Example: Correlated phase records

Let a retained perfect root control the two residual environment states

Prepare the environment with residual gates, then apply a record-controlled $Z$ to one fragment. If $k<c$ environment fragments are inaccessible, their two reduced states coincide, so $v=1$. If all $c$ fragments are inaccessible, the two environment states are orthogonal, so $v=0$. For $k=0$, the inaccessible environment is the one-dimensional trivial factor.

This differs sharply from independent redundant fragments. The relationship to environment-based classicalization Zurek2009 is architectural: fidelity measures information in the inaccessible remainder, not merely the number of fragments.