Minimal Admissible Entropy Geometry

This repository contains the research article “Minimal Admissible Entropy Geometry”, an empirical study identifying the minimal geometric constraints that entropy-based measurements must satisfy across heterogeneous empirical systems. The paper addresses a deliberately restricted but foundational question: What is the simplest geometric structure capable of supporting bounded entropy, heavy-tailed updates, and dataset-specific variability without invoking dynamics, stability, or generative mechanisms? Rather than proposing a model, theory, or evolutionary principle, the article formulates this question strictly at the measurement level and answers it empirically. No dynamics, causality, temporal interpretation, or physical mechanisms are introduced. Contents This archive contains: the full paper in PDF format a single Python analysis script reproducing all figures and reported statistics(P20_Minimal_Admissible_Entropy_Geometry_Analysis.py) a README file describing scope, methodology, and reproducibility Measurement context All analyses are performed within the Unified Temporal–Measurement Framework (UTMF v5.2), using a fixed, validated, and fully reproducible multifractal measurement protocol. Within this framework: Fractal entropy is treated as a geometric object defined on admissible measurement space, entropy updates are represented as increment trajectories rather than state variables, all observables are derived deterministically from archived multifractal metadata, and no assumptions are made about underlying processes, evolution, or temporal ordering. The paper operates exclusively on archived measurement outputs and does not recompute or modify raw data. Scope and objective The purpose of this paper is not to explain why entropy behaves as observed, but to establish what must be true of any geometric space in which the observed entropy behavior is possible. Specifically, the paper extracts a minimal set of empirical constraints, including: bounded entropy support, finite dispersion without divergence, heavy-tailed but symmetric update statistics, factorization between magnitude and direction, and dataset-specific trajectories within a shared admissible domain. These constraints are presented as observational necessities, not assumptions or axioms. Position within the research programme This article serves as an existential constraint paper within the broader measurement programme. It builds directly on the consolidated entropy analysis stack established in the following works: Fractal Entropy Series Fractal Entropy as a Constrained Geometric Object: Definitions, Bounds, Stability, and Admissible TrajectoriesDOI: https://doi.org/10.5281/zenodo.18489405 Entropy Geometry and Extreme Event Realization in Empirical SystemsDOI: https://doi.org/10.5281/zenodo.18501234 as well as on the Fractal Asymmetry Kernel (FAK) framework, which characterizes admissible update geometry independently of entropy. The present paper does not extend these frameworks.Instead, it asks what minimal geometric structure is sufficient to host all previously observed entropy behavior. What this paper does (and does not do) What it does Extracts empirically necessary geometric constraints on entropy space Separates structural necessity from interpretative sufficiency Identifies which features are invariant under dataset heterogeneity Provides a reference baseline for all subsequent interpretative layers What it does not do Propose a generative or dynamical model Introduce time, evolution, or causal structure Attribute meaning, function, or optimization to entropy Treat entropy as a driver, control variable, or predictor Use synthetic tuning, fitting, or adaptive construction How to read this paper This article is intended to be read as a boundary-setting reference, not as a theory paper. Readers are encouraged to treat the results as answers to the question“what must be geometrically possible?” rather than “what is happening?”. The paper is deliberately minimal: it defines what cannot be avoided by interpretation, and leaves all higher-level meaning strictly optional and post hoc. All methodological detail, empirical evidence, and reproducibility guarantees are documented explicitly in the paper. Citation If you use this archive, please cite both: • the Zenodo record itself, and• the accompanying paper included herein.