The Riemann Hypothesis and Hilbert-Pólya Conjecture: A Complete Proof via Prime Shift Operators.

We construct an explicit Hilbert-Pólya operator $L=M_{t}|_{ker(Z)|_{\Phi^{\prime}}}$ from prime shift operators $U_{p}\psi(x)=\psi(px)$ on $L^{2}(\mathbb{R}^{+},dx/x)$. The Euler product $Z=\prod_{p}(I-U_{p})^{-1}$ realizes the Riemann zeta function $\zeta(\frac{1}{2}+it)$ as a multiplication operator in Mellin space. We resolve two previously open steps: (1) the distributional pairing problem is closed by constructing a nuclear space with weight $e^{\pi|u|}$ and a renormalized inner product on $span\{\delta(t-\gamma_{n})\}$, proving that $\delta^{\prime}$ vectors cannot belong to any symmetric domain, hence all zeros are simple and L is essentially self-adjoint; (2) the inclusion $\mathcal{V}_{N}\subseteq\mathcal{B}$ is proved via residue synthesis, showing each $\rho_{p,k}$ lies in the Beurling-Nyman space. The Density Lemma then gives $\mathcal{B}=L^{2}(0,1)$ so $1_{(0,1)}\in\mathcal{B}$, and by the Beurling-Nyman criterion the Riemann Hypothesis follows. The operator L provides an explicit self-adjoint operator with spectrum $\{\gamma_{n}\}$ proving the Hilbert-Pólya conjecture.