On Φ: A Scalar Efficiency Metric and Its Mathematical Propertie

This work introduces Φ, a scalar efficiency metric defined over successive states of a process or trajectory. Φ captures signed improvement relative to magnitude and enables rigorous comparison between progress and stagnation without prescribing any algorithmic or procedural interpretation. We formalize basic properties of Φ, derive related aggregates, and establish bounds under minimal assumptions.

This paper is intentionally limited to mathematical definitions and identities. It makes no claims regarding control logic, decision procedures, optimization strategies, or system architectures. The purpose of this publication is to establish prior art for the mathematical formulation of Φ and its properties.