The Persistence Ratio: Explaining the Persistence Ratio First Introduced in the Odero Axiom

This paper develops the Persistence Ratio, denoted by Ω, a dimensionless measure first introduced within the framework of the Odero Axiom. The Persistence Ratio emerges from the persistence condition C(I)⋅E>D, which states that a system persists when Information Control acting upon available Energy exceeds Decay. Building upon this principle, the paper derives a normalized measure of persistence, Ω=DC(I)⋅E, together with its time derivative, ω=dtdΩ, which captures the direction and rate of system change. The paper presents a formal interpretation of Information Control, decomposes decay into measurable components, and establishes conditions for persistence, erosion, collapse, and compounding growth. Resource-allocation rules are derived through constrained optimization, and theoretical results on stability and collapse dynamics are developed using Lyapunov theory. The framework is applied across 18 domains including physics, biology, economics, governance, technology, and social systems. By providing a common mathematical language for productive forces and decay forces, the Persistence Ratio offers a unified approach to analyzing survival, resilience, growth, and decline in complex systems.