Restriction is a new extension problem

For a smaller declared solution space $\mathcal{V}$, the question is

Here ${\mathcal{E}}^{\prime }$ denotes compactly supported distributions. Every pairing is therefore defined against smooth tensors, whether or not those tensors are spacelike compact.

A residual-gauge obstruction proves failure on $\mathcal{V}$ only if its witness belongs to $\mathcal{V}$. The witness of Theorem thm:residualwitness has $(K\xi {)}_{01}{|}_0\ne 0$, so it is not a spatial-TT wave. Its full-space conclusion does not answer (eq:restrictedproblem) on a smaller TT class.

Noncompactness of one displayed conserved distribution is also not a general obstruction to an equivalent compact current. The negative proofs above used nonzero values on pure gauges. The compactness equivalence of Lemma lem:compactconditions was a separate fact specific to its advanced-potential construction.