The guard and two different location maps

Let $g_{0^{-}}$ and $g_{0^{+}}$ be the same limiting marking equipped with the incoming and outgoing one-sided data. The geometric singular-intersection map is the identity on $\{T=T_{\ast },x=o\}$.

The abstract matter-name location transport is also the identity on persistent cell labels:

These are different typed maps. For example, $e_i$ denotes $[P,A_i]$ before the event and $[B_i,A_i]$ afterward. Preserving that label does not prescribe a trajectory for a material point.

Choose integers $d\ge 1$, $r\ge 1$, and write

The exact forward guard is

It changes $m$ to one and $\nu$ to $\nu -1$, retaining the packet, locations, and internal matter state.