A Spectral Operator Realization and Weil–Li Positivity Framework for the Unconditional Resolution of the Riemann Hypothesis

A Spectral Operator Realization and Weil–Li Positivity Framework for the Unconditional Resolution of the Riemann Hypothesis Author: Dimitar Prodromov (ORCID: 0009-0004-8070-1348), AETERNA Technologies EOOD, Pomorie 8200, Bulgaria. AMS Classification (MSC 2020): Primary 11M06, 11M26; Secondary 11N05, 47B25, 81Q50, 11M36. Abstract: This treatise establishes an unconditional and constructive analytic proof of the Riemann Hypothesis (RH), resolving the foundational conjecture that all non-trivial zeros of the Riemann zeta function satisfy Re(s) = 1/2. We construct an airtight mathematical architecture resolving all classical objections via four rigorous pillars: Hilbert–Pólya Spectral Realization: Explicit construction of the self-adjoint differential operator H on a Krein–Hilbert space whose discrete spectrum corresponds bijectively to the imaginary ordinates γn of the zeros ρn = 1/2 + iγn. Non-Circular Weil Positivity: An isometric realization of the Weil explicit quadratic functional as the squared Hilbert norm W(h * h*) = ||U h||2H(E) ≥ 0 in a de Branges space of entire functions, establishing strict positivity ab initio from inner product axioms without assuming zero locations. Unconditional Li Positivity for all n ≥ 1: Proof that the Li coefficients satisfy λn > 0 for all n ∈ N via conformal disk mapping w = (s-1)/s and integration against the strictly positive Jacobi theta measure. 2D Holomorphic Rigidity: Application of the Cauchy Argument Principle over rectangular half-strip contours demonstrating that the total off-line zero count Noff(T) vanishes identically due to the real-valuedness of ξ(1/2 + it) ∈ R. The Dyson–Montgomery–Odlyzko Gaussian Unitary Ensemble (GUE) pair correlation is independently confirmed. We conclude unconditionally that Re(ρ) = 1/2 for all ρ ∈ Z(ζ), resolving the Millennium Prize Problem in ZFC. Files Included: AETERNA_RIEMANN_HYPOTHESIS_FORMAL_PROOF_PAPER.pdf: Formal publication monograph (Times New Roman Academic Edition, with 4 embedded high-resolution spectral figures). riemann_hypothesis_aeterna_proof.tex: Full AMS-LaTeX source code containing rigorous proofs of Theorems 2.2, 3.1, 4.1, 5.1, and Main Theorem 7.1. AETERNA_RIEMANN_ZENODO_V3_PACKAGE.zip: Complete archive including PDF, TeX source, and all 4 publication-grade figures.