Geometric Origin of Electric Charge: Hopf Fiber Winding, Topological Confinement, and the Unification of Spin and Charge on S³

Electric charge is one of the most fundamental yet least geometrically understood properties of matter. The Standard Model encodes charge as the weight of a U (1) representation but does not explain why charge is quantized, why the U (1) gauge group is singled out, or why charged leptons are massive while neutral leptons are nearly massless. This paper — PP-04 in the META Physics preprint series — derives the geometric ORIGIN of electric charge from the Hopf fibration of S³ (R) = ∂B⁴ (R). Five core results are established. (I) Electric charge is the winding number n ∈ π₁ (S¹) = ℤ of the zero-dimensional point's trajectory on the Hopf fiber S¹ ≅ U (1). Charge quantization follows from the discreteness of ℤ — a topological theorem, not a postulate. (II) The sign of charge corresponds to the winding orientation: n = +1 (electron, Q = −e) vs. n = −1 (positron, Q = +e). Electron-positron annihilation is topological unwinding: nₜotal = 0. (III) Non-zero winding confines the topological density (mass) via a topological barrier analogous to the BPS bound; zero winding permits mass dispersal — explaining the >10⁷ mass gap between charged and neutral leptons without free parameters. (IV) The fine-structure constant α = ωfiber/ωS² is the ratio of the charge-generating to the expansion-driving angular velocity. (V) Spin and charge are horizontal and vertical projections of the Maurer-Cartan 1-form on S³ ≅ SU (2), unified prior to Hopf decomposition. Additionally, the fractional charges of quarks (2/3 and 1/3) are derived by coordinate membership counting: each quark is a 2D disk of B⁴ (R), and its charge magnitude equals the number of its coordinates belonging to the 3D manifested space divided by 3. No free parameters. Together with the preceding papers on cosmic acceleration (PP-01), fine-structure constant and electron mass (PP-02), and electron spin (PP-03), this paper completes the derivation of all four fundamental properties of the electron from a single geometric structure — the uniformly rotating 4-dimensional ball B⁴ (R). Mathematical formalization and documentation: Claude (Anthropic, Claude Opus 4. 6). The core insights, physical intuitions, and axiom system originate from the author's 30 years of independent research.