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

Aiming at the black hole "singularity" puzzle in general relativity and the dark matter dilemma of the flattening of galaxy rotation curves, traditional theories have core limitations of cross-scale fragmentation and reliance on unobservable entities (such as higher-dimensional space, dark matter particles, etc.). Starting from the universal logarithmic asymptotic behavior of mainstream dark matter halo models, this paper proposes a minimalist gravitational potential framework with a logarithmic correction term (endowed with the core mechanism of "near-repulsion and far-attraction"). Combined with quantum vortices and nested AdS/CFT correspondence, we present a microscopic physical picture, and construct a modified Poisson equation and Schwarzschild metric. This framework makes a parameter-free a priori prediction of the black hole shadows of Sgr A* and M87* at the black hole scale, which is consistent with the Event Horizon Telescope (EHT) observations. The a priori calculation of the periastron velocities of high-velocity stars S4714 and S62 yields errors within a reasonable range. At the galaxy scale, it fits the rotation curves of various types of galaxies including the Milky Way and the Andromeda Galaxy, which is in high agreement with observations, and physically resolves the singularity through the short-range repulsive potential. This paper presents rigid a priori predictions for the shadows of 6 unobserved black holes, among which NGC315 can be used as a crucial experimental source to distinguish this theory from the standard Kerr paradigm. Without invoking dark matter or higher-dimensional hypotheses, this framework, only using the logarithmically corrected gravitational potential, makes it possible to uniformly describe the cross-scale gravitational mechanism with only the mass of ordinary matter. It provides an observable and falsifiable empirical path for quantum gravity research, and makes it possible to pull the research paradigm of pure mathematical modeling of quantum gravity (which requires conditions far from contemporary observations such as the Planck energy scale) back to the empirical research of contemporary physics.