Poor Man's Agentic Modeling: Simulating Large LLM-Agent Societies on a Laptop

# Simulating Large LLM-Agent Societies on a Laptop
Source: [https://arxiv.org/html/2608.11215](https://arxiv.org/html/2608.11215)
## Poor Man’s Agentic Modeling: Simulating Large LLM\-Agent Societies on a Laptop

\(July 2026\)

###### Abstract

Simulating societies of many large language model \(LLM\) agents is expensive, yet the questions asked of such simulations are usually macroscopic: phase behaviour, stylised facts, and scaling with the number of agentsNN, not the cognition of any single agent\. We turn a statistical\-physics observation into a method: replace each LLM agent by a low\-parameter model fitted from a few hundred to a few thousand cheap queries, then run the society at anyNNon a laptop\. Whether this works is decided before the simulation runs, chiefly by what each agent perceives\. We introduce an \[interaction order×\\timesmemory\] taxonomy that maps perception and memory to an effective theory and a predictedNN\-trend of the surrogate error\. We validate it on a faithful reimplementation of the LLM macroeconomy EconAgent and seven further named LLM simulations, with agent decisions cloned from genuine LLM elicitations \(primarily DeepSeek\) for a few dollars; the predicted error trends hold cell by cell, and the two refuted predictions, both on a strongly saturating response and traced to its curvature, are themselves matched quantitatively by the theory with no free parameters\.

Keywords:LLM multi\-agent systems; agent\-based modelling; mean\-field theory; finite\-size scaling; surrogate models; behavioural cloning; recommender systems; sociophysics\.

## 1Introduction

A growing body of work builds simulations in which each agent is a large language model: generative towns, LLM macroeconomies, social\-media societies, and epidemic models with reasoning citizens\. These simulations are valuable because they reproduce human\-like macroscopic behaviour \(business cycles, opinion polarisation, epidemic waves\) without hand\-coded behavioural rules\. They are also expensive\. A single run of a thousand\-agent society can cost tens of dollars and tens of hours in API calls, which places systematic study, and in particular the study of how behaviour scales with the number of agentsNN, out of reach for most researchers\.

The expense buys per\-agent cognition, yet the scientific questions posed to these simulations are almost always macroscopic: does a phase transition occur, what are the stylised facts of the aggregate, how does an observable scale withNN? For such questions a century of statistical physics offers a lesson: the macroscopic behaviour of a large interacting system is governed by a few collective variables, and most microscopic detail is irrelevant to it\. If that stance applies to LLM societies, then for macroscopic purposes each expensive agent can be replaced by a cheap surrogate whose few parameters are fitted from a small number of queries, and the society can be studied at anyNNon a laptop\.

Whether the stance applies is not automatic, and the contribution of this paper is a criterion for when it does\. We argue that the deciding property is*what each agent perceives*\. An agent that reacts to one population\-wide aggregate \(an inflation rate, a global trending feed\) sits in a mean\-field regime, and a scalar surrogate reproduces the macroscopic observable with an error that vanishes asN−1/2N^\{\-1/2\}\. An agent that reacts to a signal shared only within its community, or only to its graph neighbours, sits in a regime where the same scalar surrogate carries an error that does not vanish, and may even grow withNN\. The taxonomy of Section[3](https://arxiv.org/html/2608.11215#S3)makes this precise by mapping the perception and memory design of a simulation to a cell, an effective theory, and a predictedNN\-trend of the surrogate error\. Because the perception design of a social simulation is in practice set by its recommender, the recommender is the switch between the regimes in which cheap modelling succeeds and those in which it fails\.

We validate the criterion in three ways\. First, on a faithful, code\-authoritative reimplementation of the LLM macroeconomy EconAgent, we show that the surrogate reproduces the target’s macroscopic signatures and that doing so exposes what those signatures do and do not measure\. Second, we falsify the taxonomy directly: its cell assignments are pre\-registered and then tested blind, both on held\-out contact graphs and on the measured response function of a real LLM\. Third, we classify and reproduce eight named LLM simulations spanning all three perception cells: EconAgent, AgentTorch, OASIS, AgentSociety, De Marzo et al\.’s consensus game, Williams et al\.’s generative epidemic, LLMTraveler’s congestion game, and Generative Agents’ Smallville \(with TwinMarket’s financial stylised facts as a documented boundary case\), together with a cross\-domain check against a differentiable agent\-based model, using agent responses cloned throughout from genuine LLM decisions \(DeepSeek by default; six models for the De Marzo cross\-model test, and GPT\-4o alongside DeepSeek for TwinMarket\)\.

