1% paper/sections/theory.tex — UQO WP 27, "The Invisible Hand of Machines"2% Section: A stylized model of shared-model trading. Proofs in appendices/proofs.tex.3\section{A stylized model of shared-model trading}4\label{sec:theory}56This section formalizes the mechanism that the simulator is designed to test: when many traders obtain7their trading signal from the \emph{same} model, their errors stop averaging out, prices become less8informative, orders become synchronized, and leverage-driven feedback becomes harder to contain. The9model is deliberately stylized. It is a one-period, partial-equilibrium pricing exercise in the spirit of10\citet{kyle1985} and \citet{hellwig1980}, with the correlated-signal structure of11\citet{foster1996} reinterpreted as the signature of a shared foundation model or prompt12family \citep{kleinberg2021}. Order submission is mechanical (linear in the signal) and the13response intensity is exogenous; the propositions below are therefore statements about how an14\emph{aggregation} mechanism transmits correlated errors into prices, not general-equilibrium results.15We state this limitation plainly wherever it matters.1617\subsection{Setup}18\label{sec:theory:setup}1920A single risky asset pays a liquidation value $v$. Normalizing the public prior (the public present21value of the simulator) to zero, $v\sim\mathcal N(0,\sigma_v^2)$. There are $N$ traders indexed by22$i=1,\dots,N$ and a set of $G\ge 1$ ``models'' indexed by $g=1,\dots,G$. A model is any shared23signal-generating device---a foundation model, a prompt family, a shared persona. Trader $i$ is attached24to model $g(i)\in\{1,\dots,G\}\cup\{\varnothing\}$, where $\varnothing$ denotes a private (idiosyncratic)25model. Let $n_g$ be the number of traders attached to model $g$, $\phi_g=n_g/N$ its share,26$\Phi=\sum_g\phi_g\le 1$ the total share of traders on \emph{some} shared model, and27\begin{equation}28 H \;=\; \sum_{g=1}^{G}\phi_g^2,\qquad M_{\mathrm{eff}}\;=\;1/H,29 \label{eq:herfindahl}30\end{equation}31the Herfindahl concentration of model use and the implied effective number of models.3233\begin{assumption}[A1: Gaussian primitives]34\label{ass:gauss}35Trader $i$ observes $s_i=v+\varepsilon_i$ with $\varepsilon_i=\eta_{g(i)}+u_i$. Model errors36$\eta_g\sim\mathcal N(0,\sigma_\eta^2)$ are i.i.d.\ across models; a trader on a private model draws37her own $\eta_i\sim\mathcal N(0,\sigma_\eta^2)$; idiosyncratic errors $u_i\sim\mathcal N(0,\sigma_u^2)$38are i.i.d.\ across traders; noise-trader demand is $z\sim\mathcal N(0,\sigma_z^2)$. All of39$v,\{\eta_g\},\{\eta_i\},\{u_i\},z$ are mutually independent.40\end{assumption}4142Under A1 every trader's signal has the same precision---$\operatorname{Var}(\varepsilon_i)=\sigma_\eta^2+\sigma_u^2$43regardless of $g(i)$---so that the shares $(\phi_g)$ change only the \emph{correlation} of errors across44traders, never the quality of any individual signal. This is the ``holding information quality fixed''45clause of hypothesis H1.4647\begin{assumption}[A2: linear order submission]48\label{ass:linear}49Trader $i$ submits the market order $x_i=\beta s_i$ with $\beta>0$ a fixed, common response intensity.50\end{assumption}5152A2 mirrors rule-like policies: an LLM agent given a snapshot returns an order whose size scales with53the strength of the signal it extracts, and it does not condition on the price impact of other agents54using the same model. The intensity $\beta$ is a primitive here, not an equilibrium object; a55CARA/price-taking version in which $\beta$ depends on $\lambda$ and on risk aversion56\citep{hellwig1980,grossman1980} is possible but is not needed for the comparative statics57we test, and we do not pursue it.5859\begin{assumption}[A3: competitive