ALPHA SERIES

We present an analytical and numerical demonstration of the duality between the geometric Lagrangian and the Polyakov Lagrangian for a compactified bosonic string. The oscillating circle of radius is represented as a circular Matrix Product State (MPS) with bond dimension , where bounded continued fraction quotients define the virtual bond space. We elucidate the three observational levels of the system’s eigenvalue: (1) The local bare geometric action ; (2) The full MPS transfer matrix eigenvalue , incorporating the closed-loop topological winding phase and background curvature; (3) The physically projected Effective Action , obtained by topological deflation. We prove that the local generators commute for all angles (), collapsing the transfer matrix product into an exact exponential sum without Baker-Campbell-Hausdorff approximations. Furthermore, a capillary probe scanning script () identifies an exact analytical lattice resonance at steps, yielding , matching the CODATA 2022 target () with a relative precision of without empirical parameter tuning. The duality is confirmed: . The circle is not a circle; it is a string. The fine-structure constant is its projected resonance, stripped of the topological phase .