Two findings of independent interest emerge\. EconAgent’s frequently cited reproduction of Okun’s law turns out to be an accounting identity that a behaviour\-free policy already satisfies, whereas its Phillips curve is a genuine behavioural signature carried by a single labour\-cyclicality coefficient; we estimate that coefficient from cloned decisions and recover the macroscopic value as an out\-of\-sample prediction\. And when the pipeline is driven by genuine LLM decisions, a2×22\\times 2ablation isolates the reasoning step—not the wording of the prompt—as the cause of the emergent Phillips curve, so the cheap surrogate becomes an instrument: what makes the macro law appear is itself measurable\.

## 2Related work

We connect several literatures that are rarely joined; a fuller map of all 115 verified sources is given in the companion review\[[21](https://arxiv.org/html/2608.11215#bib.bib21)\]\.*Expensive LLM simulations*such as Generative Agents\[[2](https://arxiv.org/html/2608.11215#bib.bib2)\], EconAgent\[[1](https://arxiv.org/html/2608.11215#bib.bib1)\], OASIS\[[4](https://arxiv.org/html/2608.11215#bib.bib4)\], and AgentSociety\[[5](https://arxiv.org/html/2608.11215#bib.bib5)\]establish that LLM societies reproduce human\-like macroscopic phenomena and define the observables a surrogate must hit, but none is paired with a low\-parameter model whose scaling is analysed\.

*Sociophysics*supplies the off\-the\-shelf few\-parameter rules \(voter and Ising models, bounded\-confidence dynamics, kinetic opinion equations, and active\-matter flocking dynamics\[[19](https://arxiv.org/html/2608.11215#bib.bib19)\]\), leaving open which rule a given LLM agent realises\.*Coarse\-graining and equation learning*, from mean\-field reductions to equation\-free methods and archetype\-based population models, compress dynamics but stop short of the finite\-NNclosure problem and of LLM\-specific targets\.*Mean\-field game theory*\[[16](https://arxiv.org/html/2608.11215#bib.bib16)\]and*heterogeneous\-agent macroeconomics*\[[17](https://arxiv.org/html/2608.11215#bib.bib17)\]provide theN→∞N\\to\\inftylimit and three cautions we adopt: one moment is often enough but misses the tails, small\-NNexponents can agree by accident, and a shared driver can manufacture a power law without a phase transition\. Finally,*closure theory*\(the Mori–Zwanzig memory kernel\[[13](https://arxiv.org/html/2608.11215#bib.bib13)\], the BBGKY hierarchy\[[14](https://arxiv.org/html/2608.11215#bib.bib14)\], graphons\[[15](https://arxiv.org/html/2608.11215#bib.bib15)\], and inverse\-Ising methods\) supplies the machinery that our taxonomy invokes cell by cell\.

The closest prior work is MF\-LLM\[[12](https://arxiv.org/html/2608.11215#bib.bib12)\], which couples a population\-level mean field to per\-agent LLM decisions; it keeps the LLM in the loop and analyses no scaling, whereas we replace the agent and study the macroscopic observable asNNvaries\. The contribution of this paper is the join that no prior work makes: a low\-parameter surrogate per agent, fitted from genuine LLM decisions elicited on a named LLM simulation’s own prompts to reproduce its macroscopic observable, together with a classificatory layer that predicts theNN\-trend of the surrogate error and is validated blind\.

## 3The perception\-ordered taxonomy

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F1_taxonomy.png)Figure 1:The perception cell sets the scaling of the surrogate error\.A simulation’s perception and memory design places it in one cell of the \[interaction order×\\timesmemory\] taxonomy\. Each cell lists three things: its effective theory \(top, bold\), the predicted trend of the scalar\-surrogate error withNN\(middle\), and the system\(s\) that validate that cell here \(bottom, italic\)\. The tint marks whether the cheap surrogate works \(error falls withNN\), is conditional, or breaks \(error staysO​\(1\)O\(1\)or grows\), as keyed at the right\.LetΦt\\Phi\_\{t\}denote the microscopic update of the LLM society andPPthe projection onto the macroscopic observable of interest\. A cheap surrogate replacesΦt\\Phi\_\{t\}by a low\-parameter mapΦ^t\\hat\{\\Phi\}\_\{t\}acting on the projected variables\. The surrogate reproduces the observable exactly when coarse\-graining commutes with the dynamics,P​Φt=Φ^t​PP\\Phi\_\{t\}=\\hat\{\\Phi\}\_\{t\}P; in general it does not, and the size of the commutation defect∥P​Φt−Φ^t​P∥\\lVert P\\Phi\_\{t\}\-\\hat\{\\Phi\}\_\{t\}P\\rVertis what the taxonomy predicts\. We call the resulting gap in the macroscopic observable the*surrogate error*; the commutation defect is the one\-step, map\-level quantity that drives it, and the two scale together \(they coincide at the mean\-field cell\)\.

