Learned, Relied Upon, or Necessary? Separating Checkpoint Dependence from Task-Level Value in Sheaf GNNs

Abstract

Learned restriction maps in sheaf graph neural networks are often treated as proof that the model has discovered useful edge geometry. That conclusion does not follow from parameter movement or from a post-hoc ablation: both can show how one checkpoint is organized while leaving open whether learned transport still helps after the rest of the model adapts. We separate these claims with two estimands. Checkpoint reliance intervenes on the maps of a fixed predictor; protocol-relative replacement retrains matched families that remove map capacity, edge variation, or persistent edge assignment. A task-null theorem shows why the claims can diverge: labels identify only the transported classifier directions, leaving $d^2-d$ invisible degrees of freedom in every full $d\times d$ map. An exact frame model then gives the boundary at which reliance becomes unreplaced task value. Label-only training realizes the predicted separation, while audits of public NSD, DNSD, and Directed Sheaf Neural Network (DSNN) implementations recover both replaceable and unreplaced transport regimes on real graphs. All five DNSD benchmarks exhibit fixed-checkpoint reliance. After retraining, assignment-breaking or shared-map controls recover Full performance on four; Roman-Empire retains a $.0675$ advantage over continually resampled assignment and a $.0391$ advantage over a parameter-matched shared map across ten official splits. Thus, a learned map can govern a fitted computation without constituting indispensable edge geometry. Claims of learned transport should pair checkpoint interventions with matched retraining.