\section{Experiments}\label{sec:experiments}

\subsection{Protocol and evidential scope}

Before the full run, we froze the claim hierarchy, factorial interventions, sample sizes, seed counts, support thresholds, architecture, metrics, and rejection rules. The code and protocol are included in the supplementary package. All experiments are synthetic validation of the formal objects; no result is treated as causal evidence about human outcomes.

LifeGrid has $K=4$ rounds, $|\A|=4$, $|\E|=3$, and $(4\cdot3)^4=20{,}736$ complete trajectories. The exact state is the full typed history. Rewards and transition logits are generated from an environment seed, $\pi_{\refp}$ is uniform, and $\tau=1$. The behavior policy has full support. Reusing a seed across factorial cells reuses every base table.

The factorial intervention has two levels of transition entropy and two levels of event-dependent continuation dispersion. Entropy changes only the transition-logit temperature ($0.35$ versus $2.50$); dispersion changes only the scale of event and continuation features ($0.15$ versus $1.10$). Hence the high/low entropy worlds have different $\mu$ but the same reward table within a dispersion condition, while the high/low dispersion worlds have identical observational token laws but different returns.

\subsection{E1: exhaustive identity checks}

We enumerate every trajectory for \ExactSeeds{} independent worlds and all four factorial cells. Tests cover normalization and factorization in Proposition~\ref{prop:factor}, both lines of Proposition~\ref{prop:gap}, the two reference bases in Proposition~\ref{prop:audit}, Theorem~\ref{thm:pathwise}, the conditional mean of event innovations, and the KL-regret identity \eqref{eq:klregret}. The pre-specified rejection threshold was $10^{-10}$ in float64. The largest absolute residual was \ExactMaxResidual{}, so no identity test was rejected (\Cref{fig:residuals}). These computations verify the implementation of exact equalities; they are not empirical proofs of the propositions.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.82\linewidth]{identity_residuals.pdf}
    \caption{Maximum absolute residual over exhaustive identity tests. The dashed line is the rejection threshold frozen before the run.}
    \label{fig:residuals}
\end{figure}

\subsection{E2: entropy and continuation dispersion}

\Cref{fig:factorial} shows the pre-specified $2\times2$ comparison. The transition intervention changes measured event entropy while preserving it across dispersion levels, confirming that the factors are mechanically separated. With low return dispersion, the mean root gap is only \GapLowLow{} and \GapHighLow{} in the low- and high-entropy cells. Raising return dispersion increases the gap to \GapLowHigh{} and \GapHighHigh{}, respectively. The true control regret of deploying the generative action policy is \RegretLowHigh{} and \RegretHighHigh{} in the two high-dispersion cells.

The result supports the proposition's qualitative prediction: entropy alone is insufficient, while dispersion of chance continuation values creates the mismatch. It also shows interaction rather than pure additivity: a more diffuse $\mu$ exposes more of the return dispersion. This is consistent with the exact object $g_\tau$, which depends on the full conditional distribution of return, not on either scalar factor alone.

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{factorial_gap.pdf}
    \caption{Pre-specified factorial experiment. Points are means and bars are 95\% bootstrap intervals over environment seeds. Left: the entropy manipulation is invariant to return dispersion. Middle: the exact root readout gap. Right: true regret of the generative action policy evaluated with the control objective.}
    \label{fig:factorial}
\end{figure}

\subsection{E3: finite tabular estimation and support}

For ten environment seeds and five data seeds, we sample $N\in\{100,300,1000,3000,10000,30000\}$ observational trajectories in the high-entropy/high-dispersion cell. Symmetric Dirichlet smoothing ($\alpha=0.5$) estimates conditionals, and conditional reward means are learned from observations with Gaussian noise standard deviation $0.10$. We report exact trajectory KL, occupancy-weighted TV, true policy regret, and the maximum TV over each declared supported region.

