\begin{algorithm}[!t]
\caption{Multivalid Split Conformal (MVSC)}
\label{alg:mvsc}
\begin{algorithmic}[1]
\STATE \textbf{Input:} calibration set $\hat{D}$, LLM $L$, fact-level scoring function $f$, target error rate $\alpha$, split conformal threshold $q_\alpha$, max iterations $T$
\STATE Let $\mathcal{F}_{h_t}(L(X)) = \{ x \in L(X) \mid f(x) \ge h_t(X) \}$
\STATE Let $\Delta_{t, g} = P_{\mathcal{D}} (Y \in \mathcal{F}_{h_t}(L(X)) \mid g(X) = 1)$
\STATE Let $H_t(g) = P_{\mathcal{D}}(g(X) = 1) [(1 - \alpha) - \Delta_{t, g}]^2$

\STATE Initialize $h_0(X) = q_\alpha$

\FOR{$t \in \{0, 1, \ldots, T-1 \}$}
    \STATE Set
    \[
        g_t = \argmax_{g \in \mathcal{G}} H_t(g)
    \]
    \STATE Let $\hat{D}_t = \{ (X,Y) \in \hat{D} \mid g_t(X) = 1\}$
    \STATE Set $q_t$ to be the $\frac{\lceil (n+1)(1-\alpha) \rceil}{n}$th-quantile of scores $\{r(X_i, Y_i)\}$ for $X_i,Y_i \in \hat{D}_t$
    \STATE Set 
    \[
        h_{t+1}(X) = 
        \begin{cases} 
        q_t & \text{if } g_t(X) = 1 \\
        f_t(X) & \text{otherwise}
        \end{cases}
    \]
    \IF{$H_t(g_t) \ge H_{t-1}(g_{t-1})$} 
        \STATE Set t = t-1
        \BREAK
    \ENDIF
\ENDFOR
\STATE \textbf{Output:} $h_t$
\end{algorithmic}
\end{algorithm}