Input statement: XII-E: the finite remainder

For $\xi \in {\mathfrak{g}}_U$ and $P_m\varphi =0$, define

With ${\Delta }_{\xi }F=F[\gamma +K\xi ,\varphi ]-F[\gamma ,\varphi ]$,

The second term also equals ${\mathcal{A}}_1[f,g](e_{\xi };\varphi )$. On ${\mathcal{C}}_{\mathrm{sol}}$, $A\gamma =-2D^{\ast }b$, so its auxiliary form is

These are classical identities of the specified lifts, not a quantum Ward identity or an action on the quotient algebra XIIE.