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Verified signatureAQARION (Exact Observable Quotients of Finite Dynamical Systems) is a mathematically rigorous, computer-verified framework designed to measure and eliminate observable loss in state abstraction. At its core, AQARION answers a fundamental question in dynamical systems, state aggregation, and machine learning representation: When does a simplified or macroscopic view of a system preserve its exact underlying dynamics without information leakage? The Core Criterion For a finite transition system with Koopman pullback operator K and a partition projection matrix P_\Pi, AQARION introduces the Defect Operator: D_\Pi = (I - P_\Pi) K P_\Pi \text{Exact Observable Quotient} \iff K(V_\Pi) \subseteq V_\Pi \iff D_\Pi = 0 If D_\Pi = 0: The partition \Pi forms a closed, exact bisimulation (deterministic lumpable Markov chain). There is zero representation drift or macroscopic ambiguity. If D_\Pi \neq 0: \Vert D_\Pi \Vert_F measures the exact norm loss and information leakage across macroscopic s…027cb40