Manuscript VII-J-ST@1 · VII-J-ST.tex · proposition · f1f1d82c

Proposition: Composition through a reindexing or event

Causal embedding, realized routes, and robustness › Admitted implementation primitives · lines 724–741 · proof included or linked

VII-J-ST@1#prop:composition

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If $Φ = Φ_{+} R Φ_{-}$ and the three channels have predecessor maps $P_{+}, P_{R}, P_{-}$ , then

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

Causal certificates compose in the same order. Independent allocations have no model-input predecessors; discards and all remapping controls must be included.

Proof. Compose the adjoint inclusions and then the corresponding causal paths.

Displayed formulas in this passage

unlabeled · line 728
$Φ^{*} (A (B)) \subseteq A (P_{-} (P_{R} (P_{+} (B)))) .$

Nearby in this section

assumption: Assumption: Finite routing architecture
proposition: Proposition: Gather–operate–distribute
section: Admitted implementation primitives