The Hydrogen Atom from the Lagrangian Duality of the Fine-Structure Constant: Independent Derivation via a Circular Matrix Product State

We derive the hydrogen atom from first principles using a circular Matrix Product State (MPS) that encodes the duality between a geometric Lagrangian (oscillating circle of radius R=π) and the Polyakov Lagrangian of a bosonic string compactified on a circle of radius R=π. The duality yields α−1= ln⁡λmax⁡−π, where λmax⁡λmax is the dominant eigenvalue of the MPS transfer matrix with bond dimension D=45.

The vacuum persistence amplitude is then interpreted as the wavefunction of the compactified string: Ψvac = e−α−1. This dimensionless amplitude fixes the electromagnetic coupling. Using only the universal constants ℏ,c,meℏ,c,me (the lepton mass scale) and the MPS-determined $\alpha$, we compute the ground‑state properties of hydrogen: binding energy −13.6057, Bohr radius 52.92 pm, orbital velocity 2187.69 km/s, and a vacuum decay probability Pdecay= e−137.036≈3×10−60, explaining the extreme stability of ordinary matter.

All parameters are either geometric (π), topological (24 transverse modes, tension T=8), or observational (maximum continued fraction quotient q≤45). No free parameters are introduced. A complete, executable Python code is provided. The construction is open, reproducible, and falsifiable.