Corollary: Continuum operational no-signalling

Assume C1–C3. Let two compact operation/readout regions be spacelike separated at positive margin:

Replacing one experiment by another in its region does not change the unconditioned joint record law of the other.

Proof. By compactness and continuity, every point of either region is at positive distance from the causal past of every point in the other. For sufficiently small meshes, uniform C2 excludes all dependency paths in either direction between the two complete discrete regions. The discrete schedule-locality argument applies to the full set of the other experiment's reported records, not only to each marginal record separately. C3(ii)–(iii) supplies matched composites. Pass to the total-variation limit.