The Listening Coefficient: A Diagnostic Framework for Feedback Integrity

ABSTRACT Feedback loops may exist formally while failing functionally because the signal is weakly grounded, the system's internal proxy no longer represents the regulated outcome, structural updates do not reduce discrepancy or the corrective response arrives too late. This Research Note introduces the Listening Coefficient as a bounded, component‐resolved diagnostic of feedback integrity. Four functions are distinguished: signal grounding ( G ), representational fidelity ( R ), update efficacy ( U ) and timeliness ( Tτ ). The multiplicative specification is derived in two forms. When components are calibrated as conditional stage‐completion probabilities, their product Q = G × R × U × Tτ follows from the chain rule. For general bounded component scores, the scale‐normalised geometric mean L = ( GRUTτ )^(1/4) is the equal‐weight, log‐separable aggregation satisfying monotonicity, zero‐boundary, symmetry and idempotence. Q is retained as a strict joint‐completion quantity; L is the interpretable coefficient on the original 0–1 scale. A single canonical outcome‐based estimator of update efficacy is fixed, and explicit protocols are supplied for defining the environmental variable set, selecting the response horizon, testing component redundancy and calibrating interpretation. A controlled synthetic adaptive system demonstrates estimator recovery under aligned and misaligned proxy regimes, aggregation‐rule sensitivity and timeliness sensitivity. A structured retrospective application to Google Flu Trends illustrates how a real system can preserve rapid signal processing while losing representational and corrective integrity. The contribution is diagnostic and methodological rather than a claim of predictive validation. Prospective studies must test whether L predicts later adaptive outcomes beyond existing measures.