A Formal Dynamical Framework for Coherence-Preserving Dynamics on Bounded State Spaces: Applications to Cross-Domain Failure Detection and Coherence Measurement

This paper introduces a formal dynamical framework for coherence-based measurement in complex systems. Grounded in the Standard Coherence Fidelity Layer (SCFL) architecture, the framework defines a bounded state space, a nonnegative coherence functional, and admissible dynamics governing system evolution. Three core results are derived: (i) a universal sufficient condition for rupture expressed as an integral threshold, (ii) forward invariance of the rupture set under admissible dynamics, and (iii) an induced partial order on coherence level sets compatible with admissible transformations. Together these results constitute a dynamical system with invariant structure, geometric coherence stratification, and measurable early-warning boundaries. A minimal reconstruction example demonstrates constructiveness. The framework is domain-independent and applies wherever a coherence functional can be defined — including grid infrastructure, institutional systems, reinsurance, healthcare, and human flourishing. This work establishes the formal mathematical foundation of the SCFL measurement discipline and positions coherence drift and rupture as rigorously defined, instrumentally detectable phenomena.