Papers

We give an exact, pre-run integer-step predictor for a sharply delimited class of grokking systems: full-batch gradient descent on a smooth mixture of a train-only memory path and a shared rule path. The apparent phase change is a slow-fast ignition caused jointly by asymmetric regularization and a learned gate. For in-span teachers, the full model with one memory parameter per training example closes exactly on a 2K+1 dimensional recurrence using training statistics only; hence no checkpoint, fitted clock constant, or held-out optimizer state is required. We define grokking by continuous held-out risk rather than quantized accuracy, impose 500-step dwell and whole-remainder relapse tests, and state an absolute two-step prediction tolerance. Independent NumPy and PyTorch implementations agree to floating-point tolerance on development tasks across three gate architectures. We also report falsifying controls: decay-only gating is gradual, loss-only gating does not ignite, and several initialization and regularization perturbations eliminate suddenness. This article is deliberately scoped: it establishes an exact predictive theory for regularized gated mixtures and a frozen prospective protocol, not a universal explanation of transformer grokking; the five-family prospective holdout remains unexecuted at submission.