The Complete Derivation of QED from Scalar Fields Using Noncommutative Geometry: QED from the EAS Finite Spectral Skeleton by Noncommutative Geometry

This paper is archived as a speculative research work. This paper proves that the finite spectral skeleton derived from EAS scalar-field structure yields QED when inserted into the standard noncommutative-geometric construction. The preceding paper constructed the finite internal package S_F^phi = (A_Q^phi, H_F^phi, D_F^phi, J_F^phi, gamma_F^phi, rho_Q^phi), where the QED phase algebra is A_Q^phi ~= C, the finite carrier is built from scalar-point and phase-exposed association reports, the operator D_F^phi is the finite first-order report operator, J_F^phi records protected A/B closure, gamma_F^phi is the finite grading, and rho_Q^phi acts through scalar-field-derived charge classes Q_hat in {-1, 0, +1}. Here we pair this EAS-derived finite skeleton with the usual spacetime spectral triple S_M = (C^infinity(M), L^2(S), D_M, J_M, gamma_M) and form the almost-commutative product S_M x S_F^phi. The product Dirac operator D = D_M x I + gamma_M x D_F^phi admits the standard NCG inner fluctuation D -> D_A = D + A + J A J^(-1), A = sum_i a_i [D, b_i]. In the common-mode C-phase sector, this produces the QED U(1)-type gauge potential. The EAS finite charge representation determines the matter coupling: i gamma^mu (partial_mu + i Q_hat A_mu^phase), or, equivalently, i gamma^mu (partial_mu + i e Q_hat A_mu^phys). Thus same-sign A/B presentations couple with charge classes Q_hat = +/-1, while opposite-sign presentations have Q_hat = 0 and are silent under the QED phase algebra. The spectral action supplies the abelian curvature term and the coupling-normalization slot kappa_Q = 1 / (4 e^2). The result is a derivation of QED at the NCG level: EAS supplies the finite internal spectral skeleton, and the standard NCG product, inner-fluctuation, and spectral-action machinery yields the QED gauge and matter sector.