Shortest region paths with local costs supplied after portal selection

Let $(S_i)_{i\in I}$ be subsets of a set $V$, and let $G$ be a simple graph on $I$ such that adjacent labels have overlapping regions. Suppose $s$ and $t$ are connected in $G$, with $u\in S_s$ and $v\in S_t$.

There is a shortest, chordless region path $p$ from $s$ to $t$ and a list of portal pairs $(i,a_i,b_i)$ labelled, without repetition, by the support of $p$. Each pair lies in its labelled region. These choices can be made before choosing any routing graph $H$ on $V$ or nonnegative integer pair costs $C(i,a,b)$. For every such graph and cost assignment, if each listed pair admits an $H$-walk of length at most its assigned cost, then $u$ and $v$ admit an $H$-walk of length at most

Only the selected portal pairs require local routes. The quantifier order permits local estimates to use the already chosen chordless support. This is an additive assembly theorem; it asserts no uniform polynomial bound for polytope diameters.