When Multi-Domain Federated Learning Does Not Need Domain-Aware Capacity

Domain heterogeneity in federated learning does not by itself require full domain-aware capacity. When domains differ only in class proportions, adjusting class biases can suffice. For a fixed representation and an observed domain context, we compare shared predictors that ignore context, prior-aware predictors that adjust only class biases, and full domain-aware predictors that may change the feature-to-label mapping across contexts. Under log-loss, we characterize the shared-to-domain Bayes gap through conditional mutual information and derive the exact residual after canonical prior correction. Combined with family- and protocol-dependent training-quality bounds, these gaps yield a sequential one-sided rule: certify a learned shared predictor when possible, otherwise test the prior-aware predictor, and abstain when neither certificate passes, without declaring richer capacity necessary. For full-participation local SGD on frozen affine heads, we derive a recoverability bound separating optimization transient, aggregated stochastic noise, local stochastic accumulation, and client-objective drift. Controlled studies audit the reference quantities and calibrate the stopping rule under known ground truth. Real-data frozen-feature experiments show that full domain-aware heads can lower centralized log-loss, while ordinary shared aggregation often fails to recover this gain. Structure-aligned routing recovers substantially more but does not close the centralized gap. The framework separates the statistical value of domain-conditioned capacity from its federated recoverability.