Classical neural scaling laws model excess risk as a separable power law in model capacity $N$ and dataset size $D$.
At finite budgets, however, empirical learning curves and compute-optimal allocations depart from this log-linear frontier. This paper analyzes how spectral and domain heterogeneity govern these finite-scale departures.
- Along the capacity axis, Mellin--Barnes analysis of covariance resolvents yields a three-term expansion where competing spectral and regularization modes produce finite-scale curvature and logarithmic transitions.
- Because fitting multiple power laws on narrow pilot grids is ill-conditioned, we eliminate the loss floor via first-order Aitken differencing to obtain a well-conditioned closed-form initializer.
- Along the data axis, KKT duality reduces $K$-domain mixing to scalar water-filling, yielding finite-budget bottleneck bounds, an online sampling curriculum, and a simplex Hessian metric for cross-domain transfer.
- Along the iso-compute frontier $C=6ND$, we prove that the optimal allocation exponent drifts in proportion to capacity and data exponent variances, while Schur-complement reduction separates intrinsic drift from frozen-hyperparameter bias.
Across 45 operator spectra, Pythia (70M--12B), Chinchilla (245 runs), and RegMix (17 domains), our estimators achieve $5.89\%$ holdout MAPE, cut high-compute allocation regret by up to $76.4\%$, and reduce cross-scale mixture RMSE by $59.4\%$ with zero selection regret.