In time-series forecasting, rollout errors are not merely one-step fitting discrepancies: once fed back into the evolving state, they can perturb future predictions. We formalize this effect through local rollout consistency, which compares the next prediction produced under teacher forcing with that produced under free running. In principle, this consistency can be enforced directly by tracking the residual-induced perturbation of the next state. In practice, however, the residual-to-state alignment operator is often difficult to design when the input state and output block have different dimensions, heterogeneous components, or architecture-dependent layouts. We therefore use a first-order local surrogate that penalizes the Jacobian-vector product of the predictor along the actual feedback perturbation, suppressing precisely the directions through which self-generated errors remain visible to future predictions. We further give the same loss a geometric interpretation near a predictive manifold: by controlling residual-induced directions that probe the normal bundle, the objective yields theoretical conditions under which free-running trajectories remain inside a stability tube around the manifold. Experiments on time series forecasting tasks support this view, showing that the proposed regularization improves predictive accuracy. More broadly, our results suggest a training principle for time-series forecasters and sequential predictors: a model should not only make small one-step errors, but also learn not to let its own errors echo into the future. Code is available at https://github.com/Wangzzzzzzzz/Echoes-Of-Error.

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