Adaptive Recurrence as Algorithmic Time for Length Generalization in Addition

Length generalization asks whether a model can reuse a learned computation beyond the lengths seen in training. Arithmetic makes this test precise because a model trained on short additions should apply the same digit-wise rule for more steps, rather than fit patterns tied to the training range. Many strong extrapolation results, however, rely on explicit aids that tell the model how digits should be aligned. We ask whether a model can length-generalize without such aids. We train small looped transformers with adaptive halting on a controlled decimal-addition task, without positional information or arithmetic-specific side information. The models extrapolate beyond the training range across multiple training regimes. To understand why, we analyze their internal dynamics and find that recurrent depth becomes organized as algorithmic time: later answer positions are resolved at later recurrent phases. These results suggest that adaptive recurrence can turn repeated latent computation into a less hand-engineered route for applying a digit-wise rule beyond the training length.