The Depth-Dynamics Equation of the Sedimentation-Field

This chapter establishes a minimal dynamical closure for finite-resolution fields in order to explain how structures form, stabilize, decay, migrate, and generate new domains under competition and capacity constraints. By introducing local resolution occupancy, residual resolution capacity, structural overlap matching, resolution flow, structural surplus, and nucleus stabilization, the chapter unifies input, competition, diffusion, migration, decay, and nucleation within a single dynamical framework. The chapter argues that history cannot accumulate indefinitely. In finite-resolution fields, old structures release local resolution capacity through competition and decay, thereby allowing new differences to enter and new structures to emerge. Furthermore, familiar paths are shown not to be mere repetitions, but stable continuations sustained through persistent overlap between incoming structures and existing sedimented structures. Likewise, edge-penetration is described not as random fragmentation, but as the reformation of sedimentation closure in neighboring regions through resolution flow generated by sustained structural surplus. Finally, the chapter derives a system of Sedimentation-Field dynamics equations describing how structures within finite-resolution fields are preserved through competition, migrate through flow, dissipate through diffusion, release capacity through decay, and reclose into new domains after edge-penetration.