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

We introduce leaky-integrator reconstruction, a training-free method that cures the error accumulation of recursive differenced forecasting. Our first contribution is diagnostic: predicting one-step changes and integrating them by cumulative summation, the standard remedy for non-stationarity, is a discrete integrator with a pole on the unit circle, and we show this makes recursive rollout of a nonlinear model diverge, its 336-step error reaching several times that of a well-behaved forecaster (normalised MAE 1.6-3.8 versus about 0.8) across every neural architecture tested. Our second, central contribution is the fix: move the pole inside the unit circle with a leaky integrator H(z) = 1/(1 - gamma z-1), gamma < 1, which provably bounds the accumulated error variance. Applied at reconstruction time with a single fixed gamma=0.9 (no retraining, a two-line change to any deployed one-step or foundation-model forecaster), it shrinks error at every horizon, 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% (43-74% across those architectures) at H=336 (78% with an oracle pole). Crucially, it is provably inert where no pathology exists (stable or joint predictors already at the irreducible rate), making it a safe, general default.