1\section{Agent-based experiments}2\label{sec:sim}34Everything 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 instantiate the theorems of Section~\ref{sec:theory} in a system where the quantities they speak of are defined, and they search for the conditions under which institutions reduce efficiency or add fragility, as the pre-registration requires. Two things should be said before any number. First, the error structure is a parameter, not a finding: agents without a shared code misdecode with a probability proportional to their dialect distance (0.25 per unit), agents with a code with probability 0.02, and agents who can verify against a record not at all. Any statement that institutions ``raise success'' in this model is therefore a statement about those parameters, and the ablation that varies them is the one to read. What the model measures that is not assumed is the \emph{cost} side: bits, messages, queueing delay, overhead, and how they scale and fail. Second, the regimes are compared at an equal total administrative budget, so that federation can gain only by relaying less.56\subsection{Design}7\label{sec:sim-design}89$N$ 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 to its two partners (there is no broadcast). 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. Five regimes are compared. \emph{None}: raw messages of $\lceil\log_2 s\rceil$ bits per recipient, dialect errors, and a one-bit acknowledgement round that doubles the messages each agent must process. \emph{Norms}: a shared Shannon code optimal for the plan distribution (Shannon rather than Huffman lengths, so that the stale-code penalty of Theorem~\ref{thm:stale} applies verbatim), lower decoding error, 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, which removes decoding errors and replaces the acknowledgement round. \emph{Central}: records plus an eight-ary administrative tree; every task is relayed from the initiator up to the root and down to each partner, the codeword is transmitted on every hop, and every administrator on the path processes one message per hop. \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. Administrators are withdrawn from production. The total administrative budget is a multiple (0.5 or 2 in the main design; 0.25 to 4 in the ablation) of the expected relayed load of the \emph{central} design, and it is allocated to administrators in proportion to the load each one carries in a five-period warm-up with unconstrained capacity (``capacity follows load''); central and federated designs therefore face the same budget, and the budget multiple is the inverse utilization of the central design in the sense of Theorem~\ref{thm:backlog}. Shocks at period 20 of 40 either shift the plan distribution (a cyclic permutation reaching a Kullback--Leibler divergence of 1.03 bits in the non-uniform cells; in uniform cells the shift is a no-op and those cells are excluded from the shock statistics) or remove half of the links among agents (the administrative tree is not touched). 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 budget, 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. Shock effects are difference-in-differences against the unshocked run of the same cell and seed. Entropies, code lengths and divergences are computed because the plan distribution and the code are defined by the experimenter.1011\subsection{H3: the cost side}12\label{sec:sim-h3}1314Table~\ref{tab:sim-paired} pairs each institutional run with the no-institution run of the same cell and seed, and Table~\ref{tab:sim-loss} counts the design cells in which an institution does worse than nothing. Shared norms lower the operating bits per successful task in 99.7\% of pairs; once the 200-bit learning cost is included they are never cheaper within the forty-period horizon, because the coding gain of a six-bit plan space is about one bit per recipient (Table~\ref{tab:sim-ablation}: the Shannon code averages 5.44 bits at Zipf exponent 1 against 6 raw bits), exactly the break-even arithmetic of Theorem~\ref{thm:breakeven}; the ablation finds the code repaying its learning cost only with a 256-plan space, a Zipf exponent of 1.5 and a 160-period horizon (Figure~\ref{fig:sim-phase}). Records lower operating cost in 71\% of pairs and raise welfare in every pair, the acknowledgement round they replace being what saturates agents' queues at low bandwidth in the flat regimes (at bandwidth 2, flat regimes complete about half of their tasks and records 96\%; at bandwidth 8 all complete more than 90\%). The hierarchies are expensive in bits because the codeword travels on every hop: at the reference budget, central administration costs 83 bits per successful task against 25 for records and 15 for nothing, federated administration 53, and both lower welfare below the flat institutional regimes in half the cells because administrators are withdrawn from production. With a budget at half the central design's load, central administration fails almost every task (success down by 0.49 relative to nothing; welfare loss in all 108 cells), and federation, which relays fewer hops, does only somewhat better (success down by 0.29; welfare loss in 80\% of cells). Figure~\ref{fig:sim-scaling} shows the scaling with $N$: messages and bits per task grow with the depth of the tree for the hierarchies and are flat for the flat regimes; inequality of benefits is slightly higher under the hierarchies.1516\begin{table}[H]17\centering\footnotesize18\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. The budget multiple is relative to the central design's expected relayed load.}19\label{tab:sim-paired}20\resizebox{\textwidth}{!}{\begin{tabular}{l r r r r r r r r r}21\toprule22Regime & Budget & Pairs & Lower bits & Lower incl.