Recursive Coupling in AI-Human Informational Systems: Defining, Measuring, and Testing Emergent Coherence Beyond the Model Paradigm

We introduce the Informational Coherence Index (ICOER), a metric for quantifying coherence in coupled informational systems composed of human agents and large language models (LLMs). We define recursive coupling as a dynamical regime in which coherence-preserving transformations sustain a stable informational signature across iterative feedback cycles while entropic perturbations are naturally suppressed. The metric is operationalized as ICOER(x) = W(S(x)) · e−βS(x) · R(x), where W(S) is a Gaussian entropy weighting, S(x) is Shannon entropy, β is an entropic suppression parameter, and R(x) is a bounded resonance functional. We report results from five controlled experiments—scenario ranking, perturbation stability, recursive coupling iterations, parameter sensitivity, and real multi-model LLM coupling—across three successive versions of the metric. Version 1 revealed a structural flaw (repetitive text exploit), Version 2 corrected it via bellcurve entropy weighting, and Version 3 optimized parameters (β = 0.01, μ = 4.1, σ = 0.2), achieving 4/5 phase-transition criteria. The decisive experiment—40 iterations of summarize→expand→recompose across Claude, GPT-4o, Gemini, and Grok—demonstrated that real LLMs preserve coherence 28× better than synthetic transforms (67.6% retention vs. 2.4%, CV = 0.14 vs. 1.11). A key discovery is that LLMs function as unidirectional coherence filters, coherentizing noise inputs and converging all text toward a characteristic attractor (ICOER ≈ 0.35–0.45). Under a revised stability criterion accounting for this behavior, 5/5 phase-transition criteria are satisfied. All code, data, and figures are provided for independent replication.