Sequential Updates Do More Than Remember: Computation Over History in Test-Time Learning

Test-time learning is usually viewed as writing information into a fast state. We show that sequential test-time updates can do more: because each update acts on a state transformed by earlier updates, their dynamics can compute jointly and order-sensitively over history. We trace this historical computation through four stages: generation, visibility, behavioral use, and acquisition. For a broad class of smooth sequential updates with $\eta$-scaled history dependence, an interaction involving $r$ distinct histories cannot appear before order ${\eta }^r$, and its leading order-sensitive structure is determined by update chronology. In an exactly solvable DeepLinear specialization, these interactions survive in the predictor but attenuate rapidly with support; nuisance-cancelling trajectory contrasts expose rather than amplify them. The same chronology-sensitive structure persists in nonlinear learners and across two sizes of the official pretrained Test-Time Training MLP (TTT-MLP). Task supervision then recruits causal pairwise composition through the evolving fast state in pretrained TTT-MLP; separately, ControlledTTT reveals functional contributions at support two, three, and four. Finally, dense training trajectories show that deeper historical support reaches stable behavioral usefulness later, consistent with a conditional small-step timescale mechanism. Together, these results identify sequential test-time adaptation as a mechanism not only for storing history, but for computing over it.