Proposition: Inherited-root overlap

Every occurrence of the inherited first-completion branch with metric argument ${G_{\mathrm{gr}}}^{\mathrm{R}}w$ obeys

In particular,

The tree datum uses these same response and root normalizations. It is not identified with ${\mathcal{C}}_{3,\mathrm{D}}$.

Proof. Equation (eq:root-overlap) is XII-D's transposition identity with the inherited sign, not a new root construction. Since $r_g={G_{\mathrm{gr}}}^{\mathrm{R}}(g{T}_0/2)$, substitution gives (eq:stress-overlap). In (eq:tree-branches) the same metric response is contracted with the complete metric derivative of $\mathcal{O}$, and the scalar branch is its inherited switched response. Differentiating any displayed first-completion branch uses the same identity.