Spatial Unit Conservation and Dynamic Reorganization: A Unified Framework of Gravity, Cosmology and Quantum Discreteness

This paper presents a unified theoretical framework based on three fundamental axioms: the Covariance Principle, the Invariance Principle, and the Shielding Principle. Spacetime is not a continuous manifold but a discrete graph G = (V, E) of Planck-scale units, each characterized by a complex field Φ_i = √ρ_i e^{iθ_i}.Rigorously derived results (no experimental input):This paper rigorously derives from first principles: genus g = 3, spin j = 1/2, the Barbero-Immirzi parameter γ = 1/(4√3π), the Planck-scale Weinberg angle sin²θ_W(M_P) = 1/4, the color number N_c = 3, charge quantization, the lightest neutrino mass m_{ν₁} = 0, topological prohibition of fourth-generation fermions, the universal scaling ρ₀a² = 1/(2π), the Wilson coefficient r = a/4, and the gravitational wave echo structure Δt.Cross-sector validation and new discoveries in galactic dynamics (tested with observational data):(i) Using the Milky Way rotation curve (Gaia+VERA 2024) to calibrate the geometric residual parameters: r_c = 6.6 kpc, ρ₀ = 35195 km²/s², reduced χ²_ν = 1.2, perfectly replacing dark matter halos.(ii) Independent prediction of the M31 (Andromeda) rotation curve using the same parameters, in perfect agreement with observations (Carignan et al. 2006), requiring no additional adjustments.(iii) Application of the model to four dwarf galaxies (LMC, SMC, Fornax, Draco), all with reduced χ²_ν < 2.0, leading to the discovery of a new scaling relation r_c ∝ R_half (power-law exponent b = 1.02 ± 0.08, R² = 0.99), unifying the description from dwarf galaxies (R_half ∼ 0.2 kpc) to giant spirals (R_half ∼ 4.5 kpc). Dark matter theory (ΛCDM) requires independent halo parameter fitting for each galaxy and cannot explain this universal scaling law.Unified origin of dark matter and dark energy:The uniform residual of spatial unit proliferation (geometric residual) drives cosmic acceleration (dark energy), while non-uniform cascade transfer produces density gradients and density fluctuation spectra, manifesting as gravity and dark matter effects. The vacuum catastrophe is naturally resolved — uniform zero-point energy has zero gradient and does not couple to gravity. Black hole singularities are eliminated by discrete geometry.Testable predictions:No fourth-generation leptons; gravitational wave echoes (10⁻² s for stellar-mass black holes, 10³ s for supermassive black holes); neutrinoless double-beta decay m_{ββ} ≈ 0.0037 eV; rotation curve oscillations (amplitude 2-3 km/s, period 5 kpc); δ_{CP} = e/2; α_s(m_Z) = 1/(πe); superconducting phase transition radiation ν = 2Δ/h.Core conclusions:The dimensionless constants of the Standard Model are not free parameters but topologically invariant geometric lockings. Dark matter and dark energy are not independent entities but macroscopic manifestations of geometric residual effects. The scaling relation r_c ∝ R_half is the geometric fingerprint of emergent gravity.