risk-neutral market maker]60\label{ass:mm}61A competitive, risk-neutral market maker observes the aggregate order flow $Y=\sum_{i=1}^N x_i+z$ and62sets $P=\mathbb E[v\mid Y]$.63\end{assumption}6465\begin{assumption}[A4: partial equilibrium]66\label{ass:pe}67$\beta$, $\sigma_z^2$ and the shares $(\phi_g)$ are exogenous; traders do not choose whether to adopt68a shared model, and the market maker does not re-optimize $\lambda$ against the feedback demands69introduced in Sections~\ref{sec:theory:feedback}--\ref{sec:theory:spiral}.70\end{assumption}7172Under A1--A3 the projection theorem gives the linear pricing rule73\begin{equation}74 P=\lambda Y,\qquad75 \lambda=\frac{\operatorname{Cov}(v,Y)}{\operatorname{Var}(Y)}76 =\frac{\beta N\sigma_v^2}{\beta^2N^2\left(\sigma_v^2+\Omega\right)},77 \qquad78 \Omega\equiv\sigma_\eta^2\Big[H+\frac{1-\Phi}{N}\Big]+\frac{\sigma_u^2}{N}+\frac{\sigma_z^2}{\beta^2N^2}.79 \label{eq:lambda}80\end{equation}81The quantity $\Omega$ is the variance of the noise in the \emph{normalized} order flow82$\bar Y\equiv Y/(\beta N)=v+\bar\varepsilon+z/(\beta N)$, where $\bar\varepsilon=N^{-1}\sum_i\varepsilon_i$;83every result below is a statement about $\Omega$. The three objects we track are price informativeness,84the pricing error, and order-flow co-movement:85\begin{equation}86 \tau\equiv 1-\frac{\operatorname{Var}(v\mid Y)}{\sigma_v^2}=\frac{\sigma_v^2}{\sigma_v^2+\Omega},87 \qquad88 \operatorname{Var}(P-v)=\sigma_v^2(1-\tau),89 \qquad90 \rho_{ij}\equiv\operatorname{Corr}(x_i,x_j),91 \label{eq:objects}92\end{equation}93and the pricing rule can be written compactly as $\lambda=\tau/(\beta N)$, so that94$P=\tau\,\bar Y$: the price is the normalized flow shrunk toward the prior by the informativeness95coefficient. The identity $\operatorname{Var}(P-v)=\sigma_v^2(1-\tau)$ (proved in96Appendix~\ref{app:proofs}) means that, within the one-period model, informativeness and the mean97squared pricing error are two readings of the same number; the simulator's dynamic statistics98separate them.99100\subsection{Informativeness and penetration of a single shared model}101\label{sec:theory:info}102103Consider first a single shared model ($G=1$) used by a share $\phi=n/N$ of traders, all others being104on private models. Then $H=\phi^2$, $\Phi=\phi$, and105\begin{equation}106 \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},107 \qquad108 \tau(\phi)=\frac{\sigma_v^2}{\sigma_v^2+\Omega(\phi)}.109 \label{eq:tau_phi}110\end{equation}111112\begin{proposition}[Informativeness versus penetration]113\label{prop:info}114Under A1--A4 with a single shared model of share $\phi$:115\begin{enumerate}116\item[(i)] For every $N>1$, $\tau$ is strictly decreasing in $\phi$ over the admissible grid117 $\phi\in\{1/N,2/N,\dots,1\}$; in the continuous relaxation, $\partial\tau/\partial\phi<0$ for all118 $\phi>1/(2N)$. The values $\phi=0$ and $\phi=1/N$ coincide, since a model used by a single trader is a119 private model. For $N=1$ the share has no effect.120\item[(ii)] For every $N$ and every $\phi>0$,121 $\tau(\phi)\le\tau_\infty(\phi)\equiv\sigma_v^2/(\sigma_v^2+\phi^2\sigma_\eta^2)<1$, and122 $\tau(\phi)\to\tau_\infty(\phi)$ as $N\to\infty$ with $\phi$ fixed: the common error never averages123 out.124\item[(iii)] If $\phi=0$ and $\sigma_z^2/N^2\to 0$ (in particular if $\sigma_z^2$ is fixed), then125 $\tau\to 1$ as $N\to\infty$.126\end{enumerate}127\end{proposition}128129\noindent\emph{Intuition.