Stability and Instability Systems: Distance to Failure Within the Paton System

This paper establishes stability and instability systems as the domain-level expression of distance to constraint violation within the Paton System. While prior work defines admissibility as the condition for system membership, boundaries as the locations of enforcement, and constraints as the conditions that define admissibility, the degree to which a system approaches failure has not been isolated as a domain. This paper shows that stability corresponds to distance from constraint limits, while instability corresponds to proximity to or violation of those limits. Stability is therefore not a static condition but a measurable position within admissible space, and instability represents directional movement toward constraint violation and collapse. This provides a unified structural interpretation of stability across physical, biological, computational, and economic systems.