\section{Robustness}
\label{sec:robustness}

\paragraph{Bubble and crash thresholds.}
The pre-registered sensitivity grid (bubble 3/5/8\,\% for ten ticks, crash 7/10/15\,\% in twenty ticks)
is computed from the saved tick series of every run (Table~\ref{tab:threshold_sensitivity}). In the
classical design the leverage cells produce 0.3--2.3 bubble episodes per run depending on the bubble
threshold and 0--1.2 crashes depending on the crash threshold, so the leverage results are threshold-robust
in sign; in the live design every count is zero at every threshold except 0.14 bubble episodes per run
in the 25\,\%-plus-leverage cell at the 3\,\% threshold. The floor effect is therefore not an artefact of a
strict definition.

\paragraph{Cached decisions and the breaker cell.}
Every provider response is cached by the hash of its prompt, and an identical prompt is replayed rather
than re-queried. This makes matched comparisons sharper (the breaker cell is literally the 25\,\% cell
until a halt changes a snapshot) and means that provider non-determinism enters the results only once per
distinct prompt. Across the confirmatory design 7,370 of 80,031 decisions were cache hits: 4,824 in the breaker cell
(identical to the 25\,\% cell) and the rest in the first decision rounds, where the market state has not
yet diverged from a previously queried run; after that, prompts are unique to their run.

\paragraph{Realized versus assigned dose.}
Figure~\ref{fig:realized_share} shows that the realized volume share is close to the assigned slot share
up to 50\,\% and below it at 95\,\% (84\,\%), while the order share is five to ten times smaller. Re-reading
the dose--response on the realized volume share does not change any sign; we do not estimate it because
the realized share is an outcome.

\paragraph{Market mechanism.}
Under a periodic call auction (classical design) the tracking error rises by 0.7 points and the drawdown
by 2.3 points relative to continuous trading, with no change in the time to absorb; the batch mechanism
with a collar is a different economy rather than a robustness check of the same one, and we did not run
LLM cells under it.

\paragraph{Population size.}
The classical stylized facts hold at $N = 1{,}000$ and $N = 10{,}000$ with activity scaled by $100/N$
(Table~\ref{tab:scale}), and the frozen-library replay reproduces the liquidity-and-persistence signature
of the online cells at $N = 1{,}000$ (Table~\ref{tab:library}). The online LLM cells are all at $N = 100$.

\paragraph{Information quality.}
The classical information cells (precision 0.5 and 8, misleading news) bound what private information
can do in this economy: the latent shock is revealed within twenty ticks in any case, so precision moves
only the speed of absorption, and only biased public information moves the tracking error. The live
design inherits information sets from the replaced slots; an information-equalization arm in which every
LLM agent receives a signal (\code{llm.informed\_share}) is implemented and was not run within the budget.

\paragraph{Seed count and inference.}
All live contrasts are paired over seven seeds. The effects reported as significant have $d_z$ between 1
and 4 and survive Holm adjustment within their family; the null results on pricing error, absorption and
drawdown have confidence intervals that include effects of the size found in the classical leverage cells,
so they are inconclusive for small effects and conclusive only against large ones. The classical design
(30 seeds) has the power to detect $d \approx 0.5$.

\begin{table}[htbp]
\centering
\caption{Threshold sensitivity: paired effect on bubble and crash counts against the control, for every
combination of bubble (b) and crash (c) thresholds in the pre-registered grid.}
\label{tab:threshold_sensitivity}
\scriptsize
\resizebox{\textwidth}{!}{\input{tables/tab_threshold_sensitivity}}
\end{table}