SpectralTriplesArithmeticSurfacesRiemannCriticalLine

This paper constructs an explicit two-layer spectral triple: first defining a fundamental Dirac operator on the local archimedean place R_+^× × R_+^×, whose spectral determinant yields the π^ (-s/2) Γ (s/2) archimedean factor; then globally lifting this construction to GL₂ (Q) \₂ (A), introducing prime-power discrete orbits via the diagonal embedding of Q^× to fully reproduce the Euler product. We prove that the self-adjointness requirement of the global operator forms an algebraic coupling with the functional equation symmetry ξ (s) =ξ (1-s) induced by the crossed product algebra; their joint action forces the real part of every non-trivial zero of the Riemann ζ function to be identically 1/2. Furthermore, through the pairing between cyclic cohomology of the crossed product algebra and Dixmier traces, a strict Fourier duality is established between the discrete zero spectrum and the distribution of prime powers. The construction presupposes no particular value for the real part of non-trivial zeros; the critical line emerges as an inevitable corollary of self-adjoint operator spectral theory and algebraic module structure. Finally, we reveal a three-layer structure of zero fluctuations: no fluctuation at the spectral determinant level, distributional equality at the trace formula level, and scale-separated GUE fluctuations at small scales coexisting with analytic rigidity at large scales.