\begin{abstract}
Autoregressive models (ARMs) and energy-based models (EBMs) are equivalent in function space, with the autoregressive factorization obtained by a soft Bellman recursion. We ask what this equivalence means when a sequence alternates between \emph{controllable actions} and \emph{exogenous events}. A teacher-forced model fitted to observational trajectories recovers reachable action and event conditionals, but its ordinary generative readout is not a control solution. The two readouts use different chance-node operators: generation applies an exponential log-moment, whereas control averages under the fixed environment kernel. We derive an exact local-to-global recursion for their value gap, separating locally created entropic optimism from propagated downstream gap. We then give (i) a reference-measure audit showing when event typicality enters a reconstructed energy, (ii) a pathwise decomposition of realized return into soft action advantages and mean-zero event innovations, and (iii) a finite-horizon model-error and support-gated regret bound. All algebraic identities are verified by exhaustive enumeration in a 20,736-trajectory LifeGrid world. A pre-specified $2\!\times\!2$ intervention varies transition entropy independently of continuation-return dispersion. Tabular estimation studies the coverage--error--regret trade-off, and a finite causal Transformer shows that a learned observational model can support a lower-error decision readout without making the model itself a policy. The results delimit a precise bridge from sequence generation to decision support: preserve exogenous chance under its environment law, optimize only controllable actions relative to an explicit reference, and abstain outside supported regions.
\end{abstract}

\noindent\textbf{Keywords:} autoregressive models; energy-based models; soft Bellman recursion; stochastic control; exogenous chance; reference measures; decision support.
