Reciprocal Convex Costs for Ratio Matching: Axiomatic Characterization

We study ratio-induced mismatch cost functions of the form c(s,o)=JιS(s)/ιO(o) built from positive scale maps ιS:S→R>0 and ιO:O→R>0 and a penalty J:(0,∞)→[0,∞). Assuming inversion symmetry, strict convexity, coercivity, normalization at 1, and a multiplicative d’Alembert identity, we show that f(u):=1+J(eu) is continuous and satisfies the additive d’Alembert equation; hence, by a classical classification theorem, there exists a>0 such that J(x)=cosh(alogx)−1=12xa+x−a−1, x>0. We then analyze the associated argmin mapping over feasible scale sets: existence under explicit subspace-closedness assumptions, an explicit geometric-mean decision geometry for finite dictionaries with stability away from boundaries, exact compositionality for product models, and an optimal sequential mediation principle described by a geometric mean (or its log-space projection when infeasible). The paper is purely mathematical; any semantic interpretation is optional and external to theorems proved here.