Theorem: Uniform local event and cone convergence

The embedded events are locally dense in ${\mathbb{R}}^{1,2}$. For every compact source set and compact terminal set, the distance functions to both $J_{n,\mathrm{met}}^{-}(q)$ and $J_{n,\mathrm{sch}}^{-}(q)$ converge uniformly to the distance function to $J_c^{-}(q)$, uniformly over embedded terminal events in that terminal set.

The assertion uses full past sets and distance functions, not intersections with arbitrary compact sets.

Proof. A spatial point has a hub within ${\rho }_n$, and a time has a tick within $h_n$. This proves event approximation.

Fix an embedded terminal $q$, and a point $p=(t,x)\in J_c^{-}(q)$ with $0\le t_q-t\le H$. Choose a hub $H_y$ within ${\rho }_n$ of $x$.

For the metric past, Theorem thm:metric gives

Put

Then $a_n{d}_n(H_y,A)\le c(t_q-s{h}_n)$, so $(s{h}_n,z_n(H_y))\in J_{n,\mathrm{met}}^{-}(q)$. Its distance from $p$ is at most

which tends to zero uniformly for bounded $H$.

For the scheduled past, Theorem thm:routing gives a scheduled route of duration at most

Choose an earlier source tick using

in the preceding floor formula. The route then finishes no later than tick $T$. Waiting at the terminal supplies a path ending exactly there. The resulting approximation distance is at most

By Lemma lem:physicalcone, both discrete sets are contained in the continuum cone. Their distance functions are therefore no smaller than the continuum distance function.

For source and terminal points in fixed compact sets, a nearest point on the closed continuum cone lies in a uniformly bounded region: the apex itself gives a uniform distance bound. The elapsed-time parameter $H$ for those nearest points is therefore uniformly bounded. Apply the preceding approximations to those points. This supplies the reverse distance-function inequality with an error tending uniformly to zero.