Theorem: Exact schedule locality and metric cone

Let two semantically admissible runs have the same schedule over $[t_0,T]$.

If their initial snapshots on $N_{t_0}(Y)$ agree and their block channels agree whenever $b\cap N_{s+1}(Y)\ne \varnothing$, then their terminal snapshots on $Y$ agree. Equivalently,

In particular, for $Y=\{A\}$, agreement of the initial snapshots on $B(A,r(T-t_0))$ and agreement of every inner block channel imply

Every common readout at $(A,T)$ then has the same outcome law. These conclusions are reference preserving in the sense of Lemma lem:comparison.

Proof. Apply Lemma lem:onelayer backward through the ticks. Definition def:schedulepast gives (eq:adjointcone). At each step the reduced channel depends only on the blocks meeting the next scheduled slice, so the two runs have the same reduced evolution through the scheduled past.

For the metric statement, use (eq:metricenvelope) and induct forward using Lemma lem:dod. The initial snapshots agree on $I_{t_0}$, and the common inner channels give the same $K_s$ at every step. Thus the snapshots agree on $I_T=\{A\}$. Readout laws are functions of that terminal snapshot.