Fractal Geometry and Computational Complexity Reconstruction: Dimensional Phase Transitions and Polynomial Solutions to NP Problems

This paper proposes a unified framework of fractal geometry and computationalcomplexity, achieving the following breakthroughs:1. Theoretical Innovations:• Developed the Dynamic Fractal Measure Theory (DRFSMT), provingthe Dimensional Phase Transition Theorem for NP problems (criticaldimension Dc = 3.12 ± 0.03), and proposed the Fractal Church-TuringThesis.• Introduced Fractal Holographic Duality (FHEST) as an extension ofAdS/CFT, revealing an exponential compression mechanism for solutionspaces.2. Experimental Validation:• Achieved quantum acceleration T ∝ N0.81 in cold atom quantum simu?lations, with error control below 3%.• Verified the influence of fractal dimensions on quantum gate fidelity throughexperiments with fractal superconducting films.3. Application Innovations:• Defined the physical constraints for fractal chips (D ∈ (2, 4)) and outlineda three-stage technological roadmap.• Identified a computational efficiency valley at the large-scale structure ofthe universe (DH = 1.92) and proposed the hypothesis of dark energy?driven fractal optimization.—