Theorem: Directional matched relay limits

For every $x\in {\mathbb{R}}^2$, Euclidean unit vector $v$, and $u>0$, there are matched discrete relay pairs with sender events tending to $A=(0,x)$ and readout events tending to $B=(u,x+cuv)$. Their complete readout laws are ${\delta }_0,{\delta }_1$ at every sufficiently fine scale. The sender modification footprints shrink to $A$.

Proof. Choose hubs within ${\rho }_n$ of $x$ and $x+cuv$, and supply the bit at the first hub at tick zero. Their separation $L_n$ satisfies $L_n\to cu$. Use the common route and channels of Theorem thm:identity. The two runs differ only in the bit supplied at the sender.

Let the readout occur at the first delivered hub-memory event. Its elapsed time is bounded above by ${\lambda }_n{L}_n/c+2U_n+h_n$, which tends to $u$. The exact physical cone gives the lower bound $L_n/c$, which also tends to $u$. Thus the readout events converge to $B$. Reading the transported bit gives ${\delta }_0,{\delta }_1$. Preparation and any charged local readout occupy shrinking blocks, so the sender modification footprint tends to $A$.