The Relational Encoding Index (REI): A Graph-Theoretic Framework for Structural Coherence Monitoring in Autonomous AI Systems

The deployment of Large Language Models (LLMs) as persistent autonomous agents introducesa class of operational failure — relational coordination failure — that is not adequatelycaptured by existing output-probability or post-generation monitoring approaches. Wepropose the Relational Encoding Index (REI), a runtime structural observability metricderived from graph-theoretic analysis of transformer attention topology.We first establish a precise failure taxonomy distinguishing (i) relational coordinationfailure, arising from breakdown of attention-mediated context integration, from (ii) factualretrieval failure, arising from MLP-layer knowledge retrieval — and restrict the scope of REIclaims to failure mode (i). Within this scope, REI operationalises the integration–segregationbalance of the inference graph using three established network-science measures: globalefficiency (Eglob ), modularity (Q), and participation entropy (HPC).We present the metric’s formal derivation, prove that it satisfies four stated design con-straints — monotonicity, numerical stability, non-negativity, and bounded range — as formalpropositions with analytic proofs, and characterise its information-theoretic interpretation.We introduce a sink-masking protocol to address the confounding effect of attention-sinktokens on graph topology, and establish a precise temporal model for runtime deployment.We present a formula sanity check under synthetic conditions and identify a six-priorityempirical validation programme including a mandatory baseline comparison.This paper is positioned as a research proposal and theoretical framework. All operationalthreshold values are theoretical placeholders requiring empirical calibration; production-gradereliability claims require the validation programme described herein.