E4: the conditional endpoint, not applied

The lemma is not an existence theorem for $a$. Nor is the relative-coordinate ansatz imposed as an additional restriction on every extension in ${\mathfrak{C}}_{\mathrm{part}}$. It is one sufficient endpoint structure that the selected attempt would preserve if the needed limits and compatible extensions were established.

Conditionally, if the complete extended numerical graphs have the balance of Lemma lem:relative, their residual delta attachments are compatible with them: a cancellation in the product would have to vanish at each separate external slot, forcing the corresponding attachment covectors to vanish. The compact graph closure then makes the projection proper.

The proper-pushforward wavefront condition sets internal covectors to zero and leaves a sum-zero bound on the two residual covectors. This is the conditional endpoint argument, not an application to an existing joint graph distribution.

The required output would be an even, ghost-number-zero polynomial of degree two with compact kernel $t^{(1)}$ satisfying

Only after that proof could the selected coefficient $Q_H{R}_1^{(1)}\in {\mathcal{E}}_H$ be released.

The tree coefficient would remain unchanged because the new datum enters at order ${\hslash }^2$ in ${\mathsf{T}}_2$ and at order $\hslash$ after the retarded division. The inverse signs and the inherited branch overlap cannot be changed at that order.

Properness is available as a support theorem from XIII-B. It cannot define a missing distribution. Likewise, Proposition prop:finite cannot be passed to an arbitrary distributional limit without a uniform or structural wavefront argument.