} The market maker learns from the average signal. Private errors are130averaged over $N$ draws and vanish at rate $1/N$; the shared error $\eta$ is a single draw that enters131the average with weight $\phi$ and therefore survives with variance $\phi^2\sigma_\eta^2$ no matter how132many traders there are. Penetration converts a diversifiable error into a non-diversifiable one. Note133that (ii) also implies that the \emph{marginal} value of adding traders falls with $\phi$: with a134shared model, the market behaves as if it had at most $M_{\mathrm{eff}}=1/\phi^2$ independent signals.135136\subsection{Pricing error, order-flow correlation and synchrony}137\label{sec:theory:corr}138139Let $\sigma_s^2\equiv\sigma_v^2+\sigma_\eta^2+\sigma_u^2$ be the variance of a signal, and define the two140possible pairwise correlations,141\begin{equation}142 \rho_1=\frac{\sigma_v^2+\sigma_\eta^2}{\sigma_s^2}\quad(\text{same model}),\qquad143 \rho_0=\frac{\sigma_v^2}{\sigma_s^2}\quad(\text{different or private models}),\qquad144 \rho_1-\rho_0=\frac{\sigma_\eta^2}{\sigma_s^2}.145 \label{eq:rho01}146\end{equation}147Let $\bar\rho$ denote the population-average pairwise correlation over the $N(N-1)$ ordered pairs148$i\neq j$, and let the \emph{synchrony index} of a pair be the probability that the two traders submit149same-signed orders, $S_{ij}=\Pr(x_ix_j>0)$, with population average $\bar S$.150151\begin{proposition}[Pricing error, correlation and synchrony]152\label{prop:corr}153Under A1--A4 with a single shared model of share $\phi=n/N$:154\begin{enumerate}155\item[(i)] $\operatorname{Var}(P-v)=\sigma_v^2\,\Omega(\phi)/(\sigma_v^2+\Omega(\phi))$ is increasing in156 $\phi$ on the admissible grid (and for all real $\phi>1/(2N)$), and is bounded below by157 $\sigma_v^2\phi^2\sigma_\eta^2/(\sigma_v^2+\phi^2\sigma_\eta^2)$ uniformly in $N$.158\item[(ii)] $\operatorname{Corr}(x_i,x_j)=\rho_1$ if $g(i)=g(j)\neq\varnothing$ and $\rho_0$ otherwise;159 $\rho_1>\rho_0$ if and only if $\sigma_\eta^2>0$.160\item[(iii)] The average pairwise correlation is161 \begin{equation}162 \bar\rho(\phi)=\rho_0+w(\phi)\,\frac{\sigma_\eta^2}{\sigma_s^2},\qquad163 w(\phi)=\frac{n(n-1)}{N(N-1)}=\frac{\phi(N\phi-1)}{N-1},164 \label{eq:rhobar}165 \end{equation}166 which is convex in $\phi$, increasing for $\phi>1/(2N)$, and converges to167 $\rho_0+\phi^2\sigma_\eta^2/\sigma_s^2$ as $N\to\infty$.168\item[(iv)] For a pair with correlation $\rho$, $S(\rho)=\tfrac12+\arcsin(\rho)/\pi$, which is strictly169 increasing in $\rho$. The population synchrony index is170 \begin{equation}171 \bar S(\phi)=\tfrac12+\frac{1}{\pi}\Big[(1-w(\phi))\arcsin\rho_0+w(\phi)\arcsin\rho_1\Big],172 \label{eq:sync}173 \end{equation}174 which is a strictly increasing function of $\bar\rho$ whenever $\sigma_\eta^2>0$, and satisfies the175 Jensen bound $\bar S\ge\tfrac12+\arcsin(\bar\rho)/\pi$ with equality if and only if $\sigma_\eta^2=0$.176\end{enumerate}177If $\sigma_\eta^2=0$, none of $\tau$, $\operatorname{Var}(P-v)$, $\bar\rho$ or $\bar S$ depends on $\phi$.178\end{proposition}179180\noindent\emph{Intuition.} Two traders on the same model agree about the world more often than the181fundamentals justify, because they share the model's mistake. The share of such pairs is quadratic in182$\phi$ (it is the probability that two random traders both use the model), which is why co-movement183and synchrony accelerate with penetration. The last sentence is the natural placebo: a shared model184that adds no error of its own ($\sigma_\eta^2=0$) is just a precise private signal and cannot generate185any of the effects.186187\subsection{Diversity as a stabilizer}188\label{sec:theory:diversity}189190\begin{proposition}[Diversity]191\label{prop:diversity}192Under A1--A4 with $G$ shared models of shares $(\phi_1,\dots,\phi_G)$ and total share $\Phi$:193\begin{enumerate}194\item[(i)] Holding $\Phi$ fixed, $\tau$ and $\operatorname{Var}(P-v)$ depend on the shares only through195 $H=\sum_g\phi_g^2$, with $\partial\tau/\partial H<0$. When $\Phi=1$,196 $\Omega=\sigma_\eta^2/M_{\mathrm{eff}}+\sigma_u^2/N+\sigma_z^2/(\beta^2N^2)$: model error averages out at197 rate $1/M_{\mathrm{eff}}$ while private error averages out at rate $1/N$.198\item[(ii)] $\Phi^2/G\le H\le\Phi^2$. The upper bound is attained if and only if a single model carries199 the whole share $\Phi$; the lower bound if and only if the $G$ models have equal shares200 $\phi_g=\Phi/G$ (in the integer version, if and only if the $n_g$ differ by at most one).201\item[(iii)] Consequently, for any population with total shared-model share $\Phi$,202 $\tau_{\text{single}}\le\tau_{\text{mixed}}\le\tau_{\text{uniform}}$: a mixed-provider or203 diverse-prompt population is weakly more informative than a single-model population of the same size,204 strictly so whenever at least two models have positive share and $\sigma_\eta^2>0$.205\item[(iv)] The average correlation generalizes to $\bar\rho=\rho_0+w\,\sigma_\eta^2/\sigma_s^2$ with206 $w=(NH-\Phi)/(N-1)$, so that, for fixed $\Phi$, $\bar\rho$ and $\bar S$ are also increasing in $H$.207\end{enumerate}208\end{proposition}209210\noindent\emph{Intuition.} With several models, the common errors are themselves averaged over models.211The Herfindahl index is the exact sufficient statistic because the variance of a weighted average of212independent model errors is $\sigma_\eta^2\sum_g\phi_g^2$. Splitting a given LLM population evenly across213$G$ providers (or prompt families with independent errors) reduces the undiversifiable noise from214$\Phi^2\sigma_\eta^2$ to $\Phi^2\sigma_\eta^2/G$. Note that the recovery is partial unless $G$ grows with215$N$: $M_{\mathrm{eff}}\le N$ and diversity across a handful of providers cannot restore the216$1/N$ rate (hypothesis H4). In this stylized model the benefit of diversity rests entirely on the217independence of $\eta_g$ across models; models trained on similar data or distilled from each other218would have correlated $\eta_g$ and a correspondingly larger effective $H$.219220\subsection{Feedback amplification}221\label{sec:theory:feedback}222223We now add a second date and trend demand in the spirit of \citet{delong1990}. Let $P_0=0$ be the224public value before the trading round and $Y_1=\beta\sum_i s_i+z_1$ the fundamental flow of225Section~\ref{sec:theory:setup}. Trend followers submit the demand $\gamma(P_1-P_0)$, $\gamma\ge 0$.226227\begin{assumption}[A5: within-round feedback with a fixed pricing schedule]228\label{ass:feedback}229Trend demand responds to the price change of the current round, and the market maker applies the230schedule $\lambda$ of \eqref{eq:lambda} to the total flow $Y_1+\gamma(P_1-P_0)$ without re-estimating231it.232\end{assumption}233234A5 is the model counterpart of the simulator's timing: fundamental and LLM orders arrive in staggered235batches over a decision interval, while trend followers react every tick to the mid-price history, so236within a decision interval the trend response is effectively contemporaneous. Its second clause is a237restriction, not a result: a Bayesian market maker who knew that the extra flow is $\gamma\Delta P$238would filter it out (the total flow is then a deterministic multiple of $Y_1$), and there would be no239amplification. The simulator's market makers quote a mechanical inventory-skewed schedule and do not240filter, which is what A5 captures.241242\begin{proposition}[Feedback amplification]243\label{prop:feedback}244Under A1--A5, write $\Delta P=P_1-P_0$ and $\kappa\equiv 1/(1-\lambda\gamma)$.245\begin{enumerate}246\item[(i)] If $\lambda\gamma<1$ the round has the unique fixed point $\Delta P=\kappa\lambda Y_1$, which247 is the limit of the iterated response $\Delta P^{(k+1)}=\lambda\big(Y_1+\gamma\Delta P^{(k)}\big)$248 from any starting point. The variance of the price