The Rigorous Proof of the Fractal Poincar´e Duality Theorem and its Cross-Dimensional Physical Applications

Based on the geometric algebraic construction of self-similar fractal manifolds,this paper establishes a novel mathematical framework incorporating weightedchain complexes, the fractal Stokes theorem, and nonlocal spectral analysis, rig?orously proving the fractal Poincar´e duality theorem. By introducing a dynamicmeasure compensation mechanism and the theory of fractal functoriality, the frame?work resolves the invalidity of classical topological tools in fractional-dimensionalcontexts. Numerical experiments (Monte Carlo simulations and LIGO data anal?ysis) and physical models (fractal superconducting devices and AdS/CFT exten?sions) verify the reliability of the theory, offering systematic mathematical tools forcross-dimensional phenomena in quantum gravity and condensed matter physics.