The Fractal Poincar´e Duality Theorem for Manifolds with Boundary and Its Applications in Quantum Gravity

By introducing fractal boundary homology and relative cohomology theories,this paper rigorously extends the Poincar´e duality theorem to fractal manifoldswith boundary. Based on Ahlfors regular measures and the recursive constructionof fractal chain complexes, combined with a boundary-modified version of the fractalStokes theorem, the following duality relation is established:Hkµ(X) ∼= HDH−kµ(X, ∂X).This duality is induced by a modified integral:⟨α, β⟩∂ =ZXα ∧µ β −Z∂Xι∗(α ∧µ β).Through Monte Carlo grid simulations (fractal dimension DH = 2.3) and LIGO?Virgo data analysis (boundary correction factor η = 1.12 ± 0.04), the physicalvalidity of the theorem is verified. This result provides new mathematical tools foraddressing the black hole information paradox and fractal holographic duality inquantum gravity.