Restoring Component Locality in Spectral Positional Encodings

Spectral positional encodings should preserve the representation of a component when an unrelated component is added. A fixed-rank prefix of a merged Laplacian spectrum can violate this principle: the added component competes for the same eigenvector slots. We formalize Component Locality, prove that a global prefix cannot guarantee it, and provide a sufficient direct-sum criterion. Component- local selection restores the identity by assigning each component its own spectral quota. We test this mechanism with shared checkpoints and a component-local downstream path on real graphs placed in controlled disjoint unions. CL-SPE restores encoding, hidden-state, and prediction locality in an independent holdout across all seven main cells; six cells show practically significant prediction-drift reductions. BudgetMatched and ClusterComplete separate total budget, allocation, and exact-eigenspace effects. An additional USPTO study establishes the mechanism on natural reaction graphs: 49.17% of nonboundary primary training reactions exhibit displacement under BudgetMatched. Component-local selection reduces reaction-macro kernel error from 0.206 under BudgetMatched to numerical zero and restores coverage to 1. Reaction-center prediction meets the overall noninferiority criterion on all 3,512 primary test reactions. Together, controlled interventions and natural reaction graphs establish component-wise allocation as a concrete design principle for spectral positional encodings.