Emergence of Gravity, Dark Energy, and Dark Matter in the Spatial Quantum Framework

Title: Core Derivations Note: Emergence of Gravity, Black Hole Entropy, Dark Energy, and Dark Matter in the Space-Quantum Framework Abstract:This note extracts the core mathematical derivations from the full preprint "Space as the Ground State of Matter (v4.0)" (DOI: 10.5281/zenodo.22686102), focusing on strict algebraic validations. It contains four primary derivations: Gravitational Constant: Eliminating presupposed background distances, G=αGℓ∗^2 is derived based on the intrinsic spectral scale (with exact dimension M^−2). Black Hole Holographic Entropy: Substituting the microscopic G yields SBH=(β/4αG)Nhor, identifying the rigid constraint (αG=β) required to exactly recover the 1/4 area law. Dark Energy Equation of State: Through exact cancellation of ground-state node counts in energy and volume, w=−1 is strictly derived. Dark Matter Abundance: Using the standard Boltzmann equation, the analytical solution is cross-validated with ODE numerical results (Ωdh2≈0.1218, with only 1.39% error). Strict boundaries between first-principles derivations, phenomenological inputs, and unresolved P1 tasks are defined.