Proposition: Even exact final recovery does not select a radiation-entropy profile

There are finite sequential isometric processes with the same logical dimension, the same emission-register dimensions, and exact final logical recovery, whose pure-input radiation entropies are, respectively,

after three emissions, for any $k\ge 2$. Both can be appended to the same independent $\mathsf{AL}$-record.

Proof. Use outgoing registers $R_1,R_2\simeq {\mathbb{C}}^k$ and $R_3\simeq C$. In the first process, emit $|0⟩$ into each of $R_1,R_2$, then emit the logical state into $R_3$. In the second process, first create a $k$-dimensional Bell pair between $R_1$ and a retained register, emit its retained half as $R_2$, then emit the logical state as $R_3$. Every step is isometric on the admitted input with fresh pure ancillas. The first radiation prefix is mixed only in the second process. After the second emission the pair is pure inside the radiation, and final recovery is tracing $R_1,R_2$.