Abstract

We reformulate the snapshot–channel–adjoint comparison for unital local observable subalgebras without a spatial tensor decomposition. For a unital completely positive Heisenberg channel, equality of input state restrictions determines the output restriction exactly when the pulled-back output algebra is contained in the declared input algebra. The induced local channel is unique, completely positive, and stable under independent matrix references. The proof uses separation and Jordan decomposition of functionals, not a complementary-state preparation channel.

We retain central-sector probabilities, characterize the weaker conclusion available from restricted preparations, and prove the normal von Neumann algebra version. Conditional expectations are optional; their state-preserving existence is not assumed. An explicit constrained ${\mathbb{Z}}_2$ graph supplies centerful regional algebras, localized complete instruments, successful and failed containment tests, and an obstruction to independently changing shared boundary data.

For nets with local commutativity, specified internal Kraus devices give one-layer containment, algebraic ancestral factorization, metric support bounds, matched replacement invariance, and unconditioned no-signalling. A regional channel restriction alone need not determine its global device extension; this additional interface is essential. Finally, we construct a bounded-sequence observable limit and an exact subdivision limit of the finite gauge model, including compatible states, references, records, and matched devices. These are observable-level constructions, not a relativistic field-theory limit. No general gauge-theory transport compilation, Lorentz covariance, gravitational dynamics, or physical speed calibration is inferred.