Dissipative Numeral System, Paper IV Dual Periodicity and Structural Orbits — From Static Numerical Structures to Dynamic Reconstruction

The first three papers of the Dissipative Numeral System (DNS) established that numerical entities can possess structural properties beyond quantity: structural distinction between zeros (Paper 0), irreversible historical traces from decomposition (Paper II), and static cost-induced geometry (Paper III). This paper introduces the missing concept of structural evolution. We emphasize that the evolution index used here is not physical time; it records the ordering of structural operations — formation, decomposition, recombination, and reconfiguration. We extend the DNS state representation by introducing two coupled structural variables: I0(k)I0(k), which describes the integrity of the structural container, and G1(k)G1(k), which describes the aggregation state of the occupied numerical content. Unlike classical numerical systems, DNS structures possess dual periodicity: the container cycle and occupancy cycle may evolve with different characteristic periods (T0≠T1T0=T1). This creates a new class of structural trajectories in which one cycle restores one component while leaving another component partially shifted. We define the DNS Möbius process as a dual-cycle structure where a complete structural cycle corresponds to only a partial phase completion of the coupled occupancy cycle. Full restoration requires a second cycle. This framework establishes that numerical structures may preserve identity not through static persistence, but through repeated reconstruction. DNS therefore proposes a new class of numerical objects: not static quantities, but evolving structural entities.