Explicit Hamiltonian Realizing Hilbert-Pólya Conjecture Links Primes to Quantum Spectra — E8 Intelligence Research

FINDING: A proposed explicit Hamiltonian operator (arXiv:2309.00405) aims to realize the Hilbert-Pólya conjecture — that nontrivial zeta zeros are eigenvalues of a self-adjoint operator — linking prime distribution to quantum spectral theory. | MATH: Hilbert-Pólya: zeros ρ = 1/2 + iγ are eigenvalues of a Hermitian operator H. Explicit form (per Yakaboylu) involves a momentum-dependent potential derived from the prime-counting function; key identity: ζ(s) = ∏_p (1 − p^(−s))^(−1) ↔ Tr(H) ~ Σ_γ e^{iγt} = Σ_p log(p) δ(t − log(p)) (explicit trace formula, Guinand–Weil). Constants: γ_n (imaginary parts of zeros), Euler–Mascheroni γ ≈ 0.5772, prime gaps ~ log(p). | CONNECTION: The trace formula's spectral side (sum over γ) vs. geometric side (sum over primes) mirrors a duality between a 1D lattice of zeros (spacing ~ 2π/log(γ/2π)) and a multiplicative lattice of primes. The zero spacing statistics (GUE, random matrix) connect to root systems of type A_N (Weyl chamber), and the critical line R Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com