FEDBRACKET: Controlling Federated Drift through Client-Update Noncommutativity

First-order summaries of federated heterogeneity cannot distinguish client updates whose instantaneous disagreement is similar but whose local trajectories interact differently through curvature. We introduce the leading order-sensitive Hessian–gradient bracket as a control signal for local computation. FEDBRACKET estimates this interaction from sparse client pairs and selected-layer Hessian-vector products, then shortens local trajectories in high-bracket rounds. On federated CIFAR-100 with Dirichlet concentration 0.1, FEDBRACKET reaches 75.3% final accuracy in 368 rounds to 70% accuracy with normalized drift 0.52, 3.9 mean local steps, and 3.6% training overhead. A gradient-variance controller reaches 74.0% in 409 rounds with drift 0.66 and 3.8 steps, while a random controller reaches 72.9% in 461 rounds with drift 0.78. These mean-step-matched controls support update noncommutativity as decision-useful information beyond first-order variance or arbitrary step reduction.