Renormalization of Spacetime Dimension

We propose that the dimensionality of spacetime should be treated as a renormalized quantity rather than a fixed kinematical input. Within a unified framework spanning quantum cosmology and string theory, we analyze how effective spacetime dimension can vary with physical scale through mechanisms such as compactification, holographic duality, and geometric coarse-graining. Geometric structures including Calabi - Yau manifolds, Klein bottles, and Möbius strips are employed as illustrative models for dimensional reduction and transition. We derive key relations linking energy scales, characteristic lengths, and fundamental constants, clarifying how dimensional flow emerges in both quantum mechanical and quantum gravitational regimes. The role of time as a dynamical dimension is emphasized, together with implications for cosmic geometry, holography, and the large scale structure of the universe. Our results suggest that dimensionality itself admits a renormalization group like description, with fixed points corresponding to distinct physical regimes.