Part 23: Scale-Mismatched Pinning, Accelerated Dark Compression, and Bulk-Mediated Relaxation in Dual–H4 Geometry

Overview Within the framework of Origin Geometry (OG), matter is interpreted as a family of topological excitations embedded within a discrete, elastic, aperiodic, and multidimensional geometric substrate. Previous Parts developed a dual-sector architecture [13, 16, 17], represented schematically by H4 ∪ φH4, in which the visible (H4) sector and the phase-shifted (φH4) sector coexist within a shared geometric bulk while exhibiting strongly suppressed boundary-mode overlap. This suppression was previously associated with boundary phase displacement, topological pinning, Peierls–Nabarro-type barriers, and effective near-flat-band freezing. The present Part develops an intermediate dynamical mechanism connecting the pinned dark sector of Part 22 to later dark-collapse and phase-collapse regimes. The central proposal is that the (φH4) sector may differ from the visible sector not only by phase displacement, but also by an effective lattice-scale mismatch associated with golden-ratio projection [3, 13, 18–20]. Schematically, one may write: a_φ = φa where (a) denotes an effective visible-sector lattice scale and (a_φ) denotes the corresponding scale in the phase-shifted sector. Scale Mismatch and Boundary Mass Inflation The key point is not that a larger lattice scale automatically increases pinning. If the coherence width of a boundary soliton scaled synchronously with the lattice spacing, the dimensionless width (σ/a) would remain unchanged. Enhanced pinning arises only under a non-synchronous coherence condition [17, 25–29]: (σ_φ / a_φ) < (σ / a) Under this condition, a boundary-supported electron-like soliton becomes effectively narrower in lattice units. Since the Peierls–Nabarro barrier is controlled by the relative width (σ/a), the dark-sector boundary mode may experience stronger pinning than its visible-sector analogue. This enhanced pinning contributes to an increased effective inertial burden for boundary-supported modes, extending the geometric mass and boundary-pinning mechanisms developed in earlier Parts [4–11, 17]. In this Part, this effect is called boundary mass inflation. The term does not mean that a Standard Model electron mass is directly rescaled, nor that dark-sector electron-like modes become heavier than all bulk-supported modes. It means that relative to a visible-sector boundary analogue, the corresponding phase-shifted boundary mode may carry greater resistance to displacement and reconfiguration. Topological Compression and Fusion The resulting mass-inflated and strongly pinned boundary modes accelerate weakly, radiate inefficiently, lose ordinary transition accessibility, and fail to support standard atomic, chemical, and gas-like behavior. Consequently, the dark sector may lose several forms of ordinary pressure support and become more efficiently compressible under gravitational or geometric-stress conditions. Under compression, topological density may concentrate into localized structures called topological lumps. When such lumps approach one another closely enough, short-range geometric coupling may allow effective topological fusion: a reorganization of compressed topological structures into lower-energy configurations. If: ΔE = E_initial − E_final > 0 then the process releases configuration energy. Bulk-Oscillation Relaxation Channels Because electromagnetic relaxation channels are strongly suppressed in the pinned dark sector, this energy may not be efficiently transported by photon-like boundary modes. Instead, the framework proposes that a substantial fraction of the released energy may couple to collective bulk degrees of freedom of the underlying geometric substrate. These collective bulk excitations may propagate through the shared dual-sector geometry and admit, at a coarse-grained phenomenological level, a gravitational-wave-like description [14, 15, 32–35]. The present Part does not identify these modes with gravitational waves in General Relativity. It does not derive their waveform, spectrum, polarization structure, propagation law, or observational amplitude. Rather, it establishes a candidate geometric pathway through which hidden-sector structures may compress, reorganize, release energy, and relax through non-electromagnetic bulk modes. In summary, the Part proposes the conditional sequence: Scale mismatch → enhanced pinning → boundary mass inflation → electromagnetic suppression → reduced support pressure.