change is amplified by $\kappa^2=1/(1-\lambda\gamma)^2$249 relative to the no-feedback benchmark.250\item[(ii)] If $\lambda\gamma\ge 1$ the iterated response diverges, $|\Delta P^{(k)}|\to\infty$, from251 every starting point except the algebraic solution. For $\lambda\gamma=1$ no solution exists when252 $Y_1\neq0$; for $\lambda\gamma>1$ the algebraic solution $\Delta P=\kappa\lambda Y_1$ has the sign253 opposite to the flow and is repelling, so no adjustment process reaches it.254\item[(iii)] Amplification leaves $\tau$ unchanged (the price remains a deterministic multiple of255 $Y_1$) but raises the pricing error to256 $\operatorname{Var}(P_1-v)=\sigma_v^2\big[(1-\tau)+\tau(\kappa-1)^2\big]$, increasing in $\gamma$ on257 $[0,1/\lambda)$.258\end{enumerate}259\end{proposition}260261\begin{assumption}[A6: return-loading of LLM policies]262\label{ass:loading}263An LLM trader on model $g$ submits $x_i=\beta s_i+\kappa_g\,\Delta P$ with $\kappa_g\ge 0$ the264model's loading on the current price change; classical trend followers contribute an aggregate loading265$\gamma_T\ge0$.266\end{assumption}267268\begin{corollary}[Shared models raise the effective feedback coefficient]269\label{cor:llmtrend}270Under A1--A6 the effective feedback coefficient is $\gamma=\gamma_T+N\sum_g\phi_g\kappa_g$. With a single271shared model of share $\phi$ and loading $\kappa>0$, $\gamma(\phi)=\gamma_T+N\phi\kappa$ is increasing in272$\phi$, the amplification factor $\kappa^2(\phi)$ is increasing in $\phi$, and the stable regime is lost273at the critical share $\phi^\ast=(1/\lambda-\gamma_T)/(N\kappa)$. If $\kappa=0$ the share has no274effect on amplification. If the loadings $\kappa_g$ differ in sign across models (some models are275contrarian), the aggregate loading is the share-weighted average and can be smaller than that of any276single model: diversity dampens feedback, concentration does not.277\end{corollary}278279\noindent\emph{Intuition.} A price rise induced by informed flow triggers trend demand, which under a280fixed pricing schedule moves the price again, and so on; the geometric sum converges only if each281round's response is smaller than the last. LLM policies that mention momentum, ``recent trend'' or282``follow the tape'' in their rationale behave like trend followers with a loading $\kappa$ that the283simulator measures by regressing LLM orders on lagged returns. Whether $\kappa$ is positive is an284empirical matter; the corollary says what follows if it is. The variance amplification is285identical to the one obtained by DeLong et al.\ with rational speculators front-running feedback286traders, but the source here is mechanical.287288\subsection{Liquidity spiral}289\label{sec:theory:spiral}290291Finally, we add leverage, following the margin-spiral logic of \citet{brunnermeier2009}. A292cohort $L$ of $N_L$ leveraged traders holds the aggregate position $X$ (in lots) financed with equity293$W$ under a maintenance requirement of $m>0$ units of equity per lot held.\footnote{If the simulator's294maintenance margin rate is $\hat m$ and the pre-shock price is $\bar P$, then $m=\hat m\bar P$ with the295collateral valued at the pre-shock price; this linearization is exact to first order in296$|\Delta P|/\bar P$ and is the only approximation in the proposition.