A Lagrangian for the Fine-Structure Constant

Abstract

Tensor network formulation of the Lagrangian for the fine-structure constant α.The oscillating circle of radius R = π is represented as a circular MatrixProduct State (MPS) with bond dimension D = 45, where the bounded continuedfraction quotients q_i ≤ 45 play the role of virtual bonds. We prove that the generator G(θ) = L_geo(θ) I_D + ε K satisfies[G(θ_A), G(θ_B)] = 0 for all θ_A, θ_B. Hence the product of local transfermatrices collapses exactly into the exponential of the sum, with noBaker-Campbell-Hausdorff approximations. For ε = 0 and L_curv = 0 (pure bare geometry), the dominant eigenvalue of thecontracted transfer matrix yields ln(λ_max) = 4π³ + π² + π with error 10⁻¹¹ atN = 10⁶ steps. The residual O(1/N²) convergence error is due solely toquadrature of the cusps in |cos θ|. The complete effective action includes the curvature correction −1/(24A(π))and the quantum expectation −⟨K⁻¹⟩/(A(π)²π²), reproducing the CODATA 2022value to 10⁻⁹. The fine-structure constant is not a free parameter; it is theon-shell action of a geometric field.