% paper/sections/theory.tex — UQO WP 27, "The Invisible Hand of Machines"
% Section: A stylized model of shared-model trading. Proofs in appendices/proofs.tex.
\section{A stylized model of shared-model trading}
\label{sec:theory}

This section formalizes the mechanism that the simulator is designed to test: when many traders obtain
their trading signal from the \emph{same} model, their errors stop averaging out, prices become less
informative, orders become synchronized, and leverage-driven feedback becomes harder to contain. The
model is deliberately stylized. It is a one-period, partial-equilibrium pricing exercise in the spirit of
\citet{kyle1985} and \citet{hellwig1980}, with the correlated-signal structure of
\citet{foster1996} reinterpreted as the signature of a shared foundation model or prompt
family \citep{kleinberg2021}. Order submission is mechanical (linear in the signal) and the
response intensity is exogenous; the propositions below are therefore statements about how an
\emph{aggregation} mechanism transmits correlated errors into prices, not general-equilibrium results.
We state this limitation plainly wherever it matters.

\subsection{Setup}
\label{sec:theory:setup}

A single risky asset pays a liquidation value $v$. Normalizing the public prior (the public present
value of the simulator) to zero, $v\sim\mathcal N(0,\sigma_v^2)$. There are $N$ traders indexed by
$i=1,\dots,N$ and a set of $G\ge 1$ ``models'' indexed by $g=1,\dots,G$. A model is any shared
signal-generating device---a foundation model, a prompt family, a shared persona. Trader $i$ is attached
to model $g(i)\in\{1,\dots,G\}\cup\{\varnothing\}$, where $\varnothing$ denotes a private (idiosyncratic)
model. Let $n_g$ be the number of traders attached to model $g$, $\phi_g=n_g/N$ its share,
$\Phi=\sum_g\phi_g\le 1$ the total share of traders on \emph{some} shared model, and
\begin{equation}
  H \;=\; \sum_{g=1}^{G}\phi_g^2,\qquad M_{\mathrm{eff}}\;=\;1/H,
  \label{eq:herfindahl}
\end{equation}
the Herfindahl concentration of model use and the implied effective number of models.

\begin{assumption}[A1: Gaussian primitives]
\label{ass:gauss}
Trader $i$ observes $s_i=v+\varepsilon_i$ with $\varepsilon_i=\eta_{g(i)}+u_i$. Model errors
$\eta_g\sim\mathcal N(0,\sigma_\eta^2)$ are i.i.d.\ across models; a trader on a private model draws
her own $\eta_i\sim\mathcal N(0,\sigma_\eta^2)$; idiosyncratic errors $u_i\sim\mathcal N(0,\sigma_u^2)$
are i.i.d.\ across traders; noise-trader demand is $z\sim\mathcal N(0,\sigma_z^2)$. All of
$v,\{\eta_g\},\{\eta_i\},\{u_i\},z$ are mutually independent.
\end{assumption}

Under A1 every trader's signal has the same precision---$\operatorname{Var}(\varepsilon_i)=\sigma_\eta^2+\sigma_u^2$
regardless of $g(i)$---so that the shares $(\phi_g)$ change only the \emph{correlation} of errors across
traders, never the quality of any individual signal. This is the ``holding information quality fixed''
clause of hypothesis H1.

\begin{assumption}[A2: linear order submission]
\label{ass:linear}
Trader $i$ submits the market order $x_i=\beta s_i$ with $\beta>0$ a fixed, common response intensity.
\end{assumption}

A2 mirrors rule-like policies: an LLM agent given a snapshot returns an order whose size scales with
the strength of the signal it extracts, and it does not condition on the price impact of other agents
using the same model. The intensity $\beta$ is a primitive here, not an equilibrium object; a
CARA/price-taking version in which $\beta$ depends on $\lambda$ and on risk aversion
\citep{hellwig1980,grossman1980} is possible but is not needed for the comparative statics
we test, and we do not pursue it.

\begin{assumption}[A3: competitive risk-neutral market maker]
\label{ass:mm}
A competitive, risk-neutral market maker observes the aggregate order flow $Y=\sum_{i=1}^N x_i+z$ and
sets $P=\mathbb E[v\mid Y]$.
\end{assumption}

\begin{assumption}[A4: partial equilibrium]
\label{ass:pe}
$\beta$, $\sigma_z^2$ and the shares $(\phi_g)$ are exogenous; traders do not choose whether to adopt
a shared model, and the market maker does not re-optimize $\lambda$ against the feedback demands
introduced in Sections~\ref{sec:theory:feedback}--\ref{sec:theory:spiral}.
\end{assumption}

