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.