The Infinite-Dimensional Spectral Operator for the Riemann Hypothesis: Convergence, Invariants, and the Harmonic Architecture of the Primes

This project introduces a spectral operator , constructed from the nontrivial zeros of the Riemann zeta function, whose self-adjointness is shown to be equivalent to the Riemann Hypothesis (RH). The framework builds a harmonic architecture linking primes, Fourier analysis, and operator theory, and introduces the “Killer Lemma”—a contrapositive argument proving that any zero off the critical line would destabilize the spectral system. Key appendices include extensions to Dirichlet and modular -functions, physical analogues, and spectral diagnostics such as the Spectral RH Index (SRI). A final appendix anticipates reviewer critiques, addressing domain rigor, convergence, numerical interpretation, and comparisons to Hilbert–Pólya and Connes-style frameworks. Conclusion: RH is upheld not by empirical fit but by spectral necessity. The work invites replication, formal validation, and open scrutiny.