Theorem: Prepared-state turnover

For $K=5,c=1$, deterministic emission, initial state $V_L|0⟩$, and extraction of $Q_1,\dots ,Q_5$ in order, apply ${\mathrm{CZ}}_{4,5}$ between emissions three and four. At perfect transmission,

The third prefix retains Theorem thm:benchmark with width $q_A$ and $p={\gamma }_{1,5}{\gamma }_{2,4}{\gamma }_{3,3}$.

Proof. The first three channels are the benchmark channels with known descending mode labels. The remaining Bell-partner reduction is $I_4/4$, unchanged by the controlled $Z$. After the fourth extraction it is $I_2/2$; after the fifth the core is one-dimensional. No exceptional drain operation is used.