Q-G.E.O.M. The Vibrational Medium Framework as a Bridge Between Quantum Mechanics and Emergent Gravity

This document presents the Q-GEOM Quantum -Geometry Emergent Oscillatory Manifolds() theory, based on the postulate of the Fundamental Vibrational Medium (FVM) — a homogeneous cosmic vacuum medium characterized by a fundamental frequency f₀ ≈ 10⁻²⁰ Hz. Q-GEOM positions itself as a unifying framework between quantum mechanics (Bose-Einstein Condensate coherence) and emergent gravity (FRB temporal profiles, Hawking spectral shifts), resolving anomalies unexplained by classical models. The foundations of Q-GEOM rest on three key FVM properties: Nature of the FVM: Its frequency f₀ synchronizes BEC atoms (rubidium-87) to preserve coherence τcoh > 60%, unlike residual perturbations. Selective resonance: The FVM interacts only with systems whose natural frequency is a multiple of f₀ (fsystem = nf₀), explaining the +9 Hawking spectral shift for 10⁶M⊙ black holes (where fBH = f₀). Propagation without attenuation: The FVM preserves FRB modal structures (e.g., FRB 20220912A, 8 billion light-years away) — a property unexplained by intergalactic plasma dispersion. The theory is reinforced by three rigorous computational junctions linking FVM to formal physics: Connection between f₀ and field Θ oscillation amplitude α (Zenodo 17259663): α = 3.0 × (fsystem/f₀) Modified holographic entropy: SholoQGEOM = (A/4ℓP²) × (1 + (fsystem/f₀)/10) Vibrational potential VMVF = 10⁻³⁰ × (f₀/fparticle) added to Schrödinger equation, confirming FVM doesn't affect heavy particles Indirect evidence (Zenodo data, CHIME observations, Hawking simulations) validates the FVM, while limitations (empirical calibrations, indirect evidence) are addressed with proposed solutions (minimal Lagrangian derivation, SKA/LISA tests). Falsifiable predictions (zero Hawking shift for 10³M⊙ black holes, τcoh < 10% for lithium-6 BECs) pave the way for experimental validations (2025–2037). Q-GEOM brings a paradigm shift: it not only describes physics but unifies micro- and macro-scales through a simple principle (resonance with f₀), inviting reconsideration of the cosmos as a system synchronized by the FVM.