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

We organise the robustness checks by the type of claim they protect.

\paragraph{Theory.} Proposition~\ref{prop:benchmark} is verified symbolically (the wedge is an identically zero
rational function) and numerically on 60 random parameter draws over $\alpha\in(-0.95,0.95)$, $M\in\{1,\dots,6\}$ and
precisions in $[0.1,20]$; Proposition~\ref{prop:wedges}(a) is verified against its closed form at six parameter
points to machine precision; the moments of Proposition~\ref{prop:linear} are verified by Monte Carlo
(Table~\ref{tab:c_factorial} notes; \texttt{tests/unit/test\_lq\_game.py}).

\paragraph{Illustrative information parameters (Model A).} Figure~\ref{fig:A_pv_noise}(a) and
Figure~\ref{fig:A_threshold} already span model-noise standard deviations from 0.02 to 0.095 (human noise 0.10) and
compute rentals from 0.2 to 1.0; Figure~\ref{fig:A_heatmap} sweeps the effective number of models from 1 to 16.
Three conclusions survive every sweep: the private value of delegation is decreasing in $D$ (strategic
substitutability), the model-noise variance scales with $D^2/M$, and the wedge is zero in the benchmark and
positive under uninsurable wage risk. The \emph{levels} (3.3\% of consumption for the first adopter; a wage-noise
standard deviation of 7.7\% at full adoption with one model) move proportionally with $\sigma_\theta^2$ and
$1/(\gamma\sigma_R^2)$ and should be read as orders of magnitude. The compute-market crowding-out is small for every
$(K_Q,\bar q)$ we tried, for the reason given in Section~\ref{sec:results_a}; we could not find a calibration in which
inference demand reverses the automation-driven decline of the labour share by more than a few tenths of a point.

\paragraph{Savings versus portfolio mapping.} With $\gamma\in\{1,2,4\}$ and $\rho_z\in\{0.5,0.9,0.95\}$ the
partial-equilibrium savings response to news stays below 0.01 in absolute value; the conclusion that the stake lies
in allocation decisions does not depend on the baseline parameters.

\paragraph{Model B.} The ordering of Gini and top shares across ownership structures is unchanged at $n_a=100$ versus
160 grid points, three versus five Gauss--Hermite nodes, and with the borrowing limit at 0 or $-0.5$ (the
``no borrowing'' scenario). The level of the top-1\% share is too low in every scenario, as in all Aiyagari-type
models without additional return heterogeneity or discount-factor heterogeneity; the \emph{changes} across scenarios
are the object of interest.

\paragraph{Model C: designs, seeds and the competing explanation.} The H2 pattern (order-flow correlation increasing
in $D$, decreasing in $M$) holds in all three market designs with non-overlapping bootstrap intervals at $D=1$
(Figure~\ref{fig:C_herding}); it is robust to 20 versus 5 seeds and to $T=80$ versus 160. The competing explanation
flagged in the novelty audit---homogenisation reduces disagreement and therefore volatility---is \emph{also} present:
with one shared model and full delegation, trading volume collapses toward the no-trade outcome
(Figure~\ref{fig:C_volvol}) and volatility is lower in calm periods. The two effects coexist: a shared model makes the
market quieter and more correlated at the same time, which is why the stress test, not the unconditional
volatility, is the right place to look for fragility.

\paragraph{LLM laboratory.} Table~\ref{tab:llm} reports every cell at temperature 0 and 0.7 and the
prompt-variation control; the audit logs record the model fingerprints. The qualitative H2 pattern in the LLM cells
is read against the Bayesian control and the surrogate; we do not claim robustness across model versions beyond the
three models used, and we report invalid-action rates by model because they differ (Section~\ref{sec:results_c}).