Fractal, Dynamic Calabiau Manifolds, and Unified Framework for Spacetime Evolution: An Interdisciplinary Exploration from the Hodge Conjecture to Brane Cosmology

This paper proposes a unified framework to describe the multiscale fractal structure of spacetime by integrating the geometric evolution of dynamic Calabi-Yau manifolds, the rigid metric of fractal dimensions, and brane cosmology. Utilizing the modified weighted Hausdorff metric R∗(F) and dynamic modulus parameter equations, we derive an inverse relationship between the fractal dimension DH and dark energy density. Theoretical predictions are validated through cold atom simulations, fractal superconductivity experiments, and numerical compactification calculations. Key findings include: 1. The evolution of fractal dimensions from the Planck to cosmological scales satisfies the renormalization group equation: β(DH) = −0.22DH(3−DH)· 1+ 1 4 ln DH 3 ; 2. Volume fluctuations of dynamic Calabi-Yau manifolds explain residual noise in LIGO-Virgo gravitational wave spectra; 3. The coexistence of fractal dimensions DH = 3,4,6 arises from the nested fractal structure of brane cosmology. This study provides new mathematical tools and experimental paradigms for the physical realization of the Hodge conjecture.