Theorem: Regular time-slice isomorphism

Inclusion of compact tests induces

These are isomorphisms of the regular causal algebras and their formal $\kappa$ extensions.

Proof. On the shell a full smearing becomes a reduced smearing: the $b$ test contributes $D^{\mathsf{T}}{r}_b$ to the metric test. This operation preserves support. Lemma lem:timeslicetest, componentwise, shows that the reduced test-class map from $N$ to $M$ is surjective and injective.

The reduced causal form consists of the wave causal operators with the constant action fiber maps and the ghost mixing of (eq:Delta). It depends only on the reduced wave test classes and is preserved by inclusion. The causal presentation and the regular quotient theorem thus give the algebra isomorphism. Factoring inclusion through ${M_T}^{\circ }$ gives the slab statement.