Cycle-Isotropy Conditions for Local Minima in Vector-Valued Concave Network Flows

We study a finite network in which each edge carries a vector flow q_e in R^d, conservation is linear, and the edge cost is a concave radial power a_e ||q_e||^alpha with 0 = 2-alpha, so any nontrivial active cycle requires at least two flow dimensions. In two dimensions an equilateral three-cycle with directions separated by 120 degrees gives a strict cyclic local minimum. For a graph consisting entirely of a single cycle, positive definiteness of the cycle matrix together with first-order stationarity is sufficient for strict local minimality on the smooth active stratum. The result identifies a geometric mechanism by which vector dimension can stabilize cyclic support under concave economies-of-scale costs.