Part 33: Nonlinear Relaxation Regimes and Phase-Transition Dynamics in Emergent Dual–h₄ Cosmology

Part 32 established a coarse-grained dynamical framework for emergent cosmology within the dual–H₄ network of Origin Geometry, in which: geometric stress, obstruction topology, bulk collective propagation, and relaxation efficiency act as effective dynamical variables. The present work extends that framework toward: nonlinear relaxation regimes, threshold dynamics, and collective phase-transition-like behavior within the dual-sector substrate. In topology-driven non-equilibrium systems, relaxation processes often do not proceed linearly. When: obstruction density, stress concentration, or bulk excitation density exceed certain effective thresholds, the network may undergo: collective reorganization, sudden relaxation bursts, or phase-transition-like dynamics. The present framework investigates the possibility that cosmological expansion within Origin Geometry is not entirely smooth at the microscopic level, but instead emerges from a continuous sequence of nonlinear relaxation events occurring throughout the topological–geometric network. This work does not replace standard cosmology. Instead, it provides a coarse-grained dynamical interpretation of non-equilibrium cosmological evolution within the Origin Geometry framework.