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

This paper proposes an extremely simple logarithmically modified gravitational potential, whose most prominent feature is the cross-scale unity from black hole "singularities" to galactic dynamics: through the sign reversal of the gravitational potential at the microscale (r<r_*≈8.792×10^-11m), dynamics avoids any matter collapsing into "singularities". Under this mechanism, the angular diameter of black hole shadows and the orbital velocities of high-speed stars orbiting them can be a priori predicted without introducing any free parameters (such as spin, eccentricity, etc.), and finally extended to explain galactic rotation dynamics. By analyzing the mathematical asymptotic behavior of all dark matter halo models, we obtain a core finding: adding a simple logarithmic correction term to the original Newtonian gravitational potential: \( Φ(r)=-\frac{GM}{r}-\frac{(kG_h M^2 (ln⁡r+1))}{r} \) a possible framework that avoids collapse to (eliminates) singularities and explains the flattening of galaxy rotation curves under the same physical mechanism can be obtained. Among them, the logarithmic term " " is the key to realizing the cross-scale effect of "repulsion at short distances and attraction at long distances". Without introducing additional free parameters (such as spin), we a priori predict black hole shadows (Sgr A*, M87*) that are consistent with EHT observations; then, based on the same physical mechanism, a priori calculate the "perihelion" velocities of high-speed stars (S4714, S62) orbiting black holes, which are consistent with observations; finally, through this mechanism, we posteriori fit galaxy rotation curve data (Milky Way, Andromeda Galaxy, NGC2974) and other cross-scale verifications (spanning nearly 30 orders of magnitude from black hole singularities to galaxies), initially proving that the framework shows a high degree of observational consistency in both strong gravitational fields (black holes) and weak gravitational fields (galaxies) (especially the a priori prediction of black hole shadows). Based on this, we further provide almost unique quantitative a priori predictions for the angular diameters of six candidate black hole shadows (such as NGC4261, M84, etc.) that have not been observed by EHT under this theoretical mechanism (unable to adjust spin α and inclination i to match observations), as observable predictions awaiting future verification (e.g., NGC4261 is predicted to have a shadow angular diameter of 5.9 ~6.3μas, M84 is predicted to have a shadow angular diameter of 9.8 ~10.7μas, etc.). Core feature: The logarithmic correction is not introduced to address any single phenomenon. It originates from the universal result of the asymptotic mass distribution \( ρ(r)∼r^{-3} \) of dark matter halos, and is consistently reflected in: 1) the regularization of the central gravitational potential; 2) the formation of black hole shadows; 3) the dynamics of high-speed stars; 4) galactic rotation curves. These manifestations form an inseparable whole. This framework not only achieves, for the first time, a unified description of gravity from the microscopic to the macroscopic scale (requiring only ordinary matter mass) but also provides an observable and reproducible empirical framework for quantum gravity theory, potentially freeing it from the long-standing research method of pure mathematical modeling (distant from actual observations) and transitioning to physical verification.