Saturation Thresholds in Recursive Systems: A Universal Energy Field Framework for Modeling Across Scales

This paper introduces a cross-domain modeling framework based on saturation thresholds in recursive systems. Drawing on concepts from consciousness studies, complexity science, ecology, cosmology, and artificial intelligence, it proposes a Universal Energy Field (UEF) architecture governed by the Recursive Saturation Index (RSI) and a boundary function B(Mk), defined as the ratio of internal to external recursive stress. These formal tools describe how systems evolve, stabilize, or reorganize as recursive parameters approach critical thresholds. The paper presents ten demonstrative cases — from neural integration and biological aging to climate feedbacks and early galaxy formation — illustrating how recursive saturation drives systemic reorganization. By unifying cross-scale phenomena under a recursive saturation model, this framework offers a falsifiable, interdisciplinary lens for understanding emergence, boundary shifts, and phase transitions across both physical and cognitive domains. Version 3.2:  License changed to CC-BY 4.0 to enable full scientific dissemination.  No changes to content. Version 3.1 updates or corrects reference entries in the bibliography to ensure accuracy, compliance, and persistent identification. The following changes have been implemented: Authorship and Miscitation Corrections: Armstrong & Vijg (2022) corrected to Wang & Vijg (2022) Cabanela & Rummelt (2023) corrected to Lopez et al. (2021) Smith et al. (2023) (Eos cloud) corrected to Saxena et al. (2025) Chen et al. (2020) corrected to Hipp et al. (2019) BirdLife entry updated with accurate source information DOI and Identifier Updates: BirdLife (2024) — URL updated/corrected Huang et al. (2025) — DOI updated/corrected Kim et al. (2025) — DOI updated/corrected Luhmann (1995) — DOI updated/corrected Marwan et al. (2007) — DOI updated/corrected Michaud et al. (2023) — DOI updated/corrected Snell (2024) — DOI updated/corrected Wheeler (1990  — URL updated/corrected Xiao, M. (2024) — DOI updated/corrected No substantive changes were made to the main text or analysis. Revision Note (June 2025): This version includes typographic and layout refinements for clarity and consistency across the trilogy. Abstract formatting has been harmonized, figure placement and table rendering have been confirmed for visual continuity, and minor LaTeX adjustments (including the use of \sloppy in key sections) resolve previous overfull line warnings. No changes were made to the theoretical content or structure of the paper.

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Many-body entanglement in condensed matter systems can be diagnosed from equilibrium response functions through the use of entanglement witnesses and operator-specific quantum bounds. Here, we investigate the applicability of this approach for detecting entangled states in quantum systems driven out of equilibrium. We use a multipartite entanglement witness, the quantum Fisher information, to study the dynamics of a paradigmatic fermion chain undergoing a time-dependent change of the Coulomb interaction. Our results show that the quantum Fisher information is able to witness distinct signatures of multipartite entanglement both near and far from equilibrium that are robust against decoherence. We discuss implications of these findings for probing entanglement in light-driven quantum materials with time-resolved optical and x-ray scattering methods.