Proposition: Same record, opposite recovery

Every fixed metric-measure or marked-tree record admits independent completions with optimal recovery errors $0$ and $1-d^{-2}$, on the same logical and output dimensions. For every pure logical input, selected accessible entropy can be zero in both completions.

Proof. Take $R\cong B\cong C$, and define

The first accessible channel is identity and is exactly recoverable. The second is replacement. Every decoder of a replacement is a replacement, whose Bell overlap is $d^{-2}$. Thus every decoder has error at least $1-d^{-2}$. Preparation of $I_C/d$ attains the bound, as in Lemma lem:depolarizing.

For pure inputs, the accessible output in either completion is pure, so its entropy is zero. Neither isometry changes the fixed combinatorial or metric record.