Theorem: Diagonal hybrid classical realization

Suppose every $H_s=0$, preparations are diagonal in the node and scar bases, and each instrument branch acts on populations by a nonnegative substochastic matrix ${A_r}^{(s)}$, with ${\sum }_r{A}_r^{(s)}$ stochastic. Then the complete feedback history has an exact classical controlled Markov realization on node–scar pairs.

Within a sector,

Node dephasing does not affect these populations.

Proof. For $L=|i⟩⟨j|$, the gain term transfers population from $j$ to $i$, and its anticommutator removes it from $j$. The node-projector dissipators have zero diagonal contribution. Diagonal states remain diagonal.

Between observations, populations evolve by stochastic matrices. At an instrument step, the branch populations are ${A_r}^{(s)}p$, followed by the classical update $f(s,r)$. Induction gives the same unnormalized histories and probabilities.