Physics of Difference VI: Congruence Failure and Global Reduction Obstruction in Accessibility Semicategories

The present work investigates realization-dependent accessibility organization within the framework of accessibility semicategories. The central problem addressed throughout the paper was whether endpoint accessibility reduction defines a compositionally compatible reduction structure globally. More specifically, the analysis examined whether local compositional admissibility necessarily guarantees congruence-compatible endpoint reduction. The principal result demonstrates that local compositional admissibility does not in general guarantee globally coherent reduction. The paper showed that local compositional admissibility does not in general guarantee globally coherent reduction under realization composition. Explicit finite constructions demonstrated the existence of both: structurally reducible accessibility semicategories,and structurally obstructed accessibility semicategories. In the obstructed case, realization-dependent distinctions survive endpoint reduction and generate residual accessibility structure together with realization ordering dependence. Formally, the obstruction investigated here may be interpreted as congruence failure within realization-dependent accessibility composition. The main structural result of the paper is not merely the existence of obstructed examples, but the identification of congruence closure as the minimal operation required to restore globally compatible endpoint reduction. In this sense, global reduction obstruction corresponds precisely to failure of realization equivalence relations to remain stable under realization composition. An important feature of the framework is its structural minimality. The analysis requires only: distinguishable configurations, realization multiplicity, partial compositional organization, and realization equivalence relations. No assumptions concerning identity morphisms, invertibility, topology, differentiable geometry, transport structure, or algebraic closure are required. From this perspective, the present work may be interpreted as identifying minimal compositional conditions under which congruence-compatible reduction fails within accessibility semicategories. The conceptual contribution of the paper therefore does not lie in introducing a new form of geometric non-integrability. Rather, the analysis isolates a reduction-theoretic structural problem within realization-dependent compositional accessibility organization itself. At the same time, the framework remains intentionally limited in scope. The present work does not derive conventional physical theories, geometric curvature, or gauge dynamics. Instead, the objective has been to establish the mathematical possibility of globally irreducible accessibility organization within an ultra-minimal compositional framework. Several directions for future work remain open. These include: classification of structurally obstructed accessibility semicategories, necessary and sufficient conditions for reduction compatibility, relations to congruence structures in category theory, congruence closure and obstruction propagation, higher compositional structures, and possible physical realizations of realization-dependent accessibility organization. More generally, the present framework suggests that reduction compatibility itself may constitute a fundamental structural issue within relational compositional organization.