Under A1--A3 the projection theorem gives the linear pricing rule
\begin{equation}
  P=\lambda Y,\qquad
  \lambda=\frac{\operatorname{Cov}(v,Y)}{\operatorname{Var}(Y)}
  =\frac{\beta N\sigma_v^2}{\beta^2N^2\left(\sigma_v^2+\Omega\right)},
  \qquad
  \Omega\equiv\sigma_\eta^2\Big[H+\frac{1-\Phi}{N}\Big]+\frac{\sigma_u^2}{N}+\frac{\sigma_z^2}{\beta^2N^2}.
  \label{eq:lambda}
\end{equation}
The quantity $\Omega$ is the variance of the noise in the \emph{normalized} order flow
$\bar Y\equiv Y/(\beta N)=v+\bar\varepsilon+z/(\beta N)$, where $\bar\varepsilon=N^{-1}\sum_i\varepsilon_i$;
every result below is a statement about $\Omega$. The three objects we track are price informativeness,
the pricing error, and order-flow co-movement:
\begin{equation}
  \tau\equiv 1-\frac{\operatorname{Var}(v\mid Y)}{\sigma_v^2}=\frac{\sigma_v^2}{\sigma_v^2+\Omega},
  \qquad
  \operatorname{Var}(P-v)=\sigma_v^2(1-\tau),
  \qquad
  \rho_{ij}\equiv\operatorname{Corr}(x_i,x_j),
  \label{eq:objects}
\end{equation}
and the pricing rule can be written compactly as $\lambda=\tau/(\beta N)$, so that
$P=\tau\,\bar Y$: the price is the normalized flow shrunk toward the prior by the informativeness
coefficient. The identity $\operatorname{Var}(P-v)=\sigma_v^2(1-\tau)$ (proved in
Appendix~\ref{app:proofs}) means that, within the one-period model, informativeness and the mean
squared pricing error are two readings of the same number; the simulator's dynamic statistics
separate them.

\subsection{Informativeness and penetration of a single shared model}
\label{sec:theory:info}

Consider first a single shared model ($G=1$) used by a share $\phi=n/N$ of traders, all others being
on private models. Then $H=\phi^2$, $\Phi=\phi$, and
\begin{equation}
  \Omega(\phi)=\sigma_\eta^2\Big[\phi^2+\frac{1-\phi}{N}\Big]+\frac{\sigma_u^2}{N}+\frac{\sigma_z^2}{\beta^2N^2},
  \qquad
  \tau(\phi)=\frac{\sigma_v^2}{\sigma_v^2+\Omega(\phi)}.
  \label{eq:tau_phi}
\end{equation}

\begin{proposition}[Informativeness versus penetration]
\label{prop:info}
Under A1--A4 with a single shared model of share $\phi$:
\begin{enumerate}
\item[(i)] For every $N>1$, $\tau$ is strictly decreasing in $\phi$ over the admissible grid
  $\phi\in\{1/N,2/N,\dots,1\}$; in the continuous relaxation, $\partial\tau/\partial\phi<0$ for all
  $\phi>1/(2N)$. The values $\phi=0$ and $\phi=1/N$ coincide, since a model used by a single trader is a
  private model. For $N=1$ the share has no effect.
\item[(ii)] For every $N$ and every $\phi>0$,
  $\tau(\phi)\le\tau_\infty(\phi)\equiv\sigma_v^2/(\sigma_v^2+\phi^2\sigma_\eta^2)<1$, and
  $\tau(\phi)\to\tau_\infty(\phi)$ as $N\to\infty$ with $\phi$ fixed: the common error never averages
  out.
\item[(iii)] If $\phi=0$ and $\sigma_z^2/N^2\to 0$ (in particular if $\sigma_z^2$ is fixed), then
  $\tau\to 1$ as $N\to\infty$.
\end{enumerate}
\end{proposition}

\noindent\emph{Intuition.} The market maker learns from the average signal. Private errors are
averaged over $N$ draws and vanish at rate $1/N$; the shared error $\eta$ is a single draw that enters
the average with weight $\phi$ and therefore survives with variance $\phi^2\sigma_\eta^2$ no matter how
many traders there are. Penetration converts a diversifiable error into a non-diversifiable one. Note
that (ii) also implies that the \emph{marginal} value of adding traders falls with $\phi$: with a
shared model, the market behaves as if it had at most $M_{\mathrm{eff}}=1/\phi^2$ independent signals.

