Manuscript En05-B@1 · En05-B.tex · lemma · 8b626460

Lemma: Exact physical path containment

Euclidean convergence of both past constructions · lines 856–875 · proof included or linked

En05-B@1#lem:physicalcone

Every scheduled path from $(x, s)$ to $(y, T)$ satisfies

∥ z_{n} (y) - z_{n} (x) ∥_{2} \leq c (T - s) h_{n} .

Consequently

J_{n, sch}^{-} (q) \subseteq J_{n, met}^{-} (q) \subseteq J_{c}^{-} (q) .

Proof. Every transition stays in one block and crosses graph distance at most one. Its physical displacement is at most $a_{n} = c h_{n}$ . Sum the displacement bounds. The metric inclusion follows from the same graph-distance bound.

Displayed formulas in this passage

unlabeled · line 859
$∥ z_{n} (y) - z_{n} (x) ∥_{2} \leq c (T - s) h_{n} .$
unlabeled · line 863
$J_{n, sch}^{-} (q) \subseteq J_{n, met}^{-} (q) \subseteq J_{c}^{-} (q) .$

Nearby in this section

section: Euclidean convergence of both past constructions
theorem: Theorem: Uniform local event and cone convergence