The formal specification. This paper develops the mathematical operators for multi-scale recursive dynamics—connecting quantum, neural, and cosmic domains through a unified formalism.
https://doi.org/10.5281/zenodo.15784341#Mathematics #CrossScale #FormalizationPart III: Mathematical Framework for Multi-Scale Recursive Dynamics: From Quantum to Cosmic Scales
Building on convergent empirical evidence documented in Parts I and II of the Universal Recursive Dynamics Series, this paper seeks to formalize universal recursive organization through unified mathematical constructs. The mathematics provide a foundation for the empirical convergence documented in Part I while enabling the cosmic extensions explored in Part II, a unified formalism for recursive phenomena from quantum to cosmological scales. All mathematical constructs are proposed as heuristic models intended to guide theoretical development, empirical testing, and computational implementation rather than function as definitive derivations. The framework strives to maintain rigorous falsification criteria while generating testable predictions that distinguish recursive from non-recursive system dynamics. v1.3.2 License changed to CC-BY 4.0 to enable full scientific dissemination. No changes to content. v1.3.1 corrects a LaTeX error that caused the word "latex" to appear at the beginning of the abstract. Moves the Addendum in front of the Appendices for series consistency. v.1.3 corrests minor unicode quotation mark errors in text only. v.1.2 adds an Aperture Addendum and a cross-referencing footnote linking to Part IV, clarifying the recursive aperture model and bridging to the series conclusion. v.1.1 adds missing Li et al. (2012) bibliographic entry that caused (?Miroshnikova et al., 2023) citation error.
A four-part investigation into universal recursive dynamics—from empirical validation across disciplines, through cosmological extension, to mathematical formalization and implications for consciousness and AI.
https://doi.org/10.5281/zenodo.15741966#OpenScience #CrossScale #RecursiveOntologySeries Table of Contents: Universal Recursive Dynamics and the UEF Framework (Parts I - IV)
Introduces a four-part series exploring recursive dynamics as potential universal principles across science. Summarizes each paper’s focus on empirical convergence, cosmic modeling, mathematical formalization, and implications for consciousness and AI. version 1.4 License changed to CC-BY 4.0 to enable full scientific dissemination. No changes to content. version 1.3 replaces DOI for Part IV so that link displays current version. version 1.2 replaces series abstract with a series table of contents for clarity and navigation. verson 1.1 updates Zenodo title to correctly identify papers and Parts I - IV. Adds DOI hyperlinks to referenced published works incorporated.
Systems approach thresholds. Then they reorganize—or collapse. This paper formalizes saturation dynamics across scales: the core recursion equation, boundary functions, and the Recursive Saturation Index (RSI).
https://doi.org/10.5281/zenodo.15338846#PhaseTransitions #Nonequilibrium #CrossScaleSaturation 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.
Fractal Sentience