Small Enough to Know Everything: The Fully-Enumerable Transformer as an Instrument for the Science of Delayed Generalization

Tiny transformers trained on fully-enumerable tasks occupy an unusual position in the study of grokking: every input can be evaluated, every generalization ceiling can be computed exactly, and hundreds of seeds cost minutes. We argue this regime is a scientific instrument with four capabilities that approximate settings cannot offer: (a) exact, falsifiable generalization ceilings; (b) task surgery that manipulates one structural variable while provably fixing all others; (c) direct observation of every weight; and (d) survival-time statistics over many seeds that recast "does not grok" as a censored observation. The obvious objection is that laws characterized at 10^4 parameters may not mean anything beyond them. We answer it with a preregistered conservation study: three task-side laws established at 12K parameters -- a recoverability-ceiling law, a role-conflict delay law, and a weight-decay response law -- are re-measured under an identical from-scratch protocol at 12K, 1M, and 50M parameters (a 4,000x span; 360 runs plus a 44-run control arm). The ceiling law and the delay law are conserved (0/144 Holm-corrected ceiling violations; Spearman rho >= 0.75 at every scale, permutation p < 1e-4), while the weight-decay law deforms systematically, steepening with scale. Preregistered controls show the 50M role-conflict deficit survives learning-rate adjustment and a tripled budget. Conservation was tested against criteria frozen before data collection, and the third relationship fails that test even at the original scale, which is what tells us the test could have failed. The series scales the model while the tasks stay enumerable, so what it licenses is specific: the quantities this regime computes from a task keep predicting once the model has outgrown the regime itself.