A Minimal Control-Theoretic Framework for the Triadic Phase Operator ∆Φ

We present a minimal, control-theoretic formulation of the triadic phase operator ∆Φ as a deterministic diagnostic tool for stability monitoring, regime detection, and reproducible analysis in nonstationary and path-dependent systems. The framework introduces a triadic feature decomposition into state, memory, and constraint components, yielding an explicit instability vector whose norm defines an interpretable stability score. Importantly, ∆Φ is formulated as an observer-style diagnostic and does not constitute a control law or prediction mechanism. We further clarify how learning-based methods may be incorporated strictly as feature and constraint estimators without compromising auditability or Lyapunov-compatibility. The resulting framework is domain-agnostic, falsifiable, and suitable for safety-critical, human-in-the-loop environments where deterministic replay and conceptual transparency are required.