Sufficiency of the counterfactual account of Lüders' rule to rule out ontological models of quantum mechanics

Ontological models, as used in the generalized contextuality literature, play a central role in current research on quantum foundations, providing a framework for defining classicality, constructing classical analogs of key quantum phenomena, and examining the ontology of quantum states. In this work, we show that a counterfactual account of Lüders' rule—which we argue is naturally implied by the mathematical structure of the rule itself—renders such models inherently incompatible with the quantum formalism. This incompatibility arises because the counterfactual update requires ontological models to update their states according to conditional probability, which in turn renders predictions of sequential measurements order-independent. This implies that ontological models, even contextual ones, must either act differently than what we would expect given (this, typically implicitly assumed account of) quantum state update rule, or cannot model quantum behavior.