Residual instruments

The process-level comparisons recorded by Cheung in Psy02-A, Psy02-B, and En01-B distinguish sequential instruments, complete-record overlap, and residual approximation CheungPsy02A,CheungPsy02B,CheungEn01B. Equal complete classical records for a declared intervention family do not identify its hidden residual mechanism, even when every instrument is complete. Equality for one family does not imply equality after enlarging that family. Conversely, if matched accessible states evolve by the same history-dependent channels and full instruments, induction gives equal records. The precise finite statement is proved next; no conformational model or apparatus calibration is imported.

Fix a finite protocol, including all settings, failure outcomes, stopping decisions, and retained classical records. Let the unnormalized accessible branch states satisfy

where ${\Phi }_h$ is a channel and $\{{\mathcal{K}}_{w|h}{\}}_w$ a complete instrument. The declared accessible state is assumed sufficient for this continuation.

Two models with the same initial accessible state and the same ${\Phi }_h,{\mathcal{K}}_{w|h}$ after every allowed history have the same complete record law.

For any two complete laws $P,Q$, a randomized test $0\le \phi \le 1$, with $\alpha =E_P\phi$ and $\beta =E_Q(1-\phi )$, satisfies

If $P=Q$, every test has $\alpha +\beta =1$. No sample size distinguishes an identical null–alternative pair under the declared repeated-run law.