\section{Adaptive intervention policy}\label{sec:policy}

The deployed router is a \emph{value controller} that scores every nonterminal action from target-free features
and never executes a candidate edit. At state $s=(z_t,S,r,\rho)$ it computes one shared state representation
$u = \mathrm{MLP}(W_z z_t + \mathrm{ctx}(r/R_{\max},\rho/R_{\max}))$ and, for each action $a$,
\begin{equation}
\bar s_a=\frac{1}{k(a)}\sum_{j\in I(a)} S_j,\qquad
\big(\hat g(s,a),\hat c(s,a),\log\hat\sigma(s,a),\hat\ell_{\mathrm{harm}}(s,a)\big)
 = W_{\mathrm{out}}\,\mathrm{SiLU}\big(u + W_s\bar s_a + \kappa_{k(a)}\big)+b,
\label{eq:controller}
\end{equation}
so that scoring all $|\Act|$ actions costs one MLP plus $|\Act|$ projections of width $w$
(Section~\ref{sec:compute}). The four heads estimate the \emph{intervention gain} $g(s,a)$ of
Section~\ref{sec:learning}, a cost proxy, a predictive scale, and the probability that the gain is negative. Actions
are chosen greedily,
\begin{equation}
Q(s,a)=\hat g(s,a)-\lambda\,c(a)-\kappa\,\hat\sigma(s,a),\qquad Q(s,\Stop)=0,\qquad
a^\star=\arg\max_{a\in\Act\cup\{\Stop\}} Q(s,a),
\label{eq:Q}
\end{equation}
where $c(a)$ is the \emph{known} analytic cost of the action (ops of one edit plus one controller evaluation,
relative to the backbone's ops per token; Section~\ref{sec:compute}); the learned $\hat c$ is kept only as a
diagnostic regressor. The state is recomputed after every executed edit, and the loop stops when \Stop{} wins or
the cap $R_{\max}$ is reached. The penalties $(\lambda,\kappa)$ are tuned on calibration rollouts only
(Section~\ref{sec:design}); sweeping $\lambda$ traces the controller's quality--cost curve.

\paragraph{Known limitations of the rule.}
The policy is one-step greedy: it cannot represent an action whose value materialises only after a later action.
We quantify this myopia directly with a two-action lookahead oracle (Section~\ref{sec:results-oracle}) rather than
assume it away. The rule is also only as good as the calibration of $\hat g$; Proposition~\ref{prop:regret} bounds
the one-step loss of the greedy rule in terms of uniform estimation errors but says nothing about multi-step
optimality.