Stability and Attractor Topology in the Scalar Drag Emergence Framework

This work develops the stability and structural layer of the Scalar Drag Emergence Framework (SDEF), focusing on the formation, persistence, and organization of attractor configurations under transport–ancestry dynamics. Starting from the primitive generator, the paper derives the conditions under which spatial structures persist. Stability is shown to arise from the balance between transport convergence, ancestry support, and saturation, while instability occurs when this balance is disrupted. Attractors emerge through feedback between gradient amplification, ancestry accumulation, and saturation-limited localization. These structures are not imposed, but arise dynamically as persistent configurations of the underlying fields. A central result is the topological characterization of attractors in terms of nodes, corridors, and loops in gradient-aligned transport pathways. This provides a structural description of how configurations are organized, connected, and stabilized. Transitions between attractors are shown to occur through changes in transport structure, ancestry distribution, and saturation balance, leading to reconfiguration of topology and effective irreversibility under coarse-graining. This work establishes a structural layer within SDEF that complements dynamical and statistical descriptions, providing the foundation for understanding persistent configurations and the emergence of complex structure from the primitive generator.