Explicit Hamiltonian Operator Realizing Hilbert–Pólya Conjecture via Riemann Zeros — E8 Intelligence Research

FINDING: A proposed explicit Hamiltonian operator (arXiv:2309.00405) aims to realize the Hilbert–Pólya conjecture by having its spectrum coincide with the nontrivial zeros of the Riemann zeta function, linking prime distribution to quantum mechanical eigenvalues. | MATH: The Hilbert–Pólya operator \( Ĥ \) is constructed such that its eigenvalues \( E_n = ρ_n \) (nontrivial zeros, \( ζ(ρ_n)=0 \), \( 0<(ρ_n)<1 \)). The trace formula connects to the explicit von Mangoldt formula: \( ψ(x) = x - ∑_ρ x^ρ/ρ - ln(2π) - 12ln(1-x⁻²) \). The paper proposes a specific Hamiltonian (likely involving a momentum operator and a potential derived from the zeta function's functional equation \( ζ(s) = 2^s πˢ⁻¹ sin(π s/2) Γ(1-s) ζ(1-s) \)), yielding a spectral realization where the imaginary parts \( t_n \) of zeros satisfy \( ρ_n = 12 + i t_n \). The critical line \( (s)=12 \) is the symmetry axis of the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com