\label{prop:primorial-rh-protocol}
Fix primorial depth $k \geq 1$. The following protocol verifies the $\tau$-effective RH statement at depth $k$:
\begin{enumerate}[label=\textbf{Step \arabic*.}]
    \item \textbf{Construct $H_{L_k}$}: Using the definition from Chapter~23, compute the $k \times k$ matrix representation of $H_{L_k}$ in the basis $\{e_1, \ldots, e_k\}$. Entry $(i,j)$ is given by the bipolar coupling $\langle e_i, e_j \rangle_\jj$.

    \item \textbf{Certify self-adjointness}: Verify that $H_{L_k}$ is Hermitian (in the split-complex inner product). This is guaranteed by Theorem~III.T17; the verification step checks consistency.

    \item \textbf{Compute spectrum}: Using certified interval arithmetic (e.g., Arb library), compute $\operatorname{Spec}(H_{L_k})$ with rigorous error bounds. Verify that all eigenvalues have $|\Im(\lambda)| < \epsilon$ for machine tolerance $\epsilon$.

    \item \textbf{Locate zeros}: For each real eigenvalue $\lambda_j$, compute the corresponding zero $s_j = \tfrac{1}{2} + i\lambda_j$ of the finite Euler product $\zeta_{L_k}(s)$ via the determinant representation
    \[
    \det(s \cdot I - H_{L_k}) \stackrel{?}{=} C_k(s) \cdot \zeta_{L_k}(s),
    \]
    where $C_k(s)$ is a normalization polynomial. Verify that $\zeta_{L_k}(s_j) \approx 0$ within tolerance.

    \item \textbf{Check critical line}: Confirm that all zeros $s_j$ satisfy $|\Re(s_j) - \tfrac{1}{2}| < \epsilon$.

    \item \textbf{Tower coherence}: If $k > 1$, verify that the spectrum $\operatorname{Spec}(H_{L_{k-1}})$ is a subset of $\operatorname{Spec}(H_{L_k})$ up to tolerance, consistent with the nesting property of primorial towers.
\end{enumerate}
The protocol runs in $O(k^3)$ time (dominated by eigenvalue computation) and outputs a certified \textsc{pass/fail} verdict.