Joint logical observables

For any logical operator $O\in \mathcal{B}({\mathbb{C}}^d)$, define on the completed accessible sectors

Then

Thus Weyl $X$ and $Z$ coherence is accessed by key-conditioned joint observables, not by local number statistics. On a successful flag, the same expression normalized by that branch weight gives the exact logical expectation. The primary channel remains complete and unconditioned.

The decoder needs one surviving $d$-dimensional quantum logical carrier and classical key/time records. The key can be measured in its label basis and stored classically without destroying the recoverable logical coherence. This is different from measuring the occupied logical radiation in its occupation basis.