Introduction and scope

Nested-observer experiments distinguish two statistical descriptions of the same recorded quantum process: a coherent description of the entire laboratory and a model in which selected records have been dephased. The Frauchiger–Renner experiment is a prominent example FrauchigerRenner2018. The present question is operational: which accessible measurements can distinguish these descriptions?

This article consolidates the subsystem and channel developments of two earlier drafts DraftSubsystem,DraftChannel. All definitions and proofs needed for the finite-dimensional results are included here. There is no dependence on an unproved companion theorem.

The organizing chain is

Its three stages describe interference content, accessible interference, and implementable probability bias, respectively.

The channel-interface theorem is the standard operational characterization of trace distance, applied to a structured family of state differences. We do not claim a new proof of the Born rule or a new state-discrimination principle. The circuit-specific results concern record anchoring, copy screening, cross-operator cancellation, Möbius localization, and the consequences of restricting the interface.

Several distinctions are essential.

1. Internal recording is a coherent isometry, not destructive readout. A nonselective dephasing channel is obtained by discarding a fresh coherent copy.

2. Agreement of one effect is weaker than agreement of every effect on a subsystem or after a channel.

3. The copy budget is necessary, not sufficient, for detection. Relative cross-operator phases are not determined by scalar visibilities.

4. Classicality order is explicitly indexed by the object being tested: state, interface, or question.

5. Exact vanishing is not generally stable under arbitrary perturbations. Quantitative smallness and thresholded support are stable in the senses proved below.

and mathematical boundary. The Born rule and the completely positive trace-preserving (CPTP) channel formalism are assumed. The dephased model is a specified statistical comparator, not an additional physical collapse mechanism. The probability calculations for the Frauchiger–Renner state do not reproduce, resolve, or replace the original theorem concerning agents' reasoning assumptions. No claim is made about destructive-primitive uniqueness, quantum field theory, gravitational coefficient admission, or a higher-order protocol calculus.

The new finite-dimensional statements are proved below. The asymptotic quantum Chernoff and Stein theorems are standard external results, invoked with their assumptions and references in Section sec:testing.