Lemma 1, Eq.(57): path rounding

For all natural numbers $n,e$, 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 each edge. Assume that whenever two edge labels have the same endpoints in the same or reversed order, those edge labels are equal. Choose vertices $s,t\in V$ with $s\ne t$, positive natural numbers $m,k$, and a function $c:F\to\mathbb R$ with $c(a)\ge0$ for every edge. Let $N$ count all paths obtained by choosing a length $d<n$, an injective sequence $v_0,\ldots,v_d$ with $v_0=s$, $v_d=t$, and $d$ edge labels joining consecutive vertices in either orientation. Let $F_*$ be the total real supremum of the set of source outflows $\Delta_h(s)$ over functions $h:F\times\{0,1\}\to\mathbb R$ satisfying $h(a,b)\ge0$, $h(a,0)+h(a,1)\le c(a)$ for every edge, and $\Delta_h(v)=0$ for every $v\ne s,t$, 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 for the single terminal pair $(s,t)$: it has a natural number $q$ of selected paths, each with length less than $n$, injective vertex sequence from $s$ to $t$, and copied edges joining consecutive vertices in either orientation, such that distinct selected paths share no copied edge, and $(mkN)(F_*-1/k)\le q$, with natural numbers interpreted as reals in this inequality. The assumptions force $n\ge2$ but allow $e=0$ and allow $N=0$; if $N=0$, every $L(a)=0$, the only possible count is $q=0$, and the asserted bound is $0\le0$. The inequality itself does not require a positive count when its left side is nonpositive.