Proposition: Spacelike composition and no-signalling

Let ${\xi }^1,{\xi }^2\in \mathcal{X}$ have compact supports satisfying

for some $\delta >0$. Their independent composite belongs to $\mathcal{X}$, with law ${\mu }^1\otimes {\mu }^2$. Each marginal is independent of the choice of the other experiment. More generally, modifications outside the past of a chosen report subset leave that subset's joint marginal unchanged.

Proof. The inequality gives positive separation of spatial shadows. Hausdorff convergence makes the active site sets of the two families disjoint for all large $n$. Their independent product-default preparations, complete channels, and report ports factor. The product laws converge because

For the marginal statement apply Theorem thm:coarseDOD to that report subset.