Complete adaptive histories

At each node $v$ of a finite public-outcome tree, supply Kraus maps

The index $a$ is public; $\mu$ is not accessible unless separately reported. All actual memory, residuals, and failure flags belong to the node interfaces.

Each clock input is a live, unconsumed factor or a new factor returned by an explicitly specified preparation or restoration map. Reusing the spelling of a consumed name does not supply a quantum input.