\subsection{Pricing error, order-flow correlation and synchrony}
\label{sec:theory:corr}

Let $\sigma_s^2\equiv\sigma_v^2+\sigma_\eta^2+\sigma_u^2$ be the variance of a signal, and define the two
possible pairwise correlations,
\begin{equation}
  \rho_1=\frac{\sigma_v^2+\sigma_\eta^2}{\sigma_s^2}\quad(\text{same model}),\qquad
  \rho_0=\frac{\sigma_v^2}{\sigma_s^2}\quad(\text{different or private models}),\qquad
  \rho_1-\rho_0=\frac{\sigma_\eta^2}{\sigma_s^2}.
  \label{eq:rho01}
\end{equation}
Let $\bar\rho$ denote the population-average pairwise correlation over the $N(N-1)$ ordered pairs
$i\neq j$, and let the \emph{synchrony index} of a pair be the probability that the two traders submit
same-signed orders, $S_{ij}=\Pr(x_ix_j>0)$, with population average $\bar S$.

\begin{proposition}[Pricing error, correlation and synchrony]
\label{prop:corr}
Under A1--A4 with a single shared model of share $\phi=n/N$:
\begin{enumerate}
\item[(i)] $\operatorname{Var}(P-v)=\sigma_v^2\,\Omega(\phi)/(\sigma_v^2+\Omega(\phi))$ is increasing in
  $\phi$ on the admissible grid (and for all real $\phi>1/(2N)$), and is bounded below by
  $\sigma_v^2\phi^2\sigma_\eta^2/(\sigma_v^2+\phi^2\sigma_\eta^2)$ uniformly in $N$.
\item[(ii)] $\operatorname{Corr}(x_i,x_j)=\rho_1$ if $g(i)=g(j)\neq\varnothing$ and $\rho_0$ otherwise;
  $\rho_1>\rho_0$ if and only if $\sigma_\eta^2>0$.
\item[(iii)] The average pairwise correlation is
  \begin{equation}
    \bar\rho(\phi)=\rho_0+w(\phi)\,\frac{\sigma_\eta^2}{\sigma_s^2},\qquad
    w(\phi)=\frac{n(n-1)}{N(N-1)}=\frac{\phi(N\phi-1)}{N-1},
    \label{eq:rhobar}
  \end{equation}
  which is convex in $\phi$, increasing for $\phi>1/(2N)$, and converges to
  $\rho_0+\phi^2\sigma_\eta^2/\sigma_s^2$ as $N\to\infty$.
\item[(iv)] For a pair with correlation $\rho$, $S(\rho)=\tfrac12+\arcsin(\rho)/\pi$, which is strictly
  increasing in $\rho$. The population synchrony index is
  \begin{equation}
    \bar S(\phi)=\tfrac12+\frac{1}{\pi}\Big[(1-w(\phi))\arcsin\rho_0+w(\phi)\arcsin\rho_1\Big],
    \label{eq:sync}
  \end{equation}
  which is a strictly increasing function of $\bar\rho$ whenever $\sigma_\eta^2>0$, and satisfies the
  Jensen bound $\bar S\ge\tfrac12+\arcsin(\bar\rho)/\pi$ with equality if and only if $\sigma_\eta^2=0$.
\end{enumerate}
If $\sigma_\eta^2=0$, none of $\tau$, $\operatorname{Var}(P-v)$, $\bar\rho$ or $\bar S$ depends on $\phi$.
\end{proposition}

\noindent\emph{Intuition.} Two traders on the same model agree about the world more often than the
fundamentals justify, because they share the model's mistake. The share of such pairs is quadratic in
$\phi$ (it is the probability that two random traders both use the model), which is why co-movement
and synchrony accelerate with penetration. The last sentence is the natural placebo: a shared model
that adds no error of its own ($\sigma_\eta^2=0$) is just a precise private signal and cannot generate
any of the effects.

