Three-Way Quantum Gravitational Wave Model: Unified Inter-Planar Resonance and Planck-Scale Phenomenology

Abstract:We introduce the Three-Way Quantum Gravitational Wave Model, a novel framework that unites brane-world gravity, Loop Quantum Gravity holonomy corrections, and structured quanta expansion within a dual-planar Vesica Piscis wavefunction basis. By deriving a unified attractor equation that stabilizes the dimensionless coupling at ±1, we predict distinctive signatures—quantum flickering sidebands at ∼10 kHz and ultra-high-frequency cutoffs ∼10⁵ Hz—in current and next-generation gravitational-wave detectors (LIGO/LISA). Key Features: Dual-Planar Action & Field Equations: Variation of the combined 5D bulk and induced 4D brane actions yields modified Einstein equations with leading-order LQG stress–energy corrections. Resonant Energy Transfer: A structured-quanta expansion operator governs inter-planar energy exchange, constrained by unitarity. Phase-matched modes satisfy a Mathieu-type instability equation, defining resonance bands. Attractor Dynamics: The normalized coupling λ evolves via dλ/dτ = −λ + tanh(Γλ), admitting stable fixed points at λ=±1 for Γ>1. Numerical Validation: Finite-difference time-domain simulations confirm fourth-order convergence, reproduce classical dispersion within 10⁻⁵ tolerance, and map instability tongues predicted by Floquet analysis. Phenomenological Predictions: We outline data-analysis strategies for isolating Planck-scale flickering in LIGO O3 and LISA streams, quantify expected SNR, and propose dark-matter analogues via inter-brane Yukawa stress–energy. Supplementary Materials: Appendix A: Detailed Euler–Lagrange derivations, holonomy expansions, Mathieu-equation reduction, and analytic attractor solutions. Appendix B: Convergence studies, Fourier spectra, and attractor fixed-point sweeps. Appendix C: A “Three-Way” geometric toy model comparing reciprocal quartic-over-quartic maps built from integers, π, e, and the golden ratio—visualizing fixed-point intersections at |x|≈1. Usage Notes:This repository contains the full LaTeX source, auxiliary figures, and simulation notebooks. It is intended for researchers in quantum gravity, gravitational-wave phenomenology, and mathematical physics seeking a cohesive bridge between theory and observation. Keywords: quantum gravity, brane-world, Loop Quantum Gravity, gravitational waves, resonance, Planck scale, numerical relativity, dark matter analogues, Vesica Piscis wavefunctions.