Part 37: GEOMETRIC SATURATION AND PHASON-MEDIATED TOPOLOGICAL TRANSITION IN DUAL–H4 COSMOLOGY Frustration Accumulation, Projection-Window Instability, and Large-Scale Reorganization

Overview Previous Parts of the Origin Geometry program developed an emergent cosmological branch in which effective expansion arises from topological relaxation, bulk stress redistribution, void-enhanced activation, and realized network growth in a dual–H4 geometric substrate [1–4]. In this branch, expansion is not interpreted as the literal stretching of a fundamental continuum, but as a coarse-grained manifestation of increasing realized topological network count. Part 36 further proposed that expansion may function as a stability-favoring response of a non-equilibrium geometric network subject to persistent stress injection, obstruction accumulation, and collective bulk transport. The present Part asks the complementary asymptotic question: what happens when the topological-growth channel itself becomes dynamically saturated? Geometric Saturation We propose that geometric saturation occurs when the net growth rate of realized topological network count becomes strongly suppressed. This need not imply that bulk stress modes vanish. Rather, activation, damping, localization, pinning, and saturation terms may balance so that the conversion of stored or localized stress into new realized network degrees of freedom becomes inefficient. In this regime, expansion no longer functions as the dominant stress-relief channel. Residual geometric stress and frustration may then accumulate within the aperiodic dual–H4 substrate. Phason Dynamics and Topological Transition Drawing on the effective language of quasicrystal phason dynamics, we introduce a projection-space phason field w(x_∥, t) describing internal deformations of the acceptance window W that selects which parent-structure elements become realized in the effective H4-projected network [5–11]. Phason motion is not ordinary displacement in physical space. It is an internal-space reconfiguration of the projection rule itself. Because the projected network is discrete and aperiodic, phason motion is expected to experience Peierls–Nabarro-like pinning [18–22]. When stress-induced internal forcing exceeds the restoring barrier of the projection structure, the current realization may lose local stability. This produces a phason-mediated topological transition, schematically: W → W' Such a transition changes the realized geometric subset without destroying the parent structure or creating geometry from nothing. Scope and Limitations This framework provides a possible Origin Geometry interpretation of extreme cosmological regimes as projection-space reorganization rather than singular divergence. It does not claim to replace General Relativity, ΛCDM, or precision cosmology. It does not claim that large-scale topological transitions have been observationally confirmed. Its purpose is to extend the internal dynamics of Origin Geometry beyond expansion-dominated evolution and identify the mathematical conditions under which geometric saturation may give way to phason-mediated reorganization.