Learning Stable Update Rules: Iteration-Consistency Beyond the Training Horizon

We study time-generalization in neural networks by training a shared iterative cell under explicit supervision of computation length. Rather than treating depth as fixed or learned implicitly, we provide a deterministic target step schedule and penalize deviations from the prescribed execution length, enabling controlled evaluation beyond the training horizon.We perform experiments on three representative dynamical regimes: contracting (Euclidean GCD), attractor-aligned (Log-Fibonacci), and expanding additive (Log-Factorial). We find that models can generalize to larger inputs when the effective iteration depth remains within the trained regime, yet often fail when required computation length increases, even if input magnitudes are moderate. Failure modes track the stability properties of the underlying update dynamics: contraction dampens errors, attractor alignment bounds them over finite horizons, and additive accumulation induces systematic drift.These results suggest that algorithmic generalization depends not only on function approximation but on the stability of learned update rules under composition. Explicit step conditioning improves interpretability and stability of computation depth, but does not by itself guarantee robust extrapolation to longer iterative chains.