Memory Force Theory: A Modified Gravity Framework Based on Vacuum Restoring Force and Its Cross-Scale Applications

Based on the fundamental picture of vacuum medium and matter perturbations, a modified gravity framework centered on density-dependent restoring force is constructed, namely Memory Force Theory. Gravity is interpreted as the response of the vacuum to matter perturbations, introducing a scalar memory-force field and a density-dependent restoring coefficient following a power-law relation: μ(ρ) = μ₀(ρ/ρ₀)^α. Fitting the rotation curves of 15 galaxies from the SPARC database yields a universal exponent α = 0.385 ± 0.02, consistent with the QCD sum rule theoretical value α = 3/8 = 0.375 within 1σ. The framework satisfies Solar System constraints via the chameleon screening mechanism and unifies galactic dynamics, strong-field gravity, cosmic evolution, and non-perturbative QCD effects. Testable predictions include black hole shadow (0.5% shift), gravitational wave scalar polarization (Aₛ/A_T ∼ 10⁻⁶), gravitational wave dispersion (Δv/c ∼ 10⁻¹⁵), dark energy equation of state (w₀ = −0.98 ± 0.01), and laboratory fifth force (α_eff ∼ 5 × 10⁻⁸), falsifiable within 5–20 years.