Empirical Definitions of Fractal Entropy: Spectral, Dynamical, and Structural Measures in Multifractal Analysis

This release accompanies the paper Empirical Definitions of Fractal Entropy: Spectral, Dynamical, and Structural Measures in Multifractal Analysis This paper presents a fully empirical investigation of what “fractal entropy” can meaningfully denote in multifractal analysis when examined under a fixed, reproducible measurement protocol. Rather than proposing a new theoretical, thermodynamic, or information–theoretic definition of entropy, the study addresses a more basic and largely unresolved question: which operational entropy measures actually encode independent empirical structure in multifractal data, and which are redundant descriptions of the same underlying geometry? Using a composite ensemble of 186 datasets—123 empirical measurements drawn from diverse physical domains and 63 synthetic reference datasets—the paper evaluates four commonly used entropy proxies derived from multifractal analysis: • spectral width entropy• spectral area entropy• spectral curvature entropy• emergence–dimension variability All datasets are processed identically using the Unified Temporal–Measurement Framework (UTMF v5.2), ensuring strict cross-domain comparability and full reproducibility. A central result is that multiple spectral entropy constructions are empirically near-degenerate. Despite being motivated differently in the literature, spectral width and spectral curvature collapse onto an almost one-dimensional manifold within the family of spectral entropy measures, indicating strong redundancy. Spectral area entropy captures partially overlapping structure but does not define an independent spectral axis. By contrast, emergence–dimension variability behaves orthogonally to all spectral entropy measures. This separation demonstrates that static multifractal spectrum shape and subset-level emergence variability encode genuinely distinct descriptive information. In other words, spectral complexity and emergence variability cannot be reduced to one another under a fixed measurement protocol. Principal component analysis reveals that the resulting entropy landscape is effectively low-dimensional but not one-dimensional. More than 90% of the variance is captured by two dominant empirical axes, corresponding to: consolidated spectral complexity, and emergence–dimension variability. When datasets are projected into this two-dimensional entropy space, empirical and synthetic datasets separate spontaneously, without supervision, thresholding, or domain conditioning. Empirical datasets occupy a compact and highly structured region of admissible entropy space, while synthetic datasets explore a much broader range of configurations. This indicates that real physical systems are subject to empirical constraints that are not generically reproduced by stochastic reference processes. All analyses are conducted strictly at the measurement level within the empirically established Fractal Asymmetry Kernel (FAK) space. No assumptions are made regarding dynamics, causality, optimization principles, or thermodynamic entropy. The results therefore provide a model-independent empirical foundation for subsequent studies of stability, incremental structure, and temporal organization in multifractal systems. This archive contains:• the full paper in PDF format• a single self-contained Python analysis script reproducing all figures and results• a README.md describing scope, usage, and reproducibility All results are fully reproducible from a single archived UTMF v5.2 composite metadata snapshot. No raw experimental data are included, and no multifractal computations are re-run. What this paper actually does This paper does not introduce a new theory of entropy.Instead, it empirically tests which commonly used “fractal entropy” measures are genuinely independent when evaluated under a single, reproducible multifractal measurement protocol. By comparing multiple spectral and emergence-based entropy proxies across a large cross-domain dataset ensemble, the paper identifies which measures are redundant, which encode distinct structure, and how real physical datasets occupy a constrained region of admissible entropy space relative to synthetic references. Citation If you use this archive, please cite both: the Zenodo record itself, and the accompanying paper included herein.