Corollary: A proved transmission regime

For every separable reference-packet state $\tau$,

For the comparison channel,

Thus $E_{\lambda }<1/2$ proves a non-entanglement-breaking packet channel.

Proof. Local decoding preserves separability, and a separable decoded two-qubit state has Bell overlap at most $1/2$. For $J_0$, its decoded Bell overlap is at least $1-E_{\lambda }$.