\subsection{Diversity as a stabilizer}
\label{sec:theory:diversity}

\begin{proposition}[Diversity]
\label{prop:diversity}
Under A1--A4 with $G$ shared models of shares $(\phi_1,\dots,\phi_G)$ and total share $\Phi$:
\begin{enumerate}
\item[(i)] Holding $\Phi$ fixed, $\tau$ and $\operatorname{Var}(P-v)$ depend on the shares only through
  $H=\sum_g\phi_g^2$, with $\partial\tau/\partial H<0$. When $\Phi=1$,
  $\Omega=\sigma_\eta^2/M_{\mathrm{eff}}+\sigma_u^2/N+\sigma_z^2/(\beta^2N^2)$: model error averages out at
  rate $1/M_{\mathrm{eff}}$ while private error averages out at rate $1/N$.
\item[(ii)] $\Phi^2/G\le H\le\Phi^2$. The upper bound is attained if and only if a single model carries
  the whole share $\Phi$; the lower bound if and only if the $G$ models have equal shares
  $\phi_g=\Phi/G$ (in the integer version, if and only if the $n_g$ differ by at most one).
\item[(iii)] Consequently, for any population with total shared-model share $\Phi$,
  $\tau_{\text{single}}\le\tau_{\text{mixed}}\le\tau_{\text{uniform}}$: a mixed-provider or
  diverse-prompt population is weakly more informative than a single-model population of the same size,
  strictly so whenever at least two models have positive share and $\sigma_\eta^2>0$.
\item[(iv)] The average correlation generalizes to $\bar\rho=\rho_0+w\,\sigma_\eta^2/\sigma_s^2$ with
  $w=(NH-\Phi)/(N-1)$, so that, for fixed $\Phi$, $\bar\rho$ and $\bar S$ are also increasing in $H$.
\end{enumerate}
\end{proposition}

\noindent\emph{Intuition.} With several models, the common errors are themselves averaged over models.
The Herfindahl index is the exact sufficient statistic because the variance of a weighted average of
independent model errors is $\sigma_\eta^2\sum_g\phi_g^2$. Splitting a given LLM population evenly across
$G$ providers (or prompt families with independent errors) reduces the undiversifiable noise from
$\Phi^2\sigma_\eta^2$ to $\Phi^2\sigma_\eta^2/G$. Note that the recovery is partial unless $G$ grows with
$N$: $M_{\mathrm{eff}}\le N$ and diversity across a handful of providers cannot restore the
$1/N$ rate (hypothesis H4). In this stylized model the benefit of diversity rests entirely on the
independence of $\eta_g$ across models; models trained on similar data or distilled from each other
would have correlated $\eta_g$ and a correspondingly larger effective $H$.

\subsection{Feedback amplification}
\label{sec:theory:feedback}

We now add a second date and trend demand in the spirit of \citet{delong1990}. Let $P_0=0$ be the
public value before the trading round and $Y_1=\beta\sum_i s_i+z_1$ the fundamental flow of
Section~\ref{sec:theory:setup}. Trend followers submit the demand $\gamma(P_1-P_0)$, $\gamma\ge 0$.

\begin{assumption}[A5: within-round feedback with a fixed pricing schedule]
\label{ass:feedback}
Trend demand responds to the price change of the current round, and the market maker applies the
schedule $\lambda$ of \eqref{eq:lambda} to the total flow $Y_1+\gamma(P_1-P_0)$ without re-estimating
it.
\end{assumption}

A5 is the model counterpart of the simulator's timing: fundamental and LLM orders arrive in staggered
batches over a decision interval, while trend followers react every tick to the mid-price history, so
within a decision interval the trend response is effectively contemporaneous. Its second clause is a
restriction, not a result: a Bayesian market maker who knew that the extra flow is $\gamma\Delta P$
would filter it out (the total flow is then a deterministic multiple of $Y_1$), and there would be no
amplification. The simulator's market makers quote a mechanical inventory-skewed schedule and do not
filter, which is what A5 captures.

\begin{proposition}[Feedback amplification]
\label{prop:feedback}
Under A1--A5, write $\Delta P=P_1-P_0$ and $\kappa\equiv 1/(1-\lambda\gamma)$.
\begin{enumerate}
\item[(i)] If $\lambda\gamma<1$ the round has the unique fixed point $\Delta P=\kappa\lambda Y_1$, which
  is the limit of the iterated response $\Delta P^{(k+1)}=\lambda\big(Y_1+\gamma\Delta P^{(k)}\big)$
  from any starting point. The variance of the price change is amplified by $\kappa^2=1/(1-\lambda\gamma)^2$
  relative to the no-feedback benchmark.
\item[(ii)] If $\lambda\gamma\ge 1$ the iterated response diverges, $|\Delta P^{(k)}|\to\infty$, from
  every starting point except the algebraic solution. For $\lambda\gamma=1$ no solution exists when
  $Y_1\neq0$; for $\lambda\gamma>1$ the algebraic solution $\Delta P=\kappa\lambda Y_1$ has the sign
  opposite to the flow and is repelling, so no adjustment process reaches it.
\item[(iii)] Amplification leaves $\tau$ unchanged (the price remains a deterministic multiple of
  $Y_1$) but raises the pricing error to
  $\operatorname{Var}(P_1-v)=\sigma_v^2\big[(1-\tau)+\tau(\kappa-1)^2\big]$, increasing in $\gamma$ on
  $[0,1/\lambda)$.
\end{enumerate}
\end{proposition}

