FedWAM: Federating Predictive World Knowledge for Personalized Robot Control

World-action models (WAMs) can benefit from experience collected across sites, but data ownership and heterogeneous model architectures complicate joint training. We introduce FedWAM, a framework that shares predictive world knowledge while retaining local demonstrations and private action experts. Clients upload conditional chunk-end predictions in a common visual feature space. A client-balanced server teacher integrates these contributions and is broadcast to all participants. Each recipient calibrates teacher supervision against local future observations, distills it into its native world pathway, and adapts its action expert to the updated context. Repeated exchange combines shared predictive learning with personalized control. The interface accommodates both LaWAM's future features and ImageWAM's internal control representations. In the reported comparisons, FedWAM reaches 78.4% LIBERO-Plus success versus 76.3% for local training, with most gains outside the recipient's training domain, while reaching 97.9% on original LIBERO. Mixed-architecture comparisons improve both recipient families, with direction-dependent donor transfer; three Piper tasks provide a complementary physical comparison. These findings motivate assessing shared predictors through both feature accuracy and the behavior of locally adapted controllers. Project page: https://submit-anonymous.github.io/FedWAM.