Corollary: A necessary classical symmetry condition fails

An ordinary canonical BV extension generating (eq:skin), with $S_{\mathrm{ref}}$ as its antifield-independent action and with the switch rule unchanged, cannot satisfy the classical master equation for all these inputs. A correction beginning at positive formal powers of $\hslash$ cannot cancel this classical breaking.

Proof. The antifield-independent coefficient of that master equation requires $s_{\mathrm{kin}}{S}_{\mathrm{ref}}=0$. Theorem thm:matter violates this necessary condition. Formal powers of $\hslash$ have independent coefficients.