Theorem: Tree-exactness and absence of positive- terms

On the named scalar-only domain, $R_1(V_{1,g};F_c)$ is independent of $\hslash$ and of the Hadamard contraction kernel:

In particular, the first quantum correction vanishes identically.

Proof. A cross-contraction differentiates each entry once. Since ${F_c}^{(2)}=0$, no term with two cross-contractions exists. The zero-contraction product cancels in (eq:contraction-check). Its single contraction contributes one $\hslash$, removed by $i/\hslash$, and ${\Delta }_{+}$ cancels. There is no remaining positive power of $\hslash$.

The vertex is represented in a fixed free Wick ordering. Changing that ordering for a quadratic vertex changes it only by a scalar-independent term, whose first derivative is zero. Thus it cannot alter this map.