The System Is Out of Control, but Causality Isn't: Leveraging Operational Shifts for Bivariate Causal Discovery

Bivariate causal discovery often relies on subtle asymmetries in a single observational distribution, yet these signals become unreliable when causal effects are weak or masked by noise. Multiple environments can offer stronger evidence through distributional changes and invariances, but many applications provide only two regimes: a reference condition and one alternative. We study causal direction identification in this practically important but theoretically underexplored setting. We introduce Kernel Residual Displacement (KRD), which exploits a simple directional asymmetry: under the correct direction, a localized change between the two regimes leaves one residual coordinate invariant, whereas under the reverse direction the change propagates to both residual coordinates. KRD quantifies this asymmetry through kernel distances between residual distributions, requiring neither multiple auxiliary environments nor conditional-independence testing. We establish population identifiability and finite-sample recovery guarantees that explicitly connect successful identification to perturbation magnitude, regression error, sample size, and directional separation. For linear Gaussian systems where causal direction is unidentifiable from a single observational distribution, our analysis further reveals a non-monotone signal-to-noise geometry: identification is strongest when causal signal and downstream noise are balanced and deteriorates toward both near-deterministic and noise-dominated extremes. Experiments across simulations, nonlinear and non-Gaussian mechanisms, and real-world datasets show that even one additional regime can yield substantial causal information beyond conventional single-context discovery.