Complete preparation

Fix

The variable input is the whole diary $L$. An arbitrary external reference is untouched. For Choi tests use $Q\simeq L$.

Before the unknown input is used, prepare

The preparation is independent of $L$ and its references. The vault is $LHF$, of dimension

Every purifier is retained: $E$ is accessible, and $Q$ belongs to the reference side of the test, not to either receiver mouth.

A fixed encoding unitary can be absorbed into a known, input-independent vault unitary $U$. For the random model, $U$ is one Haar sample on ${\mathbb{C}}^D$. It is sampled once, not independently at every cut.