Causal State Space Models: Routing Dynamic Mechanisms for Non-Stationary Inverse Problems

State Space Models (SSMs) provide efficient representations for continuous spatial-temporal fields, yet standard global SSMs suffer from prior entanglement under non-stationary inverse scattering. A localized out-of-distribution (OOD) anomaly shifts the shared transition matrix during online adaptation, corrupting the reconstruction of stationary backgrounds. To address this, we propose the Causal State Space Model (Causal-SSM). By formulating the inverse mapping as a switched linear system via Dynamic Mechanism Routing (DOR), the model partitions continuous data into a library of independent causal mechanisms. During adaptation to physical shifts, our Targeted Mechanism Revision (TMR) confines gradient updates strictly to active anomalous mechanisms while freezing shared parameters and boundary routing. Evaluated on dynamic scattering benchmarks and Institut Fresnel measurements, Causal-SSM achieves robust zero-shot OOD generalization. It successfully assimilates localized anomalies while preserving background fidelity (BRR $\approx 1.015$), avoiding the severe degradation typical of global-update baselines.