Theorem: History-sensitive observation

Suppose observation spaces satisfy the branchwise closure

Initial agreement on ${⨁}_g{\mathcal{O}}_{0,g}$ implies:

1. equality of all geometric history probabilities;

2. agreement of every unnormalized history block on its declared terminal observation space;

3. agreement of normalized conditional expectations on every positive-probability history.

Proof. Pull a terminal $B\in {\mathcal{O}}_{n,g_n}$ backward along its fixed history. Repeated application of (eq:history-closure) places the pullback in ${\mathcal{O}}_{0,g_0}$. Pairing proves (ii); $B=I$ proves (i). Divide by the same positive probability for (iii).