Metric completion of the Bender-Brody-Müller Hamiltonian: dilation spectrum and missing eigenstates

The Bender-Brody-Müller (BBM) Hamiltonian was proposed as a non-Hermitian Hilbert-Pólya operator. We analyze the Hilbert completion induced, on the standard half-line core C_c^\infty(0,\infty)\subset L^2(\mathbbR_+), by BBM's explicit candidate metric \hat\eta=\sin^2(\hat p/2)=\Delta^\dagger\Delta/4. The bounded positive form \eta_0=\Delta^\dagger\Delta has trivial kernel but is not coercive. Completing the core in the norm \|\psi\|_\eta_0=\|\Delta\psi\| gives a Hilbert space canonically unitarily equivalent to L^2(\mathbbR_+); its free self-adjoint realization is the dilation generator, with simple, purely absolutely continuous spectrum \mathbbR. No bounded sandwich \Delta^\dagger h(D)\Delta is boundedly invertible. For the symmetric restriction transported from the stated initial core, the closure has deficiency indices (\infty,\infty) and its adjoint has every real point as an eigenvalue of infinite multiplicity, whereas the free extension is purely continuous. These domain-dependent statements are separated from the realization-independent conclusion about BBM's specified candidate functions: \Delta\psi_z=x^-z, so for \operatornameRez=1/2 they do not belong to this completed space. Thus changing the self-adjoint extension within this completion cannot make the original BBM functions and boundary condition produce point-spectrum Riemann-zero states. Other non-L^2, rigged, distributional, or unbounded-metric formulations are outside the theorem.