Finite-Volume Stability 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 and its logarithmic representation as a gradient model with potential V(t)=cosht−1. We separate general convex structure from formulas specific to this hyperbolic potential. For an oriented nearest-neighbor energy, the gradient form gives a weighted-Laplacian Hessian, while a global lower-curvature bound W″≥κ>0 yields strong convexity after removal of the constant mode, spectral-gap coercivity, unique minimizers on fixed-mean slices, unique minimizing edge representatives in fixed-flux sectors, and quadratic stability gaps. On coordinate-constant twisted tori, strict convexity already forces affine minimizers and gives the exact energy density ∑iW(ai). What is specific to the reciprocal-ratio model is the elementary hyperbolic form W′=sinh, W″=cosh: the sector equation becomes δsinhω=0, the cycle calculation is explicit, and the twisted energy density is ∑i(coshai−1). For boxes and discrete tori in Zd, explicit spectral gaps yield, in d=3, an o(L) sufficient condition for the normalized logarithmic field to vanish in averaged L2. Our analysis is carried out in finite volume on fixed graphs with prescribed boundary, mean, or flux data. At positive temperature, we formulate the corresponding height Gibbs measures on mean-fixed slices within each flux sector and describe explicitly how they transform under a change in sector representative.