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תקציר מקורי באנגליתarXiv:2605.21103v2 Announce Type: replace Abstract: Shared-state federated computations combine client-local tensor computation, mergeable aggregation into shared state, and shared-only post-processing. We introduce a typed tensor language for this class of computations. Its two tensor sorts separate client-partitioned data from globally available values, and typing tracks the partitioned axis. A virtual global tensor serves as a semantic reference for centralized evaluation. We show that typed one-round programs factor through shared tensors whose shapes depend on the program but are independent of client and sample counts. The converse applies to typed-realizable factorizations: each encoder component is represented by an allowed aggregation or contraction with its valid merge, and the d