Theorem: Complete retained algebraic incidence

The full formal coefficient on the retained mixed attachment/output germ is obtained by contracting ${{\mathcal{V}}_g^{\prime \prime }}_{\gamma \varphi }$ with the complete kernel

against the grouped line difference in the first line of (eq:species).

All other branch contributions are zero on this germ for the reasons proved above. No collision contact defines or changes this germ while the attachments are retained.

Proof. Use Proposition prop:Hessian, Lemma lem:ledger, Proposition prop:reduction, Lemma lem:channel, and Lemma lem:contact.