Adelic Hilbert–Pólya Hamiltonian for Zeta Zeros — E8 Intelligence Research

FINDING: The Hilbert–Pólya conjecture is being actively pursued via adelic GL(1) automorphic forms and a proposed explicit Hamiltonian (arXiv:2309.00405), aiming to realize the nontrivial zeros of ζ(s) as eigenvalues of a self-adjoint operator. | MATH: Hilbert–Pólya: zeros ρ = 1/2 + iγ ↔ eigenvalues of self-adjoint operator. Adelic idèle class group: GL(1,𝔸)/GL(1,ℚ) ≅ ℝ₊ × ∏ₚ ℤₚ* — the compact part ∏ₚ ℤₚ* is profinite, with Haar measure normalized to 1. The proposed Hamiltonian (Yakaboylu) is constructed via a trace formula matching the explicit formula: ∑_γ h(γ) = h(i/2) + h(−i/2) − 2∑_n Λ(n)/√n ĥ(log n) + ... (Riemann–von Mangoldt form). The spectral side requires the operator to have discrete spectrum with γ real. | CONNECTION: The idèle class group's compact factor ∏ₚ ℤₚ* is a profinite abelian group — its dual (Pontryagin) is ℚ/ℤ, which is the torsion part of the circle group. The eigenvalues of the Frobenius at p in the adelic picture relate to pⁱᵗ — these are roots of unity s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com