The Cactus Criterion: When Nonlinear Hodge Theory Reduces to Linear on Graphs

Let $G$ be a finite connected simple graph with a chosen orientation of its edges. For the edge potential $\psi (t)=\cosh t-1,$ we minimize ${\sum }_{e\in E^{\to }}\psi (z_e)$ over each affine class $\omega +d{C}^0(G)\subset C^1(G)$. The minimizer is the unique representative satisfying the nonlinear coclosed equation $\delta \sinh z=0,$ and hence defines a nonlinear selector $\text{Picc}:C^1(G)\to C^1(G)\mathrm{.}$ We show that $\text{Picc}$ is real analytic, identify its image as $\text{imop}\text{Picc}=\text{Mcc}=\mathrm{arsinh}(\ker \delta ),$ and compute its differential as a weighted Hodge projector. In particular, $\text{Picc}$ agrees with the ordinary Hodge projector $\text{PiH}$ to first order at the origin, and the first nonlinear correction is cubic. Our main global theorem is a graph-theoretic criterion: for every admissible edge potential – even, $C^2$, strictly convex, and non-quadratic – the associated nonlinear selector coincides with $\text{PiH}$ on all of $C^1(G)$ if and only if $G$ is a cactus graph. Finally, we work out the two-triangle graph, the smallest connected simple obstruction, and record a self-concordant Newton method for computing $\text{Picc}$.