On the Nature of Curvature: A Transport-Based Ontology for Closure Theory

Closure Theory begins from the proposition that the primitive language ofNature is geometry. This conceptual foundations paper proposes that curvaturemay be interpreted as the local organisation of geometry and that CanonicalTransport governs the infinitesimal evolution of that organisation. The paper derives the established hyperbolic dual-curvature recurrence fromthe Canonical Transport Equation by exponentiating a generator satisfyingK^2=I. The hyperbolic recurrence therefore appears as the integrated transporthistory rather than as an independent ansatz. The resulting ontology is economical: curvature is organised geometry;transport governs its evolution; Closure selects globally admissible histories;stable matter is recurrent transported curvature; and projection determineshow intrinsic organisation becomes observable as mass, charge, fields andmotion. The work is presented as a conceptual and mathematical contribution to anactive theoretical research programme. It distinguishes established algebraicresults from proposed physical interpretation and identifies the principalopen requirements concerning global Closure, Stationarity, particleadmissibility, projection and empirical falsifiability.