Proposition: Discard and nested access

Readout and forgetting of a residual factor equal its partial trace, including every untouched reference:

More generally, any unreported TP local operation on $B$ leaves the already-returned $A$ channel unchanged. For the nested release schedule, ${\epsilon }_{\mathrm{rec}}({{\mathcal{N}}_k}^U)$ cannot increase with $k$.

Proof. Expand an operator as ${\sum }_{i,j}{X}_{ij}\otimes |i⟩⟨j{|}_B$. Readout and discard return ${\sum }_i{X}_{ii}$. For an arbitrary TP residual channel, the adjoint fixes the identity, giving the same accessible reduction.

A later accessible channel has the earlier channel as postprocessing by discard of the additional cells. An earlier decoder can therefore be used at the later cut.