Co-transformations and the absence of auxiliary slots

For a proposed multiplet rule

the full-coefficient infinitesimal interface is

The lift in (eq:full-object) has no independent auxiliary or ghost argument. Hence a rule with ${Y_{\xi }}^{\gamma }={Y_{\xi }}^{\varphi }=0$ changes it by zero. A pure auxiliary or ghost variation cannot cancel the copied nonzero value.

A scalar variation is different: it also differentiates the scalar-sourced slot. Explicitly,

Thus the zero metric variation of that slot does not license dropping it from a scalar co-transformation calculation.

A correction containing $b$ explicitly requires a declared action on $b$, matching parity, and its own admitted kernel. The displayed shell relation alone supplies none of those transformation laws. No correction is absorbed into $Q_H$, the inherited equation ideal, or an unlabelled modification of $r_{\mathrm{D}}$.

The witness excludes the original coefficient, and it excludes proposals whose added variation is demonstrably zero at that witness. A finite-dimensional class of corrections would require its generators and variations to be frozen and tested. An undefined space of all completions cannot be excluded by one scalar pairing.