Conditional shortest-path versus two-path threshold tradeoff

There exist a natural number $N$, an injective placement $x_i\in\mathbb Z^2$ of the vertices $V=\{0,\ldots,N-1\}$ defining a grid network $G$, distinct vertices $a,b\in V$, a finite edge set $H$, three encoded routes $r,s,t$, and a real number $\theta$ with the following properties. The set $H$ is contained in the physical edge set consisting of pairs $(i,j)$ with $i<j$ whose positions have Manhattan distance $1$. Each route $u\in\{r,s,t\}$ has a length $1\le m(u)\le N^2$ and a vertex sequence $(v_0,\ldots,v_{m(u)})$ starting at $a$ and ending at $b$, with distinct canonical consecutive edge pairs belonging to $H$ and with no intermediate vertex equal to either endpoint; additionally, each of these three entire vertex sequences is injective. Write $E(u)$ for its set of canonical consecutive edge pairs. Among all encoded routes satisfying the endpoint, distinct-edge, edge-membership, and intermediate-vertex conditions in $H$—without requiring those other routes to have injective vertex sequences—$r$ has minimum length. After removing $E(r)$ from $H$, there is no encoded route satisfying those same admissibility conditions from $a$ to $b$. Furthermore, $E(s)\cap E(t)=\varnothing$, $m(r)<m(s)$, $m(r)<m(t)$, and $0<\theta<1$. For every real $q$ with $0<q<\theta$,

and for every real $q$ with $\theta<q\le1$,

The minimum-length condition does not require $r$ to be the unique shortest route, and no comparison is asserted at $q=0$ or $q=\theta$. The distinct endpoints ensure these witnesses cannot have $N=0$ or $N=1$.