Circuit Synchronization Precedes Generalization: A Causal Precursor to Grokking

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.

Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar $(1-β)/(ηλ)$ scaling in the weak-regularization regime. The theory further predicts distinct effects of optimizer choice, distinguishing coupled $L_2$ regularization from decoupled weight decay, and yields causal predictions for interventions that modify the grokking component. We verify all theoretical identities without fitted parameters in a synthetic model where every subspace and relaxation rate is computable in closed form. We further observe genuine delayed generalization in modular addition, where the measured delay follows the predicted scaling and the late-time relaxation agrees closely with the theoretical clock.

We give the first quantitative prediction of grokking delay under AdamW. Treating the delay as a first-passage time, we derive a closed-form law T_grok - T_mem = (1 / 2 kappa_LL eta lambda) log(V_mem / V_star), where V_t = ||theta_t||^2 is the squared parameter norm, V_star is an architecture-dependent threshold, and kappa_LL absorbs the AdamW correction to the clean-SGD contraction rate 2 eta lambda. Calibrating (kappa_LL, V_star) on a single hyperparameter cell predicts grokking delays on 26 held-out runs with MAPE 17.7% over a 41x delay range; the law generalises to MLPs (MAPE 18.0%, N=34) and degrades to 23.3% on cross-task extension (N=46, 43.5x range), with a structured residual in which V_star / V_mem stays comparatively stable within architecture (CV about 14% on the 1L transformer). First-passage of V_t is necessary but not sufficient. A quantile-margin theorem establishes that positive delay requires both norm separation V_mem > V_post and angular reachability of a threshold alpha_star = arcsin(C / V_T_mem^(1/2)), where C is computable from the empirical NTK feature map and the validation-margin quantile. Calibrating C on modulus p=89 predicts alpha_star = 47.2 degrees at p=97 (observed 47.8 degrees, error 1.3%) as a prior cross-cell prediction. Causal interventions that freeze the norm or remove weight decay at memorisation eliminate grokking (0/6 vs. 3/3 baseline), trapping the angular displacement near 12 degrees. kappa_LL is empirically measured per architecture rather than derived from (beta_1, beta_2, epsilon); within-architecture CV stays at most 15% across four architectures, but values differ by about 2x between architectural variants beyond depth alone. Empirical scope is algorithmic tasks (modular arithmetic, sparse parity) under AdamW; whether the law transfers to natural-language scale models is open.