In plain terms, the surrogate replaces every agent by a single average agent, so it can be right only to the extent that the population is well summarised by its average, and what decides that is what each agent looks at\. When every agent reacts to the same population\-wide quantity \(an inflation rate, a global trending feed\), the agents move together, the mean is a sufficient statistic, and the only error is sampling noise that washes out as1/N1/\\sqrt\{N\}\. When each agent instead reacts to a private draw or to its own neighbourhood, the agents no longer share an input; averaging discards the structure that actually drives them, and the error stops falling withNN: it plateaus, or even grows\. Perception thus decides whether averaging is legitimate at all, and memory whether the past must be tracked\.

We organise this defect along two coarse\-graining axes: interaction order and memory\.

The first is the*interaction order*: how many other agents feed into a single agent’s decision\. A global aggregate feed is order zero \(every agent sees the same population statistic\) and yields a mean\-field theory in which the scalar surrogate error is set by sampling noise and vanishes asN−1/2N^\{\-1/2\}\. A community feed, shared within each of a fixed number of blocks, is a heterogeneous mean field: the error no longer vanishes but falls only with the number of blocks, leaving anO​\(1\)O\(1\)floor inNN\. A local feed, restricted to graph neighbours, is genuinelykk\-body, and the error is controlled by the degree structure rather than byNN\. The second is*memory*: whether an agent’s decision depends only on current inputs or on an accumulated internal state\. Long memory makes the dynamics non\-Markovian and requires a memory kernel in the closure\.

These two axes define the cells of Figure[1](https://arxiv.org/html/2608.11215#S3.F1), and each cell names both a predictedNN\-trend of the surrogate error and the minimal closure that removes it: a scalar mean field for the global cell, a block or graphon mean field for the community cell, and a*pair approximation*\(a moment closure that tracks two\-agent correlations rather than only single\-agent means\[[26](https://arxiv.org/html/2608.11215#bib.bib26)\]\) with a memory kernel for the local, long\-memory cell\.

Two further axes refine the picture and are demanded by data presented later\. A shared driver that itself fluctuates makes the mean field random, so its fluctuations fall more slowly than the naive rate\. And a strongly curved per\-agent response makes coarse\-graining fail via Jensen’s inequality even under a private feed, because the average of a nonlinear response is not the response at the average\. Collecting these,∥P​Φt−Φ^t​P∥\\lVert P\\Phi\_\{t\}\-\\hat\{\\Phi\}\_\{t\}P\\rVertis bounded, heuristically, by a sum of an interaction\-order term, a memory term, and a response\-curvature term\. The first two are the classical BBGKY and Mori–Zwanzig contributions, and the third is the curvature axis that Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)isolates on a real LLM\. We treat this decomposition as an organising heuristic rather than a theorem: the terms are not derived and the constant is not bounded\.

