Residual Orthogonality as an Online Monitor of Structural Change: Detection, Localization, and a Delay Law

When a system changes, most monitors can only tell you that something moved. We introduce CR-CUSUM, a passive online detector that identifies which structural relationship changed. The method fits a structural model once on a reference window, freezes it, and monitors each equation’s residual for out-of-sample dependence on its regressors. For an unchanged equation, the frozen residual remains its exogenous noise and stays orthogonal to its regressors despite upstream distribution shifts; under sparse change, only the changed equation develops an orthogonality defect. CR-CUSUM therefore localizes the changed mechanism, requires only a topological order rather than the full graph, and can detect structural changes that standard marginal-variance monitors miss. We characterize the detectable class, including variance-preserving coefficient changes and covariance-preserving nonlinear changes, while showing that distribution-preserving changes are unobservable to any detector. We also derive a detection-delay law with a matching lower bound and establish finite-sample localization guarantees. The same frozen-residual principle extends to learned sequence representations, allowing a frozen model embedding to be monitored without handcrafted features. Experiments on synthetic structural systems, real server telemetry, nonlinear dynamics, and Bitcoin limit-order-book microstructure show that CR-CUSUM matches strong detectors on raw change detection while uniquely providing mechanism-level localization.