Partial Spectral Closure: Fractals, Recursive Geometry, and Incomplete Manifold Completion

This paper develops a closure-theoretic reading of fractal geometry. Rather than treating fractals as anomalous or merely irregular sets, we interpret them as partial closure structures recursive domains that approach closure without fully resolving into integer-dimensional manifolds. Within this framework, Hausdorff dimension measures the degree of achieved geometric closure, while spectral dimension tracks admissible mode organization on therecursive domain. The manuscript further argues that many physically significant fractals arebest understood through the coupling of self-similar geometry and spectral scaling, linkingfractality to Laplacian eigenvalue growth, anomalous transport, localization, and generalized quantization. A review section positions the closure view relative to classical fractal geometry, spectral-fractal analysis, fractal spacetime models, and technical closure schemes in turbulence and combustion. The central thesis is that fractality is not a geometric defect but the stable residue of incomplete yet organized manifold completion. This places recursivegeometry, spectral closure, and shell organization on a common ontological ladder and recasts fractal dimensionality as a closure residue rather than an isolated anomaly.