Theorem: Conditional boundary identification

Under Specification spec:boundary, the exact operational probabilities determine ${B_{g,A}}^{\Gamma }$. If ${\mathsf{U}}_{\mathcal{C}}$ is additionally established, they determine $[g,A]$ in that coefficient class.

The implication is proved. Existence of a general physical package satisfying Specification spec:boundary is not proved here.

Proof. Invert the response:

Polarization gives $B(f,h)$ for $f,h\in \mathcal{T}$. Boundedness and density extend this uniquely to ${\mathcal{H}}_{\Gamma }$.

If two coefficient pairs produce the same probabilities, the extensions of their forms are equal. Hypothesis ${\mathsf{U}}_{\mathcal{C}}$ then identifies their equivalence classes. The compatibility clause makes the conclusion independent of the permitted data presentations.