Germs and two different transports

Let $g_0^{-}$ be the limiting slice at $s=0$ equipped with the incoming one-sided formulas and labels, and let $g_0^{+}$ be the same limiting slice equipped with the outgoing one-sided formulas and labels. The additional outgoing cells are inactive germ slots at $s=0$.

The geometric particle realization transport is

with both points equal to $(0,0,0)$ on the fixed worldline.

The matter-name transport is a function on cell labels. Retain the labels

and the six exterior labels. Define

for every incoming label $\ell$. The geometric realizations of some persistent labels change: for example, $e_i$ is $[P,A_i]$ before the event and $[B_i,A_i]$ afterward. Cell-label transport is abstract location transport, not a trajectory for a material point.

Thus particle realization transport and matter-name transport are different typed maps, even though both preserve the particle-cell label $P$.