Lemma: Affine slot identity

For all smooth $H,k,r,\varphi$,

Proof. Linearity of $a_1,b_1$ gives $\mathcal{W}[H,k+r]x^{\mu }=\mathcal{W}[H,k]x^{\mu }+{\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \mu }[r])$, hence $X_2[H,k+r]=X_2[H,k]+X_1[r]$ by (eq:legs); and $V_{2,g}[H,k+r]=V_{2,g}[H,k]+V_{1,g}[r]$, hence ${\varphi }_2[H,k+r;\varphi ]={\varphi }_2[H,k;\varphi ]+{\varphi }_1[r;\varphi ]$. No other term of (eq:Asc) contains $k$. The difference of integrands is ${\varphi }_1[r;\varphi ]-{X_1}^{\mu }[r]{\partial }_{\mu }\varphi$, which is (eq:Aone).