Part 18: Dynamic Completion of Origin Geometry: Variational Dynamics, Spectral Localization, and Infrared Stability on Aperiodic H4 Substrates

Previous works within the Origin Geometry (OG) program established a discrete geometric substrate derived from aperiodic H₄ projection and demonstrated the emergence of spectral structure, geometric hierarchy, boundary–bulk separation, localization phenomena, mass generation mechanisms, and algebraic closure through E₈ completion. Despite these advances, a central limitation remained unresolved. The framework lacked an explicit dynamical principle capable of generating the observed spectral structures from a variational foundation. In this work, we provide the minimal dynamical completion of OG by constructing a nonlinear lattice field theory defined on weighted aperiodic H₄-derived substrates. A scalar order parameter is introduced solely as a geometric deformation proxy, allowing the formulation of a variational action whose linearized spectrum reproduces previously observed OG spectral structures. Building upon this variational foundation, we demonstrate that localization width is not an arbitrary parameter but a dynamically selected quantity determined by competition between edge stiffness and onsite potential energy. The resulting equilibrium width is shown to be controlled by the spectral gap of the underlying substrate. Substituting the spectrally determined localization width into the Peierls–Nabarro pinning mechanism yields an emergent mass scale expressed entirely in terms of geometric spectral data. Finally, we construct an explicit coarse-graining procedure for weighted aperiodic H₄-derived graphs and show that the low-spectrum sector remains stable under iterative scale transformation. Infrared dynamics preserve their Lorentz-like structure, and the previously established mass hierarchy remains robust across scales. The present work therefore provides the first complete dynamical foundation of Origin Geometry, linking geometry, dynamics, localization, mass emergence, and infrared stability within a single variational framework.