Joint residuals and coherence

The two terms in (eq:residualembedding) are coherent parts of one message-bearing residual branch. Tracing missed modes gives memory leakage $c_{d}q(T)$; tracing memory gives missed-field leakage $c_d{\ell }_N(T)$. The joint result is proved directly, not inferred from those marginals.

In general, individual residual marginals can be constant while their joint channel carries information. For a classical bit $a$, output a uniform random bit $k$ and $k\oplus a$ to two registers. Both marginals are uniform, but the joint output reveals $a$. Prepending code dephasing turns this into a channel counterexample to marginal-based joint constancy. This illustrates the supplied Hawk02-A warning CheungHawk02A ({prop:jointresidual}).

Later collectors and decoders can add residual systems. Equation (eq:packeterrors) applies precisely to (eq:AB), before such additions.