Uniqueness of the Canonical Reciprocal Cost

Jonathan Washburn; Milan Zlatanović

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arXiv · 2602.05753

Uniqueness of the Canonical Reciprocal Cost

Jonathan Washburn·Milan Zlatanović

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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) a d'Alembert-type composition law on $\R_{>0}$, and (ii) a single quadratic calibration at the identity (in logarithmic coordinates), we prove that $F$ is uniquely determined. The composition law implies the normalization $F(1)=0.$ 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.

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Jonathan Washburn, Milan Zlatanović. 2026-02-05. Uniqueness of the Canonical Reciprocal Cost. https://arxiv.org/abs/2602.05753

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