Theorem: Conditional acquisition of the named physical observable

Suppose a physical package satisfying Definition def:physical-package is supplied. Then the records of its complete acquisition instrument estimate

with Theorem thm:calibrated, using

The resulting radius is

If the additional competing-family package has a positive complete-record margin, Theorem thm:competition supplies the corresponding physical discrimination certificate.

The theorem identifies one promised source observable. It does not reconstruct the source quantum state, establish a Galactic source law, or validate its own physical-package hypotheses.

Proof. For each source state,

by the complete channel comparison. Trace-distance contraction under the CPTP map ${\mathcal{A}}_{\eta }$ gives

Their triangle inequality bounds the distance from the ideal record law by $e_{\mathrm{acq}}+e_{\mathrm{src}}$. For classical outputs that distance is total variation. The source-observable identity follows from the two template normalizations in (P1). The assumptions now match those of Theorem thm:calibrated, proving (eq:physical-radius).

The complete-record comparison and statistical proof apply to each supplied competing family in exactly the same way. Theorem thm:competition then supplies the conditional discrimination conclusion.