} Let $b\equiv W-mX\ge0$ be the297initial margin buffer. An exogenous price shock $\Delta P_0<0$ hits the cohort's side of the market;298after a total price change $\Delta P$ the cohort's equity is $W+X\Delta P$, so it must liquidate299\begin{equation}300 F(\Delta P)=\max\Big\{0,\;X-\frac{W+X\Delta P}{m}\Big\}=\max\Big\{0,\;\frac{-X\Delta P-b}{m}\Big\}301 \label{eq:forced}302\end{equation}303lots. Under A5 the market maker prices the forced sales as order flow, so the total price change solves304$\Delta P=\Delta P_0-\lambda F(\Delta P)$. Define the spiral coefficient $a\equiv\lambda X/m$.305306\begin{proposition}[Liquidity spiral]307\label{prop:spiral}308Under A1--A5 and \eqref{eq:forced}:309\begin{enumerate}310\item[(i)] If $\Delta P_0\ge -b/X$ the shock is absorbed by the buffer: $F=0$ and $\Delta P=\Delta P_0$311 is a fixed point at which the cascade stops at once. (When $a>1$ and $b>0$ a second, self-fulfilling312 fixed point with $F>0$ also exists; it is not reached by the cascade and we select the cascade limit313 throughout.)314\item[(ii)] If $\Delta P_0<-b/X$ and $a<1$, the unique fixed point is315 \begin{equation}316 \Delta P^\ast=\frac{\Delta P_0+\lambda b/m}{1-a},\qquad\text{so that with } b=0:\quad317 \Delta P^\ast=\frac{\Delta P_0}{1-\lambda X/m},318 \label{eq:spiral_fp}319 \end{equation}320 with amplification multiplier $1/(1-\lambda X/m)>1$. The fixed point is the limit of the321 round-by-round cascade in which each round's forced sales respond to the previous round's price.322\item[(iii)] If $\Delta P_0<-b/X$ and $a\ge 1$, no finite fixed point exists and the cascade decreases323 without bound: in a market with a price band, the price reaches the band.324\item[(iv)] Let $X=\big|\sum_{i\in L}x_i\big|$ be the absolute net position of the cohort and $\bar\rho_L$325 its average pairwise order correlation. Then $X$ is the absolute value of a centered Gaussian with326 standard deviation $\sigma_X=\beta\sigma_s\sqrt{N_L[1+(N_L-1)\bar\rho_L]}$, so that327 $\mathbb E[X]=\sigma_X\sqrt{2/\pi}$ and the spiral probability $\Pr(\lambda X/m\ge1)=2\bar\Phi(m/(\lambda\sigma_X))$328 are both strictly increasing in $\bar\rho_L$, hence (Proposition~\ref{prop:corr}) in the shared-model329 share of the cohort. (Here $\bar\Phi$ is the standard normal survival function.)330\end{enumerate}331\end{proposition}332333\noindent\emph{Intuition.} The forced sale of one lot per $m/X$ of price decline moves the price by334$\lambda$ per lot; when $\lambda X/m\ge1$ each round of liquidation generates at least as large a335decline as the one that triggered it. A shared model does not change the expected position of any336single leveraged trader, but it makes positions line up on the same side, which fattens the337distribution of the cohort's net exposure; the spiral threshold $m/\lambda$ is crossed more often. The338net position is a conservative base for forced sales---the gross exposure on the shocked side is339weakly larger---so (iv) understates the mechanism. Since by \eqref{eq:lambda} $\lambda$ itself is340decreasing in $\Omega$, there is a countervailing channel (a less informative market is also a less341price-sensitive one); in the simulator, however, $\lambda$ is set by the mechanical market makers' depth,342so we treat it as a parameter.343344\subsection{Safeguards}345\label{sec:theory:safeguards}346347Two safeguards from the simulator have direct counterparts. A \emph{position limit} caps the cohort's348exposure at $L$ lots (a per-trader cap $\ell$ with $N_L\ell=L$). A \emph{trading halt} of length $h\ge1$349rounds is triggered when the price change reaches a threshold $\theta>0$; during the halt no forced350sales execute. For the halt we need a reduced-form statement of what changes while trading is stopped.351352\begin{assumption}[A7: halt effectiveness]353\label{ass:halt}354After a halt of length $h$, the cascade resumes with spiral coefficient $\psi(h)\,a$, where355$\psi:\{0,1,2,\dots\}\to(0,1]$ is nonincreasing with $\psi(0)=1$.356\end{assumption}357358A7 summarizes two mechanisms, margin top-ups (fewer lots must be sold per unit of price decline) and359the arrival of patient liquidity (smaller impact per lot sold), without modelling