A 2-Categorical Framework for Proof Systems with Endofunctorial Struc

This work introduces a minimal 2-categorical framework in which logical systems are modeled as connection-like structures equipped with higher-dimensional transport data. In this formulation, proofs are interpreted as morphisms generated by parallel transport, while cut elimination corresponds to flatness of the connection and normalization corresponds to a gauge-theoretic reduction process. We define a 2-category equipped with a 2-endofunctor and associated 2-cells that play the role of connection and transport data. The failure of 2-naturality of the transport structure is interpreted as curvature, providing a geometric measure of non-classical logical behavior. Classical logic, intuitionistic logic, and non-classical (dialectical) logics are classified as different geometric phases of the same underlying structure, distinguished by curvature and holonomy properties. Furthermore, metatheoretical phenomena such as incompleteness are interpreted as obstructions to global gauge fixing within this framework. The resulting model provides a unifying perspective in which proof theory is reformulated as differential-geometric behavior on a higher-categorical space, linking logical deduction with connection theory and curvature.