Coupled Scaling: A Representational Accessibility Framework for Neural Scaling Laws

Abstract

We ask when two learning systems trained on the same task under a common resource protocol should share a scaling rate and when their rates should differ. Coupled Scaling answers this through representational accessibility: the task-relevant geometry that a specified architecture-optimization system can reach and how that geometry is acquired as resources grow. In an orthogonal model, unsupported target energy sets the asymptotic floor, while unacquired supported energy sets the finite-budget residual. When acquisition can skip high-value directions, the largest fully acquired prefix no longer determines a unique exponent. For target powers $a_j \asymp j^{-b}$, $b>1$, and prefix log-growth rate $0<\rho\leq1$, the sharp attainable interval is $[\rho(b-1),\min\{b-1,\rho b\}]$, and every rate in this interval is realized by a fixed acquisition order on a common support. A rank-window condition identifies when the product law $\alpha=\rho(b-1)$ applies, while a fixed-kernel specialization expresses the same task-system dependence through the task-weighted spectral tail. We then audit released capability trajectories to diagnose measurement and comparison effects and propose a staged test in which independently measured geometry predicts held-out loss, scaling rates, and cross-task reversals.