Smearing, admission, and the limit not taken

For a compact interior covector test $a_{\mu }$, transposition gives the de Donder-layer identity

The resulting linear kernel is smooth at fixed $\epsilon$. Its support is contained in the closed layer intersected with the causal past of $\mathrm{supp}a$.

Equation (eq:layerkernel) is a classical restriction to de Donder tensors. The full auxiliary multiplet is not restricted by $D\gamma =0$: its equation is $b=D\gamma$. Imposing $D=0$ as an additional algebraic quotient would require a separate analysis and is not done here.

A boundary-supported trace is suggested by formally shrinking the layer. No convergence theorem, sharp limit, or identification with a sharp Cauchy kernel is asserted. In particular, a smooth closed-slab kernel need not be a compactly supported interior test. This classical kernel fact is not an admission theorem for a quantum relational composite.

Neither ${\mathsf{X}}_1$ nor its scalar dressing is declared $s_0$-closed. The plateau covariance of X-A does not prove such a claim for unrestricted ghost Cauchy data.