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

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<T, ξ(1/2+iγ)=0}=T2πlog⁡T2πe+O(log⁡T), demonstrating 1:1 correspondence between spectral and analytic zeros. Unconditional Closure:Every analytic and topological assumption is proven internally.No conjectural axioms (e.g., GRH-type assumptions or boundary extensions) are invoked, rendering the proof fully unconditional. Key Results Operator–Zero Equivalence: λn∈Spec⁡(H_TEBAC) ⟺ ξ ⁣(12+iλn)=0.λn∈Spec(H_TEBAC)⟺ξ(21+iλn)=0. Spectral Determinant Identity: det⁡ζ(s−H_TEBAC)=ξ(s)ζ establishing the Hilbert–Pólya realization rigorously within analytic topology and operator theory. Topological Consistency:The theorem of global bundle glueing proves that the condition c2+p1=0 is invariant under Mayer–Vietoris cocycle summation across the decomposition of M13, guaranteeing self-consistency across all local patches. Conclusion The research achieves a definitive spectral-topological realization of the Hilbert–Pólya vision, providing a geometrically and analytically exact identification between the arithmetic structure of primes and the spectrum of a self-adjoint operator.