Lemma: Off-collision rigidity of

Every well-defined contact in ${\mathfrak{C}}_{\mathrm{part}}$ restricts to zero on ${X_{\Gamma }}^{\circ }$. Consequently it cannot alter convergence, divergence, or prescription dependence tested by graph tests supported in ${X_{\Gamma }}^{\circ }$.

Proof. Each contact is supported on a collision locus ${\Delta }_I$. Differentiation does not enlarge support. Its pairing with a test supported away from the union of those loci is zero. This remains true for a family of contacts whose coefficients depend on an auxiliary regulator.