Roadmap: Applying a Spiral-Time (Phase-Memory) Operator to Hysteresis / Path-Dependent Dynamics

This note outlines a concrete, testable procedure to apply a phase-memory operator (Spiral-Time Operator) to engineering systems that exhibit hysteresis, path dependence, and drift under timing constraints. The operator serves as a mathematical coordinator of regime stability rather than a new physical actuator. By extracting the instantaneous phase and utilizing a triadic state model (phase deviation, drift rate, and memory), a stability index (J(t)) is defined, which acts as an early warning indicator for regime shifts and control failures. Validation is performed empirically by analyzing the predictive performance regarding hysteresis loops in time-series data.