Scalar completion and its equation term

Define

For $\xi =c/2$ supported in $U$,

where $D_{\xi }u={\partial }_{\mu }({\xi }^{\mu }u)$ and $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$. Zero data for $X_{\xi }\varphi$ and the preflight identity give

Thus it vanishes for a free scalar.

If one names an external scalar antifield slot and the single Koszul–Tate rule ${\delta }_{\mathrm{KT}}{\varphi }^{‡}=-P_m\varphi$, the last expression is

This identifies the equation term without constructing a full antibracket or an interacting differential.

The relation between (eq:plateau) and (eq:KT) is explicit: the variation of the scalar response supplies $X_{\xi }\varphi$ and an equation term, whereas the coordinate word displays its scalar co-transformation separately. No full quantum solution-map intertwiner is inferred from this coefficient comparison.