Remark: The information target matters

Theorem thm:exact concerns identity recovery of the entire code. For the smaller target ${\mathrm{Tr}}_S$ on $C_L\otimes C_S$, the channel $\mathcal{N}={\mathrm{Tr}}_S$ returns $C_L$ exactly while a complementary channel returns the generally nonconstant state on $C_S$. Full-code constancy must not be imposed as a necessary condition for that different task.