, and error budgets are carried along the same lineage. The word selected matters too. Existence of a large distributor of possible passages does not choose one represented physical continuation.
Example (illustrative)
Suppose a coarse regulator and a fine regulator each contain several candidate source sectors. A comparison that sends the chosen coarse sector into one fine sector, preserves its selected form domain, transports the full ground projection, and fits the next refinement comparison belongs to a coherent selected branch. Merely knowing that some fine-sector realization exists does not.
Scope and limits
A coherent selected branch is not a global Hilbert space containing every vacuum lineage. It does not create a global scattering operator. Every operation remains supported on the Hilbert
Source
Phenomenological Implications I, §2.1, Definition 2.1 and Remark 2.2.