Lemma 2, Eq.(107): worst-case concurrent rounding

For all natural numbers $n,e,r$, let $G$ have vertex set $V=\{0,\ldots,n-1\}$ and edge set $F=\{0,\ldots,e-1\}$, with distinct endpoints $\ell(a),\rho(a)$ for every edge, and assume that two edge labels with the same endpoints in the same or reversed order must be equal. Assume $r>0$, choose terminal maps $s,t:\{0,\ldots,r-1\}\to V$ with $s_i\ne t_i$ for every index $i$, choose positive natural numbers $m,k$, and choose nonnegative real capacities $c(a)$. For each $i$, let $p_i$ count all choices of a length $d<n$, an injective vertex sequence from $s_i$ to $t_i$, and $d$ edge labels joining consecutive vertices in either orientation, and put $N=\max_i p_i$. Define $W$ to be the total real supremum of all real $z$ for which there exist functions $h_i:F\times\{0,1\}\to\mathbb R$ satisfying $h_i(a,b)\ge0$ for all indices, edges, and direction indices; $\sum_i(h_i(a,0)+h_i(a,1))\le c(a)$ for each edge; $\Delta_{h_i}(v)=0$ for every $v\ne s_i,t_i$; and $z\le\Delta_{h_i}(s_i)$ for every $i$, where $\Delta_h(v)=\sum_{a\in F}\bigl(\mathbf1_{\ell(a)=v}(h(a,0)-h(a,1))+\mathbf1_{\rho(a)=v}(h(a,1)-h(a,0))\bigr)$. Replace each edge $a$ by $L(a)=\lfloor(mkN)c(a)\rfloor_{\mathbb N}$ labeled copies with the same endpoints. Then there exists a packing with a natural count $q_i$ for each terminal index and $q_i$ selected paths from $s_i$ to $t_i$, each having length less than $n$, an injective vertex sequence, and copied edges joining consecutive vertices in either orientation, such that any two distinct selected occurrences share no copied edge, whether their terminal indices agree or differ, and $(mkN)(W-1/k)\le q_i$ for every $i$, with natural numbers interpreted as reals in these inequalities. Terminal pairs may repeat or share vertices. The assumptions force $n\ge2$ but permit $e=0$ and $N=0$; if $N=0$, there are no copied edges, all counts are zero, and every asserted inequality is $0\le0$. A nonpositive left side does not require a positive count.