### 3\.1An exactly solvable case

For a tractable class of agents, all three terms in the commutation bound∥P​Φt−Φ^t​P∥\\lVert P\\Phi\_\{t\}\-\\hat\{\\Phi\}\_\{t\}P\\rVertare exact rather than heuristic, and working out the perception term yields a quantitative prediction we confirm later on a real LLM \(Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)\)\. LetA​\(g\)=𝔼x​f​\(x,g\)A\(g\)=\\mathbb\{E\}\_\{x\}f\(x,g\)be the infinite\-population aggregate response to a perceived signalgg, and suppose the population is partitioned intoBBequal communities, communitybbperceivingg∗\+δbg^\{\\ast\}\+\\delta\_\{b\}withδb\\delta\_\{b\}independent and𝒩​\(0,σ2\)\\mathcal\{N\}\(0,\\sigma^\{2\}\)\. WriteD=A​\(g∗\+δ\)−A​\(g∗\)D=A\(g^\{\\ast\}\+\\delta\)\-A\(g^\{\\ast\}\)for a single community, with meanm1=𝔼​Dm\_\{1\}=\\mathbb\{E\}Dand variancev1=Var⁡Dv\_\{1\}=\\operatorname\{Var\}D\. Since𝔼​δ=0\\mathbb\{E\}\\delta=0, Jensen’s inequality\[[23](https://arxiv.org/html/2608.11215#bib.bib23)\]makesm1=𝔼​\[A​\(g∗\+δ\)\]−A​\(g∗\)m\_\{1\}=\\mathbb\{E\}\[A\(g^\{\\ast\}\+\\delta\)\]\-A\(g^\{\\ast\}\)nonzero for smallσ\\sigmawheneverA′′​\(g∗\)≠0A^\{\\prime\\prime\}\(g^\{\\ast\}\)\\neq 0\(to leading orderm1=12​A′′​\(g∗\)​σ2m\_\{1\}=\\tfrac\{1\}\{2\}A^\{\\prime\\prime\}\(g^\{\\ast\}\)\\sigma^\{2\}\); we call it the*Jensen bias*\. The scalar mean\-field surrogate predictsA​\(g∗\)A\(g^\{\\ast\}\), and the realisedBB\-community aggregate is1B​∑bA​\(g∗\+δb\)=A​\(g∗\)\+1B​∑bDb\\tfrac\{1\}\{B\}\\sum\_\{b\}A\(g^\{\\ast\}\+\\delta\_\{b\}\)=A\(g^\{\\ast\}\)\+\\tfrac\{1\}\{B\}\\sum\_\{b\}D\_\{b\}, so its error is\|1B​∑bDb\|\\bigl\\lvert\\tfrac\{1\}\{B\}\\sum\_\{b\}D\_\{b\}\\bigr\\rvert\.

###### Proposition 1\(Community floor\)\.

IfAAis affine, then

Floor​\(B\)=𝔼​\|1B​∑bDb\|=𝔼​\|𝒩​\(m1,v1/B\)\|;\\mathrm\{Floor\}\(B\)=\\mathbb\{E\}\\Bigl\\lvert\\tfrac\{1\}\{B\}\\textstyle\\sum\_\{b\}D\_\{b\}\\Bigr\\rvert=\\mathbb\{E\}\\bigl\\lvert\\mathcal\{N\}\(m\_\{1\},\\,v\_\{1\}/B\)\\bigr\\rvert;that is, the scalar mean\-field surrogate error floor equals exactly the mean of a*folded normal*distribution\[[24](https://arxiv.org/html/2608.11215#bib.bib24)\]\. More generally, for nonlinearAAthe same expression gives the leading\-order asymptotic approximation to the floor\.

###### Consequence 1\(Symmetric versus curved regimes\)\.

If the response is symmetric about the operating point, thenDDis odd in the mean\-zero perturbationδ\\delta, som1=𝔼​D=0m\_\{1\}=\\mathbb\{E\}D=0andFloor​\(B\)=2​v1/\(π​B\)\\mathrm\{Floor\}\(B\)=\\sqrt\{2v\_\{1\}/\(\\pi B\)\}decays asB−1/2B^\{\-1/2\}\. If the response is curved, thenm1≠0m\_\{1\}\\neq 0, the decay stalls atFloor​\(∞\)=\|m1\|\\mathrm\{Floor\}\(\\infty\)=\\lvert m\_\{1\}\\rvert, and theB−1/2B^\{\-1/2\}law breaks\.

A private feed is the limitB=NB=N: every agent draws its own misperception, so the same expression withB=NB=Ngives the mean\-field error as a function of population size\. It decreases only until the sampling spreadv1/N\\sqrt\{v\_\{1\}/N\}falls to the Jensen bias\.

