Proposition: Complete prefix channels

For $t=0,1,2$,

where $I_{2^0}=1$. At $t=3$, the channel is (eq:N3), equivalently (eq:normal). These are equalities on all operators, including off-diagonal message matrix units.

Proof. The first two final radiation marginals follow from (eq:N3). Future gates act only on the remnant, so they cannot change already emitted marginals. In particular, the remaining ${\mathrm{CZ}}_{2,3}$ after the first emission does not alter the first carrier's law or reference correlations. At $t=0$, there is no accessible quantum factor.