Proposition: Tree-exactness and the local leg count

The maps (eq:second-map) and (eq:iteration) contain no positive power of $\hslash$ and no Hadamard contraction kernel. They are tree-exact in the specified linear rooted prescription.

Proof. Algebraically this follows from (eq:moller): its coefficients are ordinary test operators, with neither $\hslash$ nor a Hadamard kernel. The quantum statement concerns these coefficients, not the $\hslash$ in the field commutator.

The local contraction count confirms the scope. Use the fixed polynomial Wick presentation of X-B-v3. In that presentation each cross-contraction differentiates both entries once and supplies one factor of $\hslash$ FR (Sections 4–5). A quadratic vertex with a linear insertion admits only one cross-contraction, as in X-B-v3's first-map verification.

For two quadratic vertices and one linear insertion there are five scalar legs. A connected rooted graph joining the three entries needs two edges, consumes four legs, and leaves one scalar leg. Since the insertion has degree one, the graph is a chain ending at that insertion. Three contractions would require six legs and are impossible. The two possible vertex orderings give the bookkeeping polarization (eq:polarization).

The two contraction factors are removed by the second-order retarded prefactor. A vacuum component between the two vertices disconnected from the insertion is not part of the rooted relative response. Self-contractions are absent in the fixed Wick presentation. Thus the rooted coefficients have no positive-$\hslash$ remainder.