A Quantum Isomorphism Framework for the Riemann Hypothesis: An Operator-Theoretic Program with an Open Spectral Conjecture

We construct a self-adjoint Hamiltonian operator HREX on L2(R,dy) consisting of a harmonic oscillator backbone H0 perturbed by a prime-comb potential encoding the von Mangoldt function. We rigorously establish: (i) self-adjointness via Kato–Rellich, (ii) compact resolvent yielding purely discrete spectrum, (iii) trace-norm convergence of the Duhamel–Dyson expansion for the heat semigroup, (iv) explicit archimedean heat trace structure fromH0, and (v) that the first-order spectral correction Θ1(s) equals the logarithmic derivative−ζ′(s)/ζ(s), with integration and exponentiation reproducing the Euler product. These results establish a Hilbert–P´olya-type operator whose construction is fully rigorous and whose first-order spectral behavior matches the zeta function’s prime-counting structure. The framework does not, however, constitute a proof of the Riemann Hypothesis. The central remaining step — a proposed Resolvent Trace–Zeta Logarithmic Derivative Identity connecting the renormalized resolvent trace of HREX to Ξ′/Ξ(1/2 + iλ) — is posed here as an open conjecture, not a proven theorem. If this identity could be established from the explicit operator construction, the self-adjointness of HREX (which does hold rigorously) would immediately imply the Riemann Hypothesis. Establishing that identity is precisely the content of the decades-old Hilbert–P´olya program and remains unsolved in this work, as it does in the broader literature (Berry–Keating, Connes, Sierra, and others). This paper should therefore be read as a structural research program — a concretely defined, partially rigorous candidate operator and a precise statement of the one conjecture that would complete the argument — rather than as a claimed proof.