Geometric Diversification Hypothesis (GDH): Trajectory Ensemble Heterogeneity as a Universal Early Indicator of Dynamical State Transitions

This restricted-access preprint presents the Geometric Diversification Hypothesis (GDH), a proposed framework for understanding the earliest stages of dynamical state transitions in complex systems. The central hypothesis is that major transitions are preceded by an increase in trajectory-ensemble heterogeneity before a measurable displacement of the ensemble mean state. In this view, the initial signature of an approaching transition is not necessarily a shift in average system behavior, but a progressive geometric diversification of trajectories within the population. The manuscript develops the conceptual foundations of the hypothesis, places it in the context of existing transition theory and early-warning research, and formulates a set of explicit, testable, and falsifiable predictions. Pilot analyses across multiple independent systems are presented as preliminary evidence supporting the proposed mechanism. The framework is intended as a model-agnostic approach applicable to a broad range of physical, biological, and engineered systems in which collective dynamical transitions occur. This record is deposited on Zenodo to establish a verifiable and citable archival version of the work prior to journal publication. The file is maintained as a restricted-access research preprint while the manuscript undergoes further scientific evaluation and development. Access may be granted to editors and reviewers for the purposes of scholarly assessment. The concepts, terminology, hypothesis structure, geometric observables, validation framework, and associated methodological formulations presented in this work constitute original scholarly contributions of the author and are intended for scientific evaluation and future empirical testing. Author: Josef Piskač ORCID: 0009-0004-1434-0724