Lemma: Contact support in the common comparison space

For each inventory graph, retain its original two attachment variables $a,b$ and the interaction vertex $y$. Every connected collision contact involving $y$ is supported in

Thus every defined inherited contact vanishes on $Y\times A\times B\times Z$ while these attachment variables are retained.

Proof. The only edges incident to $y$ are its two cross-lines, whose observable endpoints are $a$ and $b$. A connected vertex subset containing $y$ and at least one further vertex must therefore contain at least one of those endpoints. Collapsing the subset forces $y=a$ or $y=b$.

Intersections and the corresponding contracted loci retain that equality when expressed in the original attachment variables. Differential operators do not enlarge support. Projection over other internal variables preserves the displayed equality because $y,a,b$ are not projected out.

The three neighborhoods have disjoint closures, proving the vanishing.