Causal Orbitals: Order-Sensitive Probing for Certifiable Test-Time Adaptation

Test-time adaptation (TTA) typically assumes that an unlabeled target batch provides sufficient evidence for deciding how a model should adapt. However, distinct causal shift mechanisms can produce nearly indistinguishable observations while inducing incompatible task-risk gradients, making adaptation fundamentally ambiguous. We study meta-trained, label-free TTA under causal mechanism uncertainty and ask whether additional test-time measurements can make a safe adaptation direction identifiable. We introduce Causal Orbitals, an operator-valued representation of unresolved causal mechanisms, and model self-supervised test-time probes as sequential measurement instruments that update this ambiguity state. Unlike a single-shot mechanism posterior, the proposed formulation captures order-sensitive probe responses, where applying the same probes in different sequences can yield different evidence about the underlying shift. We derive spectral certificates that determine when a candidate update guarantees local risk reduction and abstain when no common descent direction can be certified. Our analysis characterizes the classical boundary of the framework: it reduces to an ordinary posterior under commuting measurements, while selected noncommuting probe sequences can distinguish mechanisms that share identical single-probe marginals. We further develop certifiability-directed probing, which selects probe sequences according to their expected guaranteed adaptation value rather than generic uncertainty reduction. We evaluate the framework on controlled causal-shift benchmarks and realistic domain shifts, measuring sequential identifiability, certificate validity, negative adaptation, certified coverage, and probe efficiency.