StatTS-FM: Statistical Structure as an Inductive Bias for Time-Series Foundation Models

Pointwise forecasting accuracy does not determine whether a prediction preserves the statistical structure of a time series. We study statistical fidelity of the predictive mean through its empirical value distribution, spectral composition, and lag dependence. We introduce StatTS-FM, a framework for supervised foundation-model adaptation that connects statistical representation, learning, and prediction. Trend, seasonal, remainder, and ACF/PACF-based tokens represent history; statistical losses constrain the forecast mean; and historical drift conditions predictive location and scale. We adapt a frozen TimesFM-3 backbone through low-rank adapters and these modules. We show how marginal supervision controls projected location and spread, and derive a statistical-constraint-dependent generalization bound under stationary $\beta$-mixing and uniform loss-moment conditions. We evaluate point forecasting across twelve benchmarks and examine the statistical design on ETTh1, Electricity, and Weather at two prediction horizons. Against a supervised raw-patch adaptation of the same backbone, StatTS-FM reduces macro-average MSE by 7.88% while lowering marginal, spectral, and lag-dependence errors. Without the statistical losses, the structured model improves MSE over raw-patch adaptation but worsens all three fidelity metrics; adding these losses improves both with the architecture fixed.