Proposition: Tree-exactness on the specified sector

The tests $s_c,i_c$, and the completed tests constructed below, contain no $\hslash$ and no Hadamard contraction kernel. They have no positive-$\hslash$ remainder in the directly defined linear rooted prescription.

Proof. All coefficients in (eq:moller) are compositions of ordinary Green and differential operators on tests. The same is true after substituting the bare test polynomial $c[q]$. This proves the assertion algebraically.

The fixed Wick presentation provides a consistency count, not an alternative construction. A quadratic vertex and one linear insertion have three scalar legs, permitting one rooted cross-contraction and leaving one leg. Two quadratic vertices and a linear insertion have five legs. A connected rooted three-entry graph needs two edges, consumes four legs, and leaves one leg. Since the insertion has only one leg, that graph is a chain ending at the insertion. Three contractions would require six legs and are impossible.

The relevant contraction powers are removed by their first- and second-order response normalizations. A vertex-only vacuum component disconnected from the insertion is not part of the rooted prescription; self-contractions are absent in the fixed Wick presentation FR. The count explains the quantum order of the direct formulas, but does not define any missing singular product.