Complete protocols, including off-support inputs

Set

three-slot protocol. Read and consume labels. Retain the first occupied rail through the second arrival. If both early labels are occupied, apply $B_{12}$, read both rails as $t,s$, and consume them. If all three labels are occupied, apply $C_{ts}{R}_3$ to rail three and return it after the final arrival. Otherwise consume all remaining data and prepare $I/2$.

two-slot protocol. The identity $B_{12}F=(H_1{S_1}^{†})H_2$ allows separate early readouts. Consume each early rail before the next arrival. On the all-occupied history apply the same final correction; otherwise prepare $I/2$ after consuming all packets.

two-slot protocol. Read $Z,X,Z$ on the occupied rails, retaining only classical outcomes. On the all-occupied history prepare the logical $X$ eigenstate of negative parity; otherwise prepare $I/2$.

The identities

verify the successful corrected channel. Equivalently, substituting the three logical Pauli representatives into the four early-readout Kraus maps gives the logical Pauli operators at the corrected output. The early pair is $I_4/4$, so all four outcomes have probability $1/4$. The complete decoded channels are $a\mathrm{id}+(1-a)\mathcal{T}$ and $a{\Delta }_X+(1-a)\mathcal{T}$, respectively.

To extend each protocol to every physical packet input, mixed vacuum/occupied histories are failures. Invalid binary packet states are also failures. All such histories are consumed before fixed-output preparation. Cross-history coherence is removed by the complete label-readout instrument.

accounting. Single-qubit gates, CNOT, CZ, allocation, destructive computational readout, classical control, and final return are admitted. Preparing $I/2$ uses one recyclable slot: allocate zero, apply $H$, destructively read a random bit, retain it classically, reallocate zero, conditionally apply $X$, and return. Preparation of an $X$ eigenstate uses zero allocation, conditional $X$, and $H$. These events are included in the ledger; a final direct return is not recorded as a destructive measurement.

The even reference state is locally Clifford-equivalent to Smolin's unlockable state Smolin. Indeed, for $R_0=H{S}^{†}$, conjugation by $Y_R\otimes R_{0,1}\otimes I_2\otimes R_{0,3}$ maps it to $(I+X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes 4})/16$. The unlocking mechanism is inherited.