Arithmetic Power Geometry as an Interpretable Out-of-Distribution Detector for Materials Discovery: A Pilot Study on Matbench Crystal Benchmarks

Machine-learning models for materials discovery are increasingly accurate on standard property-prediction benchmarks, but their reliability may degrade when candidate materials move outside the structural and chemical regimes represented in training data. This study introduces a pilot Arithmetic Power Geometry (APG) framework for interpretable out-of-distribution (OOD) screening in materials discovery. Given a positive material descriptor vector, APG maps squared coordinate mass into a probability distribution and computes normalized entropy, concentration, and exponent-deformation defects. The method is motivated by prior APG Shannon–Rényi control laws and by empirical APG anomaly studies in which entropy collapse and concentration growth are treated as representation-dependent structural signatures rather than universal laws. We evaluate APG descriptors on three Matbench datasets: matbench_mp_gap, matbench_mp_e_form, and matbench_perovskites. A pilot sample of 1,000 materials per dataset is analyzed using composition-derived and structure-augmented APG descriptors. Strong structural separation is observed across the three datasets. When matbench_mp_e_form is treated as the in-distribution reference, the APG Mahalanobis one-vs-rest OOD AUROC is 0.8426; the perovskite reference yields an AUROC of 0.9210, reflecting stronger descriptor separability from the two Materials Project tasks. These results should not be interpreted as state-of-the-art materials prediction. Instead, they support APG as a compact, interpretable structural monitoring layer that can be attached to Matbench, Matbench Discovery, graph neural networks, and high-throughput materials screening pipelines.