Reciprocal Convex Costs for Ratio Matching: Functional-Equation Characterization and Decision Geometry

Abstract

We study ratio-induced mismatch costs of the form $c(s,o)=J({\hat{I}}_S^1(s)/{\hat{I}}_O^1(o))$, built from positive scale maps ${\hat{I}}_S^1:S\to (0,\infty )$ and ${\hat{I}}_O^1:O\to (0,\infty )$ and a penalty $J:(0,\infty )\to [0,\infty )$. Assuming inversion symmetry, strict convexity, normalization $J(1)=0$, and a multiplicative d'Alembert identity, we show that $f(u):=1+J(e^u)$ satisfies the additive d'Alembert equation and hence $J(x)=\cosh (a\log x)-1=\frac{1}{2}(x^a+x^{-a})-1$ for some $a>0$. We then analyze the associated argmin mapping over feasible scale sets: existence under explicit subspace-closedness hypotheses, geometric-mean decision boundaries for finite dictionaries with stability away from boundaries, exact compositionality for product models, and an optimal sequential mediation principle given by a geometric mean (or its log-space projection when infeasible). The paper is purely mathematical; any semantic interpretation is optional and external to the theorems proved here.

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