Viability Gradient: A Structural Bridge from Recursive Continuation to Distance-to-Failure

Systems that generate candidate states under constraint must remain within admissible regions in order to continue operating. The Paton System describes this continuation process through a recursive architecture in which candidate states are generated, evaluated against admissibility constraints, and allowed to persist only while those constraints remain satisfied. This paper introduces the concept of a viability gradient: a structural quantity describing how system states move relative to their admissible region. The viability gradient provides a way to describe directional change in system viability and the distance-to-failure that arises as systems approach admissibility boundaries. The framework does not replace existing domain models or governing equations. Instead, it provides a structural description of how recursive systems move within admissible space and how proximity to failure can be understood as directional movement toward admissibility collapse. The result forms a bridge between recursive continuation architecture and operational diagnostics such as Paton Assist and Paton Compass within the broader Paton System.