Theorem: Spectator consistency for stochastic histories

Under Assumption ass:blind and (eq:block-extension), extend every branch by ${{\Psi }^{(t)}}_{h|g}\otimes {\mathrm{id}}_N$. Then, for every finite history,

Consequently:

1. the complete geometric history distributions agree;

2. the reduced history–matter outputs agree;

3. on every supported history, the conditional original-matter reductions agree;

4. the nonselective spectator marginal is unchanged.

Proof. Apply Lemma lem:partial-trace to each map in (eq:path-block). This proves (eq:path-spectator) without normalizing any branch. Taking traces proves (i); taking the direct sum proves (ii); division by a common positive trace proves (iii).

For (iv), the sum of all branches of a one-step transition is trace preserving on each input label. Pair the spectator reduction with $B_N$. Its expectation after the step is

Iteration proves nonselective invariance.