Stability region for the reversible four-cycle flow network ($\max ({\rho }_1,{\rho }_3)+\max ({\rho }_2,{\rho }_4)<1$)

Group the normalizing sum by its odd and even flow counts. The coefficient sums are $h_p({\rho }_1,{\rho }_3)={\sum }_{i=0}^p{\rho }_1^i{\rho }_3^{p-i}$ and the analogous $h_q$ for even routes. If $a$ and $b$ are the two maximum loads, then $a^p\le h_p\le (p+1)a^p$ and similarly for $b$. The generating identity ${\sum }_{p,q}\left(\genfrac{}{}{0ex}{}{p+q}{p}\right)a^p{b}^q=(1-a-b{)}^{-1}$ proves necessity and sufficiency of the displayed strict inequality. It is equivalent to strictly subunit offered load on each of the four resources.