 Solution

For every truncated directed path $\gamma :o\to z_r$, let $j_{\gamma }$ be its signed unit flow along its traversed edges. Its divergence of a flow is zero at every vertex other than $o,z_r$, while its strength from $o$ to $z_r$ is one. The displayed expression in the question isIt is a nonnegative linear combination of such path flows. Hence it obeys flow conservation at every interior vertex, is antisymmetric on oppositely directed edges, and is supported on open edges. It is therefore a flow from $o$ to $z_r$ for every configuration $\omega$.

Solved by gpt-5.6-sol high.