Corollary: Full-shell separation

A compact full-multiplet kernel $w$ annihilates all smooth shell tuples if and only if

with compact distributional primitives. Smooth $w$ admits smooth primitives. Such a kernel annihilates every smooth full-shell multisolution.

Proof. Use the invertible differential change of variables $\Phi \mapsto (Y,\rho )$, $\rho =b-D\gamma$. Expand each slot using $\Phi =JY+e_b\rho$. Terms containing a $\rho$ slot already factor through that slot's equation $b-D\gamma$. The remaining reduced kernel is $(J^{\mathsf{T}}{)}^{\otimes n}w$ and satisfies Lemma lem:reducedseparation. Integration by parts expresses the resulting terms as (eq:fullseparation). The transformations are differential and preserve compact support and smoothness. The converse and the multisolution statement follow by transposition.