Uniqueness of the Canonical Reciprocal Cost
Jonathan Washburn, Milan Zlatanović
TL;DR
This paper addresses the rigidity of a ratio-cost function $F:\mathbb{R}_{>0}\to\mathbb{R}_{\ge0}$ with $F(1)=0$, under a multiplicative composition law and a single quadratic calibration at the identity. By moving to logarithmic coordinates $t=\ln x$ and setting $H(t)=F(e^t)+1$, the authors recast the problem as a d'Alembert equation $H(t+u)+H(t-u)=2H(t)H(u)$ and use a unit log-curvature condition $\kappa(F)=1$ to invoke the classical classification of continuous solutions, selecting the hyperbolic-cosine branch. They prove that the unique calibrated solution is $F(x)=J(x)=\frac{x+x^{-1}}{2}-1$, equivalently $H(t)=\cosh t$, and they establish the necessity of each assumption via counterexamples and pathologies, along with a stability result under bounded defect. The paper also develops the canonical cost's structural properties, including its relation to the Bregman divergence, the induced Riemannian metric, Chebyshev structure, connections to the golden ratio, and an energy-like interpretation, highlighting the practical significance of the canonical reciprocal cost in multiplicative-ratio penalties.
Abstract
We study a rigidity problem for functions \(F:\R_{>0}\to\R_{\ge 0}\) that penalize deviation of a positive ratio from equilibrium \(x=1\). Assuming (i) normalization \(F(1)=0\), (ii) a d'Alembert-type composition law on \(\R_{>0}\), and (iii) a single quadratic calibration at the identity (in logarithmic coordinates), we prove that \(F\) is uniquely determined. The unique solution is called the canonical reciprocal cost, namely the difference between the arithmetic and geometric means of \(x\) and its reciprocal. Our proof uses the logarithmic coordinates \(H(t)=F(e^t)+1\), where the composition law becomes d'Alembert's functional equation on \(\R\). The calibration provides the minimal regularity needed to invoke the classical classification of continuous solutions and fixes the remaining scaling freedom, selecting the hyperbolic-cosine branch. We also establish necessity of each assumption: without calibration the composition law admits a continuous one-parameter family, without the composition law the calibration does not determine the global form, and without regularity the composition law admits pathological non-measurable solutions. Finally, we establish a stability estimate for approximate solutions under bounded defect and characterize some properties of the canonical cost.