Theorem: Formal super-cone cycle construction

Assume Hypothesis hyp:boost. Suppose lossless classical-bit transport is available in both directions between $P,Q$, in one original tick, with $\rho =d(P,Q)>r$. Then an admitted composition has a directed closed chain through one event.

Proof. Put $G=\lfloor \rho /r\rfloor \ge 1$ and define

If $d(x,y)\le r$, the two ceiling terms differ by at most one. Taking the positive part preserves that bound, so $\phi$ is compatible. Also $\phi (P)=G$, $\phi (Q)=0$.

The ${\beta }_{\phi }$-image of the forward transport and the ${\beta }_{-\phi }$-image of the reverse transport both have duration $1-G$. By the closure hypothesis, they are independently realizable in the common substrate.

Starting at $(P,s_0)$, use the forward image, wait at $Q$ for $2(G-1)$ ticks, and use the reverse image. The final event is $(P,s_0)$, because the total duration is

The devices are composed from two different retiming images; this is not the application of one coordinate change to an original network.