Fractal Entropy as a Constrained Geometric Object: Definitions, Bounds, Stability, and Admissible Trajectories

This repository contains a single standalone research article that consolidates and closes an empirical research programme on fractal entropy. The work is complete, citable, and archival in nature, and is not a preprint or a journal submission. Fractal Entropy as a Constrained Geometric ObjectDefinitions, Bounds, Stability, and Admissible Trajectories Fractal entropy is widely used across the sciences as a compact descriptor of multifractal structure, complexity, or variability. Yet despite its ubiquity, its empirical meaning is often unclear: multiple operational definitions coexist, stability behavior is inconsistent, and entropy magnitude is frequently conflated with noise, disorder, or randomness. This paper provides a consolidated, measurement-level account of fractal entropy as it is empirically observed under a fixed, fully reproducible multifractal measurement protocol. Rather than proposing new entropy definitions, models, or theoretical interpretations, the paper synthesizes the results of four previously released empirical studies into a single coherent geometric description. Together, these studies establish fractal entropy as a constrained geometric object, whose informative content lies not in magnitude alone, but in admissible bounds, stability structure, and trajectory behavior under systematic measurement variation. What this paper shows Across a large and heterogeneous ensemble of empirical datasets, and under identical measurement conditions, fractal entropy is shown to be: Low-dimensional and partially redundantMultiple commonly used spectral entropy measures collapse empirically onto a single dominant axis of spectral complexity, while emergence-related variability forms an orthogonal and independent component. Sharply bounded and resolution-dependentEntropy values occupy restricted, nontrivial regions of measurement space. These bounds persist across resolutions and are not artifacts of estimator saturation or finite sample size. Stability-qualified rather than convergentStability emerges as a scale-dependent, measure-specific property. No empirical system exhibits a resolution at which all entropy measures stabilize simultaneously. Structurally distinct from noiseSynthetic and noise-based control datasets routinely access higher entropy magnitudes than empirical systems, but fail to reproduce their bounded geometry, stability structure, or admissible entropy trajectories. High entropy is therefore not a proxy for noise or disorder. Trajectory-based rather than staticUnder admissible subset variation, entropy responds through constrained, anisotropic trajectories in entropy space. These trajectories are continuous, directionally organized, and globally confined, establishing entropy as a path-dependent geometric response rather than a fixed scalar property. What this paper is not It does not introduce new entropy measures. It does not propose dynamical, causal, thermodynamic, or informational interpretations. It does not treat entropy as a universal complexity scale. It does not rely on model fitting, learning, or optimization. All results are strictly measurement-level, derived deterministically from archived multifractal outputs, and fully reproducible. Research Lineage and Component Papers This paper is part of a tightly integrated empirical research programme focused on multifractal measurement geometry, structural variability, and the empirical interpretation of fractal entropy.All component studies are fully reproducible, measurement-level analyses released as standalone papers or software archives on Zenodo. Below, the development of the programme is outlined chronologically, with each contribution positioned by its empirical role rather than by theoretical intent. 1. Multifractal Measurement Foundations jedi_mfdfa (software)A minimal and robust implementation of Multifractal Detrended Fluctuation AnalysisDOI: https://doi.org/10.5281/zenodo.17982899 This software provides a numerically defensive, failure-explicit implementation of classical MFDFA.It prioritizes reproducibility, explicit invalid states, and minimal assumptions over estimator flexibility. jedi_mfdfa (paper)A Minimal and Robust Implementation of Multifractal Detrended Fluctuation AnalysisDOI: https://doi.org/10.5281/zenodo.17990463 (published 19-12-2025) Describes the design philosophy, numerical safeguards, and validation strategy underlying the software.This estimator is embedded verbatim into all subsequent analyses. 2. Unified Measurement Framework UTMF-Core (software)DOI: https://doi.org/10.5281/zenodo.17808853 UTMF-Core (paper)UTMF-Core: A Universal Multifractal Measurement Framework for Heterogeneous Physical Time SeriesDOI: https://doi.org/10.5281/zenodo.17905073 (published 04-12-2025) Introduces UTMF as a measurement instrument, not a model: a fixed, domain-agnostic pipeline that applies identical multifractal analysis rules across heterogeneous datasets.This establishes the principle of measurement-level comparability. 