Proposition: Ambient inclusions and preserved relations

${\mathcal{V}}_{\mathrm{K}}\subseteq {\mathcal{V}}_{\mathrm{adm}}$ and ${\mathcal{V}}_{\mathrm{graph}}\subseteq {\mathcal{V}}_{\mathrm{adm}}$. Moreover, on ${\mathbb{R}}^4\setminus \{0\}$:

1. Edge interchange. ${q^{\circ }}_{\alpha \beta }={q^{\circ }}_{\beta \alpha }$.

2. Constant-coefficient traces / Leibniz. ${\partial }_{\gamma }(S^2)=2S{\partial }_{\gamma }S$ and ${\partial }_{\gamma }(S{\partial }_{\alpha }S)={\partial }_{\gamma }S{\partial }_{\alpha }S+S{\partial }_{\gamma }{\partial }_{\alpha }S$.

3. Wave operator. ${\eta }^{\alpha \beta }{q^{\circ }}_{\{0,e_{\alpha }+e_{\beta }\}}=S\square S=0$.

All three are inherited from XVI-K, Lemma lem:grading, and live already in ${\mathcal{V}}_{\mathrm{K}}$, hence in ${\mathcal{V}}_{\mathrm{adm}}$.

Proof. Every XVI-K unordered placement $\mathsf{k}=\{p,q\}$ occurs in the $C_2$ block of ${\Lambda }_{\mathrm{adm}}$ (source slots $a=b=0$, vertex placements with $|p|+|q|\le 2$; Lemma lem:L3comp(I2)); thus every generator of ${\mathcal{V}}_{\mathrm{K}}$ is a generator of ${\mathcal{V}}_{\mathrm{adm}}$, and ${\mathcal{V}}_{\mathrm{K}}\subseteq {\mathcal{V}}_{\mathrm{adm}}$. Separately, ${\Lambda }_{\mathrm{graph}}\subseteq {\Lambda }_{\mathrm{adm}}$ by definition, so ${\mathcal{V}}_{\mathrm{graph}}\subseteq {\mathcal{V}}_{\mathrm{adm}}$. (1)–(3) are XVI-K, Lemma lem:grading(2)–(3), which use only Lemma lem:Lprod and $\square S=0$ off the origin; they do not depend on the source sector.