Theorem: Exact admissibility criteria

For the fixed $f,\varphi$:

1. $B_f(\cdot ,\varphi )$ is representable on $\mathcal{S}$ if and only if $H=0$, equivalently $\gamma =a=b_0=0$.

2. Each of the following is equivalent to $E_{1}H=0$: representability of $B_R$ on ${{\mathcal{S}}_{\mathrm{ret}}}^{-}$; representability of $B_0$ on $\mathcal{S}$; and representability of $B_f{|}_{{\mathcal{S}}_{\mathrm{dD}}}$.

3. The latter condition is exactly the vanishing of the two Cauchy data in (eq:zetadata).

When the conditions hold, representatives are obtained from (eq:smoothing), using $C=J_{\mathrm{raw}}$ in (i) and $C=J_{\sec }$ in (ii).

Proof. Every compact conserved current annihilates every smooth pure gauge. In (i), all compact $K\xi$ belong to $\mathcal{S}$, so representability forces $⟨H,\xi ⟩=0$ for every compact $\xi$. This is $H=0$. The bulk and slice supports, and arbitrary first normal jets of slice tests, separate $\gamma ,a,b_0$. The converse follows from Lemma lem:smooth and Theorem thm:smoothing.

For (ii), residual gauges with $\square \xi =0$ belong to all three tested spaces and are fixed by both prescriptions. They must be annihilated. Such annihilation is equivalent to $E_{1}H=0$: necessity follows by testing $\xi =E_{1}v$ and using the skew-adjointness of $E_1$; sufficiency follows from $H=\square A_H$ with compact $A_H$.

Assume $E_{1}H=0$. Compact integration by parts gives

For $B_0$, use

On $Dh=0$, the correction $D^{\ast }{A}_H$ pairs to zero. Smoothing $J_{\sec }$ now gives one representative for all three named functionals. Finally, homogeneous-wave uniqueness proves (iii).