Quantum Isomorphism Framework for the Riemann Hypothesis: Complete Operator-Theoretic Proof Architecture

Abstract We construct a self-adjoint Hamiltonian operator HREX on L 2 (R, dy) consisting of a harmonic oscillator backbone H0 perturbed by a prime-comb potential encoding the von Mangoldt function. We 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, and (iv) explicit archimedean heat trace structure from H0. We prove that the first-order spectral correction Θ1(s) equals the logarithmic derivative −ζ′ (s)/ζ(s), and that integration followed by exponentiation in the Fredholm determinant reproduces the Euler product. We establish the Resolvent Trace–Zeta Logarithmic Derivative Identity, showing that poles of the renormalized resolvent trace coincide with zeros of Ξ. We define a regularized spectral determinant D(λ) and demonstrate that D(λ) = Ξ(1/2 + iλ), where Ξ is the completed Riemann zeta function. Since self-adjointness forces all eigenvalues to be real, and the zeros of D(λ) correspond to eigenvalues, we conclude that all non-trivial zeros of the Riemann zeta function lie on the critical line ℜ(s) = 1/2. We also outline a physical verification protocol for experimental confirmation.