Spectral Proof of the Riemann Hypothesis – A Quantum Model and Spectral Analysis

This version represents the original formal submission of the spectral proof of the Riemann Hypothesis, deposited and timestamped via public notary on May 7, 2025. In this work, a self-adjoint quantum operator H^ζ\hat{H}_\zetaH^ζ is constructed from a potential derived numerically from the modulus of the Riemann zeta function. Using inverse spectral theory and Hadamard's uniqueness theorem, we demonstrate that the spectrum of H^ζ\hat{H}_\zetaH^ζ reproduces the imaginary parts of the non-trivial zeros of ζ(s)\zeta(s)ζ(s), leading to the conclusion that all such zeros lie on the critical line ℜ(s)=12\Re(s) = \frac{1}{2}ℜ(s)=21. This formal version includes: theoretical foundation based on spectral inversion (Borg–Marchenko), functional equation alignment, numerical evidence of spectral matching, declaration of falsifiability, and the first complete demonstration that a quantum spectrum encodes the critical zeros of ζ(s)\zeta(s)ζ(s). This version was later refined into version 5, which expands the formal argument and adds an analytic autonomous model (Appendix E) and the final statement of absolute proof (Appendix G). Nonetheless, this notarized version remains the first historically verified demonstration, and is preserved here for archival and citation purposes.