Theorem: Coarse Poincare covariance under T

Assume T. Then

defines an action of ${\mathcal{P}}^{\uparrow }$ on $\mathcal{T}$, for every $D\ge 1$. It preserves realizability, outcome laws, witnessed modification relations, spacelike independent composites, and conditioning of laws.

Proof. Lemma lem:diagonal gives realizability of $g_{\#}\xi$. Apply it simultaneously to matched families using T3–T4 to preserve the modification relation.

A fixed Poincare map preserves strictly spacelike separation. On compact positively separated supports, continuity gives a positive transformed margin. Independent composites of the transformed objects exist by Proposition prop:coarseNS; their product law is the transformed product law.

Conditioning acts on the unchanged law. On object data, $(gh{)}_{\#}=g_{\#}{h}_{\#}$, and $g_{\#}^{-1}$ supplies the inverse. Preservation of the existential relations in both directions follows by applying T to the inverse map.