Manuscript VII-J@1 · VII-J.tex · proposition · d8c01b04

Proposition: Support composition through a transition

Localization and the exact physical-containment theorem › Composing relations through an update · lines 715–739 · proof included or linked

VII-J@1#prop:eventcomposition

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Let

Φ = Φ_{+} \circ R \circ Φ_{-},

where the three channels may act on differently indexed tensor products. Suppose their support predecessors are $P_{-}$ , $P_{R}$ , and $P_{+}$ , respectively: each output algebra pulls back into the algebra of its declared predecessor set. Then

Φ^{*} (A (B)) \subseteq A (P_{-} (P_{R} (P_{+} (B)))) .

If all three predecessor relations have physical causal certificates in one common event space, the composite does also.

Proof. Compose the three adjoint inclusions. Compose the corresponding causal chains and use transitivity.

Displayed formulas in this passage

unlabeled · line 718
$Φ = Φ_{+} \circ R \circ Φ_{-},$
unlabeled · line 727
$Φ^{*} (A (B)) \subseteq A (P_{-} (P_{R} (P_{+} (B)))) .$

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section: Composing relations through an update