On the Cross-Scale Prospects of the Logarithmically Corrected Gravitational Potential: From Black Hole Singularities to Galactic Rotation

We derive an effective gravitational potential \( Φ_{halo} (r)∼-[ln⁡( r/r_*)+1]/r \) from the asymptotic behavior of dark matter halo models. At microscopic scales, the logarithmic term changes sign, producing repulsion that prevents matter from collapsing into a singularity. The corresponding logarithmically corrected Schwarzschild metric yields parameter-free, a priori predictions for the shadows of Sgr A* and M87* that agree with Event Horizon Telescope observations. Six falsifiable predictions for unobserved black holes, particularly NGC315, can discriminate this metric from the Kerr solution. On galactic scales, the same logarithmic term fits rotation curves of the Milky Way, Andromeda, and NGC2974 using only ordinary matter, and passes the Bullet Cluster lensing test. Tidal effects in the Solar System are far below current experimental limits, ensuring consistency with the equivalence principle and parameterized post-Newtonian tests. We further derive the modified field equations via coarse-grained variation (Appendix B) from the effective action of a quantum vortex background, thus providing a more complete theoretical bridge to the modified Poisson equation and metric used in the main text. This effective theoretical framework indicates that various gravitational phenomena from black holes to galaxies may share a common quantum topological origin. It provides a unified, testable alternative to the dark matter problem, and also points out a potential path for the observable detection of quantum gravity effects.