The Long Delay to Arithmetic Generalization: When Learned Representations Outrun Behavior

Grokking -- the abrupt transition from memorization to generalization after prolonged training -- has been linked to confinement on low-dimensional execution manifolds in modular arithmetic. Whether this mechanism extends beyond arithmetic remains open. We study two sequence-learning benchmarks: SCAN compositional generalization and Dyck-1 depth prediction. Across both tasks and a wide range of learning rates, the commutator defect -- a curvature measure derived from non-commuting gradient updates -- rises well before generalization, with lead times following a superlinear power law (alpha approximately 1.18 for SCAN, approximately 1.13 for Dyck), consistent with prior results on modular arithmetic. Weight-space PCA reveals that spectral concentration is not a universal precursor; the commutator defect is. Causal interventions demonstrate a mechanistic role: amplifying non-commutativity accelerates grokking (roughly 32% on SCAN, roughly 50% on Dyck), while suppressing orthogonal gradient flow delays or prevents it. The three task families form a spectrum of causal sensitivity -- modular arithmetic is rigid, Dyck is responsive, SCAN is intermediate -- yet suppression delays or prevents grokking in all cases, establishing necessity as a universal finding. These results identify the commutator defect as a robust, architecture-agnostic, causally implicated early-warning signal for delayed generalization in transformers.

Grokking -- generalization long after training-set interpolation -- has been accelerated by structure-agnostic interventions (gradient filtering, weight-norm clamping, geometric penalties). Whether the delay specifically measures the time to form task-structured representations has remained observational. We test it causally by injecting representational priors of varying content into a one-layer transformer learning modular addition, via a supervised-contrastive loss whose positives encode (i) the task's true structure ($(a+b) \bmod p$), (ii) a coherent-but-wrong sibling ($(a-b) \bmod p$), or (iii) a random partition -- all with identical loss form, strength, class sizes, and geometry. Whether generalization occurs follows a clean gradation: true 22/30 runs, sibling (same periodic features, wrong combination) 14/15, random (only memorizable) 0/20 (Fisher $p=1.3\times10^{-7}$). A weight-norm-matched control replaying the norm trajectory onto plain cross-entropy generalizes 0/15, ruling out the norm as mediator. Probes show structure formation precedes and predicts generalization in all runs. Only the true structure also accelerates grokking (up to $2.75\times$), but this is dose-dependent and bimodal. We then confirm the mechanism by prediction: because the acceleration is gated by a weight-norm side-effect, clamping the norm during training yields a reliable, standalone accelerator with a median $8.6\times$ speedup (up to $22\times$ on the fastest seeds, under 1000 epochs), growing monotonically as the norm is held lower; the residual stalls also vanish, though significant only pooled over the two mitigations run at both strengths ($0/40$ vs $6/20$, $p=7.7\times10^{-4}$), not per method. The grokking delay is, causally, the time to form the right representational structure -- decided at the level of features, not labels.

Grokking -- the delayed onset of generalization long after a network has fit its training set - -is usually studied in models too large to read completely and reported from single training runs. We instead study a publicly released ~11,856-parameter Llama-style transformer (Glimmer-1-Base) on modular arithmetic, small enough to enumerate its weights, attention, and full input-output map, and we measure grokking as a multi-seed rate rather than a single outcome. In this fully-tractable regime grokking is a conditional, fragile phase transition. It is gated by training-set coverage, whose threshold tracks output cardinality (the modulus) more than task structure, an ordering that holds above the transition and across a ten-fold change in domain size. Weight decay reproduces the Omnigrok inverted-U at 12K parameters, a positive control on the rate measurement. Grokking also sits on a numerical knife-edge: two perturbations of the floating-point environment -- CPU thread count (reduction order) and CPU-versus-GPU execution -- each flip a minority of same-seed outcomes without a detectable shift in the aggregate rate. Decomposition into sub-task specialists helps chiefly by making coverage cheap rather than by adding supervision. Methodologically, multi-seed control under a fixed numerical environment overturns three dramatic single-run narratives in our own data, each a seed confound. The unit of evidence for grokking must therefore be a multi-seed rate under a pinned numerical environment, checked where possible against a direct reading of the model.