either. It is an360assumption about the halt, not a result; the case $\psi(h)=1$ is a halt that changes nothing but the361clock.362363\begin{proposition}[Safeguards]364\label{prop:safeguards}365Under the assumptions of Proposition~\ref{prop:spiral}:366\begin{enumerate}367\item[(i)] (Position limit.) If $L<m/\lambda$, then for every shared-model share and every realization368 of signals, $\lambda X/m\le\lambda L/m<1$: the spiral region is empty and the multiplier is bounded by369 $1/(1-\lambda L/m)$. The cost is informational: if the limit binds on informed orders, each capped370 order $\tilde x_i=\min\{\max\{\beta s_i,-\ell\},\ell\}$ has loading371 $\operatorname{Cov}(v,\tilde x_i)=\Pr(|\beta s_i|<\ell)\operatorname{Cov}(v,x_i)<\operatorname{Cov}(v,x_i)$ on the372 fundamental.373\item[(ii)] (Halt: depth.) Under A7, if the halt is triggered by the shock itself ($|\Delta P_0|\ge\theta$)374 and $\psi(h)a<1$, the post-halt cascade converges to375 $\Delta P^\ast_h=(\Delta P_0+\lambda b/m)/(1-\psi(h)a)$, which is finite even when $a\ge1$ and satisfies376 $|\Delta P^\ast_h|\le|\Delta P^\ast|$ when $a<1$, strictly if $\psi(h)<1$.377\item[(iii)] (Halt: time-to-absorb.) Let $T_\epsilon$ be the first round at which the price is within378 $\epsilon$ of its terminal value. Without a halt and $a<1$,379 $T_\epsilon(a)=\max\{0,\lceil\log(\epsilon(1-a)/|\Delta P_0|)/\log a\rceil-1\}$ (for $b=0$; for $b>0$380 replace $|\Delta P_0|$ by $|\Delta P_0|-b/X$); with the halt,381 $T^{h}_\epsilon=h+T_\epsilon(\psi(h)a)$. Hence $T^h_\epsilon\ge h$: no information---neither the shock382 itself nor public news arriving during the halt---is incorporated before $h$ rounds have elapsed,383 and if $\psi(h)=1$ the halt is pure delay, $T^h_\epsilon=T_\epsilon(a)+h$.384\end{enumerate}385\end{proposition}386387\noindent\emph{Intuition.} A position limit works by construction: it bounds the base of forced sales388and therefore the slope of the feedback loop, at the cost of truncating the informed demand that makes389prices informative. A halt is more ambiguous. It reduces the depth of the spiral only through what390changes during the halt (A7), and it always costs at least its own length in delayed price discovery;391whether the trade-off is favourable is the empirical question H6 asks of the simulator.392393\subsection{Mapping to the simulator}394\label{sec:mapping}395396Table~\ref{tab:mapping} maps each theoretical primitive to the simulator parameter that manipulates or397measures it. Two points deserve emphasis. First, $\phi$ is \emph{not} the LLM share: it is the share of398traders on one and the same model, i.e.\ \texttt{llm\_share} times the share of LLM slots on a given399provider--prompt combination (\texttt{provider\_mix}, \texttt{prompt\_set}). A population with400\texttt{llm\_share} $=0.5$ split evenly across two providers with diverse prompts has a small $H$ and401should behave, in the model, like a population of private models. Second, $\beta$ and $\lambda$ are402measured, not set: the LLM order aggressiveness is the slope of submitted quantity on the private signal403(or on the public-value gap for uninformed LLM agents), and $\lambda$ is the Kyle lambda estimated by404regressing mid-price changes on signed order flow.405406\begin{table}[htbp]407\centering408\caption{Theoretical primitives and their simulator counterparts.