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 & 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.