Part 37: Large-Scale Topological Transitions in Dual–h₄ Geometry

Previous parts of the Origin Geometry (OG) program developed an emergent cosmological framework in which cosmological expansion is interpreted as a coarse-grained consequence of: topological relaxation, geometric stress redistribution, nonlinear collective dynamics, and bulk propagation within the dual–H₄ network. Within this framework, expansion acts as a mechanism through which accumulated geometric stress can be redistributed and reduced through the growth and reorganization of the underlying geometric substrate. A central unresolved question, however, concerns the asymptotic behavior of the system. What occurs when the internal energy source driving topological growth becomes progressively weaker? How does the geometric network evolve once expansion is no longer capable of efficiently relieving accumulated stress? The present work investigates this asymptotic regime. We argue that when topological growth becomes dynamically suppressed, the dual–H₄ network enters a state of geometric saturation in which stress can no longer be dissipated primarily through expansion. Instead, geometric frustration begins to accumulate throughout the aperiodic structure. Drawing inspiration from quasicrystal physics, we propose that phason degrees of freedom may become the dominant dynamical channel in this regime. Large-scale phason activation can modify the projection structure that connects the physical H₄ network to its higher-dimensional E₈ parent space. When accumulated geometric stress exceeds a critical threshold, the system may undergo large-scale topological reorganization through substantial shifts of the projection window in internal space. This framework provides a possible geometric description of extreme cosmological regimes without invoking singularities of infinite density. It further suggests the possibility of cyclic geometric dynamics driven by successive phases of: topological growth, stress accumulation, geometric saturation, and topological restructuring. The present work does not claim that such cyclic behavior has been established observationally. Rather, it identifies a coherent dynamical pathway through which asymptotic cosmological evolution may be interpreted within the Origin Geometry framework.