Manuscript VII-E@1 · VII-E.tex · theorem · b713cf3c

Theorem: Prescribed schedule locality

Prescribed kinematics and schedule dependence › Exact backwards support · lines 673–706 · proof included or linked

VII-E@1#thm:locality, VII-E@1#eq:support-bound, VII-E@1#eq:metric-bound

For $A \in A (Y)$ and $0 \leq s \leq t$ ,

supp ((Φ_{s + 1}^{*} \circ \dots \circ Φ_{t}^{*}) (A)) \subseteq N_{s, t} (Y),

and

N_{s, t} (Y) \subseteq {x : d (x, Y) \leq (t - s) Δ} .

The output reduction on $Y$ depends on the state at tick $s$ only through its reduction to $N_{s, t} (Y)$ .

Proof. Repeated application of lem:one-layer in reverse tick order proves eq:support-bound.

If $x \in N_{s, t} (Y)$ , its block meets $N_{s + 1, t} (Y)$ at some $y$ . Hence

d (x, Y) \leq d (x, y) + d (y, Y) \leq Δ + (t - s - 1) Δ.

Induction proves eq:metric-bound.

Finally, every output observable on $Y$ pulls back to an observable on $N_{s, t} (Y)$ . Equal input reductions there give equal expectations for all output observables, and therefore equal output reductions.

Displayed formulas in this passage

VII-E@1#eq:support-bound · line 676
$supp ((Φ_{s + 1}^{*} \circ \dots \circ Φ_{t}^{*}) (A)) \subseteq N_{s, t} (Y),$
VII-E@1#eq:metric-bound · line 682
$N_{s, t} (Y) \subseteq {x : d (x, Y) \leq (t - s) Δ} .$
unlabeled · line 697
$d (x, Y) \leq d (x, y) + d (y, Y) \leq Δ + (t - s - 1) Δ.$

Theorem: Prescribed schedule locality

Displayed formulas in this passage

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