}409\label{tab:mapping}410\small411\begin{tabular}{p{0.26\linewidth}p{0.68\linewidth}}412\hline413Primitive & Simulator parameter or statistic \\414\hline415$\phi$, $(\phi_g)$, $H$ & \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 \\416$\sigma_\eta^2$ & 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) \\417$\sigma_u^2$ & private-signal noise $\sigma^2/\texttt{precision}$ (\texttt{population.precision}, \texttt{llm.precision}) and \texttt{llm.informed\_share} \\418$\sigma_v^2$ & dividend-innovation variance $\sigma^2$ scaled by $\rho/(1+r-\rho)$ (\texttt{fundamentals.sigma}, \texttt{rho}, \texttt{r\_period}) \\419$\beta$ & LLM order aggressiveness: measured slope of order size on the signal (\texttt{llm.max\_order\_qty} caps it) \\420$\lambda$ & estimated Kyle lambda (mid change on signed volume over \texttt{div\_interval} windows); depth set by market-maker slots (5\,\%) \\421$\gamma$, $\kappa_g$ & \texttt{population.mix[trend]}, \texttt{mix[leveraged]}; LLM return loading estimated from decisions on lagged returns \\422$m$, $W$, $X$ & \texttt{population.maintenance\_margin}, \texttt{leverage\_cap}, endowments; $X$ = realized net position of leveraged slots \\423$L$, $\ell$ & position limit (per-agent order/position cap; \texttt{allow\_short}, \texttt{max\_short\_lots}) \\424$\theta$, $h$ & \texttt{exchange.halt\_threshold}, \texttt{halt\_window}, \texttt{halt\_duration} (circuit breaker) \\425$z$, $\sigma_z^2$ & noise-trader slots (\texttt{mix[noise]}, activity) \\426$N$ & \texttt{population.n\_traders} ($100$, $1{,}000$, $10{,}000$) \\427$\tau$ & 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 \\428$\operatorname{Var}(P-v)$ & mean squared pricing error $(P_t-V_t)^2$ \\429$\bar\rho$, $\bar S$ & average pairwise correlation of signed order flow across agents; share of agent pairs with same-signed orders within a decision interval \\430$\Delta P^\ast$, $T_\epsilon$ & 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}) \\431\hline432\end{tabular}433\end{table}434435\subsection{Testable predictions and what would falsify them}436\label{sec:predictions}437438Table~\ref{tab:predictions} lists the hypotheses of the pre-analysis plan together with the439proposition that generates each, the predicted sign and the simulator statistic that tests it. Each440line also states what would count against the model. We stress once more that these are predictions of441a stylized, partial-equilibrium aggregation model; the simulator is not an implementation of the model442but an environment in which the same mechanism should, if it is real, leave the same signature.443444\begin{table}[htbp]445\centering446\caption{Hypotheses, generating propositions, predicted signs and tests.}447\label{tab:predictions}448\small449\begin{tabular}{p{0.04\linewidth}p{0.33\linewidth}p{0.13\linewidth}p{0.38\linewidth}}450\hline451 & Prediction (source) & Sign & Test statistic; falsifying pattern \\452\hline453H1 & 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 \\454H2 & 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 \\455H3 & 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 $\phi$ \\456H4 & 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) \\457H5 & 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 \\458H6 & 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 \\459\hline460\end{tabular}461\end{table}462463Three further implications are useful as placebo or dose--response checks. (a) By464Proposition~\ref{prop:corr}, an LLM treatment whose decisions carry no common component465($\sigma_\eta^2=0$, e.g.\ the mock provider with independent seeds) should produce none of H1--H3; a466positive effect there would indicate a channel outside the model (for instance, a change in $\beta$ or in467latency). (b) By Proposition~\ref{prop:info}(ii), the loss of informativeness should not shrink as468$N$ grows from $100$ to $10{,}000$ at fixed $\phi$, whereas by (iii) it should vanish in the private-model469benchmark. (c) By Corollary~\ref{cor:llmtrend}, amplification should respond to $\phi$ only in treatments470where the measured return loading $\kappa$ of LLM decisions is positive.471