From $N=\TabularNFirst{}$ to $N=\TabularNLast{}$, mean exact trajectory KL decreases from \TabularKLFirst{} to \TabularKLLast{}, while occupancy-weighted event TV decreases from \TabularTVFirst{} to \TabularTVLast{} (\Cref{fig:tabular}). Control-value error falls from \TabularCtrlErrorFirst{} to \TabularCtrlErrorLast{}. In contrast, generative-value error moves from \TabularGenErrorFirst{} toward \TabularGenErrorLast{}, nearly the exact high/high structural gap \GapHighHigh{}. Better estimation therefore does not make the generative readout a control readout; it makes the model converge more accurately to the different generative objective. These averages assess predictive learning but cannot certify Theorem~\ref{thm:error}; the theorem requires a maximum error on the entire deployment region.

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{tabular_learning.pdf}
    \caption{Tabular learning curves. Lines are means and bands are 95\% bootstrap intervals over environment/data seeds. KL is evaluated exactly by trajectory enumeration. The right panel shows structural rather than estimation error: as $\hat\mu$ improves, the generative readout converges toward the wrong chance operator for the control target.}
    \label{fig:tabular}
\end{figure}

A visit-count gate exposes the missing trade-off (\Cref{fig:support}). At $N=\TabularNLast{}$, the supported chance mass changes from \SupportAtLow{} at the lowest threshold to \SupportAtHigh{} at the highest. Stronger gating can reduce the maximum error among retained nodes but raises the coverage-restriction cost; in sparse full-history states, that cost can dominate. The computed worst-case bound remains conservative and often vacuous, which is itself an important negative result: a theorem whose premises require full-tree uniform accuracy does not become an operational certificate merely because average prediction is good.

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{support_tradeoff.pdf}
    \caption{Coverage--error--regret trade-off at $N=\TabularNLast{}$. A count gate improves the declared error region by excluding data-poor nodes, but exclusion can create substantial opportunity cost.}
    \label{fig:support}
\end{figure}

\subsection{E4: finite causal Transformer and two readouts}

We train a two-layer causal Transformer (width 32, four heads, feed-forward width 128) by teacher forcing on $30{,}000$ sequences for 1,200 AdamW updates. Typed-token masking permits only actions at action positions and events at event positions. There are five environment seeds and five training seeds per entropy condition. Because dispersion changes only rewards, the same trained observational model is reused for the two dispersion conditions. Evaluation enumerates every valid prefix and every complete trajectory; it is not based on sampled test rollouts.

The learned model reaches mean exact trajectory KL \NeuralKLlow{} and \NeuralKLHigh{} in the low- and high-entropy conditions. We then use the same estimated $\hat\mu$ in two planners: the chance-corrected control recursion and the naive generative recursion. In the high-entropy/high-dispersion cell, mean control-value error is \NeuralCtrlErrorHH{} versus \NeuralGenErrorHH{} for the generative readout; true policy regret is \NeuralCtrlRegretHH{} versus \NeuralGenRegretHH{} (\Cref{fig:neural}). The comparison isolates the readout: both planners receive the same finite model and reward.

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{neural_readouts.pdf}
    \caption{Finite Transformer results. L/L, H/L, L/H, and H/H denote transition entropy / return dispersion. Bars are means and error bars are 95\% bootstrap intervals. The right panel reports exact trajectory KL of the observational ARM.}
    \label{fig:neural}
\end{figure}

The neural experiment does not show that teacher forcing solves control. It shows the more limited and useful statement: a teacher-forced model can estimate observational conditionals, after which a structurally correct readout can materially reduce decision error. Approximation and optimization error remain visible in nonzero KL and maximum prefix TV.

\subsection{Summary of claim support}

\begin{table}[t]
\centering
\caption{What each experiment can and cannot establish.}
\label{tab:evidence}
\begin{tabularx}{\linewidth}{@{}lYY@{}}
\toprule
Experiment & Supported conclusion & Not established \\
\midrule
Exact enumeration & Code satisfies stated equalities to tolerance & Generality of the proofs (established analytically) \\
Factorial intervention & Gap tracks chance-return dispersion; entropy modulates exposure & A universal scalar law in entropy or variance \\
Tabular estimation & Prediction improves; coverage and uniform error conflict & A non-vacuous certificate for sparse full histories \\
Transformer & Same learned model benefits from a control readout & Causal identification, human utility, real-world safety \\
\bottomrule
\end{tabularx}
\end{table}
