Why the failure merger and output boundary matter

The raw even-parity probabilities are

They generally vary under the collective gauge phase. The invariant boundary is therefore the declared coarse record $F,+,-$, with no exported raw failure log and no returned signal qubit.

The merger is an explicit classical coarse-graining: two selective maps are added, and no branch is removed. It is not normalized postselection. If a further accessible apparatus output retains the raw distinction, that output must be included, and the larger boundary need not be invariant.

The protocol is informative about $\cos \theta$, not globally faithful to the real number $I_{\mathrm{rel}}$. Periodicity and evenness produce aliases. Physical informativeness requires admissible invariant inputs with different cosines.

The circuit is a constructed finite channel at its abstract input interface. Its application to the convolution model remains conditional: no second coefficient, acquisition map, or whole-algebra matching certificate has been supplied.