Finite-Volume Rigidity and Flux Sectors in a Reciprocal Ratio Gradient Model on Graphs

We study the finite-volume nearest-neighbor energy generated by the symmetric reciprocal-ratio penalty, or equivalently the gradient potential \( V(t)=\cosh t-1 \) in logarithmic variables. Uniform convexity places the model in the standard class of noncompact height theories and yields weighted-Laplacian Hessians, strict convexity on fixed-mean slices, ground-state characterization, and coercive finite-graph bounds. The specific \( \cosh \) form gives several closed-form results. In each cohomology class of oriented edge fields there is a unique minimizing representative, characterized by the nonlinear coclosed equation \( \delta\sinh\omega=0 \), and the sector energy admits a quadratic gap. On twisted discrete tori this representative is explicit, so affine fields are the unique minimizers modulo constants and the finite-volume energy density equals \( \sum_{i=1}^d(\cosh a_i-1) \), independent of torus size. For boxes and discrete tori in \( \mathbb{Z}^d \), the spectral gaps are explicit, giving in \( d=3 \)an \( o(L) \) sufficient condition under which the normalized logarithmic field vanishes in averaged \( L^2 \).