Separating Information Limits from Realization Gaps in Local Graph Distillation

Graph distillation reports one teacher–student discrepancy, mixing unavailable teacher-relevant information with excess relative to the best predictor under the declared observation. We separate them using $Q_k=\mathbb{E}[P\mid A,Z_k]$, where $A$ declares training-side information and $Z_k$ is the radius-$k$ observation. This standard expected-KL decomposition yields the information term ${\Delta }_k$; our contribution is to define and diagnose it under an explicit information set and compatible-world law. The observable pair $(A,Z_k)$ does not determine that law, and one complete realization cannot uniformly recover a law-dependent ${\Delta }_k$ without additional assumptions. We assign exact values to newly observed shells and characterize the information boundary through predictive Fisher curvature and conditional Green covariance. On Cora, PubMed, and Roman-empire, matched audits estimate all KL components under post-encoder graph-correlated completion laws. Conditioning on realized training-side logits changes radius-zero information shares from 12.5%, 17.9%, and 80.9% to 2.3%, 1.7%, and 23.7%, reversing the Roman-empire diagnosis. Predictive checks assess the adequacy of these model-conditional values, while qualitative conditioning-set diagnoses remain stable across retained laws. A contractive nonlinear teacher further tests the accounting beyond linear diffusion. The diagnosis is therefore relative to the declared information set and law, not intrinsic to one graph.