Scaling Gains Have Shape: Structural Response in Time-Series Foundation Models

Scaling a time-series foundation model is usually summarized by the error reduction from a smaller checkpoint to a larger one. This number identifies whether an upgrade works, but collapses the data conditions that support its gain. We introduce Structural Scaling Response (SSR): the upgrade's log-risk gain on the original series minus its gain on a matched surrogate that preserves the power spectrum while randomizing Fourier phases. Positive SSR means that the original temporal phase structure amplifies the upgrade gain; negative SSR means that it attenuates the gain. Restoring the original phases from low to high frequencies produces an SSR curve that locates the timescales at which the difference emerges. We characterize when this response vanishes and bound the risk perturbation needed to change its sign. On untouched PEMS08, all four larger checkpoints improve accuracy, with channel-aggregated error reductions of 10.02–38.54%. Their SSRs nevertheless separate: the two Toto upgrades and the Chronos upgrade are positive ($0.286$, $0.209$, and $0.113$), while the Moirai upgrade is negative ($-0.143$). Matched surrogates, alternative losses, blocked resampling, mean aggregation, and channel-resampling analyses preserve all four signs. Across 16 upgrade–dataset combinations, SSR is specific to the pair: the same upgrade changes across datasets, and one dataset can contain both signs. On PEMS04 and PEMS08, each positive SSR reaches half its endpoint after phases at periods down to 8–15 hours have returned. Temporal phase structure is therefore a second coordinate of scaling. SSR provides a training-free protocol for measuring where scaling gains arise and comparing upgrades under deployment-relevant temporal structure.