Clock Phases and Signed Feedback in Adaptive Optimizer Memory

Adam computes its first and second moments from the same minibatch. We show that this detail governs how Adam responds to the order of its training data. Linearizing a step, the adaptive contribution to order dispersion splits into a non-negative preconditioner term $V_p$ and a numerator–denominator covariance $C$. We prove that under same-sample coupling $C$ is negative whenever momentum agrees in sign with the current gradient, and that it outweighs $V_p$ when ${\beta }_2$ is close to one; an independent second-moment sample makes $C$ vanish, so decoupled memory can only amplify order dispersion. Coupling also gives Adam a per-coordinate step bound, a known consequence of Cauchy–Schwarz that we show holds if and only if the second-moment input dominates the numerator input. We test both consequences by changing the pairing and nothing else. In a small adapter, manipulating the alignment between numerator and second-moment fluctuations flips the sign of the adaptive correction in all 18 units. At mature ResNet-18 checkpoints, replacing each second-moment sample by an independent draw with the same task and marginal law increases order dispersion on three task pairs, at every checkpoint for parameter and prediction variance. On pretrained Pythia-160M the same replacement drives a small set of coordinates $10^2$–$10^5$ times past the step bound at the first update and the model diverges within a few updates, even when fifteen of sixteen sequences in the second-moment minibatch are shared.