\section{Agent-based experiments}
\label{sec:sim}

Everything in this section is \SIM. The simulation exists to measure what the historical record cannot: messages, bits, queues, overhead and failures under institutional regimes that are \emph{imposed} by the experimenter. Nothing emerges in these experiments, and nothing in them is evidence about history; they test the theorems of Section~\ref{sec:theory} and search for the conditions under which institutions reduce efficiency or add fragility, as the pre-registration requires.

\subsection{Design}
\label{sec:sim-design}

$N$ agents are placed at random in the unit square and linked on a lattice-like nearest-neighbour graph, a small-world rewiring of it, or a random graph of the same degree. Each period, tasks arrive at rate $\lambda = 0.5$ per producing agent; a task is initiated by one agent and requires $k=3$ participants---the initiator and two network neighbours---to hold the same plan, drawn from a set of $s=64$ plans with a Zipf distribution of exponent 0 (uniform, entropy 6 bits) or 1 (entropy 4.86 bits). The initiator must communicate the plan. Each agent can process a bounded number of messages per period (2 or 8); excess messages queue first-in-first-out, and a task whose last message is served more than three periods after arrival fails. Agents differ in a continuous ``dialect'' that is partly spatial; without a shared code, a message is misdecoded with probability proportional to the dialect distance (0.25 per unit), and misdecoded tasks fail. Five regimes are compared. \emph{None}: raw messages of $\lceil\log_2 s\rceil$ bits per recipient, dialect errors, one acknowledgement round. \emph{Norms}: a shared Shannon code optimal for the plan distribution, lower decoding error (0.02), a one-time learning cost of 200 bits per agent, and, after a shift in the plan distribution, a stale code until the institution re-learns it ten periods later. \emph{Records}: norms plus a written record per task (four bits of administrative overhead) against which recipients verify, eliminating decoding errors and the acknowledgement round. \emph{Central}: records plus an eight-ary administrative tree through whose root every task is relayed; administrators have their own bandwidth and are withdrawn from production. \emph{Federated}: the same tree, but a task is relayed only up to the lowest common ancestor of its participants, so that local tasks never reach the root. Administrative bandwidth is set as a ratio (0.5 or 2) of the expected relayed load. Shocks at period 20 of 40 either shift the plan distribution (a cyclic permutation chosen to reach a target Kullback--Leibler divergence; the achieved divergence, 0.52 bits, is recorded) or remove half of the links. The main design crosses $N\in\{50,200,800\}$, three networks, two bandwidths, two entropies, three shock conditions and five regimes with eight seeds (6,048 runs); the ablations (584 runs) vary verification, adaptation speed, horizon, learning cost, heterogeneity, administrative capacity, branching and the plan space. Measured outcomes are task success, messages and bits per task, bits per successful task with and without set-up, latency, failure shares by cause, administrative overhead share, welfare (task value minus a cost per bit), the Gini coefficient of benefits among producers, and root load. Entropies, code lengths and divergences are computed because the plan distribution and the code are defined by the experimenter.

\subsection{H3: institutions lower the cost of coordination---with exceptions}
\label{sec:sim-h3}

Table~\ref{tab:sim-paired} pairs each institutional run with the no-institution run of the same cell and seed. Shared norms lower the operating bits per successful task in 99.7\% of pairs and raise success in 99.3\%, by eliminating most decoding failures between heterogeneous agents; records do so in 71\% and 100\% of pairs, and raise welfare in every pair. Once set-up costs are included, however, norms are never cheaper than nothing within the forty-period horizon: the coding gain of about one bit per recipient on a six-bit plan space does not repay 200 bits of learning per agent, exactly as Theorem~\ref{thm:breakeven} says it should not, and the ablation confirms that the code pays for itself only with a 256-plan space, a Zipf exponent of 1.5 and a 160-period horizon (Table~\ref{tab:sim-ablation}, Figure~\ref{fig:sim-phase}). Centralized administration with capacity twice the expected load is indistinguishable from records in success and bits, adds messages (relay), and lowers welfare in half the cells because administrators are withdrawn from production; with capacity at half the expected load it is worse than no institution in 99\% of pairs, success falls by 0.44 on average, and welfare by 28 units per agent (Table~\ref{tab:sim-loss}). Federated routing at the same insufficient capacity keeps success higher than no institution in 84\% of pairs, because most tasks never reach the root: the root-load bound of Theorem~\ref{thm:routing} is what separates the two hierarchies (Table~\ref{tab:sim-capacity}, Figure~\ref{fig:sim-phase}). Figure~\ref{fig:sim-scaling} shows the scaling with $N$: messages per task rise with the depth of the tree for the hierarchies and are constant for the flat regimes, bits per success are flat in $N$ for all regimes, and inequality of benefits is slightly higher under the hierarchies.

