The affine metric-slot target

A possible interface is

For a degree-matched candidate, $r_{\xi }$ should be even, have ghost number zero, be linear in the even parameter and linear in the chosen even field slots, and have no scalar dependence. A retarded candidate may be required to have zero past data and support in $J^{+}(\mathrm{supp}\xi )\cap M_T$ for each fixed smooth field input.

Those conditions on smooth outputs are not sufficient for admission. The composed functional must also have a compact polarized degree-two kernel, with its wavefront set disjoint from the two all-causal cones. For example, a covector-sum-zero bound would be a sufficient kernel condition, but no such new kernel is constructed here. The existence of a source pairing on smooth fields is not substituted for this obligation.

Equation (eq:slot-target) records a range and compatibility problem; it does not assume an inverse of ${\mathcal{A}}_1$ on arbitrary auxiliary pairings. Even a pointwise assignment satisfying this slot equation would have to be realized as the variation of one declared correction, with consistent support, field degree, and parameter composition. It is that observable-completion problem which remains unresolved.

An infinitesimal solution must also be integrated into the finite coefficient law. The finite term $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ cannot be recovered by merely naming a linear ghost substitution. If a correction operates on a lower coefficient of the relational expansion, its coupling order must be printed; it is not added to the already filled second metric slot a second time.