Federated hypergraph learning with latent cross-client hyperedge correspondence poses fundamental identifiability and computational challenges, as the same local release can correspond to multiple global hypergraph structures. We characterize the tractability of recovering these latent correspondences, showing tractable pairwise assignment but generally intractable multi-client hyperedge matching. Building on this characterization, we derive a cross-entropy decomposition that reveals when local prediction can outperform predictors fitted with additional cross-client information. Motivated by this principle, we introduce Federated Adaptive Spectral Training with Representation Alignment (FedASTRA), which learns correspondence-invariant diffusion representations through adaptive heat diffusion and masked latent prediction while communicating model parameters only. We establish its asymptotic optimality with respect to the permitted local information and prove global population-risk convergence under explicit conditions on recovery, representation, and update alignment. Experiments on eight real-world datasets demonstrate the effectiveness of FedASTRA against identifier-free federated graph and hypergraph methods. On ogbn-papers100M, FedASTRA reaches the performance-communication Pareto frontier, achieving 50.07 Accuracy and 29.73 Macro-F1 while communicating only 59.60 MiB per round.

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