We present a fully rigorous operator-theoretic framework targeting the spectralrealization of the nontrivial zeros of the Riemann zeta function. By constructingprime-localized perturbations of the self-adjoint dilation generator on L2(R+)and employing KLMN form-sum techniques, we establish well-defined Hamiltonianswhose domains are stable under infinite-prime limits. Finite-rank resolvent differencesdefine spectral shift functions consistent with Birman–Krein theory. Complexdilation-analytic scaling unblocks the continuous spectrum, allowing arithmeticinducedresonances to emerge. Integration of the Standing/Sitting Band Framework(SSBF) provides deterministic weighting of primes, ensuring convergence anda staircase-like spectral shift corresponding to the critical line. Obstruction theoremsprevent compact-resolvent Hilbert–P´olya realizations, highlighting the necessityof the present construction.