###### Proposition 2\(The kneeN∗N^\{\\ast\}\)\.

Under a private feed the scalar surrogate error equals𝔼​\|𝒩​\(m1,v1/N\)\|\\mathbb\{E\}\\lvert\\mathcal\{N\}\(m\_\{1\},v\_\{1\}/N\)\\rvert\. It decreases asN−1/2N^\{\-1/2\}up to the knee

N∗=v1m12≈4​A′​\(g∗\)2A′′​\(g∗\)2​σ2,N^\{\\ast\}=\\frac\{v\_\{1\}\}\{m\_\{1\}^\{2\}\}\\;\\approx\\;\\frac\{4\\,A^\{\\prime\}\(g^\{\\ast\}\)^\{2\}\}\{A^\{\\prime\\prime\}\(g^\{\\ast\}\)^\{2\}\\,\\sigma^\{2\}\},and plateaus beyond it at the curvature floor\|m1\|≈12​\|A′′​\(g∗\)\|​σ2\\lvert m\_\{1\}\\rvert\\approx\\tfrac\{1\}\{2\}\\lvert A^\{\\prime\\prime\}\(g^\{\\ast\}\)\\rvert\\sigma^\{2\}\.

###### Consequence 2\(Finite versus infinite knee\)\.

A curved response therefore has a finite knee, computable from the fitted response before any simulation is run; a near\-linear response hasN∗→∞N^\{\\ast\}\\to\\inftyand improves asN−1/2N^\{\-1/2\}indefinitely\.

The remaining term \(interaction order\) and the memory term are exact in the opposite sense: they are*zero*at the mean\-field cell\. Take a conditionally linear societyxi,t\+1=a​xi,t\+f​\(gt\)\+ξi,tx\_\{i,t\+1\}=a\\,x\_\{i,t\}\+f\(g\_\{t\}\)\+\\xi\_\{i,t\}withξi,t\\xi\_\{i,t\}independent, mean zero, varianceσ2\\sigma^\{2\}, wheregt=1N​∑jxj,tg\_\{t\}=\\frac\{1\}\{N\}\\sum\_\{j\}x\_\{j,t\}is the global mean perceived identically by every agent, and letPPproject onto the population mean\.

###### Proposition 3\(Exact commutation at the mean\-field cell\)\.

For*any*responseff,𝔼\[gt\+1∣gt\]=agt\+f\(gt\)=:Φ^\(gt\)\\mathbb\{E\}\[g\_\{t\+1\}\\mid g\_\{t\}\]=a\\,g\_\{t\}\+f\(g\_\{t\}\)=:\\hat\{\\Phi\}\(g\_\{t\}\)\. The commutation defect∥P​Φt−Φ^t​P∥\\lVert P\\Phi\_\{t\}\-\\hat\{\\Phi\}\_\{t\}P\\rVerttherefore vanishes asN→∞N\\to\\infty, and the finite\-NNdefect equals theO​\(N−1/2\)O\(N^\{\-1/2\}\)fluctuation of the mean noise1N​∑iξi,t\\frac\{1\}\{N\}\\sum\_\{i\}\\xi\_\{i,t\}\.

###### Consequence 3\(Mean\-field exactness and curvature irrelevance\)\.

In conditional expectation the mean\-field surrogate is therefore exact, and the curvature offfcontributes nothing: every agent perceives the identicalgtg\_\{t\}\(contrast the heterogeneous perception of Proposition[1](https://arxiv.org/html/2608.11215#Thmproposition1)\)\.

Propositions[1](https://arxiv.org/html/2608.11215#Thmproposition1)–[3](https://arxiv.org/html/2608.11215#Thmproposition3)are proved in Appendices[A](https://arxiv.org/html/2608.11215#A1)–[C](https://arxiv.org/html/2608.11215#A3); Proposition[3](https://arxiv.org/html/2608.11215#Thmproposition3)makes the global\-feed defectC/NC/\\sqrt\{N\}identically for an affine and for a curved response, while adding a heterogeneous misperception reinstates theO​\(1\)O\(1\)Jensen floor of Proposition[1](https://arxiv.org/html/2608.11215#Thmproposition1); both are borne out on the real LLM in Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)\. The first two propositions are confirmed in Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6): the near\-linear consumption head hasN∗≈2\.5×104N^\{\\ast\}\\approx 2\.5\\times 10^\{4\}and keeps improving, while the saturating work head hasN∗≈29N^\{\\ast\}\\approx 29and its private\-feed error is already flat at the predicted floor0\.0180\.018acrossN=100N=100to32003200\. Together the three propositions make the schematic bound exact at the mean\-field cell and along its first step off each axis: the defect isO​\(N−1/2\)O\(N^\{\-1/2\}\)under a global feed, acquires the curvature floor of Propositions[1](https://arxiv.org/html/2608.11215#Thmproposition1)–[2](https://arxiv.org/html/2608.11215#Thmproposition2)under heterogeneous perception, and would acquire the memory and interaction\-order terms under long memory and a local feed respectively\.

