Smooth Client–Instance Adapter Fields for Personalized Federated Learning under Compound Heterogeneity

Personalized federated learning (PFL) commonly binds one model to each client or selects among finite experts, creating artificial boundaries between nearby distributions and reducing a multimodal client to one identity. We introduce FedSPLINE (Federated Smooth Personalization via Layerwise Implicit Neural Experts), a tensor-product cubic Bernstein field of low-rank adapters over learned client–instance coordinates. A distribution signature anchors each client, a bounded router refines instance queries, and sixteen nonnegative analytic weights blend shared controls while offsets and heads remain private. This explicit convex geometry gives partition-of-unity and Lipschitz properties, a cold-start decomposition, and supporting conditional shared-parameter analysis. Under source-identity-disjoint evaluation on three compound-shift benchmarks and twelve controlled baselines, FedSPLINE has the highest observed mean accuracy (83.11%) and label-free cold-start mean (63.23%). Its 0.14-point three-seed and 0.19-point five-seed HyperAdapter gaps are descriptively unresolved ($p=0.658$ and $p=0.390$), while it remains Pareto-relevant through explicit geometry and slightly lower communication than the capacity-matched generator. On unseen interior coordinates, its 0.49-point advantage over that generator is nonsignificant ($p=0.150$), and extrapolation is weaker. The evidence supports an inspectable geometry–utility trade-off, not a universally dominant PFL rule.