Abstract

We give a self-contained finite-dimensional theory of access to interference in nested-measurement circuits with undisturbed records. Internal recording is coherent copying, and established records are subsequently used only as copy sources or computational-basis controls. For a record set $J$, the discrepancy between a coherent state $\rho$ and its record-dephased model is the traceless Hermitian operator $X_J=\rho -{\Delta }_J\rho$. For every quantum channel $\Lambda$, agreement on all available effects is equivalent to $\Lambda (X_J)=0$, and the maximal probability bias is $\frac{1}{2}\Vert \Lambda (X_J){\Vert }_1$. Subsystem and algebraic interfaces follow as special cases.

A branch cross-operator calculus separates perfect-copy screening, residual distinguishability, operator-level cancellation, and question-specific agreement. Pairwise visibility is the fidelity of the inaccessible branch states. This gives exact product laws for independent noisy fragments and all-or-nothing behavior for correlated phase records. Operator-valued Möbius elements distinguish state, interface, and question orders. State and subsystem supports are down-closed; arbitrary channel supports need not be, and every prescribed support family is realizable by a binary measurement channel on a full-support tower. For the symmetric Frauchiger–Renner state, order two belongs to Wigner's question, not to the state.

We prove sequential agreement on the record-invariant algebra, diamond-norm perturbation bounds, a same-data counterexample to scalar visibility characterizations, and the codimension-two geometry of the complex-amplitude Frauchiger–Renner order-two stratum. The read-chain bias $2^{-c}$ is established for single product effects, while local parity measurements attain the unrestricted bias $1/2$. Standard hypothesis-testing theorems give a trace-distance Chernoff sandwich and a Pinsker lower bound for the Stein exponent. An embedded, hash-identifiable regression program is supplied without claiming an execution record.