Emergence of Quantum Structure from a Dynamical Geometric Instability in Extra Dimensions: A Consistent Model with Testable Predictions

We propose a dynamical field theory in a (4+1)-dimensional spacetime where the foundational principles of quantum mechanics–wave-particle duality, the Heisenberg uncertainty principle, and quantum superposition–emerge as low-energy phenomena from a **geometric self-localization mechanism**. The core innovation is a rigorously defined feedback loop between a matter field (Φ) and a geometric order field (Ψ) in a compact extra dimension of radius R. This interaction causes the confining potential on our 3-brane to self-consistently depend on the local energy density of Φ. We demonstrate with mathematical rigor that: 1. An **effective Planck constant** (ℏeff) emerges naturally from the geometry and coupling: ℏeff = αρ0R2 c, where α is a dimensionless parameter determined by the model’s coupling constants, and ρ0 is a characteristic mass density of the bulk. We show that ℏeff can match the observed ℏ without fine-tuning. 2. Dimensional reduction and the non-relativistic limit yield an **energy dependent Schr¨odinger equation** for brane-bound observers that preserves uni tarity to all orders, with nonlinear corrections suppressed by (E/Ec) 2 where Ec = ℏeffc/R. 3. Quantum fluctuations lead to a **Geometrically Generalized Uncertainty Principle (G-GUP)** derived rigorously from the curved brane embedding, with correction terms scaled by ∆X/R and ∆P/(ℏeff/R). This G-GUP is distinct from quantum gravity GUPs and becomes significant at macroscopic scales. 4. The model predicts a **de Broglie wavelength-dependent correction** to the phase in neutron interferometry (COW experiment) of order ∆ϕ/ϕ0 ∼ β(λdB/2πR) 2 , with β = 0.247±0.008 determined from numerical integration of the full equations. 5. The framework naturally incorporates **Dirac fermions**, where spin-1/2 emerges from the localization of chiral zero modes, leading to an energy-dependent effective fermion mass with controlled backreaction. The model preserves chiral symmetry in the massless limit. 6. Nonlinear analysis reveals stable **solitonic solutions** in the bulk, which correspond to particle-like states on the brane, offering a geometric interpretation of particles as topological defects. We prove the stability of these solutions against radial perturbations. 7. Additional testable predictions include energy-dependent shifts in the **Comp ton wavelength** and the **fine-structure constant**, accessible via precision spec troscopy and atomic clock comparisons. We provide quantitative estimates for next-generation experiments. 8. **Novel prediction**: A modified wave-particle complementarity relation P 2 + V 2 = 1 − β(d/2πR) 2 that deviates from the standard quantum mechanical equality, testable in matter-wave interferometry with sensitivity 10−6 . Using the tightest experimental bound on large extra dimensions (R < 44 µm), the predicted effect for ultracold neutrons is ∆ϕ/ϕ0 ∼ 1.2×10−8 , potentially within reach of next-generation instruments. The theory reduces exactly to standard quan tum mechanics when the de Broglie wavelength is much smaller than R and provides a clear, extensible geometric foundation for fermions and gauge fields. All results are derived from first principles without ad hoc assumptions