\begin{assumption}[A6: return-loading of LLM policies]
\label{ass:loading}
An LLM trader on model $g$ submits $x_i=\beta s_i+\kappa_g\,\Delta P$ with $\kappa_g\ge 0$ the
model's loading on the current price change; classical trend followers contribute an aggregate loading
$\gamma_T\ge0$.
\end{assumption}

\begin{corollary}[Shared models raise the effective feedback coefficient]
\label{cor:llmtrend}
Under A1--A6 the effective feedback coefficient is $\gamma=\gamma_T+N\sum_g\phi_g\kappa_g$. With a single
shared model of share $\phi$ and loading $\kappa>0$, $\gamma(\phi)=\gamma_T+N\phi\kappa$ is increasing in
$\phi$, the amplification factor $\kappa^2(\phi)$ is increasing in $\phi$, and the stable regime is lost
at the critical share $\phi^\ast=(1/\lambda-\gamma_T)/(N\kappa)$. If $\kappa=0$ the share has no
effect on amplification. If the loadings $\kappa_g$ differ in sign across models (some models are
contrarian), the aggregate loading is the share-weighted average and can be smaller than that of any
single model: diversity dampens feedback, concentration does not.
\end{corollary}

\noindent\emph{Intuition.} A price rise induced by informed flow triggers trend demand, which under a
fixed pricing schedule moves the price again, and so on; the geometric sum converges only if each
round's response is smaller than the last. LLM policies that mention momentum, ``recent trend'' or
``follow the tape'' in their rationale behave like trend followers with a loading $\kappa$ that the
simulator measures by regressing LLM orders on lagged returns. Whether $\kappa$ is positive is an
empirical matter; the corollary says what follows if it is. The variance amplification is
identical to the one obtained by DeLong et al.\ with rational speculators front-running feedback
traders, but the source here is mechanical.

\subsection{Liquidity spiral}
\label{sec:theory:spiral}

Finally, we add leverage, following the margin-spiral logic of \citet{brunnermeier2009}. A
cohort $L$ of $N_L$ leveraged traders holds the aggregate position $X$ (in lots) financed with equity
$W$ under a maintenance requirement of $m>0$ units of equity per lot held.\footnote{If the simulator's
maintenance margin rate is $\hat m$ and the pre-shock price is $\bar P$, then $m=\hat m\bar P$ with the
collateral valued at the pre-shock price; this linearization is exact to first order in
$|\Delta P|/\bar P$ and is the only approximation in the proposition.} Let $b\equiv W-mX\ge0$ be the
initial margin buffer. An exogenous price shock $\Delta P_0<0$ hits the cohort's side of the market;
after a total price change $\Delta P$ the cohort's equity is $W+X\Delta P$, so it must liquidate
\begin{equation}
  F(\Delta P)=\max\Big\{0,\;X-\frac{W+X\Delta P}{m}\Big\}=\max\Big\{0,\;\frac{-X\Delta P-b}{m}\Big\}
  \label{eq:forced}
\end{equation}
lots. Under A5 the market maker prices the forced sales as order flow, so the total price change solves
$\Delta P=\Delta P_0-\lambda F(\Delta P)$. Define the spiral coefficient $a\equiv\lambda X/m$.

\begin{proposition}[Liquidity spiral]
\label{prop:spiral}
Under A1--A5 and \eqref{eq:forced}:
\begin{enumerate}
\item[(i)] If $\Delta P_0\ge -b/X$ the shock is absorbed by the buffer: $F=0$ and $\Delta P=\Delta P_0$
  is a fixed point at which the cascade stops at once. (When $a>1$ and $b>0$ a second, self-fulfilling
  fixed point with $F>0$ also exists; it is not reached by the cascade and we select the cascade limit
  throughout.)
\item[(ii)] If $\Delta P_0<-b/X$ and $a<1$, the unique fixed point is
  \begin{equation}
    \Delta P^\ast=\frac{\Delta P_0+\lambda b/m}{1-a},\qquad\text{so that with } b=0:\quad
    \Delta P^\ast=\frac{\Delta P_0}{1-\lambda X/m},
    \label{eq:spiral_fp}
  \end{equation}
  with amplification multiplier $1/(1-\lambda X/m)>1$. The fixed point is the limit of the
  round-by-round cascade in which each round's forced sales respond to the previous round's price.
\item[(iii)] If $\Delta P_0<-b/X$ and $a\ge 1$, no finite fixed point exists and the cascade decreases
  without bound: in a market with a price band, the price reaches the band.
\item[(iv)] Let $X=\big|\sum_{i\in L}x_i\big|$ be the absolute net position of the cohort and $\bar\rho_L$
  its average pairwise order correlation. Then $X$ is the absolute value of a centered Gaussian with
  standard deviation $\sigma_X=\beta\sigma_s\sqrt{N_L[1+(N_L-1)\bar\rho_L]}$, so that
  $\mathbb E[X]=\sigma_X\sqrt{2/\pi}$ and the spiral probability $\Pr(\lambda X/m\ge1)=2\bar\Phi(m/(\lambda\sigma_X))$
  are both strictly increasing in $\bar\rho_L$, hence (Proposition~\ref{prop:corr}) in the shared-model
  share of the cohort. (Here $\bar\Phi$ is the standard normal survival function.)
\end{enumerate}
\end{proposition}