\begin{table}[H]
\centering\footnotesize
\caption{\SIM\ Paired comparison of each institutional regime with the no-institution run of the same cell and seed (main design, 864 pairs per row). Shares of pairs in which the regime has lower bits per successful task (operating; including set-up), higher success and higher welfare, and mean differences.}
\label{tab:sim-paired}
\resizebox{\textwidth}{!}{\begin{tabular}{l r r r r r r r r r}
\toprule
Regime & Cap.\ ratio & Pairs & Lower bits & Lower incl.\ set-up & Higher success & Higher welfare & $\Delta$ bits/success & $\Delta$ success & $\Delta$ welfare \\
\midrule
\input tables/tab_sim_paired 
\bottomrule
\end{tabular}}
\end{table}

\begin{table}[H]
\centering\footnotesize
\caption{\SIM\ Fragility search: share of the 108 design cells (averaged over seeds) in which the regime has lower welfare, higher bits per success including set-up, or lower success than no institution.}
\label{tab:sim-loss}
\begin{tabular}{l r r r r r}
\toprule
Regime & Cap.\ ratio & Cells & Welfare loss & Bits loss & Success loss \\
\midrule
\input tables/tab_sim_loss 
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/fig_sim_scaling.pdf}
\caption{\SIM\ Scaling by regime without shocks (administrative capacity ratio 2): bits per successful task including set-up (log scale), success rate and messages per task against $N$; means and 95\% confidence intervals over cells and seeds.}
\label{fig:sim-scaling}
\end{figure}

\subsection{Theorems in silico}
\label{sec:sim-theorems}

The coding-gain sweep reproduces Theorem~\ref{thm:coding} cell by cell: the expected code length lies in $[H(p), H(p)+1)$ for every plan space and exponent, and the code is shorter than raw description whenever $H(p) \le \log_2 s - 1$; in the one cell where the distribution is nearly uniform (exponent 0.5, entropy 5.78 bits) the Shannon code is \emph{longer} than the six raw bits (6.24), which is the exception the theorem allows and a reminder that standardization is not free when there is little regularity to exploit. After a distribution shock, a non-adapting code costs 1.90 extra bits per recipient against a divergence of 1.98 bits, within the one-bit rounding of Theorem~\ref{thm:stale}; a code that re-learns after ten periods pays half of that, and an instantly adapting code almost nothing (Table~\ref{tab:sim-ablation}). Because decoding success does not depend on which plan was drawn, distribution shocks raise cost without lowering success in this model (Table~\ref{tab:sim-shocks}); link removal, by contrast, concentrates load on surviving neighbours and lowers success by about 0.08 under the hierarchies and records and by almost nothing under flat regimes (difference-in-differences against the unshocked run of the same cell). The capacity sweep traces the saturation of Theorem~\ref{thm:backlog}: central success collapses as the capacity ratio falls below one, while federated routing degrades gracefully (Figure~\ref{fig:sim-phase}).

\begin{table}[H]
\centering\footnotesize
\caption{\SIM\ Shock responses by regime: difference-in-differences of success and of bits per task (post minus pre, shocked minus unshocked run of the same cell and seed); standard error of the mean in parentheses; achieved divergence for distribution shocks.}
\label{tab:sim-shocks}
\begin{tabular}{l l r r r r}
\toprule
Shock & Regime & Cap.\ ratio & $\Delta$ success (SE) & $\Delta$ bits/task & $D_{KL}(q\|p)$ \\
\midrule
\input tables/tab_sim_shocks 
\bottomrule
\end{tabular}
\end{table}

\begin{table}[H]
\centering\footnotesize
\caption{\SIM\ Hierarchy saturation ($N=200$, lattice): success, latency failures, mean latency and root load against the administrative capacity ratio.}
\label{tab:sim-capacity}
\begin{tabular}{l r r r r r}
\toprule
Regime & Cap.\ ratio & Success & Latency failures & Latency & Root load \\
\midrule
\input tables/tab_sim_capacity 
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}\centering\includegraphics[width=\textwidth]{figures/fig_sim_phase.pdf}\caption{Saturation of hierarchies.}\end{subfigure}\hfill
\begin{subfigure}[t]{0.48\textwidth}\centering\includegraphics[width=\textwidth]{figures/fig_sim_breakeven.pdf}\caption{Break-even of a shared code.}\end{subfigure}\\[2mm]
\includegraphics[width=0.75\textwidth]{figures/fig_sim_shocks.pdf}
\caption{\SIM\ Phase diagram of success against administrative capacity (left), ratio of bits per success with and without a shared code against the horizon (right), and difference-in-differences shock responses (bottom).}
\label{fig:sim-phase}
\end{figure}

\begin{table}[H]
\centering\footnotesize
\caption{\SIM\ Ablations ($N=200$, lattice, eight seeds) and theorem checks.}
\label{tab:sim-ablation}
\begin{tabular}{L{5.2cm} L{10.2cm}}
\toprule
Ablation & Result \\
\midrule
\input tables/tab_sim_ablation 
\bottomrule
\end{tabular}
\end{table}

\subsection{What the experiments say about the hypothesis}
\label{sec:sim-verdict}

H3 is supported in the operating sense and refuted in the total-cost sense within short horizons, which is the content of Theorem~\ref{thm:breakeven}; the experiments also show that the benefit of institutions in this model comes at least as much from eliminating misunderstanding between heterogeneous agents as from shortening messages, since the coding gain on a six-bit plan space is small while the decoding gain is large (with heterogeneity at 0.6, success is 0.81 without a code and 0.96 with one; with heterogeneity at 0.05 the ranking reverses, 0.98 against 0.96, because the code's own error rate is then the larger one). The fragility search found what the framework predicts: institutions hurt when their fixed cost is not amortized, when a centralized hierarchy is asked to relay more than it can process, and when there is too little regularity for a code to compress. None of these cases is exotic. All of them are described in the historical literature on over-extended administrations and rigid bureaucracies, and none of them is measurable with the indicators available in Seshat.