Proposition: The two residuals cannot be identified

The pairing $W$ is not the classical master-equation residual in (eq:swCME). In particular, the statement that the unrestricted master equation fails “exactly by $W$” is not valid for the frozen data.

Proof. The master residual is odd with ghost number one. The pairing $W$ is an even real number of ghost number zero, bilinear in two free metric probes. No coefficient-extraction or comparison map identifying those arguments has been specified.

There is also a direct separation. In the natural reference action, set the scalar and ghosts to zero and all antifields to zero. The master residual is then zero, including in the compact onset background of (eq:recipdata). The independent reciprocity experiment in that same background still has $W>0$. The pairing detects the absence of a symmetric Hessian reproducing a different, source-switched map; it is not evaluation of a master action that has not been constructed.