This eighth and final installment completes the Prime–Resonance Operator (PRO) programme. Building on the Gaussian explicit-formula framework developed in PRO I–III and the analytic positivity bounds established in PRO VII, this paper embeds the PRO kernel into a continuum quadratic-form setting aligned with the classical Weil explicit formula. For each scale parameter \sigma>0, we construct a limiting continuum operator \mathcal{P}_{\sigma} obtained as the M\to\infty limit of the windowed, truncated Gaussian–Riemann kernels introduced in PRO IV–VII. We prove convergence of the associated quadratic forms Q_{\sigma,M,W}[f] \;\longrightarrow\; Q_{\sigma}[f] \qquad (f\in\mathcal{A}) with uniformity in \sigma, where \mathcal{A} denotes the Gaussian–Schwartz test class fixed throughout the PRO series. We then identify Q_{\sigma}[f] with a Gaussian instance of the Weil explicit-formula quadratic form and show that the sign of Q_{\sigma}[f] is governed entirely by deviations of the nontrivial zeros \rho=\beta+i\gamma from the critical line. Using this identification and the sign-detection mechanism of PRO II, we prove the Equivalence Theorem: \text{RH} \;\Longleftrightarrow\; Q_{\sigma}[f] \ge 0 \text{ for all } \sigma>0, f\in\mathcal{A} \;\Longleftrightarrow\; Q_{\sigma}[f] \ge \varepsilon\|f\|^{2} \text{ for some } \varepsilon>0. Thus the PRO framework provides a complete explicit-formula reformulation of the Riemann Hypothesis in the Gaussian test-function setting. This installment is fully analytic; numerical plots from PRO IV–VI are not required for the equivalence theorem.