PART 29: EMERGENT GEOMETRIC EXPANSION FROM TOPOLOGICAL RELAXATION IN DUAL–H₄ GEOMETRY

Within the Origin Geometry (OG) framework, effective spacetime is not treated as a fundamental metric continuum but rather as a discrete multi-sector topological–geometric network governed by collective non-equilibrium dynamics. Previous works within the OG research program have investigated bulk collective modes, topological obstruction, phase compression near black holes, cross-sector leakage, and geometric relaxation processes. The present work extends these concepts to cosmological scales and explores the possibility that cosmological expansion may emerge as a coarse-grained consequence of topological relaxation within a dual–H₄ geometric substrate. Unlike approaches in which expansion is postulated directly as a geometric property of a continuum metric, the present framework proposes that stress redistribution, obstruction relaxation, collective rearrangement, and evolving connectivity structures may collectively generate an increase in the effective large-scale separation between excitations. The framework does not assume the literal creation of new space at the microscopic level. Instead, cosmological expansion is interpreted as a large-scale manifestation of geometric restructuring arising from the relaxation dynamics of a dual-sector topological network. A central component of the proposal is the special role played by low-density regions. Within Origin Geometry, voids possess lower obstruction densities, reduced collective scattering, and greater accessibility for bulk propagation. Consequently, void-dominated regions may act as preferred relaxation environments in the emergence of effective large-scale expansion. The present framework does not replace Friedmann–Robertson–Walker cosmology or the ΛCDM paradigm. Rather, it explores the possibility that part of the observed phenomenology of cosmological expansion may admit a microscopic geometric interpretation rooted in stress relaxation within a discrete topological substrate.