Flow decomposition theorem

Fix a network of $m$ directed arcs on nodes $\{0,\dots,n-1\}$ with no self-loops, and a vector $f:\{0,\dots,m-1\}\to\mathbb{R}$ with $f\ge 0$ componentwise, $Nf=0$ where $N$ is the incidence matrix ($N_{ik}=[\mathrm{tail}_k=i]-[\mathrm{head}_k=i]$; i.e. $f$ is a circulation), and $f\ne 0$. The conclusion is a conjunction. (a) There exist $\kappa\in\mathbb{N}$, step-lists $\mathrm{cyc}_1,\dots,\mathrm{cyc}_\kappa$ and reals $a_1,\dots,a_\kappa$ such that: each $\mathrm{cyc}_i$ is a cycle at some vertex $v$ — a nonempty walk from $v$ back to $v$ (each step = arc index + direction flag) whose visited-node list with its final entry removed has no repeats, and whose arc indices are pairwise distinct (an arc cannot be reused even in the other direction); every step of every $\mathrm{cyc}_i$ is a forward traversal; each $a_i>0$; and $f_e=\sum_{i=1}^{\kappa}a_i\,\chi^{(i)}_e$ for every arc $e$, where $\chi^{(i)}_e=[(e,\text{fwd})\in\mathrm{cyc}_i]-[(e,\text{bwd})\in\mathrm{cyc}_i]$; since all steps are forward with distinct arcs, $\chi^{(i)}$ is the $0/1$ indicator of the arcs of $\mathrm{cyc}_i$. (b) If moreover every $f_e$ equals an integer, the same decomposition exists with strictly positive integer coefficients $a_i$. There is no bound claimed on the number $\kappa$ of cycles.