Geometric Origin of the Fine-Structure Constant and Electron Mass: Hopf Fibration, Topological Density, and the α⁻¹⁸ Bridge between Cosmology and Quantum Mechanics

The fine-structure constant α ≈ 1/137. 036 has been regarded for a century as physics' deepest unexplained number. This paper — PP-02 in the META Physics preprint series — demonstrates that α is a proportionality constant: the ratio of the Hopf fiber angular velocity to the Hopf base angular velocity within the uniformly rotating 4-dimensional hypersphere B⁴ (R). α = ωfiber / ωS² = δΩ/ (4Ω̄) The Hopf holonomy confirms this identification exactly. In the Hopf projection, δΩ completely disappears from the S² base and hides within the fiber, explaining why WS₃ observers cannot directly detect the non-isoclinicity but measure its trace as the electromagnetic coupling constant. Building on this geometric ORIGIN, the electron mass is derived through topological density (stacking number). The uniform rotation endows each zero-dimensional point on S³ (R) with a fundamental energy ε₀ = ℏΩ̄. The DRAIN mechanism compresses B⁴ (R) onto its boundary, creating topological density N (a) at each point, which manifests as mass: m (a) = N (a) ·ℏΩ̄/c². The electron stacking number Nₑ = R/λ̄ₑ ≈ α⁻¹⁸ to within 0. 44%, where 18 = C (4, 2) × (4−1) = 6 × 3 is forced by the four-dimensionality of B⁴ (R). The 0. 44% residual is fully resolved by the B⁵ (R) whole rotation (Ω₅/Ω̄ = 0. 0044), with the two hidden rotations connected by Ω₅ = δΩ/ (4√e), yielding: mₑ = α⁻¹⁸ · ℏΩ̄/c² This formula independently predicts H₀ = 67. 84 km/s/Mpc with zero free parameters (Planck +0. 9σ), establishing a quantitative bridge between cosmology (R), quantum mechanics (mₑ), and quantum electrodynamics (α) that is absent from the Standard Model. Among the broader implications, the Euler identity e^ (iπ) + 1 = 0 is identified as the algebraic self-portrait of the coalesced zero-dimensional point, and the quaternion non-commutativity ijk = −1 ≠ kji = +1 is identified as the ORIGIN of time asymmetry and the second law of thermodynamics. 10 established results (ER-1 through ER-10). 41 pages. 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.