Federated learning over graph-structured data exposes a fundamental mismatch between standard aggregation mechanisms and the operator nature of graph neural networks (GNNs). While federated optimization treats model parameters as elements of a shared Euclidean space, GNN parameters induce graph-dependent message-passing operators whose semantics depend on underlying topology. Under structurally and distributionally heterogeneous client graph distributions, local updates correspond to perturbations of distinct operator manifolds. Linear aggregation of such updates mixes geometrically incompatible directions, producing global models that converge numerically yet exhibit degraded relational behavior. We formalize this phenomenon as a geometric failure of global aggregation in cross-domain federated GNNs, characterized by destructive interference between operator perturbations and loss of coherence in message-passing dynamics. This degradation is not captured by conventional metrics such as loss or accuracy, as models may retain predictive performance while losing structural sensitivity. To address this, we propose GGRS (Global Geometric Reference Structure), a server-side aggregation framework operating on a data-free proxy of operator perturbations. GGRS enforces geometric admissibility via directional alignment, subspace compatibility, and sensitivity control, preserving the structure of the induced message-passing operator.

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