Theorem: Complete recovery programme

A deterministic protocol has

and complete decoded channel

Thus its achieved error is $3(1-P_{\mathrm{all}}(t))/4$. At perfect transmission $P_{\mathrm{all}}(t)=1-s_t$.

Proof. Measure each surviving-set label, consume padding, and retain only actual transmitted rails. Each mode is emitted at most once. Collecting $K$ rails therefore certifies completion and no loss. Only then do observed vacua certify no-emission opportunities.

The complete history is reconstructible. Its collected map is $\sqrt{{\alpha }_h}{B}_{h}J$, with a known unitary $B_h$ on $K$ factors. Apply ${B_h}^{†}$, undo initialization, and consume the $K-1$ ancillary factors. Every corrected success is the identity channel. On incomplete collections, consume everything and prepare $I/2$.

Before an arrival there are at most $K-1$ undecoded rails or one decoded output. The atomic arrival adds $q_{\mathrm{loss}}$ slots. Label and padding consumption leave at most $K$ real rails. The inverse uses these existing slots. Early completion leaves one live output through subsequent arrivals; $q_{\mathrm{loss}}+1\le q_{\mathrm{loss}}+K-1$.

The total success effect is a scalar identity, so fixed-output failure has input-independent total probability. This proves the complete channel, including references.