Domain Mathematics A Structural-Dynamic Framework for Cognitive Synchronization, Stabilization, and Fragmentation

Building upon Field Mathematics, this chapter investigates the formation, stabilization, synchronization, and fragmentation of domains within the Sedimentation-Field. It proposes that memory, personality, emotion, and perceptual continuity are not pre-given entities, but dynamic outcomes arising from long-term synchronization, historical accumulation, and fragmentation competition among localized stable continuation structures. By introducing structural-dynamic quantities such as InFlow, MaintainFlow, Lock, Sync, and FragmentRate, the chapter develops a domain dynamics framework that departs from traditional symbolic logic and classical cognitive models, treating cognition instead as a process of historical bias stabilization, synchronization competition, and compressive flow organization within the Sedimentation-Field. The framework further proposes that conceptual stability and generalization emerge from long-term synchronization and compressive stabilization among domains, thereby offering unified structural accounts of perceptual stability, emotional dynamics, and cognitive fragmentation.