Causal Fisher-Information Inequalities: Classical Causal Model Falsification and Metrological Advantage

Abstract

Fisher-information inequalities have recently been used as operational witnesses of nonclassical metrological behavior, but their physical meaning is often tied to a particular narrative, such as, segmented dynamics or discrete trajectories. We show that a broader interpretation is available and, in fact, more natural: once an experiment is assumed to admit a classical causal model specified by a directed acyclic graph, conditional independences, and modular parameter dependence, the corresponding Fisher informations are forced to obey causal Fisher-information inequalities (CFIIs). The backbone result is a causal-path series law: for an additive causal parameter that propagates through a classical path $A \to C \to B$, the inverse Fisher information behaves as an information resistance and must add in series. Consequently, any CFII violation is a rigorous falsification of the entire classical causal model class. We then show that the violation is automatically a metrological resource certificate, because it implies a precision that no member of the classical causal class can attain. The gain mechanism is identified as Fisher-information synergy, i.e. off-diagonal score correlations that classical modularity forbids. A single-qubit coherent-rotation example demonstrates the deterministic CFII violation, estimator-level achievability of the resulting gain, robustness against split-optimized classical benchmarks, and a chain-amplified advantage in long causal decompositions. Finally, an AI-assisted adversarial finite-data stress test shows that the witness remains certifiable under the realistic visibility loss and readout error, while optimized modular classical causal models saturate but do not cross the CFII frontier.