Representation-Invariant Analysis of Elastic Tensors via Riemannian Geometry: Crystal Symmetry Classification, Property Prediction, and Anomaly Detection on 11,265 Materials

We demonstrate that Euclidean analysis of elastic stiffness tensors is fundamentally flawed: distances between materials change by up to 96,913× depending on whether stiffness or compliance representation is used. The affine-invariant Riemannian metric eliminates this representation dependence exactly. Using proprietary Riemannian descriptors, we achieve 94.4% crystal system classification accuracy from only 5 features (vs. 72.4% for 21D Voigt baseline) and 96.9% with combined features on the de Jong et al. (2015) dataset (1,181 DFT-computed materials). Scaling to 11,265 materials from the Materials Project, we identify 2,018 entries (15.2%) with non-physical elastic tensors that fail positive-definiteness validation. Property prediction achieves R² = 0.989 for bulk modulus and R² = 0.977 for shear modulus. Precomputed Riemannian features for all 11,265 materials are available via commercial API at materials.omnisciences.io.