Construct a canonical relative/renormalized spectral zeta \(Z_x(s)\) for the full Morse Hamiltonian by combining the bound-state contribution with the scattering spectral-shift contribution relative to the dissociation/free reference Hamiltonian and subtracting explicit semiclassical counterterms. Require the normalization to satisfy \[ Z_x(s)\longrightarrow \zeta(s) \] for \(\Re s>1\) as \(x\to0^+\). Prove that, after the explicit renormalization, this convergence extends meromorphically and locally uniformly to a connected domain containing the critical strip \(0<\Re s<1\).

For every simple nontrivial Riemann zero \(\rho\), prove that for all sufficiently small \(x>0\) there is a unique nearby zero \(s_\rho(x)\) of \(Z_x\) with \[ s_\rho(x)\to\rho\qquad(x\to0^+), \] and determine the first nonzero asymptotic term in \[ s_\rho(x)-\rho \] explicitly from the exact Morse bound-state spectrum and scattering phase. Under RH and simplicity of the nontrivial Riemann zeros, determine whether the horizontal displacements \[ \operatorname{Re}(s_\rho(x)-\rho) \] take both signs infinitely often as \(|\operatorname{Im}\rho|\to\infty\); a quantitative lower bound for both signs or a limiting distribution for suitably normalized displacements is an acceptable stronger result.