Fractal Cohomological Shadows: Renormalization, Spectral Profiles, and Functoriality

We develop the theory of Fractal Cohomological Shadows (FCS), a renormalization-based invariant of compact metric spaces that couples cohomological persistence under metric scaling with multifractal spectral asymptotics. Building on coarse cohomology and analysis on fractals, we introduce the shadow exponent spectrum, filtered shadow complexes, and a shadow Laplacian that encodes the interaction between harmonic scaling and cochain-level renormalization. We prove foundational structural results (finite generation, stability, and functoriality under coarse maps), compute new examples (Cantor sets, post-critically finite fractals, boundaries of hyperbolic groups, and self-similar carpets), and formulate a web of conjectures tying shadow exponents to Hausdorff and spectral dimensions. The resulting framework resists classical simplifications while opening several concrete pathways for computation and comparison, suggesting a deep invariant with long-term consequences across geometric analysis and algebraic topology.