A Unified Quantum Gravity Framework: Derivation of Mass-Energy Equivalence (E=mc^2) and Heisenberg's Uncertainty Principle via Scale-Dependent Phase Factor \eta(\rho) and Critical Decoherence Threshold \gamma_c

This research presents a unified mathematical framework that resolves cosmological singularities and accounts for emergent space-time from quantum wave states. By introducing a scale-dependent structural phase factor \eta(\rho) = 1 - e^{-\rho/\rho_{\text{Planck}}}, the effective gravitational stress-energy tensor vanishes (T_{\text{net}} \to 0) as density approaches the Planck threshold (\rho \to \infty), preventing non-physical gravitational singularities. The framework successfully recovers foundational classical and quantum limits through two rigorous derivations: Mass-Energy Equivalence (E=mc^2): Derived in the localized particle limit (\vert{}c_p\vert{}^2 \to 1), where stored anti-energy (E_{\text{anti}}) couples back into rest-mass dynamics, yielding exact standard rest-mass energy calculations (e.g., E_0 \approx 0.511 \text{ MeV} for localized electrons). Heisenberg Uncertainty Principle (\Delta x \cdot \Delta p \ge \hbar/2): Derived from the critical vacuum decoherence threshold \gamma_c = \frac{2\Delta E_{\text{vacuum}}}{\hbar}, defining the minimum momentum-position uncertainty during wave-particle phase collapse. Furthermore, the model provides candidate mechanisms for the Black Hole Information Paradox, vacuum negative pressure driving Dark Energy (P_{\text{net}} < 0), and pre-Big Bang timelessness (\mathcal{T} = 0). Computer simulations confirm density-dependent gravity suppression and temporal emergence curves.