Cross-Scale Fractal Coherence: Hierarchical Multifractal Analysis of Urban Form and Environmental Stress in Four U.S. Megacities

Urban morphology has traditionally been studied through single-layer Euclidean descriptors, yet the cross-scale architecture of cities — how parcels, buildings, and blocks cohere hierarchically — remains poorly quantified. This paper develops a hierarchical fractal framework that simultaneously characterizes self-similarity, multifractality, and cross-scale coherence at three nested morphological scales. Using a 2020 geodatabase of 134,863 census blocks across four U.S. megacities (New York, Chicago, Houston, Los Angeles) containing 7.3 million parcel polygons and 4.4 million building footprints, we compute box-counting fractal dimensions, gliding-box lacunarity, and the multifractal spectrum f(α) at block, parcel, and building scales. We introduce three novel indices — the Multifractal Heterogeneity Score (MHS), the Fractal Harmony Index (FHI), and Cross-Scale Consistency (CSC) — to operationalize hierarchical coherence. Pooled and city-stratified models reveal that hierarchical-fractal descriptors explain up to 78% of crash risk variance and 70% of PM2.5 variance, with threshold non-linearities consistent with the Edge-of-Chaos hypothesis. City-specific analyses expose a Simpson’s paradox masked in pooled correlations and demonstrate that effective morphological interventions must be city-tailored. The findings argue for a shift from single-layer Euclidean planning to complexity-informed design anchored in cross-scale fractal balance.