Uniqueness of the Canonical Reciprocal Cost

TL;DR

This paper proves the uniqueness of a specific cost function called the canonical reciprocal cost, characterized by a composition law and a quadratic calibration, using logarithmic coordinates and classical functional equation classification.

Contribution

It establishes the unique form of the canonical reciprocal cost function under specified composition and calibration assumptions, with a detailed proof and necessity of each assumption.

Findings

The canonical reciprocal cost is uniquely determined by the composition law and calibration.

Without calibration, the composition law admits a family of solutions.

The paper provides a stability estimate for approximate solutions.

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.