Recursion-Based Structural Physics (RBSP): A Formal Framework for Fundamental Physics

KOGNETIK formulates a domain-general operator law for recursive systems: Ψ=∂𝑆/∂𝑅 where R denotes recurrence (the activation count of a system’s generative mechanism), S denotes the structural rule-class of the system, and Ψ quantifies structural sensitivity to recurrence. This paper develops, from that operator law, a recursion-based structural physics (RBSP) in which time, space, mass, quantum behavior, gravitation, entropy, and cosmological structure emerge from the dependence 𝑆(𝑅). We provide: • Formal definitions of the operators in metric spaces,• An additional layer of theorems (existence, invariance, phase sequence),• An explicit unit and scaling structure for Δ𝑅, Δ𝑆, and Ψ,• Model classes suitable for numerical implementation (two-state system, structural oscillator),• A functional definition of physical “particles” as structurally stable regions in 𝑆,• Qualitative examples of Ψ-profiles in different regimes,• Falsifiable predictions for quantum experiments, gravitational drift, and cosmological phase alignment. RBSP thus appears as a mathematically precise, empirically approachable and theoretically integrative framework, treating classical and quantum physics as limiting cases of recursion-driven structural evolution. --- Interpretative Scope Statement All derivations of time, space, gravitation, quantum collapse, and cosmological structure presented in this work are structural descriptions of system behavior under recurrence. They do not claim ontological reduction, physical causation, or the replacement of existing state-level dynamics, including but not limited to field equations, Hamiltonians, Lagrangians, or spacetime metrics. The framework operates exclusively at the level of operator-defined structural relations, as formalized by the recurrence–structure operator Ψ = ∂S/∂R. Any ontological, metaphysical, or causal interpretation of these derivations exceeds the formal scope of the present work and constitutes an external reading not implied by the theory itself. --- Intellectual Property & Licensing The KOGNETIK Research Series is released under the Creative Commons Attribution–NonCommercial 4.0 International License (CC BY-NC 4.0). All scientific works within the series may be cited, shared, and adapted for non-commercial research purposes with proper attribution. Commercial use—including consulting, advisory services, integration into commercial platforms, monetized training, certification, or system-level deployment—is not permitted under this license and requires a separate written agreement. Full license text:https://creativecommons.org/licenses/by-nc/4.0/ For licensing, partnerships, translations, or applied development inquiries:research@kognetik.dehttps://www.kognetik.de ORCID: https://orcid.org/0009-0000-8544-4847 Kognetik Series Information KOGNETIK — Minimal Operator Definition of Reflexivity (Ψ = ∂S/∂R) Reflexivity as structural rate-of-change:Ψ = ∂S/∂R measures structural drift under recurrence. Process, not state:Reflexivity specifies a transformation rule rather than a content or level. Domain-independent operator:Applicable across biological, cognitive, artificial, social, industrial, and geophysical systems. Non-ascriptive and empirically testable:Ψ enables comparative analysis of systems via observable structure and recurrence. Higher-order phenomena as specifications:Learning, adaptation, consciousness, governance, and identity are structured regimes of Ψ.