Foundations of the Scaling Law

Tianyu Zhao, Zhixiang Ren

Empirical scaling laws relate validation loss to model size, training exposure and compute, providing a quantitative basis for allocating resources. This technical note explains these relationships through three linked analyses. Geometric approximation connects spatial resolution, intrinsic dimension and smoothness to function error; the loss function determines how that error transfers to task risk. Spectral analysis combines feature visibility, covariance eigenvalues and target alignment to describe how additional data reduce unresolved predictive mass. Constrained optimization then gives the parameter-data balance and the training-inference trade-off. We also examine emergence and grokking, separating scoring thresholds from learning dynamics, and test-time scaling, separating candidate coverage from selected-answer accuracy. Applications to vision and molecular prediction show how domain-specific coordinates change the resource comparison. An $\mathrm{SE}(3)$ case study quantifies the dimension dependence of quotient-space approximation gains and compares optimized compute frontiers, alongside separate analyses of conformational coverage and correlated-sample precision.