\label{def:tau-effective-rh}
For every primorial depth $k \geq 1$, let $L_k = \{p_1, \ldots, p_k\}$ denote the first $k$ primes, and let $H_{L_k}$ denote the lemniscate operator on the $k$-dimensional space $\mathcal{H}_{L_k} = \bigoplus_{j=1}^k \mathbb{C} \cdot e_j$ defined in Chapter~23. The \textbf{$\tau$-effective RH statement at depth $k$} asserts:
\begin{enumerate}[label=(\roman*)]
    \item \textbf{Spectral reality}: All eigenvalues $\lambda \in \operatorname{Spec}(H_{L_k})$ are real, i.e., $\Im(\lambda) = 0$.
    \item \textbf{Zero reality}: The finite Euler product
    \[
    \zeta_{L_k}(s) = \prod_{p \in L_k} \frac{1}{1 - p^{-s}}
    \]
    has all zeros in the $\tau$-effective window $\Re(s) \in [0, 1]$, $|\Im(s)| \leq T_k$ (for computable cutoff $T_k$) located on the critical line $\Re(s) = \tfrac{1}{2}$.
    \item \textbf{Spectral-zero correspondence}: The real eigenvalues $\lambda_j$ correspond bijectively to zeros $s_j = \tfrac{1}{2} + i\lambda_j$ via the determinant representation (Conjecture O3).
\end{enumerate}
The statement is \textbf{$\tau$-effective} because each depth $k$ involves only finite-dimensional linear algebra and finite products, both computable with certified interval arithmetic.