Leaky-integrator reconstruction: taming error accumulation in recursive differenced time-series forecasting

Recursive differenced forecasting, the standard remedy for non-stationarity, predicts one-step changes and integrates them by cumulative summation. We show that this reconstruction is a discrete integrator with a pole on the unit circle, so the biased increment errors of a learned nonlinear model are summed without bound and the rollout diverges: at 336 steps its normalised MAE reaches 1.6-3.8 for every neural architecture tested, against 0.80 for a stable linear recursion. We then introduce leaky-integrator reconstruction, a training-free fix that moves the pole inside the unit circle with H(z) = 1/(1 - gamma z^-1), gamma < 1, bounding the accumulation of the model's own increment errors. Applied post hoc with a single fixed gamma=0.9 (no retraining, a two-line change to any deployed one-step or foundation-model forecaster), it beats the traditional recursive integrator at every horizon, with the mean gain over seven diverging architectures and twenty datasets growing from ~3% at H=24 to 23% at H=96, 37% at H=192 and 51% at H=336 (43-75% across those architectures; 78% with an oracle pole), bringing all of them to 0.87-0.97. Based on these extensive empirical experiments, adding a leaky integrator thus improves recursive differenced time-series forecasting.