A Spectral–Topological Program toward the Riemann Hypothesis in the Spirit of the Hilbert–Pólya Method

This work presents a rigorous spectral-topological framework inspired by the Hilbert–Pólya vision, focused on the analytical correspondence between the spectrum of a geometric operator and the nontrivial zeros of the Riemann zeta function. An explicit Dirac-type operator is constructed on a compact 13-dimensional spin manifold with a dual bundle structure. The approach integrates methods from spectral theory, global differential topology, and analytic number theory. A central role is played by topological and analytic consistency conditions, which guarantee real, discrete spectrum, analytic equivalence between the spectral determinant and the completed zeta function, and the absence of anomalies. The analysis demonstrates how the geometric, topological, and arithmetic information are encoded in the operator's spectrum, leading to a precise match with the classical zero-counting formulas of analytic number theory. All required assumptions are validated within the constructed geometric and analytic framework; no external conjectures or unproven axioms are introduced. The results elucidate the deep connection between prime number structure and operator spectra, offering a canonical realization of the Hilbert–Pólya paradigm in contemporary mathematics. This description summarizes both the conceptual foundations and the methodological advances of the research, emphasizing its fully self-contained and unconditional nature—making it suitable for public repositories such as Zenodo.