Definition: Finite classical-environment null

Fix:

1. carrier dimension $d$;

2. classical environmental alphabet $\mathcal{E}=\{1,\dots ,L\}$;

3. a finite intervention schedule;

4. allowed instruments, feedback, controller storage, and complete record access.

The joint state is classical–quantum:

An environmental update is specified by CP maps ${{\mathcal{T}}_{e^{\prime }\mid e}}^{(t)}$ satisfying

with

Past-record and setting dependence is represented by the declared controller.

The environment has no additional direct access to the fresh input unless such access is expressly admitted. Its initial joint preparation is independent of the fresh input. No coherence between the classical environmental labels is permitted.