Density-Dependent Vacuum Recovery Scalar (DVRT) Model: A Static Weak-Field Phenomenological Modified Gravity Framework for Galactic Dynamics

We propose a two-parameter scalar field modified gravity model, the Density-Dependent Vacuum Recovery Scalar (DVRT) model, which reproduces the SPARC galactic rotation curves with a reduced chi-squared χ^2/dof ≈ 1.1 without introducing dark matter. A static scalar field Φ is introduced to describe the vacuum perturbation induced by matter density, with the recovery coefficient following an empirical power law μ(ρ) = μ_0(ρ/ρ_0)α. Under static, spherically symmetric, and weak-field approximations, we derive an analytical rotation velocity formula based on the integral-form scalar field solution. Fitting to 15 SPARC galaxies yields α = 0.385 ± 0.02 and μ_0 = (2.13 ± 0.06) × 10⁻⁴ \, kpc⁻¹. Combined with a chameleon-like screening mechanism and force-range analysis, we demonstrate that the model fully satisfies Solar System gravitational constraints. Quantitative predictions and falsification criteria are presented within the model's range of validity. This framework is strictly limited to static, weak-field, low-velocity astrophysical systems. Open issues including Lagrangian formulation, fully covariant extension, tachyon instability in ultra-low density regions, and cluster-scale tests are identified for future investigation. Version 2 - Revised and updated version. Supersedes v1 (2026-06-17).