Deep Connections Between Fractal Cores and Calabi-Yau Manifolds in Terms of Their Mathematical Structures and Physical Mechanisms

This paper systematically explores the deep connections between fractal coresand Calabi-Yau manifolds in terms of their mathematical structures and physicalmechanisms. By constructing a fractalized iterative function system (IFS) and afractional-order geometric framework, we demonstrate that the self-similarity offractal cores can discretely and recursively modify the compactification processof Calabi-Yau manifolds. Furthermore, by introducing Hausdorff measures andfractal curvature corrections, we rigorously derive an existence theorem that ensuresRicci-flatness of fractalized manifolds under extreme conditions. Physically, wereveal the synergistic role of fractal cores in smoothing black hole singularitiesand mediating string compactification mechanisms. An experimental pathway isproposed to validate theoretical predictions through gravitational wave modulationsignals and fractal perturbations in the cosmic microwave background (CMB). Thisstudy provides a new mathematical paradigm for unifying discrete and continuousspacetime views in quantum gravity theories