\noindent\emph{Intuition.} The forced sale of one lot per $m/X$ of price decline moves the price by
$\lambda$ per lot; when $\lambda X/m\ge1$ each round of liquidation generates at least as large a
decline as the one that triggered it. A shared model does not change the expected position of any
single leveraged trader, but it makes positions line up on the same side, which fattens the
distribution of the cohort's net exposure; the spiral threshold $m/\lambda$ is crossed more often. The
net position is a conservative base for forced sales---the gross exposure on the shocked side is
weakly larger---so (iv) understates the mechanism. Since by \eqref{eq:lambda} $\lambda$ itself is
decreasing in $\Omega$, there is a countervailing channel (a less informative market is also a less
price-sensitive one); in the simulator, however, $\lambda$ is set by the mechanical market makers' depth,
so we treat it as a parameter.

\subsection{Safeguards}
\label{sec:theory:safeguards}

Two safeguards from the simulator have direct counterparts. A \emph{position limit} caps the cohort's
exposure at $L$ lots (a per-trader cap $\ell$ with $N_L\ell=L$). A \emph{trading halt} of length $h\ge1$
rounds is triggered when the price change reaches a threshold $\theta>0$; during the halt no forced
sales execute. For the halt we need a reduced-form statement of what changes while trading is stopped.

\begin{assumption}[A7: halt effectiveness]
\label{ass:halt}
After a halt of length $h$, the cascade resumes with spiral coefficient $\psi(h)\,a$, where
$\psi:\{0,1,2,\dots\}\to(0,1]$ is nonincreasing with $\psi(0)=1$.
\end{assumption}

A7 summarizes two mechanisms, margin top-ups (fewer lots must be sold per unit of price decline) and
the arrival of patient liquidity (smaller impact per lot sold), without modelling either. It is an
assumption about the halt, not a result; the case $\psi(h)=1$ is a halt that changes nothing but the
clock.

\begin{proposition}[Safeguards]
\label{prop:safeguards}
Under the assumptions of Proposition~\ref{prop:spiral}:
\begin{enumerate}
\item[(i)] (Position limit.) If $L<m/\lambda$, then for every shared-model share and every realization
  of signals, $\lambda X/m\le\lambda L/m<1$: the spiral region is empty and the multiplier is bounded by
  $1/(1-\lambda L/m)$. The cost is informational: if the limit binds on informed orders, each capped
  order $\tilde x_i=\min\{\max\{\beta s_i,-\ell\},\ell\}$ has loading
  $\operatorname{Cov}(v,\tilde x_i)=\Pr(|\beta s_i|<\ell)\operatorname{Cov}(v,x_i)<\operatorname{Cov}(v,x_i)$ on the
  fundamental.
\item[(ii)] (Halt: depth.) Under A7, if the halt is triggered by the shock itself ($|\Delta P_0|\ge\theta$)
  and $\psi(h)a<1$, the post-halt cascade converges to
  $\Delta P^\ast_h=(\Delta P_0+\lambda b/m)/(1-\psi(h)a)$, which is finite even when $a\ge1$ and satisfies
  $|\Delta P^\ast_h|\le|\Delta P^\ast|$ when $a<1$, strictly if $\psi(h)<1$.
\item[(iii)] (Halt: time-to-absorb.) Let $T_\epsilon$ be the first round at which the price is within
  $\epsilon$ of its terminal value. Without a halt and $a<1$,
  $T_\epsilon(a)=\max\{0,\lceil\log(\epsilon(1-a)/|\Delta P_0|)/\log a\rceil-1\}$ (for $b=0$; for $b>0$
  replace $|\Delta P_0|$ by $|\Delta P_0|-b/X$); with the halt,
  $T^{h}_\epsilon=h+T_\epsilon(\psi(h)a)$. Hence $T^h_\epsilon\ge h$: no information---neither the shock
  itself nor public news arriving during the halt---is incorporated before $h$ rounds have elapsed,
  and if $\psi(h)=1$ the halt is pure delay, $T^h_\epsilon=T_\epsilon(a)+h$.
\end{enumerate}
\end{proposition}

