Proposition: Operator balance and resource balance

For declared Hermitian observables $H_g$, preservation of their expected value for every hybrid input is equivalent to

The branch intertwinings $H_h{K}_{h|g,\nu }=K_{h|g,\nu }{H}_g$ are sufficient. A uniform operator defect at most $ϵ$ gives a one-step expectation error at most $ϵ$, and at most $nϵ$ after $n$ steps for one common observable.

Separately, if a permission function and edge costs satisfy

on every supported branch, then each history obeys

Proof. Pair ${\mathcal{F}}^{\ast }H-H$ with every state supported in one block. The branch sufficiency follows by summing $K^{†}{H}_{h}K=K^{†}K{H}_g$. For a normalized hybrid state, the absolute defect expectation is bounded by the largest block operator norm. Iteration proves the accumulated bound. The resource identity telescopes pathwise.