This work presents a theoretical formulation of a dual-system dynamic model based on the Y–X equation, focusing on operator-level structure and field admissibility. The manuscript is released as a theoretical preprint. No empirical validation is claimed at this stage. The contribution lies in the construction and internal consistency of the proposed formalism, rather than experimental confirmation. The work explores the conditions under which coupled systems converge onto a closed, self-sustaining dynamical loop, and introduces a Navier–Stokes–type field formulation as a minimal continuous representation of the underlying operator structure.Version 3.0 note. Major structural updates were introduced in this release: • Added Appendix B, formalizing operator-level admissibility criteria underlying dual-system coupling. • Relocated the Dimension Spectrum to Appendix A for improved conceptual coherence and alignment with the main mathematical framework. • Added Section 5.0, providing the formal Filter–Damp Transition between System Y (subcritical, selective damping) and System X (supercritical, selective amplification). • Corrected naming invariants, harmonized terminology across sections, and fixed multiple cross-references. • Performed minor structural refinements to improve internal consistency.