Cycle Closure: Exact Potentials, Weighted Defects, and Scale Compatibility

We characterize when local increments on the edges of a finite graph arise from a global potential. Exactness is equivalent to the vanishing of all cycle periods, to vanishing on any cycle-space basis, and to path independence. For positive edge weights, the nearest exact form is obtained by a weighted Poisson problem. If the columns of C form a basis of the cycle space and p = CTω is the period vector, then the squared distance to exactness is distW(ω, im BT)2 = pT(CTW−1C)−1p. This identity has a basis-independent cohomological form and separates data defect from cycle geometry. We also prove stability under perturbations, a directed cycle criterion for dominating potentials, and scale-compatible subdivision laws. Series conductance preserves the full weighted defect under edge subdivision.