Anatomy of Capability Emergence: Scale-Invariant Representation Collapse and Top-Down Reorganization in Neural Networks

Abstract: Capability emergence during neural network training remains mechanistically opaque. We track five geometric measures across five model scales (405K-85M parameters), 120+ emergence events in eight algorithmic tasks, and three Pythia LLMs (160M-2.8B). We find: (1) training begins with a universal representation collapse to task-specific floors that are scale-invariant across a 210X parameter range (e.g., modular arithmetic collapses to RANKME ~ 2.0 regardless of model size); (2) collapse propagates top-down through layers (32/32 task X model consistency), contradicting bottom-up feature-building intuition; (3) a geometric hierarchy in which representation geometry leads emergence (75-100% precursor rate for hard tasks), while the local learning coefficient is synchronous (0/24 precursor) and Hessian measures lag. We also delineate prediction limits: geometric measures encode coarse task difficulty but not fine-grained timing (within-class concordance 27%; when task ordering reverses across scales, prediction fails at 26%). On Pythia, global geometric patterns replicate but per-task precursor signals do not -- the precursor relationship requires task-training alignment that naturalistic pre-training does not provide. Our contribution is the geometric anatomy of emergence and its boundary conditions, not a prediction tool.