Theorem: Compact scalar transmission and recovery

For the compact ordered model and decoder ${\mathcal{D}}_{\lambda }$,

The decoded Bell and worst-case entanglement fidelities are at least $1-E$, and

For every desired positive recovery allowance, finite compact ordered source parameters exist with $E$ below that allowance, within the stated growing-resource family.

In particular, finite compact parameters exist for which the identified outgoing scalar packet channel is not entanglement breaking.

Proof. Postcompose Theorem thm:stability with the same complete decoder and use Theorem thm:matchedrecovery. Effect contraction proves the fidelity and witness bounds.

First choose finite $\lambda$ so that $E_{\lambda }$ is small. For that fixed $\lambda$, $\mu$ is finite and positive. Lemma lem:bridge permits finite $L_1,L_2$ making $e_{\mathrm{src}}$ arbitrarily small, while the time slabs remain ordered. If $E<1/2$, the witness is negative.