Emergence Depth: A Hierarchical Coordinate in Multifractal Asymmetry Spaces

We introduce Emergence Depth (ED), a new empirical scalar that quantifies the hierarchical position of a dataset within multifractal asymmetry space. ED is defined using hierarchical clustering on Fractal Asymmetry Theory (FAT) feature vectors derived from the Unified Temporal–Measurement Framework (UTMF) v5.2. Using a fully reproducible archive comprising ten independent UTMF v5.2 runs over 114 heterogeneous physical datasets, we show that ED is stable across runs and insensitive to dataset ordering. ED values span a numerically wide but structurally constrained range, covering several orders of magnitude while remaining non-degenerate. Datasets associated with stochastic or maximally decorrelated sources occupy shallow emergence depths, while highly structured measurements exhibit deeper hierarchical embedding. Emergence Depth is entirely data-driven, requires no physical modeling assumptions, and depends only on archived multifractal outputs. It therefore provides a compact and reproducible descriptor of structural emergence in multifractal asymmetry spaces and establishes a natural intermediate layer between Fractal Asymmetry Theory and higher-level relational analyses.