Proposition: Failure of highest-degree reconstruction

The array (eq:null-array) belongs to the defined degree-at-most-four space and has zero reconstruction:

For a suitable $b$, its degree-four part reconstructs to the nonzero distribution

Thus $A\mapsto \mathsf{Rec}(A_{[4]})$ does not descend.

Proof. The identity $q{\delta }^{(j)}=-j{\delta }^{(j-1)}$ follows by applying the derivative distribution to $q\phi (q)$. Take $j=4$, tensor with $R(r)$, and multiply by $b$.

Choose $b=1$ near a product region meeting the support of $R$. A test with a nonzero third $q$-jet and nonzero retarded-kernel pairing detects the degree-four part. Proposition prop:descent then excludes the rule.