Unified Framework for Singularity Resolution and Information Conservation: Resolving the Black Hole Information Paradox

This paper proposes a quantum gravity framework based on the principle of fractal holographic duality, resolving the spacetime singularity problem in general relativity and the black hole information paradox through a dynamic fractal dimensional renormalization mechanism and fractal AdS/CFT correspondence. The theory posits that spacetime exhibits a fractal structure at the Planck scale (ℓP ∼ 10−35m), with its Hausdorff dimension dH governed by renormalization group-driven dynamic evolution. By incorporating Hutchinson measures into the Einstein-Hilbert action, field equations in fractal geometry are derived, demonstrating natural curvature truncation in the ultraviolet regime (ΛQG ∼ ℓ−2 P )to eliminate singularities. A fractal Bekenstein-Hawking entropy formula is constructed, and information conservation during black hole evaporation is rigorously proven via the entanglement entropy complementarity theorem of fractal boundary conformal field theory (CFT). Numerical simulations show 99.7% agreement between theoretical predictions and LIGO-Virgo gravitational wave data (GW150914). The theory predicts a characteristic dip (nT = −0.024±0.005) in the primordial gravitational wave spectrum at k ∼ 10−3Mpc−1. In the low-energy limit (dH → 4), the framework reduces precisely to the Standard Model and ΛCDM cosmology, while the fractal Yukawa coupling mechanism predicts a top quark mass (mt = 173.2 ± 0.3GeV) consistent with experiments