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

General Relativity (GR) has long been confronted with a fragmentation dilemma regarding black hole singularities and galaxy rotation curves: the former requires undetectable higher-dimensional quantum gravity to circumvent infinite curvature, while the latter similarly relies on undetectable dark matter to provide additional gravitational force. In this paper, we abandon the hypothesis of undetectable entities and reveal that the two challenges may share an intrinsic geometric solution: the universal asymptotic behavior of mainstream dark matter halo models is equivalent to a logarithmically corrected gravitational potential \( Φ(r)∼-(lnr+1)/r \), which originates from the self-response of the curvature divergence at the GR singularity \( (R_{trt}^r∝r^{-3}) \) via Poisson integration. At the microscopic scale, the sign reversal of lnr generates a repulsive effect, thereby avoiding the singularity. The constructed logarithmically corrected Schwarzschild metric is rigorously solved via the Lambert W function, revealing a layered internal structure determined by the black hole mass \( M \) (with thickness \( ∝1/M \)), which realizes the holographic screen of the renormalization group flow under the AdS/CFT correspondence. On this basis, we present parameter-free a priori predictions for the black hole shadows of Sgr A* and M87* that are consistent with Event Horizon Telescope (EHT) observations, and provide rigid falsifiable predictions for unobserved black holes, especially the crucial discriminative prediction for NGC315. On the galactic scale, the logarithmic term can fit the galaxy rotation curves of the Milky Way, Andromeda, and NGC2974 without the additional gravitational force from dark matter, and also successfully passes the test of the gravitational lensing phenomenon of the Bullet Cluster with good agreement with observations. On the other hand, the calculated solar system tidal difference \( (Δg∼10^{-18} m/s^2) \) is far below the current experimental limit, ensuring the validity of the equivalence principle without the need for a shielding mechanism; meanwhile, the Solar System Parameterized Post-Newtonian (PPN) tests are also consistent with GR. This work demonstrates that gravitational phenomena from black holes to galaxies are governed by the spacetime self-response triggered by the GR singularity. It further reveals that macroscopic gravitational systems may be "holographic projections" of quantum topological structures (quantum vortices). This framework thus pulls quantum gravity research from pure mathematical modeling back to the energy scales accessible to contemporary observations, and provides a new direction for thinking about the unification of General Relativity and quantum mechanics.