A Falsifiable Criterion for Understanding: Commensurability of Learned Step Sequences

The question whether a trained model has understood a rule or merely memorized its training set is currently answered with output-side measurements: held-out accuracy, or at best a hand-designed mechanistic probe. We argue that all such measurements share a defect — they are defined after the correct answer is known, and therefore explain rather than predict. We propose a criterion defined on the model's internal step sequence instead of its outputs. A system has understood a structure of period p if and only if the period of its internal step sequence is commensurable with p — that is, if the two sequences possess a common closure. Concretely: write the model's learned frequency as omega-hat and define the reduced frequency kappa = p times omega-hat / 2 pi. The criterion is that the dominant spectral components of the model satisfy kappa in the integers (or a simple rational) within a tolerance fixed by the closure condition, not chosen by us. We show that this criterion re-reads the three phases of grokking (memorization, circuit formation, cleanup; Nanda et al., 2023) as a single trajectory of a commensurability defect delta. It gives a structural reason for the abruptness of generalization: frequency converges continuously, but landing on the integer lattice is a discrete property. We state four predictions, two of which are constructively falsifiable on publicly available setups, and we specify in each case what result would refute the criterion.