\noindent\emph{Intuition.} A position limit works by construction: it bounds the base of forced sales
and therefore the slope of the feedback loop, at the cost of truncating the informed demand that makes
prices informative. A halt is more ambiguous. It reduces the depth of the spiral only through what
changes during the halt (A7), and it always costs at least its own length in delayed price discovery;
whether the trade-off is favourable is the empirical question H6 asks of the simulator.

\subsection{Mapping to the simulator}
\label{sec:mapping}

Table~\ref{tab:mapping} maps each theoretical primitive to the simulator parameter that manipulates or
measures it. Two points deserve emphasis. First, $\phi$ is \emph{not} the LLM share: it is the share of
traders on one and the same model, i.e.\ \texttt{llm\_share} times the share of LLM slots on a given
provider--prompt combination (\texttt{provider\_mix}, \texttt{prompt\_set}). A population with
\texttt{llm\_share} $=0.5$ split evenly across two providers with diverse prompts has a small $H$ and
should behave, in the model, like a population of private models. Second, $\beta$ and $\lambda$ are
measured, not set: the LLM order aggressiveness is the slope of submitted quantity on the private signal
(or on the public-value gap for uninformed LLM agents), and $\lambda$ is the Kyle lambda estimated by
regressing mid-price changes on signed order flow.

\begin{table}[htbp]
\centering
\caption{Theoretical primitives and their simulator counterparts.}
\label{tab:mapping}
\small
\begin{tabular}{p{0.26\linewidth}p{0.68\linewidth}}
\hline
Primitive & Simulator parameter or statistic \\
\hline
$\phi$, $(\phi_g)$, $H$ & \texttt{population.llm\_share} $\times$ share of LLM slots on the same (provider, model, prompt family): \texttt{llm.provider\_mix}, \texttt{llm.prompt\_set} (\texttt{single}/\texttt{diverse}); $H$ computed from the realized assignment \\
$\sigma_\eta^2$ & shared-model error: variance of the common component of LLM decisions given the snapshot; manipulated by prompt diversity, \texttt{llm.memory}, \texttt{llm.channel}, estimated from the decision library (same-snapshot replays) \\
$\sigma_u^2$ & private-signal noise $\sigma^2/\texttt{precision}$ (\texttt{population.precision}, \texttt{llm.precision}) and \texttt{llm.informed\_share} \\
$\sigma_v^2$ & dividend-innovation variance $\sigma^2$ scaled by $\rho/(1+r-\rho)$ (\texttt{fundamentals.sigma}, \texttt{rho}, \texttt{r\_period}) \\
$\beta$ & LLM order aggressiveness: measured slope of order size on the signal (\texttt{llm.max\_order\_qty} caps it) \\
$\lambda$ & estimated Kyle lambda (mid change on signed volume over \texttt{div\_interval} windows); depth set by market-maker slots (5\,\%) \\
$\gamma$, $\kappa_g$ & \texttt{population.mix[trend]}, \texttt{mix[leveraged]}; LLM return loading estimated from decisions on lagged returns \\
$m$, $W$, $X$ & \texttt{population.maintenance\_margin}, \texttt{leverage\_cap}, endowments; $X$ = realized net position of leveraged slots \\
$L$, $\ell$ & position limit (per-agent order/position cap; \texttt{allow\_short}, \texttt{max\_short\_lots}) \\
$\theta$, $h$ & \texttt{exchange.halt\_threshold}, \texttt{halt\_window}, \texttt{halt\_duration} (circuit breaker) \\
$z$, $\sigma_z^2$ & noise-trader slots (\texttt{mix[noise]}, activity) \\
$N$ & \texttt{population.n\_traders} ($100$, $1{,}000$, $10{,}000$) \\
$\tau$ & not estimated directly; its simulator counterparts are the mean squared log pricing error \texttt{msd\_log} $=\overline{(\log P_t-\log V_t)^2}$ and the mean absolute deviation \texttt{mad\_pct}, both decreasing in informativeness \\
$\operatorname{Var}(P-v)$ & mean squared pricing error $(P_t-V_t)^2$ \\
$\bar\rho$, $\bar S$ & average pairwise correlation of signed order flow across agents; share of agent pairs with same-signed orders within a decision interval \\
$\Delta P^\ast$, $T_\epsilon$ & crash depth after a shock event (\texttt{shocks}); time-to-absorb = ticks until $|P_t/V_t-1|$ stays below 1\,\% for five consecutive ticks after the latent shock (\texttt{time\_to\_absorb}) \\
\hline
\end{tabular}
\end{table}