### 3\.2From one\-step defect to observable error

Propositions[1](https://arxiv.org/html/2608.11215#Thmproposition1)–[3](https://arxiv.org/html/2608.11215#Thmproposition3)bound the*one\-step closure defect*εt:=∥P​Φt−Φ^t​P∥\\varepsilon\_\{t\}:=\\lVert P\\Phi\_\{t\}\-\\hat\{\\Phi\}\_\{t\}P\\rVert\(the commutation defect of Section[3](https://arxiv.org/html/2608.11215#S3)\), whereas Section[5](https://arxiv.org/html/2608.11215#S5)measures the error of a macroscopic observable read off a whole surrogate trajectory\. These are three distinct quantities, and a standard propagation bound fixes when the first controls the last\. Writez^t\\hat\{z\}\_\{t\}for the surrogate trajectory \(z^t\+1=Φ^t​\(z^t\)\\hat\{z\}\_\{t\+1\}=\\hat\{\\Phi\}\_\{t\}\(\\hat\{z\}\_\{t\}\),z^0=P​x0\\hat\{z\}\_\{0\}=Px\_\{0\}\),et:=∥P​xt−z^t∥e\_\{t\}:=\\lVert Px\_\{t\}\-\\hat\{z\}\_\{t\}\\rVertfor the*trajectory error*, andOOfor anLOL\_\{O\}\-Lipschitz observable, so that the*observable error*equals\|O​\(P​xT\)−O​\(z^T\)\|\\lvert O\(Px\_\{T\}\)\-O\(\\hat\{z\}\_\{T\}\)\\rvertand is at mostLO​eTL\_\{O\}\\,e\_\{T\}\.

###### Lemma 1\(Error propagation\)\.

Suppose the surrogate mapΦ^t\\hat\{\\Phi\}\_\{t\}isLL\-Lipschitz for everyttand the one\-step defect satisfiesεt≤ε\\varepsilon\_\{t\}\\leq\\varepsilonfor everyt<Tt<T\. Then the trajectory error obeys

eT≤∑t=0T−1LT−1−t​εt≤ε​LT−1L−1,e\_\{T\}\\ \\leq\\ \\sum\_\{t=0\}^\{T\-1\}L^\{\\,T\-1\-t\}\\,\\varepsilon\_\{t\}\\ \\leq\\ \\varepsilon\\,\\frac\{L^\{T\}\-1\}\{L\-1\},and the observable error is at mostLO​eTL\_\{O\}\\,e\_\{T\}\. In particular, a contractive surrogate \(L<1L<1\) haseT≤ε/\(1−L\)e\_\{T\}\\leq\\varepsilon/\(1\-L\)uniformly in the horizon\. A neutral one \(L=1L=1\) haseT≤T​εe\_\{T\}\\leq T\\varepsilon\. An expanding one \(L\>1L\>1\) may amplify the defect geometrically, so the one\-step floor controls the observable only over horizonsT≲1/log⁡LT\\lesssim 1/\\log L\.

The bound is the discrete Grönwall recursionet\+1≤εt\+L​ete\_\{t\+1\}\\leq\\varepsilon\_\{t\}\+L\\,e\_\{t\}\(the triangle inequality plusLL\-Lipschitz continuity ofΦ^t\\hat\{\\Phi\}\_\{t\}\), unrolled frome0=0e\_\{0\}=0\.

###### Consequence 4\(When the one\-step floor is what the observable sees\)\.

Reading a macroscopic observable’s accuracy off the one\-step floor and knee of Propositions[1](https://arxiv.org/html/2608.11215#Thmproposition1)–[2](https://arxiv.org/html/2608.11215#Thmproposition2)is legitimate whenever the coarse dynamics are non\-expanding: there the observable errors of Section[5](https://arxiv.org/html/2608.11215#S5)inherit the sameNN\-trend as the one\-step defect\. They can depart from it only where the surrogate map is locally expanding, which is why the near\-critical cells of Figure[1](https://arxiv.org/html/2608.11215#S3.F1)are the ones on which the trend must be read over a longer run rather than a single step\.

