\section{Model error, global regret, and support gating}\label{sec:error}

Let $M=(\mu,r)$ and $\hat M=(\hat\mu,\hat r)$ share action sets, $\pi_{\refp}$, and $\tau$. Rewards at action and event positions are bounded by $R_{\max}$ in absolute value. Suppose
\[
 |\hat r-r|\le\epsilon_r,\qquad
 \TV(\hat\mu(\cdot\mid s'),\mu(\cdot\mid s'))\le\epsilon_\mu
\]
uniformly on the relevant region. Let $V_{M,h}^{\pi}$ be the KL-regularized value of policy $\pi$ with $h$ rounds remaining.

\begin{theorem}[Uniform simulation and policy regret]\label{thm:error}
For every Markov policy $\pi$ and $h\le K$,
\begin{align}
\|V_{M,h}^{\pi}-V_{\hat M,h}^{\pi}\|_\infty&\le C_h,\label{eq:sim-bound}\\
C_h&=2h\epsilon_r+\epsilon_\mu\sum_{j=1}^{h}
\left[2R_{\max}+(j-1)(4R_{\max}+\tau B_{\refp})\right].\label{eq:ch}
\end{align}
The same bound holds between optimal values. If $\hat\pi$ is $\eta$-optimal in $\hat M$, then
\begin{equation}\label{eq:regretbound}
 J_M^\tau(\pi_M^*;x)-J_M^\tau(\hat\pi;x)\le2C_K+\eta.
\end{equation}
For a uniform reference over at most $A_{\max}$ actions, the order is
$O(K\epsilon_r+K^2(R_{\max}+\tau\log A_{\max})\epsilon_\mu+\eta)$.
\end{theorem}

The theorem controls a complete rollout, not merely one-step $Q$ error. It is intentionally conservative: the uniform maximum error prevents a new policy from exploiting a rare but poorly estimated state. Weighted prediction error may be operationally useful, but it cannot replace $\epsilon_\mu$ in \eqref{eq:ch}.

\subsection{Gating and abstention}

Let $G(s)\subseteq\A(s)$ be a support gate and $\Pi_G$ the policies that put no mass outside $G(s)$. Let $\pi_G^*$ optimize the true model over $\Pi_G$ and $\hat\pi_G$ be $\eta$-optimal for the learned model over $\Pi_G$.

\begin{corollary}[Support-gated regret]\label{cor:gate}
If the assumptions of Theorem~\ref{thm:error} hold on all nodes reachable by $\Pi_G$, then
\begin{align}
J_M^\tau(\pi_M^*;x)-J_M^\tau(\hat\pi_G;x)
&\le \underbrace{J_M^\tau(\pi_M^*;x)-J_M^\tau(\pi_G^*;x)}_{\text{coverage-restriction cost}}\nonumber\\
&\quad+\underbrace{2C_K(G)+\eta}_{\text{within-gate estimation/planning cost}}.\label{eq:gated}
\end{align}
\end{corollary}

Abstention is therefore not free. Increasing a count threshold can lower maximum model error but may exclude valuable actions or leave no supported route to parts of the tree. A deployed system should report coverage and error together, and should switch from ranking to scenario exploration when the gate is not credible.

\paragraph{Proof strategy.} A backward induction bounds every fixed policy uniformly. The optimal-value bound follows by taking suprema, and \eqref{eq:regretbound} follows from a model-comparison sandwich. Corollary~\ref{cor:gate} inserts and subtracts the true gated optimum. Complete proofs are in Appendix~\ref{app:proofs}.
