Corollary: Implementation agreement through histories

Suppose both implementations satisfy the same branchwise inclusions (eq:time-branch-inclusion), and their branch adjoints agree on each ${\mathcal{O}}_{t,h}$. Starting from the same input, their complete histories have equal probabilities and equal accessible history-conditioned residual expectations.

Proof. In the backward expectation formula from Theorem thm:histories, replace the final adjoint by the other implementation's adjoint. The inclusion keeps its image in the previous observation space, where the next replacement is valid. Continue backwards to the initial state.