Lemma: Midpoint-event causality

For a fixed-wire experiment at scale $j$, let $e=(s,b)$ and $e^{\prime }=(s^{\prime },b^{\prime })$, $s^{\prime }<s$ excluded, be consecutive participations of one register, with $s^{\prime }>s$. Let $m(b)$ be the midpoint of a pair or the site of a singleton. Then

The midpoint segment is within $O({\epsilon }_j)$ of the declared worldline-closed support.

Proof. The register is static at a site $p$ between its participations. Its deterministic output position from $b$ and input position in $b^{\prime }$ are both $p$. Thus $p\in b\cap b^{\prime }$. Each midpoint is within ${\epsilon }_j/2$ of $p$. The time difference is at least one tick.

The retained dwell segment at $p$ belongs to the declared support. Interpolating the midpoint offsets bounds the midpoint segment's distance from that dwell segment by at most ${\epsilon }_j$.