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 \( \hat{H} \) is constructed such that its eigenvalues \( E_n = \rho_n \) (nontrivial zeros, \( \zeta(\rho_n)=0 \), \( 0<\Re(\rho_n)<1 \)). The trace formula connects to the explicit von Mangoldt formula: \( \psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln(2\pi) - \frac12\ln(1-x^{-2}) \). The paper proposes a specific Hamiltonian (likely involving a momentum operator and a potential derived from the zeta function's functional equation \( \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) \)), yielding a spectral realization where the imaginary parts \( t_n \) of zeros satisfy \( \rho_n = \frac12 + i t_n \). The critical line \( \Re(s)=\frac12 \) is the symmetry axis of the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com