Rule-Space Dynamics under Recurrence: A Minimal Structural Theory of Adaptive Systems

Adaptive systems operate under recurring conditions. Machine learning models encounter repeated training data, organisms repeatedly interact with environmental conditions, and organizations repeatedly solve operational problems. Such systems therefore evolve under recurrence. Most theoretical descriptions of adaptation model trajectories in state space, where system parameters change while the governing rules remain fixed. However, many adaptive processes eventually encounter limits that cannot be overcome through further optimization within the existing rule structure. At such points structural change becomes necessary. This paper introduces a minimal structural framework describing adaptive systems as evolving simultaneously in state space and rule space. Rule objects define the objectives, constraints, and update operators governing system dynamics and induce rule classes that determine reachable regions in state space. Persistent mismatch between task requirements and structural capability produces structural load under recurrence. When load exceeds admissibility thresholds, rule-space operators may modify rule objects and thereby produce structural transitions. The framework formalizes three structural properties: structural inertia, load-coupled structural transition, and rule-space path dependence. Together these results describe how adaptive systems generate trajectories not only in state space but also in rule space. Finally, the framework provides an operational interpretation of the operator Ψ = ∂S / ∂R which measures the structural sensitivity of rule classes to changes in recurrence regimes. 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.