Fractal and Quantum Corrections to Black Hole Entropy: Unified Theoretical Framework, Numerical Simulations, and Multi-Messenger Observational Validation

Black hole entropy, as a central concept linking thermodynamics, quantummechanics, and general relativity, is classically described by the Bekenstein?Hawking formula SBH =A4G, which assumes a smooth spacetime. However,at the Planck scale, quantum fluctuations and spacetime discreteness mayintroduce fractal geometric structures. In this paper, we propose a modifiedblack hole entropy formula by integrating fractal measures dkHx, higher-ordercurvature correction terms R(k), and a generalized gravitational constantGH:SBH =14GHZ∂HfR(k) dkHx,where the fractal dimension k and GH are rigorously derived from field the?ory and dynamically evolve under renormalization group (RG) flow. Themodel is further extended to the early universe, predicting a local-type non?Gaussianity parameter fNL ∝ k2induced by scalar perturbations. By com?bining numerical simulations (box-counting methods, correlation dimensionanalysis) and multi-messenger observations (Event Horizon Telescope, gravi?tational waves, CMB anisotropies), the physical effects of fractal correctionsare validated. Comparisons with loop quantum gravity and string landscapetheories highlight the unique advantages of this model in continuously tun?ing degrees of freedom and directly linking to macroscopic observations. Thisstudy offers new perspectives on the black hole information paradox, the na?ture of spacetime, and the unification of quantum gravit