Operational Noncommutativity in Sequential Metacognitive Judgments

Metacognition, understood as the monitoring and regulation of one's own cognitive processes, is inherently sequential: an agent evaluates an internal state, updates it, and may then re-evaluate under modified criteria. Order effects in cognition are well documented, yet it remains unclear whether such effects reflect classical state changes or reveal a deeper structural non-commutativity. We develop an operational framework that makes this distinction explicit. In our formulation, metacognitive evaluations are modeled as state-transforming operations acting on an internal state space with probabilistic readouts, thereby separating evaluation back-action from observable output. We show that order dependence prevents any faithful Boolean-commutative representation. We then address a stronger question: can observed order effects always be explained by enlarging the state space with classical latent variables? To formalize this issue, we introduce two assumptions, counterfactual definiteness and evaluation non-invasiveness, under which the existence of a joint distribution over all sequential readouts implies a family of testable constraints on pairwise sequential correlations. Violation of these constraints rules out any classical non-invasive account and certifies what we call genuine non-commutativity. We provide an explicit three-dimensional rotation model with fully worked numerical examples that exhibits such violations. We also outline a behavioral paradigm involving sequential confidence, error-likelihood, and feeling-of-knowing judgments following a perceptual decision, together with the corresponding empirical test. No claim is made regarding quantum physical substrates; the framework is purely operational and algebraic.