Why Empirical Transition Classification in Recursive Systems Is Undecidable Without Ψ = ∂S/∂R

This paper assumes the operator law hypothesis Ψ = ∂𝑆/∂𝑅as formalized in Autological Recursion as a Law Hypothesis: Scope, Decidability, and Falsifiability Conditions (Elbasan, latest published version). No re-justification is attempted and no empirical validation is claimed. The paper provides a single contribution: it identifies a class of research questions—transition and regime classification under recurrence—that remains undecidable under state-only descriptions, and cannot be resolved by semantic labeling. Under the Rule–State Separation Axiom (RSSA), the decisive ambiguity is whether recurrence changes only system state or the system’s admissible structure. This is treated as a measurement-architecture limitation. The paper derives a minimal requirement: any method that claims to classify structural transitions under recurrence must implement a structural response operator equivalent to ∂𝑆/∂𝑅(up to admissible reparameterization). An explicit Ψ-regime declaration (Ψ=0, Ψ≠0, Ψundefined), a counterfactual undecidability analysis (excluding structural response operators), and local falsifiability conditions are provided. --- Intellectual Property & ContactKOGNETIK® is a registered trademark of Serkan Elbasan (Germany).The KOGNETIK Research Series is released under the Creative Commons Attribution 4.0 International License (CC BY 4.0). All scientific works within the series are open for citation and derivative research under proper attribution.For partnerships, translations, or applied development inquiries:✉️ research@kognetik.de · 🌐 https://www.kognetik.de --- 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 is a transformation rule, not a content or level. Domain-independent operator:Valid across biological, cognitive, artificial, social, industrial, and geophysical systems. Non-ascriptive, empirically testable:Ψ compares systems by observable structure and recurrence. Higher-order phenomena as specifications:Learning, adaptation, consciousness, governance, and identity are structured regimes of Ψ.