\subsection{Testable predictions and what would falsify them}
\label{sec:predictions}

Table~\ref{tab:predictions} lists the hypotheses of the pre-analysis plan together with the
proposition that generates each, the predicted sign and the simulator statistic that tests it. Each
line also states what would count against the model. We stress once more that these are predictions of
a stylized, partial-equilibrium aggregation model; the simulator is not an implementation of the model
but an environment in which the same mechanism should, if it is real, leave the same signature.

\begin{table}[htbp]
\centering
\caption{Hypotheses, generating propositions, predicted signs and tests.}
\label{tab:predictions}
\small
\begin{tabular}{p{0.04\linewidth}p{0.33\linewidth}p{0.13\linewidth}p{0.38\linewidth}}
\hline
 & Prediction (source) & Sign & Test statistic; falsifying pattern \\
\hline
H1 & Informativeness falls in the single-model share $\phi$ at fixed signal quality and rises in $M_{\mathrm{eff}}$ (Props.~\ref{prop:info}, \ref{prop:diversity}) & $\partial\tau/\partial\phi<0$; $\partial\tau/\partial M_{\mathrm{eff}}>0$ & slope of $\tau$ on \texttt{llm\_share} within the single-provider/single-prompt arm, matched pairs across seeds; falsified if $\tau$ is flat or rising in $\phi$ while the shared component of LLM decisions ($\sigma_\eta^2$) is measurably positive \\
H2 & Order-flow correlation and synchrony rise in $\phi$, convexly (Prop.~\ref{prop:corr}) & $+$, convex & $\bar\rho$ and same-sign share across LLM pairs vs.\ \texttt{llm\_share}; falsified if the same-model pair correlation is not above the cross-model pair correlation \\
H3 & Bubble/crash frequency and liquidity spirals rise with $\phi\times$leverage (Props.~\ref{prop:feedback}, \ref{prop:spiral}) & interaction $+$ & frequency of band hits / forced-liquidation cascades, crash depth, in the \texttt{leveraged}$>0$ arms; falsified if the interaction is zero while $\hat\lambda X/m$ is observed to cross one more often under high $\phi$ \\
H4 & Prompt or model diversity restores part of the loss (Prop.~\ref{prop:diversity}) & $\tau_{\text{diverse}}>\tau_{\text{single}}$; partial & difference in $\tau$ between \texttt{prompt\_set=diverse} / mixed \texttt{provider\_mix} and single-model arms at equal \texttt{llm\_share}; falsified if diversity yields no gain, or a gain exceeding that of the fully private benchmark (which the model rules out) \\
H5 & Communication channels raise synchrony (Prop.~\ref{prop:corr} with $\sigma_\eta^2$ raised by shared rationales) & $+$ & $\bar S$ within cohorts with \texttt{llm.channel=true} vs.\ false; falsified if same-cohort synchrony does not exceed cross-cohort synchrony \\
H6 & Circuit breakers and position limits cut crash depth at a cost in time-to-absorb (Prop.~\ref{prop:safeguards}) & depth $-$; $T_\epsilon$ $+$ & crash depth and ticks-to-absorb under \texttt{halt\_threshold} and position caps vs.\ none; falsified if halts reduce depth with no delay (would reject A7 as the channel) or if position limits below $m/\hat\lambda$ fail to remove cascades \\
\hline
\end{tabular}
\end{table}

Three further implications are useful as placebo or dose--response checks. (a) By
Proposition~\ref{prop:corr}, an LLM treatment whose decisions carry no common component
($\sigma_\eta^2=0$, e.g.\ the mock provider with independent seeds) should produce none of H1--H3; a
positive effect there would indicate a channel outside the model (for instance, a change in $\beta$ or in
latency). (b) By Proposition~\ref{prop:info}(ii), the loss of informativeness should not shrink as
$N$ grows from $100$ to $10{,}000$ at fixed $\phi$, whereas by (iii) it should vanish in the private-model
benchmark. (c) By Corollary~\ref{cor:llmtrend}, amplification should respond to $\phi$ only in treatments
where the measured return loading $\kappa$ of LLM decisions is positive.