Global apparatus is the surviving implementation boundary

The normal instruments are explicitly defined on $\mathcal{B}({\mathcal{H}}_i)$ or its trace-class predual, and admit a qubit controlled-unitary dilation. Those facts are proved.

Compact support of $f$ or $J_{\psi }$ identifies a smearing in a classical or form-valued functional. It does not show that the resulting controlled unitary is generated by a compact relativistic system–probe interaction. Likewise, finite active-mode support does not mean spacetime-local apparatus access.

No external background-anomaly calibration or two-probe mixed-mode disturbance is needed for any of the three proofs. Those experiments concern different measured quantities.