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

Year: 2025
Type: preprint
Abstract: The proof is realized in the spirit of the Hilbert–Pólya program, through the construction of a self-adjoint spectral operator H_TEBAC whose eigenvalues coincide exactly with the non-trivial zeros of the completed zeta function ξ(s). The argument unites global differential topology, spectral analysis, and analytic number theory.It introduces an explicitly defined Dirac-type operator on a compact 13-dimensional spin manifold M13, endowed with a dual bundle structure SU(2)⊕SO(4).The central topological constraint c_2+p_1=0 guarantees essential self-adjointness and real discrete spectrum, while the geometric–arithmetic balance condition A_1=0 establishes analytic equivalence between the zeta-regularized spectral determinant and ξ(s). Scientific Contribution Topological–Spectral Construction:The paper develops a Hilbert–Pólya operator H_TEBAC built on topological bundles with characteristic classes satisfying c_2+p_1=0, ensuring the global absence of anomaly and index obstruction. Self-Adjointness and Symmetry:Through index-theoretic and microlocal arguments, it is proved that the operator is essentially self-adjoint on L2(M13,E) and possesses a real spectrum symmetric about zero, enforcing the reflection law corresponding to the symmetry ξ(s)=ξ(1−s). Analytic Equivalence:Using Seeley–DeWitt heat-kernel formulas, Atiyah–Singer index theory, and Kontsevich–Vishik regularization, it is shown that det⁡ζ(s−H_TEBAC)=ξ(s). The cancellation of the global heat coefficient A1 confirms geometric–arithmetic balance and eliminates all regulator poles. Spectral Trace and Zero Count:A trace formula of Guillemin–Uribe/Berry–Keating type connects the global spectral trace of H_TEBAC with the Riemann–von Mangoldt zero-counting law: N(T)=#{γ:0<γ
DOI: 10.5281/zenodo.17402051
Language: English
Open Access status: green
OpenAlex ID: W7092623801
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