Target-Specific Causal Recovery under Federated Support Fragmentation

A federation can have enough information overall to resolve a causal target, such as an edge direction, while no client can form some required conditional-independence (CI) answers. We study a fixed set of answers sufficient to determine that target. Some losses leave it recoverable; others admit a graph with a different target answer. When unavailable queries abstain and available answers are correct, we identify the inclusion-minimal groups whose joint loss permits such an alternative. Recovery fails exactly when one such group is entirely missing. Queries with shared conditioning requirements rely on the same client counts. These counts determine when destructive losses occur and distinguish insufficient pooled support from failure caused solely by data placement. For $K$ clients with independently sampled data, we derive an exact failure probability accounting for dependent query losses. Individual missing rates generally do not determine this probability. Holding the same global counts fixed isolates risk caused only by repartitioning observations across clients; averaging these conditional risks recovers the overall failure probability. Controlled enumeration shows different risks with each query's missing probability held fixed. Paired $Alarm$ and $Child$ experiments isolate pure fragmentation. Added evidence improves $Alarm$ recovery through alternatives that clients can support, and the exact law closely predicts the transition. In $Child$, these alternatives share a support bottleneck and yield no observed recovery gain.