Closure Theory: An Introductory Résumé of Papers 0–IV

Closure Theory: An Introductory Résumé of Papers 0–IV This short paper provides an introductory overview and architectural map of the first five papers in the Closure Theory research series. It summarises the progression from canonical intrinsic transport to the Completed Intrinsic Closure, the elimination of observable charge sign as an invariant of an isolated intrinsic object, and the subsequent relocation of charge sign to projected relational geometry. The series begins with the Canonical Transport Equation and the demonstration that its minimal associated frame transport is induced rather than independently postulated. It then explores the circular, hyperbolic and potentially helical geometry suggested by canonical transport, develops the Completed Intrinsic Closure as the fundamental recurrent transported object, and distinguishes local transport, recurrence and global admissibility. The later papers investigate whether observable electric charge sign can be identified with an intrinsic binary invariant of the Completed Intrinsic Closure. The failure to establish such an invariant provides an important architectural constraint and motivates the interpretation developed in Paper IV: intrinsic closure supplies identity, Transformation establishes relational structure, and Projection produces observable quantities. Within this framework, observable charge is separated into a magnitude and a projected relational orientation, qᵢ=sᵢ|e|, while mass remains associated with the transported intrinsic invariant. The overview identifies the principal purpose and result of each paper, explains the logical connections between them, distinguishes established mathematical results from open hypotheses, and provides recommended reading routes for readers interested in transport theory, intrinsic geometry, Completed Intrinsic Closures, charge or Projection. The five-paper sequence may be summarised as: Paper 0 — Foundation: Canonical frame transport is induced by the Canonical Transport Equation. Paper I — Exploration: Canonical exponential transport suggests a wider circular–hyperbolic and potentially helical geometry. Paper II — Architecture: The recurrent transported fibre and frame form a Completed Intrinsic Closure subject to local, recurrent and global admissibility conditions. Paper III — Constraint: No satisfactory intrinsic binary invariant is found that can be identified with observable electric charge sign. Paper IV — Interpretation: Charge sign is relocated to projected relational orientation, while mass and charge contribute to a single observable geodesic. This document is intended as the principal introductory gateway to Papers 0–IV and to the first completed stage of the Closure Theory programme.