We uncover ELR collapse in language model pretraining: learning rate (LR) and parameter norm govern loss dynamics primarily through their ratio, the effective learning rate (ELR). When ELR is matched across runs, their loss trajectories collapse throughout training despite substantially different LRs and parameter norms. Across optimizers, architectures, datasets, and model scales, mean collapse errors are typically a few x 10^-3, below the seed-to-seed variation measured in a representative configuration. Systematic ablations identify normalization design and the timescale of LR-norm variation as key determinants of collapse precision. Controlled interventions further show that weight decay and Hyperball shape loss dynamics primarily through the ELR schedules they induce. Replacing LR with ELR enables a fitted functional scaling law (FSL) to transfer across norm-control methods. The resulting ELR-based FSL also explains delayed acceleration, a recurring effect of norm control. Together, these results establish ELR as a common coordinate linking LR scheduling, norm control, and loss dynamics.
A trained transformer's weight magnitudes can be summarized by a two-parameter Weibull distribution whose shape $k \approx 1.2$ is stable across layers and models, so the scale $λ$ carries most training-induced movement. What corpus property sets how much $λ$ grows? Using the bigram conditional entropy $D = H(\text{next} \mid \text{prev})$, a training-free statistic computed before training, we find across controlled corruption families a learning-rate-conditioned law, $λ^2 - λ_0^2 = C_0(η) + C_1(η)(H_r - D)^{0.59}$, where $H_r$ is a matched-budget shuffle baseline. The convex exponent is inherited from an independently measured data-side saturation relation rather than fitted directly to the growth curve. After removing the two per-$η$ coefficients, 23 runs spanning an order of magnitude in learning rate collapse onto $(H_r - D)^{0.59}$ with unit slope ($R^2 = 0.941$; direct per-$η$ fits are weaker, $R^2 \approx 0.82$). Because $D$ is computed before training, the law is a forward predictor: an end-to-end self-validation recovers held-out within-family weight growth with 5.7% relative error. The readout holds at model and per-layer resolutions and across two tested architectures, with the functional form preserved and only the coefficients changing. It also marks its boundary: cross-corpus prediction over-predicts code, implicating redundancy as a second axis of a broader $Φ(D,R,A,H)$ data-to-weight framework.