Theorem: Harmonic-prescription classification

For fixed $f,\varphi$, the following are equivalent:

1. $B_{\mathrm{raw}}(K\xi )=0$ for every smooth spacelike-compact residual wave $\square \xi =0$;

2. $\zeta ={\mathsf{E}}_1\mathbf{H}=0$;

3. both equations

   $$b={{\int }_0}^T{S}_{\mathrm{w}}(s)\gamma (s)\mathrm{d}s,a=-{{\int }_0}^T{C}_{\mathrm{w}}(s)\gamma (s)\mathrm{d}s$$

   hold;

4. $B_R$ is FH-representable on ${\mathcal{S}}_{\mathrm{ret}}$;

5. $B_0$ is FH-representable on $\mathcal{S}$;

6. $B_{\mathrm{raw}}{|}_{{\mathcal{S}}_{\mathrm{dD}}}$ is FH-representable on ${\mathcal{S}}_{\mathrm{dD}}$.

When they hold, one common representative is

It represents each of the three functionals on its stated domain.

Proof. The equivalence of (i)–(iii) follows from Proposition prop:threegauges and (eq:zetadata). Any of (iv)–(vi) implies (i), because residual pure gauges belong to ${\mathcal{S}}_{\mathrm{dD}}\subset {\mathcal{S}}_{\mathrm{ret}}$, are fixed by both prescriptions, and must be annihilated by compact conserved tests.

Assume (ii). By Lemma lem:compactconditions, $A$ and $J_{\sec }$ are compact. Equation (eq:hybrid) now permits unrestricted compact integration by parts:

For the Cauchy prescription,

so $B_0(h)=⟨J_{\sec },h⟩$ on $\mathcal{S}$. If $h\in {\mathcal{S}}_{\mathrm{dD}}$, the $D^{\ast }A$ term pairs to zero and $⟨J_{\sec },h⟩=B_{\mathrm{raw}}(h)$. Finally apply Lemma lem:smoothprop and Theorem thm:smoothing to $J_{\sec }$.