Spectral Stability of Gaussian-Smoothed Mellin Reconstructions and Structural Constraints on Zeta Zeros

This manuscript develops a rigorous theory for a Gaussian-smoothed inverse Mellin reconstruction operator built from the nontrivial zeros of the Riemann zeta function. The Gaussian weight yields absolute convergence of the spectral sums for each fixed evaluation point x > 1, enabling a clean separation between distributional stability of the reconstruction and structural properties of zeros at the supremal real part. Main results include quantitative Mellin-decay bounds, absolute convergence of the Gaussian-weighted spectral expansion, the unconditional implication RH ⇒ distributional stability, and the converse implication distributional stability ⇒ RH under two explicit, checkable structural hypotheses. The paper contains full proofs and discusses connections to inverse spectral problems, PDE spectral stability, and operator-theoretic trace formulae. No numerical experiments are included.