Definition: Matched refinement of frozen operational data

Fix a complete finite target channel $\Phi :\mathcal{B}({\mathcal{H}}_{\mathrm{in}})\to {\mathcal{A}}_{\mathrm{out}}$, including its preparation, calibration, clock Enter instrument, all retained records, residual ownership, and any acceptance or rejection rule.

An admissible refinement supplies, for each scale $j$:

1. an implemented channel ${\tilde{\Phi }}_j$ on these same external comparison interfaces;

2. an ideal factor chain on compatible intermediate interfaces whose complete composition is exactly $\Phi$;

3. corresponding implemented factors, compared with those ideal factors on the same intermediate interfaces;

4. an explicit assignment of factor errors to encoding, flow, event, classical record processing, and decoding;

5. any needed padding, identification, or finite classical coarse-graining maps, included as factors rather than suppressed.

Intermediate physical registers may depend on $j$. The external comparison algebra does not.