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

Proposition: Zero-duration remote SWAP is excluded

The event transition: exact macro fidelity and buffered locality › What an instantaneous remapping would get wrong · lines 1803–1821 · proof included or linked

VII-J@1#prop:instant

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Two distinct points on any one slice $Σ_{s}$ are spacelike separated. A same-slice remote SWAP has unit signalling between them and is not a causally faithful transition for the cone relation.

Proof. The slice estimate gives

∣ F_{s} (x) - F_{s} (y) ∣ \leq λ_{0} d_{C} (x, y) < d_{C} (x, y)

for distinct points. Lemma lem:conecausal therefore excludes either causal direction. The replacement-state SWAP test gives unit signalling, as in Theorem thm:eventbenchmark.

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$∣ F_{s} (x) - F_{s} (y) ∣ \leq λ_{0} d_{C} (x, y) < d_{C} (x, y)$

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section: What an instantaneous remapping would get wrong