Introduction and precise scope

The comparison lemmas in Cheung En05-A, En05-B, and En06-A identify three descriptions of finite hybrid-channel locality: factorization through a snapshot, determinacy by that snapshot, and containment of the Heisenberg pullback CheungEn05A,CheungEn05B,CheungEn06A. Their complementary-state proof uses a spatial tensor interface. The present paper removes that assumption from the comparison theorem itself.

Gauge constraints make this distinction useful. A physical regional observable algebra can have a center and need not define a tensor factor of the global physical algebra. Moreover, assigning an observable algebra to a region involves a boundary convention CasiniHuertaRosabal. Introducing boundary variables or edge modes is a different construction; it is not required by the theorem proved here DonnellyFreidel.

The elementary operator-algebra ingredients are standard: Hahn–Banach separation, Jordan decomposition of bounded Hermitian functionals, inherited matrix positivity of a $C^{\ast }$-subalgebra, and completely positive maps Pedersen,Arveson,Stinespring. We do not claim a new general separation theorem. The purpose is to give a complete operational formulation, its gauge-compatible implementation interface, explicit counterexamples, and compatible observable-limit constructions.