Proposition: Spatial route control is not spacetime control

A route lying exactly on a spatial segment, with endpoints fired at their correct nominal times and no late arrival, need not stay near the intended spacetime segment.

Proof. In one dimension take $0<v<c$, nominal endpoints $(0,0)$ and $(T,vT)$, and intended path $x=vt$. Travel instead at speed $c$ until $q=(vT/c,vT)$, then wait until $T$. The spatial route is exactly $[0,vT]$, and the destination is reached early, not late.

Write $\ell =vT$. The distance in $d_c$ from $q$ to the intended segment is

Indeed minimizing $\max (c|t^{\prime }-\ell /c|,|\ell -v{t}^{\prime }|)$ balances the two terms at $t^{\prime }=2\ell /(c+v)\in [0,T]$. The positive distance does not disappear when the digital mesh tends to zero.