Proposition: Residual readout and forgetting

For the declared residual interface,

including untouched references and all surviving receiver factors. Thus residual readout and forgetting leave $\mathcal{N}$ unchanged. The same is true for any unreported TP local residual operation.

Proof. Expand an operator as

Readout and forgetting return ${\sum }_i{X}_{ii}={\mathrm{Tr}}_{B}X$. The general TP assertion follows from the adjoint identity ${\Lambda }_B^{\ast }(I_B)=I_B$.