Universal Quantitative Abstraction: Categorical Duality and Logical Completeness for Probabilistic Systems

A unified theory of quantitative abstraction is presented for probabilistic systems that links category theory, optimal transport, and quantitative modal logic. At its core is a canonical $ \varepsilon $-quotient endowed with a universal property: among all $ \varepsilon $-abstractions, it is the most informative one that respects a prescribed bound on value loss. This construction induces an adjunction between abstraction and realization functors $ (Q_{\varepsilon} \dashv R_{\varepsilon}) $, established via the Special Adjoint Functor Theorem, revealing a categorical duality between metric structure and logical semantics. A behavioral pseudometric is characterized as the unique fixed point of a Bellman-style operator, with contraction and Lipschitz properties proved in a coalgebraic setting. A quantitative modal $ \mu $-calculus is introduced and shown to be expressively complete for logically representable systems, so that behavioral distance coincides with maximal logical deviation. Compositionality under interface refinement is analyzed, clarifying how abstractions interact across system boundaries. An exact validation suite on finite Markov decision processes corroborates the contraction property, value-loss bounds, stability under perturbation, adversarial distinguishability, and scalability, demonstrating both robustness and computational feasibility. The resulting framework provides principled targets for state aggregation and representation learning, with mathematically precise guarantees for value-function approximation in stochastic domains.