Towards the Geometric Origins of Mass and Charge: Projection, Geodesics and Relational Sign in Closure Theory

Towards the Geometric Origins of Mass and Charge: Projection, Geodesics and Relational Sign in Closure Theory This paper develops the architectural consequences of the intrinsic Closure Theory programme by investigating the emergence of observable mass and electric charge through Transformation and Projection. Building upon the Completed Intrinsic Closure established in the preceding papers, and following the elimination of an intrinsic charge invariant, the analysis examines how observable properties may instead arise from projected relational geometry. A central result of the paper is the separation of observable charge into two distinct components: an observable magnitude determined through the Projection programme and a primitive relational orientation that classifies interactions between projected closures. This distinction permits the familiar attractive and repulsive behaviour of electric charge to be interpreted without requiring charge sign to exist as an intrinsic property of the transported closure itself. The paper further develops the concept of the projected observable geodesic, demonstrating how mass, charge and interaction may be understood within a single geometric framework while preserving the distinction between intrinsic transport, relational dynamics and observable projection. Throughout, the analysis maintains the architectural hierarchy of Closure Theory, in which the intrinsic T-domain generates completed transported structures, Transformation preserves intrinsic information, and Projection produces the observable quantities accessible to experiment. Rather than introducing additional primitive assumptions, the work shows how the elimination of the intrinsic charge invariant naturally leads to a simpler and more coherent interpretation of charge as a projected relational property. The resulting framework provides a unified architectural setting for future investigations into gravitation, electromagnetism and the geometric origin of observable interactions. This paper forms Paper IV of the Closure Theory research series and concludes the first stage of the programme by relocating observable charge from intrinsic transported geometry to projected relational structure.