Theorem: The strong closure also excludes ordinary relays

Hypothesis hyp:boost and E7(a) are incompatible with any bidirectional lossless one-tick transport between distinct sites, even when their distance is at most $r$.

Proof. Let $P\ne Q$ be such sites. Define $\phi (P)=1$ and $\phi (x)=0$ for $x\ne P$. This function is $r$-compatible for every metric because every difference of values is at most one.

The ${\beta }_{\phi }$-image of the $P$-to-$Q$ step has duration zero. The ${\beta }_{-\phi }$-image of the $Q$-to-$P$ step also has duration zero. Unrestricted composition makes $(P,s)\to (Q,s)\to (P,s)$ an admitted dependency cycle. Proposition prop:cyclecontradiction contradicts E7(a).