Beyond Indices: A General Theory of Predictive Information Loss under Scientific Measurement Compression

Scientific practice routinely replaces rich measurements with ratios, scores, categories, stages, and composite indices. These summaries simplify communication, but they also define which underlying states become indistinguishable to every downstream analysis. We develop Beyond Indices, a decision-theoretic and information-theoretic framework for auditing deterministic scientific measurement compression. Information Compression Loss measures target information removed by a summary, while Predictive Compression Loss measures the corresponding increase in Bayes risk. Compression fibres and target-relevant collisions localize failure to states merged despite requiring different predictions. We establish conditional-information, sufficiency, proper-loss, multistage, coarsening, no-recovery, environment-dependence, and canonical target-sufficient representation results, with explicit regularity conditions and a representation-estimation decomposition. Controlled simulations include a lossless falsification regime and recover increasing loss as target dependence enters discarded dimensions. In a real anthropometric audit, nasal index and its conventional three-level category produce AUC compression gaps of 0.272 and 0.326, respectively, with paired uncertainty intervals. A high-dimensional breast-cancer stress test and environment-shift experiments further expose representation-specific and distribution-specific failure. Matched audits hold models and resampling fixed, separating representational damage from ordinary finite-sample error. The framework therefore changes the scientific question from which downstream algorithm is strongest to whether the retained representation still contains the distinctions the task requires.