\ set-up & Higher success & Higher welfare & $\Delta$ bits/success & $\Delta$ success & $\Delta$ welfare \\23\midrule24\input tables/tab_sim_paired25\bottomrule26\end{tabular}}27\end{table}2829\begin{table}[H]30\centering\footnotesize31\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.}32\label{tab:sim-loss}33\begin{tabular}{l r r r r r}34\toprule35Regime & Budget & Cells & Welfare loss & Bits loss & Success loss \\36\midrule37\input tables/tab_sim_loss38\bottomrule39\end{tabular}40\end{table}4142\begin{figure}[H]43\centering44\includegraphics[width=\textwidth]{figures/fig_sim_scaling.pdf}45\caption{\SIM\ Scaling by regime without shocks (budget multiple 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.}46\label{fig:sim-scaling}47\end{figure}4849\subsection{Theorems in silico}50\label{sec:sim-theorems}5152The coding-gain sweep reproduces Theorem~\ref{thm:coding}: the expected code length lies in $[H(p), H(p)+1)$ for every plan space and exponent, and it is below the raw length 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 longer than the six raw bits (6.24). As the remark after Theorem~\ref{thm:coding} notes, this is a property of the suboptimal Shannon code, not of coding: a Huffman code is never longer than the fixed-length code. We keep the Shannon code because it is the one Theorem~\ref{thm:stale} is stated for, and the sweep then reproduces that theorem too: after a shift in the plan distribution, a non-adapting code costs 1.90 extra bits per recipient against a divergence of 1.98 bits, within the one-bit rounding; a code that re-learns after ten periods pays half, an instantly adapting code almost nothing. 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}), which is a modelling choice and not a finding. Link removal lowers success by 0.07--0.08 under records and the hierarchies at the reference budget and by almost nothing under the flat regimes; the latter is partly a floor effect, since at bandwidth 2 the flat regimes are already failing half of their tasks before the shock (the replication package reports the decomposition by bandwidth). The budget sweep traces the saturation of Theorem~\ref{thm:backlog} and the routing bound of Theorem~\ref{thm:routing} together (Table~\ref{tab:sim-capacity}, Figure~\ref{fig:sim-phase}): central success is 0.03 at a budget of half the load, 0.48 at a budget equal to the load, and 0.94 at one and a half times; federated success, with the same budgets, is 0.29, 0.99 and 1.00, because its root carries a tenth of the central root's load.5354\begin{table}[H]55\centering\footnotesize56\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 (non-uniform cells only).}57\label{tab:sim-shocks}58\begin{tabular}{l l r r r r}59\toprule60Shock & Regime & Budget & $\Delta$ success (SE) & $\Delta$ bits/task & $D_{KL}(q\|p)$ \\61\midrule62\input tables/tab_sim_shocks63\bottomrule64\end{tabular}65\end{table}6667\begin{table}[H]68\centering\footnotesize69\caption{\SIM\ Hierarchy saturation ($N=200$, lattice): success, latency failures, mean latency and root load against the administrative budget multiple, at equal total budgets for the two designs.}70\label{tab:sim-capacity}71\begin{tabular}{l r r r r r}72\toprule73Regime & Budget & Success & Latency failures & Latency & Root load \\74\midrule75\input tables/tab_sim_capacity76\bottomrule77\end{tabular}78\end{table}7980\begin{figure}[H]81\centering82\begin{subfigure}[t]{0.48\textwidth}\centering\includegraphics[width=\textwidth]{figures/fig_sim_phase.pdf}\caption{Saturation of hierarchies.}\end{subfigure}\hfill83\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]84\includegraphics[width=0.75\textwidth]{figures/fig_sim_shocks.pdf}85\caption{\SIM\ Success against the administrative budget multiple (left), ratio of bits per success with and without a shared code against the horizon (right), and difference-in-differences shock responses (bottom).}86\label{fig:sim-phase}87\end{figure}8889\begin{table}[H]90\centering\footnotesize91\caption{\SIM\ Ablations ($N=200$, lattice, eight seeds) and theorem checks.}92\label{tab:sim-ablation}93\begin{tabular}{L{5.2cm} L{10.2cm}}94\toprule95Ablation & Result \\96\midrule97\input tables/tab_sim_ablation98\bottomrule99\end{tabular}100\end{table}101102\subsection{What the experiments say about the hypothesis}103\label{sec:sim-verdict}104105On the cost side, which is the side the model actually measures, H3 is supported for operating cost and refuted for total cost within short horizons: a code saves bits per task in proportion to the regularity of the task environment, and it repays its learning cost only when that regularity is large and the horizon long (Theorem~\ref{thm:breakeven}). Records are the regime that wins most often, and they win by replacing a round of acknowledgements with a lookup, not by compressing messages. Hierarchies buy routing at the price of relaying, and they fail abruptly when the budget falls below the load they relay; federation halves that load and is the only hierarchical design that survives a tight budget. On the success side, the ranking of regimes is the ranking of their error parameters: with heterogeneity at 0.6 the no-code regime succeeds in 81\% of tasks against 96\% with a code, with heterogeneity at 0.05 the ranking reverses (98\% against 96\%) because the code's own error rate is then the larger one. The fragility search therefore finds what the framework predicts and nothing else: institutions hurt when their fixed cost is not amortized, when a hierarchy is asked to relay more than it can process, and when there is too little regularity for a code to compress. These are the cases the historical literature on over-extended administrations and rigid bureaucracies describes, and none of them is measurable with the indicators available in Seshat.106