Quantum Isomorphism Framework: An Architectural Approach to the Riemann Hypothesis

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. This work presents a novel architectural framework based on quantum isomorphism and self-adjoint operators. It does not constitute a completed classical proof of the Riemann Hypothesis but supplies a coherent structural scaffolding within which such a proof (or verification) becomes feasible.