Free-shell class and the unresolved admission test

The compact kernel permits a free-shell class to be defined. This does not determine its admission or its relation to the frozen XI-G class.

Let $\mathcal{K}$ be the matching space of compact three-slot kernels with the stated tensor symmetry. Define

Since $\mathcal{P}$ is invertible and constant, $-\square \mathcal{P}a=0$ is equivalent to $\square a=0$. Compactness makes these pairings with smooth solutions well defined.

The class of the constructed kernel is

Comparison with a matching frozen kernel $u_0$ requires determining whether

If that condition holds, equality of the affine classes follows by linearity; the applicable frozen obstruction then transfers. A different-class proof would remove only that transfer, not establish admission XVIA,XVIB.

The inherited target ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ uses compact graded-symmetric polynomial kernels satisfying

Where the XI-G separation theorem applies with its complete hypotheses, it tests the trisolution

through

The propagators here are ambient causal propagators, not the slab retarded inverses by themselves XIG.

Under the inherited compact-distribution propagator interface, ${\mathsf{W}}_{t_{\mathrm{D}}}$ is an abstractly defined distributional trisolution. No explicit expression, wavefront estimate, or exclusion test for it is supplied in this note. Neither $\mathrm{WF}(t_{\mathrm{D}})$ nor $\mathrm{WF}({\mathsf{W}}_{t_{\mathrm{D}}})$ is calculated.