Beyond Missing Values: How Imputation Changes Observed Representations in Time Series Foundation Models

Imputation is commonly evaluated by reconstruction error at missing coordinates. In time series foundation models with input-dependent normalization, filling a gap also changes the statistics used to transform the history. It can therefore alter the representations of observed values that remain unchanged in the raw history. Reconstruction scores alone do not describe the changes actually presented to the forecaster. Assessing downstream utility requires understanding how these changes arise and under which observation conditions they are amplified. We analyze the propagation of imputation error through normalization in frozen forecasters. Preserved observations constrain the variance of every feasible completion within each statistical support used by normalization. For positive observed dispersion, we derive a uniform bound on normalized error under ordinary standardization, with a coefficient determined before imputation and sharp when at least two coordinates are missing. Experiments with three frozen forecasters separate reconstruction error, normalized input change, forecast displacement, and forecast loss. In 10% of the 190 baseline conditions, selection by reconstruction score gives forecast RMSE more than 10.14% above the best candidate. For TimesFM, perturbations of $10^{-5}$ observed-history standard deviations yield mean forecast displacement above $0.01$ in the same scale in 454 of 518 constant-patch configurations; fixing historical statistics reduces the mean displacement in every configuration below $4.31\times 10^{-5}$. Imputation evaluation should report observed dispersion and completed scale within the model's statistical supports, and measure forecast loss directly.