Corollary: Necessary master-action condition

An ordinary canonical BV extension generating (eq:nonlinear) and having $S_{\mathrm{ref}}$ as its antifield-independent action cannot satisfy the classical master equation with the scalar switch and transformations unchanged. A correction starting strictly at $\hslash$ cannot cancel this classical breaking.

Proof. The antifield-independent coefficient of the classical master equation requires $s{S}_{\mathrm{ref}}=0$. Theorem thm:matter violates this necessary condition. Coefficient extraction at ${\hslash }^0$ proves the final assertion.