Theorem: Exact observable descent and nonzero detection

Fix a reference response ${\mathcal{R}}_{\ast }$ in the matched family.

1. The pairing $⟨\mathcal{R},F⟩$ is independent of every prescription in ${\mathfrak{P}}_0$ if and only if $F\in {{\mathfrak{A}}_{\mathfrak{N}}}^{\perp }$.

2. There exists $F\in {{\mathfrak{A}}_{\mathfrak{N}}}^{\perp }$ with $⟨{\mathcal{R}}_{\ast },F⟩\ne 0$ if and only if

   $${\mathcal{R}}_{\ast }\notin {\overline{{\mathfrak{A}}_N}}^{\sigma ({\mathcal{E}}^{\prime },\mathcal{E})}.$$

3. Every invariant test defines a continuous linear functional on the weak Hausdorff quotient

   $${\mathcal{E}}^{\prime }/{\overline{{\mathfrak{A}}_N}}^{\sigma ({\mathcal{E}}^{\prime },\mathcal{E})}.$$

Proof. For (i), prescription independence is exactly the vanishing of the pairing on every generating difference in (eq:complete-ambiguity). Linearity extends it to their span.

If a test annihilates ${\mathfrak{A}}_N$, weak continuity makes it annihilate its weak closure. A nonzero pairing with ${\mathcal{R}}_{\ast }$ therefore implies (eq:detectability).

Conversely, a weak neighbourhood separating ${\mathcal{R}}_{\ast }$ from the closure is determined by finitely many tests $F_1,\dots ,F_m$. Define

Then $T({\mathcal{R}}_{\ast })\notin T({\mathfrak{A}}_N)$; otherwise an ambiguity would match all those pairings and belong to the separating neighbourhood. The linear subspace $T({\mathfrak{A}}_N)\subset {\mathbb{R}}^m$ is closed. Choose $w$ orthogonal to it with $w^{\mathsf{T}}T({\mathcal{R}}_{\ast })\ne 0$. The test $F={\sum }_i{w}_i{F}_i$ has the required properties.

The pairing is continuous in the weak topology and vanishes on the closed subspace. The universal property of the quotient proves (iii).