Proposition: The stated equal-time separation need not exist

TRANSPORT's lane clause requiring equal-time meetings to be separated by $2\nu$, while each spatial offset has norm at most $\nu$, is false in general, including $D=1$.

Proof. Take three simultaneous singleton events at sites $-h_j,0,h_j$, each with one register. Then $B_{\xi }=M_{\xi }=3$ and the source's prescribed offset budget is $\nu ={\epsilon }_j/360$. Choose a nested source scale with $L_j>360$, so $h_j<\nu$. For $g=\mathrm{id}$, every adjusted position lies in $[-h_j-\nu ,h_j+\nu ]$, an interval of length $2h_j+2\nu <4\nu$. Three points with pairwise separation at least $2\nu$ need a span at least $4\nu$. Such offsets do not exist.