What Survives Higher-Order Signal Reduction? Harmonic Unobservability at Vertices

Higher-order models represent signals across vertices, edges, faces, and higher-rank cells, yet collapsing these representations to a lower rank raises a fundamental question: which higher-order information remains observable after reduction? We characterize this information pathway for rank-shared boundary-mediated reductions and show that cross-rank transfer necessarily factors through boundary operators. Under the Hodge decomposition, the retained lower-rank representation can access boundary-visible gradient components, whereas harmonic components are structurally unobservable. We further show that this loss is caused by the observation interface itself rather than by the numerical treatment of singular higher-rank blocks. Motivated by this characterization, we introduce HARP, a HArmonic Readout Pipeline that explicitly exposes the missing harmonic information. HARP constructs a topology-preserving Morse complex, computes its harmonic basis, lifts the resulting representatives through a chain-compatible map, and projects them onto the harmonic subspace of the original complex. The resulting basis yields compact, sample-dependent harmonic coordinates that complement the information accessible through lower-rank reduction. Experiments on synthetic and real higher-order signals validate the predicted information pathway and show that HARP recovers useful information specifically when harmonic components are task-relevant. Together, these results identify a precise blind spot of rank-shared higher-order reduction and provide a compact interface for preserving the information hidden beyond it. Code will be released upon acceptance.