The TDI Atlas: Topological Deformation as a Cross-Domain Signature of Learned Neural Representations

Neural networks transform data through a sequence of learned representations, yet most empirical analyses of these transformations rely on linear probes, activation statistics, or dimensionality estimates. This paper introduces the Topological Deformation Index (TDI), a persistent-homology-based measure of how strongly a network deforms the topology of a dataset as representations propagate through successive layers. We define TDI as a layer-wise sum of Wasserstein distances between persistence diagrams computed from activation point clouds, and evaluate it across a cross-domain empirical atlas of 373 datasets spanning 25 scientific and applied domains. The atlas uses two filtrations, Vietoris-Rips and Alpha complexes, and two homology dimensions, H0 and H1, to construct a topology fingerprint for each dataset-model pair. We further introduce two derived diagnostics: the signal ratio, which compares trained-model deformation against random-label deformation, and purity gain, which quantifies class-local reorganisation in the learned representation. Across the atlas, topology fingerprints reveal recurring domain-level patterns and seven empirical topology archetypes, including pre-separated manifolds, high-dimensional sparse manifolds, loop-rich geometries, and topological amplifiers. The results suggest that topology is not only a property of data, but also a measurable property of how small neural models reorganise data. TDI provides a practical framework for diagnosing representation complexity, selecting small-model architectures, evaluating topology-aware lifts, and identifying datasets where structure rather than scale is the dominant modelling constraint.