Efficiency Phase Calculus (EPC): A Mathematical Framework for Dynamic Efficiency Analysis We introduce the Efficiency Phase Calculus (EPC), a unified mathematical framework for analysing efficiency as a dynamic function rather than a static ratio. Framework Overview: Given any differentiable input function g(x) and output function f(x), the EPC defines: - Marginal Efficiency: F(x) = f'(x)/g'(x) — the instantaneous rate of output per unit input - Average Efficiency: A(x) = f(x)/g(x) — the cumulative output per unit input Core Contributions: 1. Foundational Properties (10 Laws): A complete set of mathematical relationships governing efficiency dynamics, including the Efficiency Gap Law, Reconstruction Formula, Scaling Invariance, Composition Law, Duality Law, Curvature Law, and the Integral Conservation Law. 2. Main Theorems: Eight provable theorems including Uniqueness of Reconstruction, Characterisation of Constant Efficiency, Efficiency Phase Transition, Superposition, Cascade Product Rule, Monomial Classification, Fixed-Point Theorem, and the Fundamental Theorem of Efficiency Calculus. 3. Applications: The framework is demonstrated across three domains — business profit maximisation, biological optimal foraging, and engineering motor efficiency — showing universal applicability. Significance: The EPC transforms efficiency from a static retrospective point-value into a dynamic, predictive mathematical object with its own calculus, structure, and classification. It unifies concepts scattered across economics, engineering, biology, and optimisation theory into a single coherent language. This working paper establishes the theoretical foundation of EPC. Companion tools and extended applications are under development.
Ashley Woods (Thu,) studied this question.