Proposition: Two explicit zero-evaluation subclasses

The tree evaluation vanishes on either of the following subsets of ${\mathcal{C}}_W$:

Also, under {(UCP${}_W$)}, the independent inherited identity ${\Delta }_{\xi }\mathcal{O}(u)=0$ remains valid for all $u\in {\mathcal{C}}_W$.

Proof. At $\gamma =0$, ${{\mathcal{O}}^{\prime }}_{\gamma }=0$ because its metric sector is quadratic, and $p_1[\gamma ;\varphi ]=0$. Thus both terms of (eq:tree-branches) vanish.

At $\varphi =0$, both $r_g[\varphi ]$ and $p_1[\gamma ;\varphi ]$ vanish. The same formula gives zero. The last assertion is the inherited restriction theorem, applied only to classical evaluation.