Obstructions to the Spectral Proof of the Riemann Hypothesis via the Yakaboylu Hamiltonian

We analyze the spectral approach to the Riemann Hypothesis based on the Hamiltonian construction of Yakaboylu (arXiv:2408.15135), which realizes the nontrivial zeros of the Riemann zeta function as eigenvalues of a non-self-adjoint operator on a Hilbert space. We identify three fundamental obstructions to completing this program: (1) the transformed Dirichlet domain is not dense in the Hilbert space, rendering the self-adjointness gap formulation ill-posed; (2) the bounded positive intertwining operator gives only conjugate pairing of eigenvalues rather than reality, due to a directional mismatch in the intertwining relation; and (3) the unbounded intertwining operator that would yield reality has a domain that excludes the eigenfunctions. We further show that the natural completeness condition required to close the gap via pseudo-Hermitian operator theory is equivalent to the Nyman-Beurling criterion, itself a known reformulation of the Riemann Hypothesis. Our results provide a precise obstruction map for the Hilbert-Polya spectral program in this framework.