Towards an Explicit Hilbert–Pólya Operator

This working paper addresses the Hilbert–Pólya conjecture — the search for a self-adjoint operator whose eigenvalues coincide with the imaginary parts of the non-trivial zeros of the Riemann zeta function. Measuring the three principal candidates in the literature (Connes' adelic class space, the Berry–Keating Hamiltonian, and the Bost–Connes C*-dynamical system) against two structural conditions — a discrete spectral hierarchy anchored on the primes 2, 3, 5, 7 with a 1/p amplitude law, and a KMS phase structure with temperature-driven symmetry breaking — the paper proposes the Bost–Connes system over the cyclotomic field ℚ (ζ₄₂₀) as a candidate satisfying both. The ramified primes of ℚ (ζ₄₂₀) /ℚ are exactly 2, 3, 5, 7; the maximal real subfield has Galois group of order 48, matching the admissible residue classes modulo the first primorial. Three empirical signatures are presented (GUE level statistics, the harmonic distribution of higher primes in twin-prime midpoints, and the exactness of the four-axis sieve below 11²), and the resulting conjecture is formulated with an explicit verification programme. No proof of the Riemann hypothesis is claimed; the spectral identification remains an explicitly stated open problem.