\section{Reference-measure audit and pathwise attribution}\label{sec:audit}

\subsection{What a reconstructed energy measures}

Let $b(z\mid s)$ equal $\pi_{\refp}$ at a decision node and $\mu$ at a chance node. Suppose a full-support conditional $\hat p$ is represented by local score $q_b$:
\begin{equation}\label{eq:localscore}
\hat p(z\mid s)=b(z\mid s)\exp\{[q_b(s,z)-V_{q_b}(s)]/\tau\},\quad
V_{q_b}(s)=\tau\log\Exp_{z\sim b}\exp\{q_b(s,z)/\tau\}.
\end{equation}
Define the inverse edge score $r_b(s,z)=q_b(s,z)-V_{q_b}(s\oplus z)$.

\begin{proposition}[Relative-energy audit]\label{prop:audit}
For every complete trajectory,
\begin{equation}\label{eq:audit}
\sum_{(s,z)\in\xi}r_b(s,z)
=\tau\log\frac{\hat\rho(\xi\mid x)}{m_{\refp}(\xi\mid x)}+V_{q_b}(x).
\end{equation}
For the observational law $\rho_\beh=\pi_\beh\times\mu$, event terms cancel under the $\mu$-aware base:
\begin{equation}\label{eq:eventcancel}
\tau\log\frac{\rho_\beh(\xi\mid x)}{m_{\refp}(\xi\mid x)}
=\tau\sum_{t=1}^{K}\log\frac{\pi_\beh(a_t\mid s_t)}{\pi_{\refp}(a_t\mid s_t)}.
\end{equation}
Under a counting base, the reconstructed energy instead contains both $\sum_t\log\pi_\beh(a_t\mid s_t)$ and $\sum_t\log\mu(e_t\mid s'_t)$.
\end{proposition}

Both bases are mathematically legitimate, but they answer different questions. Counting-base energy measures joint trajectory typicality. Relative energy under \eqref{eq:base} removes event typicality when interpreting action deviations from $\pi_{\refp}$. Conflation arises only when joint likelihood is labeled merit, choice quality, or intentionality. Typicality is also distinct from favorability: $\log\mu(e\mid s')$ asks whether an event was expected, not whether it helped.

\subsection{Choice and event components}

Using the control readout, define the soft action advantage and event innovation
\begin{align}
A_\ctrl(s,a)&=Q_\ctrl(s,a)-V_\ctrl(s)
=\tau\log\frac{\pi^*_\ctrl(a\mid s)}{\pi_{\refp}(a\mid s)},\label{eq:advantage}\\
I_{\mathrm{event}}(s',e)&=r(s',e)+V_\ctrl(s'\oplus e)-W_\ctrl(s').\label{eq:eventimpact}
\end{align}
By construction, $\Exp_{e\sim\mu}I_{\mathrm{event}}(s',e)=0$. Other action-impact baselines may be useful: for any declared $\nu(\cdot\mid s)$,
\begin{equation}
I_{\mathrm{action}}^\nu(s,a)=Q_\ctrl(s,a)-\Exp_{a'\sim\nu(\cdot\mid s)}Q_\ctrl(s,a').
\end{equation}
The baseline must be reported. Using $\pi_\beh$ describes deviation from comparable observed behavior; using a user's default policy describes deviation from that user's tendency.

\begin{theorem}[Pathwise choice--event decomposition]\label{thm:pathwise}
For any realized trajectory $\xi$ and terminal $V_\ctrl(s_{K+1})=0$,
\begin{equation}\label{eq:pathwise}
R(\xi)-V_\ctrl(s_1)
=\sum_{t=1}^{K}A_\ctrl(s_t,a_t)
+\sum_{t=1}^{K}I_{\mathrm{event}}(s'_t,e_t).
\end{equation}
The identity holds regardless of the policy that produced the realized actions.
\end{theorem}

The first sum records the realized sequence of controllable deviations from the soft baseline; the second records favorable and unfavorable chance innovations. The theorem is algebraic and predictive. A causal reading of either component still requires Assumption~\ref{ass:invariance} and the usual longitudinal identification conditions.

\subsection{A policy-level regret identity}

For any full-support Markov policy $\pi$, let $d_t^\pi$ be its state occupancy under $\mu$. The KL-regularized performance difference specializes to
\begin{equation}\label{eq:klregret}
J_M^\tau(\pi^*_\ctrl;x)-J_M^\tau(\pi;x)
=\tau\sum_{t=1}^{K}\Exp_{S_t\sim d_t^\pi}
\KL\bigl(\pi(\cdot\mid S_t)\,\|\,\pi^*_\ctrl(\cdot\mid S_t)\bigr).
\end{equation}
This identity supplies an exact implementation test and connects local action divergence to global regret.

