Mathematical Architecture of Coherence in Complex Systems

This manuscript develops the mathematical foundations of coherence dynamics within the SIUC (Systemic Integration Unified Coherence) framework. It introduces the coherence space C as an abstract structure for representing invariants, tensions, transitions, and multiscale patterns in complex systems. The article defines the operators governing coherence propagation—including differential, pseudodifferential, nonlocal, multiscale, and memory-based mechanisms—and analyzes how local and global coherence interact across hierarchical levels. Together, these components establish SIUC as a unified mathematical architecture for coherence in physical, biological, cognitive, and collective systems.