Part 31: Spectrum and Detectability of Ultra-High-Frequency Bulk Gravitational-Wave-Like Modes in Dual–H4 Geometry Bulk Stress Dynamics, Nonlinear Mode Transfer, and Experimental Outlook

Overview Within the Origin Geometry framework, previous Parts developed a dual–H4 cosmological architecture in which dark-sector topological relaxation, phase compression, obstruction release, and bulk stress redistribution play central roles in large-scale geometric evolution [1–10]. Part 29 proposed that cosmic expansion may emerge from void-driven lattice proliferation, while Part 30 identified observable signatures of such geometric expansion through environment-dependent residuals, void dynamics, redshift–distance deviations, luminosity–distance correlations, and high-frequency bulk stress backgrounds. The present Part focuses on the spectral and detectability aspects of that bulk background. Gravitational-Wave-Like Bulk Modes In Origin Geometry, the relevant excitations are not assumed to be ordinary gravitational waves in the strict General Relativistic sense. They are not, at the microscopic level, defined as tensor perturbations of a smooth continuum metric [11–13]. Rather, they are collective bulk stress modes of a dual-sector topological–geometric substrate. Nevertheless, after coarse-graining, such modes may admit a gravitational-wave-like phenomenology because they propagate through bulk geometry, carry geometric stress, redistribute configuration energy, and may project weakly onto effective metric-like observables. The Ultra-High-Frequency Source Spectrum The central claim of this Part is that dark-sector relaxation processes generically source a broad stochastic spectrum of bulk stress excitations, with most source energy expected to lie in an ultra-high-frequency regime [14–17, 21]. This follows from the small characteristic length and time scales associated with topological relaxation, defect cancellation, phase barrier compression, and localized network reconfiguration. If a characteristic topological length scale is ℓ_topo, then the corresponding angular frequency satisfies schematically: ω_bulk ~ c_bulk / ℓ_topo or, for ordinary frequency: f_bulk ~ c_bulk / (2π ℓ_topo) When ℓ_topo is much smaller than astrophysical length scales, the resulting bulk spectrum is naturally shifted far above the frequency ranges targeted by conventional gravitational-wave observatories. Nonlinear Mode Transfer and Detectability However, the observable spectrum need not be identical to the source spectrum. Nonlinear interactions in the geometric substrate may allow a small fraction of ultra-high-frequency bulk stress energy to transfer into lower-frequency modes. This down-converted component is expected to be weak, but it may provide the most realistic near-term observational window into the bulk background. Consequently, current non-detections of high-frequency signals do not directly falsify the primary ultra-high-frequency OG background, but upper bounds on low-frequency stochastic components can constrain nonlinear transfer efficiency [15–18, 21]. This Part develops a phenomenological spectral language for the bulk background, introduces a distinction between source spectrum and effective observable spectrum, discusses nonlinear mode transfer, analyzes stochastic accumulation, and outlines possible detection strategies. These include high-frequency gravitational-wave searches, resonant systems, electromagnetic cavities, quantum sensors, optomechanical concepts, standard stochastic-background constraints, and indirect cosmological signatures linked to voids and compact-object environments. Scope and Limitations The framework remains exploratory. It does not claim detection, does not replace General Relativity, and does not derive a complete microscopic field theory. Its purpose is to establish a falsifiable bridge between the internal bulk dynamics of Origin Geometry and possible future experimental or observational tests.