The Number 137 in ZEBTS‑SUPER Theory: Rigorous chi‑Spectral Reconstruction of the Fine‑Structure Constant

The fine‑structure constant 𝛼 has no derivation within the Standard Model and remains one of physics’ most persistent mysteries. ZEBTS‑SUPER resolves this problem by demonstrating that 𝛼 is a spectral invariant of the vacuum. The theory is built on a self‑adjoint χ‑operator 𝐷 𝜒 with compact resolvent acting on the χ‑Hilbert space 𝐻 𝜒 . Its discrete spectrum { 𝜆 𝑘 } contains a unique mode, 𝑘 = 137 , selected by simultaneous minimisation of information tension, modular stabilisation, and gauge‑vacuum coherence. This mode defines the fine‑structure constant via 𝛼 − 1 = 𝜆 137 , providing the first operator‑spectral derivation of 𝛼 . The mathematical framework includes the construction of 𝐻 𝜒 , the rigorous self‑adjointness of 𝐷 𝜒 , the spectral information‑tension tensor, and χ‑modular dynamics. The analytical and numerical evaluation of 𝜆 137 yields a stable value consistent with CODATA. Physical consequences follow directly: hydrogen spectroscopy, the electron g‑factor, strong‑field QED, and high‑Z relativistic collapse all emerge from the χ‑Boundary eigenvalue. Three independent experimental channels—hydrogen fine‑structure, electron anomalous magnetic moment, and Schwinger pair‑production threshold—each reconstruct 𝜆 137 with precision 10 − 6 . Their agreement confirms that 137 is not a fit parameter but a spectral invariant. ZEBTS‑SUPER thus provides the first fundamental resolution of the 137‑problem.