## 4Methods

The method is one procedure, applied unchanged to every target in Section[5](https://arxiv.org/html/2608.11215#S5)\(Algorithm[1](https://arxiv.org/html/2608.11215#alg1)\): classify the simulation’s perception cell, which*predicts*theNN\-trend of the surrogate error before any fitting; screen the observable; elicit a few hundred to a few thousand genuine LLM decisions on the target’s own prompts; clone a low\-parameter surrogate; read off its error floor and knee from the fitted response; and run the surrogate society to largeNNon a laptop, validating the macroscopic observable and checking the error trend against the cell’s prediction\. The remainder of this section states the problem formally and details each component; the parenthetical step numbers refer to Algorithm[1](https://arxiv.org/html/2608.11215#alg1)\.

Algorithm 1The poor\-man’s recipe: replace each LLM agent by a low\-parameter surrogate, then scale the society\.1:a target LLM simulation

SSwith its perception and memory specification; a macroscopic observable

MM; the published value

M⋆M^\{\\star\}at conditions

CC
2:a low\-parameter surrogate reproducing

MMat any size

NNwhere the cell permits, and a prediction \(made from the cell, before any fitting\) of how the surrogate error scales with

NN
3:Classifythe perception cell:

\(order,memory\)←classify\_cell​\(S\)\(\\text\{order\},\\text\{memory\}\)\\leftarrow\\textsc\{classify\\\_cell\}\(S\)⊳\\trianglerightpredicts the error trend before any fitting

4:Screen

MM: if a behaviour\-free policy already reproduces it, reject

MM⊳\\trianglerightit is then an accounting identity

5:Elicitdecisions:

𝒟←\\mathcal\{D\}\\leftarrowquery the LLM on

SS’s own prompts over a small state grid⊳\\trianglerightcached; a few dollars

6:Clonethe surrogate:

θ←clone​\(𝒟\)\\theta\\leftarrow\\textsc\{clone\}\(\\mathcal\{D\}\)⊳\\trianglerightfit the22–1212parameters by behavioural cloning

7:Readthe floors:

\(floor,N∗\)←perception\_floors​\(θ\)\(\\text\{floor\},N^\{\\ast\}\)\\leftarrow\\textsc\{perception\\\_floors\}\(\\theta\)⊳\\trianglerighterror floor and kneeN∗N^\{\\ast\}from the fitted response

8:Sweep

NNgeometrically with

N≫NtargetN\\gg N\_\{\\text\{target\}\}: simulate the surrogate society and measure

M​\(N\)M\(N\)
9:Validate: compare

M​\(N\)M\(N\)to

M⋆M^\{\\star\}at

CCand test the error trend against the cell’s prediction⊳\\trianglerightpre\-registered

Problem statement\.We are given a target LLM society with microscopic updateΦt\\Phi\_\{t\}, a macroscopic observable fixed by the projectionPPof Section[3](https://arxiv.org/html/2608.11215#S3), its published valueM⋆M^\{\\star\}at conditionsCC, and a target sizeNN\. We seek a low\-parameter surrogate mapΦ^t\\hat\{\\Phi\}\_\{t\}with parameter vectorθ∈ℝd\\theta\\in\\mathbb\{R\}^\{d\},ddsmall, whose per\-agent policyπ^θ\\hat\{\\pi\}\_\{\\theta\}is fitted from real LLM decisions, such that coarse\-graining commutes with the surrogate dynamics: the objective is to minimise the finite\-size surrogate errorε​\(N\)=∥P​Φt−Φ^t​P∥\\varepsilon\(N\)=\\lVert P\\Phi\_\{t\}\-\\hat\{\\Phi\}\_\{t\}P\\rVertread out through the observable\. Crucially, the parameters are fitted by behavioural cloning of the per\-agent policy,θ=arg⁡minθ​∑\(s,a\)ℓ​\(a,π^θ​\(s\)\)\\theta=\\arg\\min\_\{\\theta\}\\sum\_\{\(s,a\)\}\\ell\\bigl\(a,\\hat\{\\pi\}\_\{\\theta\}\(s\)\\bigr\)over a transfer set of teacher decisions\(s,a\)\(s,a\),*not*by matchingM⋆M^\{\\star\}; the macroscopic observable is therefore never fitted and is always an out\-of\-sample test, and the taxonomy of Section[3](https://arxiv.org/html/2608.11215#S3)predicts the trend ofε​\(N\)\\varepsilon\(N\)from the design’s cell alone\.