3. Stability, Resolution, and Coherence UTMF v5.2 (software)DOI: https://doi.org/10.5281/zenodo.18096777 Subset Design and Stability Regimes in Multifractal Detrended Fluctuation Analysis (paper)DOI: https://doi.org/10.5281/zenodo.18098538 (publisched 30-12-2025) Uses: UTMF v5.2 MFDFA Stability Archive v1DOI: https://doi.org/10.5281/zenodo.18090981 Establishes how subset design and resolution affect multifractal estimates, showing that stability is empirical, scale-dependent, and measure-specific. UTMF v5.1 Coherence (software)DOI: https://doi.org/10.5281/zenodo.17926485 Multifractal Coherence Across Physical Measurement Domains (paper)DOI: https://doi.org/10.5281/zenodo.17927721 (published 14-12-2025) Demonstrates cross-domain coherence of multifractal structure under a fixed protocol, supporting the existence of a shared measurement geometry. 4. Asymmetry Geometry and Emergence Structure Fractal Asymmetry Theory (FAT) (paper)An Empirical Asymmetry Kernel Observed Across Physical DomainsDOI: https://doi.org/10.5281/zenodo.17993683 (published 20-12-2025) Introduces an empirical asymmetry kernel describing normalized multifractal fluctuations, establishing a shared statistical geometry across domains. Emergence Depth: A Hierarchical Coordinate in Multifractal Asymmetry Spaces (paper)DOI: https://doi.org/10.5281/zenodo.18124466 (published 02-01-2026) Defines Emergence Depth as a relational, hierarchical coordinate capturing how datasets integrate structurally within the ensemble. Subsequent papers extend this framework: Structural Memory in Emergent Systems (paper)DOI: https://doi.org/10.5281/zenodo.18149464 (published 05-01-2026) Emergence Is Confined (paper)DOI: https://doi.org/10.5281/zenodo.18192204 (published 09-01-2026) Incremental Emergence in Multifractal Asymmetry Spaces (paper)DOI: https://doi.org/10.5281/zenodo.18210508 (published 11-01-2026) Together, these establish emergence as incremental, confined, and geometrically constrained. 5. Fractal Asymmetry Kernel (FAK) The Fractal Asymmetry Kernel (FAK) (paper)DOI: https://doi.org/10.5281/zenodo.18227029 (published 15-01-2026) FAK-II: Tail Structure, Moments, and Energetic Regimes (paper)DOI: https://doi.org/10.5281/zenodo.18275832 (published 17-02-2026) FAK-III: Contextual Isolation of Extreme Asymmetry Updates (paper)DOI: https://doi.org/10.5281/zenodo.18319688 (published 20-01-2026) These papers establish FAK as a universal empirical constraint on admissible multifractal asymmetry updates, including tail structure, energetic dominance of extremes, and the contextual isolation of rare events. 6. Capstone Measurement Geometry A Universal Measurement Geometry for Multifractal Analysis (paper)Empirical Coherence, Hierarchical Structure, and the Fractal Asymmetry KernelDOI: https://doi.org/10.5281/zenodo.18407377 (published 30-01-2026) Uses: UTMF_CAPSTONE_COMPOSITE_v1.jsonDOI: https://doi.org/10.5281/zenodo.18407143 This capstone paper unifies coherence, emergence, and asymmetry geometry into a single measurement-level reference frame.It defines the fixed empirical geometry within which fractal entropy is subsequently analyzed. 7. Fractal Entropy Series Empirical Definitions of Fractal Entropy (paper)DOI: https://doi.org/10.5281/zenodo.18437352 (31-01-2026) Defines operational entropy proxies and demonstrates dimensional collapse and redundancy. Bounds and Stability of Fractal Entropy (paper)DOI: https://doi.org/10.5281/zenodo.18445836 (01-02-2026) Establishes admissible bounds, resolution dependence, and the absence of global simultaneous stability. Fractal Entropy vs Noise (paper)DOI: https://doi.org/10.5281/zenodo.18452739 (02-02-2026) Uses synthetic controls to show that high entropy magnitude does not imply structure and that noise deforms entropy geometry. Fractal Entropy as a Constrained Trajectory (paper)DOI: https://doi.org/10.5281/zenodo.18463644 (03-02-2026) Recasts entropy as a path-dependent geometric response under admissible subset actualization. 8. Present Paper Fractal Entropy as a Constrained Geometric ObjectDefinitions, Bounds, Stability, and Admissible TrajectoriesDOI: https://doi.org/10.5281/zenodo.18489405 (05-02-2026) This paper synthesizes the entropy series into a single geometric description, closing the entropy layer of the programme.It introduces no new data or analyses, but consolidates what is empirically established—and what is ruled out—about fractal entropy under a fixed multifractal measurement protocol. Synthetic datasets are explicitly included in several component studies as internal reference controls and are treated identically to empirical data at the measurement level. Their systematic geometric divergence from empirical entropy behavior constitutes a key empirical result. Why this matters The results impose concrete empirical constraints on how fractal