The Slice and the Field: Toward a Geometric Derivation of the Fine Structure Constant

If particles are observational artifacts; if they are what fields look like when sliced by finite subsystems; then the equations governing their interactions should be constrained by the geometry of the slicing process. We show that modelling observation as a dimensional reduction on real projective space yields five results and one structural prediction. First, the one-loop QED beta function coefficient equals Vol(RP²)/(3 × Vol(RP³)) = 2/(3π) exactly. Second, the self-interaction coefficient 1/24 is the unique convergence of spectral, combinatorial, and analytic structures in three dimensions; the only dimension where 2n = n!. Third, the self-referential correction series contains only odd powers, as required by the reflexivity constraint on self-referential observation. Fourth, using the cut constant P = 4π³ + π² + π (the total geometric content of the observation act) as input, the self-consistent equation yields α⁻¹ = 137.035999191, within 1.4 standard deviations of the Morel et al. (2020) rubidium measurement [2]. The prediction is not a truncated decimal; it is the exact irrational solution to a closed-form equation, and every digit is determined. Fifth, the tree-level weak mixing angle equals d/λ₁(RP³) = 3/8, arising because the projective identification that makes observation self-referential raises the spectral gap from 3 to 8. The structural prediction: the cut constant is the irreducible empirical input. The framework derives the form of the equation and all of its coefficients, but not the boundary condition. The polynomial P was first observed in Wolfgang Pauli’s “World Clock” dream during his analysis with Carl Jung [5]; the present work derives its geometric origin as the total content of dimensional reduction on RP³.