Entropy Collapse and Concentration Growth as Structural Signatures of Anomalous Behavior: Evidence from Financial Fraud and Blockchain Transaction Networks

Modern fraud and anomaly detection systems are commonly optimized for predictive performance, yet their internal decision mechanisms often provide limited insight into the structural organization of anomalous behavior. This paper develops and evaluates an interpretable descriptor framework derived from Arithmetic Power Geometry (APG) for studying entropy, concentration, and exponent-defect patterns in financial and blockchain transaction data. The central question is not whether APG outperforms state-of-the-art classifiers, but whether APG descriptors expose stable structural tendencies that help explain anomalous activity. We formulate two empirical principles: the Entropy Collapse Principle, which states that anomalous behavior may occupy fewer effective dimensions than normal behavior, and the Concentration Growth Principle, which states that anomalous behavior may be dominated by fewer structural components. Experiments are reported on the Credit Card Fraud dataset and the Elliptic Bitcoin transaction network. In the credit-card benchmark, fraud transactions show lower normalized APG entropy than normal transactions (0.6605 versus 0.7274) and higher concentration (0.3249 versus 0.2696), with statistically significant descriptor differences. In the Elliptic graph setting, graph-derived APG descriptors also show entropy collapse and concentration growth for illicit nodes; however, APG descriptors computed directly on raw Elliptic transaction features show the opposite directional pattern, demonstrating that the entropy-collapse hypothesis is representation-dependent rather than universal in all feature spaces. Predictive experiments show that APG descriptors provide complementary information to simple baselines but do not replace strong supervised classifiers such as logistic regression, XGBoost, or graph neural networks. The study positions APG as an interpretable structural and feature-engineering framework for anomaly analysis, rather than as a standalone state-of-the-art fraud detector.