An executable complete decoder

On valid successful binary words, the two key bits and the ciphertext data bit implement recovery by:

1. controlled $X$ from key bit $a$ to the data bit;

2. controlled $Z$ from key bit $b$ to the data bit;

3. return the data bit and discard the count, occupancy, and key metadata.

The order applies $Z^b{X}^a={P_{ab}}^{†}$.

For every invalid word or vacuum ciphertext, swap the data bit with a fresh independent qubit prepared in $I_2/2$, then discard the other systems. This gives the failure output $I_2/2$.

The validity projector is diagonal and depends on count ranges and payload occupancy, not on the successful ciphertext data bit. A projector-controlled swap with the fresh qubit is generated by

with pulse area $\pi$. This is a specified bounded multi-register control. It is not claimed to be a free single-qubit operation or an unpriced efficient compilation into a more restrictive gate library.

All metadata is discarded without logical destructive readout. On arbitrary receiver-register inputs this implements the complete extension of $\mathcal{D}$: invalid words are consumed and replaced by $I_2/2$.

Let the receiver decoder's fixed fresh-bath preparation error be $e_D$, and let its bounded generator deviations have total integral $I_D$. Put

This is a complete-channel certificate, including invalid inputs and references.