Proposition: Diagonal-preserving complete processes have classical record realizations

Fix a basis on every retained carrier. Suppose initial states are diagonal and every selective map of every admitted instrument sends diagonal states to diagonal states. Suppose controls use only declared classical records.

Then every complete record law has a classical stochastic realization. If the complete input-bearing state on the storage cut has at most $K$ basis values, the classical cut factorization applies with that $K$.

Proof. For a selective map ${\mathcal{I}}_o$, define

These numbers are nonnegative. The complete instrument condition gives

On diagonal populations the selective map is precisely multiplication by this substochastic matrix. Products of the matrices reproduce the unnormalized history populations and probabilities. Induct through the finite controlled experiment.