Theorem: Directional geometric and coarse relays

For every Euclidean unit $v$, duration $\sigma >0$, and source position $x$, there are matched full-tail experiments with source modifications approaching $A=(0,x)$, report events approaching $B=(\sigma ,x+c\sigma v)$, and laws exactly ${\delta }_0,{\delta }_1$ at every scale. Their declared supports converge to the closed segment from $A$ to $B$.

Proof. Choose a long generator $s_n$ with direction within ${\delta }_n$ of $v$, a source site $x_n\to x$, and $k_n$ with $k_n\Delta t_n\to \sigma$. A singleton source payload writes bit $0$ or $1$. Relay the bit along $k_n$ successive $s_n$-edges using the matching phases of Proposition prop:matchings, then read it locally. The constant finite number of preparation/readout ticks adds $O(\Delta t_n)$, which vanishes.

The displacement is $k_n{h}_n{s}_n$, with length between $(1-{\delta }_n)k_n{\epsilon }_n$ and $k_n{\epsilon }_n$, and direction tending to $v$. It tends to $c\sigma v$. The footprints and paths lie within vanishing neighborhoods of the segment, and all source differences lie in a shrinking neighborhood of $A$. The report laws remain deterministic and distinct.