This paper constructs a candidate Hilbert–Pólya operator $L$ built from prime shift operators $U_p$ on $L^2(\mathbb{R}^+, dx/x)$. The construction utilizes the Mellin transform to realize the Riemann zeta function as a multiplication operator, where nontrivial zeros emerge naturally as the spectrum. While the work proves the Density Lemma unconditionally and establishes a convergent sum over the zeros, it identifies two specific open steps—rigorous distributional pairing and Beurling-Nyman inclusion—required to complete a proof of the Riemann Hypothesis.