Manuscript VII-J-ST@1 · VII-J-ST.tex · corollary · f1f1d82c

Corollary: Partitioned metric envelope

The complete timed network and exact scheduled support · lines 588–614 · proof included or linked

VII-J-ST@1#cor:metric-envelope

For layers $Φ_{j} = ⨂_{B \in B_{j}} Φ_{j, B}$ on a finite metric site set $(X, d_{comp})$ , assume $diam B \leq Δ$ . Define backward sets by

N_{t, t} (Y) = Y, N_{s, t} (Y) = B \in B_{s + 1} B \cap N_{s + 1, t} (Y) \neq = \emptyset ⋃ B .

Then

supp (Φ_{s + 1}^{*} \dots Φ_{t}^{*} (O_{Y})) \subseteq N_{s, t} (Y), N_{s, t} (Y) \subseteq {x : d_{comp} (x, Y) \leq (t - s) Δ} .

Proof. The support assertion is Theorem thm:schedule. Each backward layer adds at most one block diameter. Apply the triangle inequality inductively.

Displayed formulas in this passage

unlabeled · line 595
$N_{t, t} (Y) = Y, N_{s, t} (Y) = B \in B_{s + 1} B \cap N_{s + 1, t} (Y) \neq = \emptyset ⋃ B .$
unlabeled · line 602
$supp (Φ_{s + 1}^{*} \dots Φ_{t}^{*} (O_{Y})) \subseteq N_{s, t} (Y), N_{s, t} (Y) \subseteq {x : d_{comp} (x, Y) \leq (t - s) Δ} .$