Structural Learning Theory: A Metric-Topology Factorization Approach

Learning in structured, multi-context, or non-stationary environments involves two orthogonal difficulties. The first is metric: once the correct context is known, how hard is prediction within it? This is the domain of Statistical Learning Theory (SLT). The second is structural: how many local contexts are required, and how can they be discovered from data? This paper develops Structural Learning Theory (StrLT) for the structural axis. We introduce width, the minimum number of jointly contractive and low-risk cells needed to cover a learning problem. Width is incomparable with VC dimension: either can diverge while the other remains bounded. We show that width induces a phase transition: if the allocated number of cells (K<w), learning suffers an irreducible structural error floor; if (K≥ w), the problem reduces to ordinary within-cell statistical learning. To estimate width, we introduce the contractive-similarity (CS) operator, a task-adaptive graph kernel combining geometric locality with predictive compatibility. Its CS Laplacian exposes contractive basins through spectral separation. We further develop the metric slingshot, which reuses low-dimensional latent contraction maps to reduce funnel-learning cost. Together, width, CS estimation, and the slingshot decompose learning into trap discovery and funnel generalization, with deep implications for continual and lifelong learning in an open-ended environment.