The primary target that instantiates this problem is EconAgent\[[1](https://arxiv.org/html/2608.11215#bib.bib1)\]\(step 6’s environment\), a macroeconomy in which each household is a GPT agent that decides monthly whether to work and how much to consume; the released code reproduces business\-cycle signatures including a Phillips and an Okun relation\. We reimplement its market mechanics in NumPy so that a surrogate can be dropped into an identical environment, and the reimplementation is code\-authoritative: it chains the four published components \(labour, progressive taxation, consumption, and savings\) and reproduces the sign and magnitude of the released non\-LLM baseline\.

Before an observable is used as a validation target it must pass an a\-priori screen \(step 2\), because not every macroscopic relation measures behaviour: we ask whether a behaviour\-free policy already produces it, and if so the observable is an accounting identity that validates nothing\. This discriminator is fixed in advance and applied uniformly\.

The behavioural content that survives is captured by cloning \(steps 3–4\)\. A surrogate agent is a twelve\-parameter student: two logistic heads, one for the work decision and one for the consumption propensity, each linear in six features \(a bias, three standardised state variables, the interest rate, and the macroscopic signalgg\), fitted from a transfer set of \(state, action\) decisions produced by a teacher\. The teacher may be a hand\-specified policy or a real LLM, and a single loader makes the two kinds of trace interchangeable, so the same pipeline serves the controls and the LLM experiments\. For the LLM experiments the teacher is DeepSeek \(deepseek\-chat, queried through its Anthropic\-compatible endpoint\), prompted on EconAgent’s own monthly household prompts; decisions are cached to disk, so a fitted result re\-runs at no cost, every paid run carries a hard budget guard, and the total DeepSeek spend across the whole study is a few dollars\.

The classification and finite\-size steps \(1 and 5\) are carried by a small reusable module:classify\_cellmaps a perception and memory spec to a cell, andperception\_floorsreturns the mean\-field error floors of a fitted response\. We validate these primitives against systems whose scaling is known exactly \(the Minority Game and network epidemic models\) before trusting them on LLM targets\.

## 5Results

The experiments answer three questions\.RQ1 \(prediction\):does a simulation’s perception cell predict the sign of the surrogate error’sNN\-trend \(vanishing, plateauing, or growing\)*before*any agent is cloned?RQ2 \(reproduction\):can a two\- to twelve\-parameter surrogate, fitted from a few dollars of genuine LLM decisions, reproduce the published macroscopic behaviour of named LLM simulations across all three cells: the number within noise where matching it does not require the target’s own model, and the mechanism and functional form where it does?RQ3 \(mechanism\):for a given macroscopic observable, is it behavioural at all, and which single microscopic ingredient carries it? RQ1 is tested by the blind, pre\-registered graph and perception experiments \(Sections[5\.5](https://arxiv.org/html/2608.11215#S5.SS5)–[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)\) and by the finite\-size trend in every cell; RQ2 by EconAgent \(Section[5\.1](https://arxiv.org/html/2608.11215#S5.SS1)\) and the remaining named targets \(Appendix[E](https://arxiv.org/html/2608.11215#A5)\); RQ3 by the a\-priori observable test and the identifiability and De Marzo analyses \(Sections[5\.1](https://arxiv.org/html/2608.11215#S5.SS1),[5\.2](https://arxiv.org/html/2608.11215#S5.SS2),[5\.7](https://arxiv.org/html/2608.11215#S5.SS7)\)\.

### 5\.1EconAgent in the mean\-field cell

EconAgent’s published targets are a Phillips correlation of−0\.619\-0\.619and an Okun correlation of−0\.918\-0\.918\. The a\-priori test separates them\. In EconAgent real GDP is, by construction, an affine\-invertible function of the number of working agents \(anR2=0\.996R^\{2\}=0\.996fit\), so Okun’s law relates a quantity to an affine image of itself; a behaviour\-freeBernoulli​\(0\.5\)\\mathrm\{Bernoulli\}\(0\.5\)work policy already yields an Okun correlation of−0\.998\-0\.998\. Reproducing Okun therefore validates nothing\.

The Phillips curve is different: wage inflation is driven by goods\-market imbalance with no direct employment\-to\-wage channel, so a negative unemployment–inflation relation is a genuine behavioural signature\. Its mechanism is a single procyclical\-labour coupling: work propensity rising with the price signal\. Fitting the twelve\-parameter student to a procyclical teacher recovers that coupling, and the macroscopic Phillips correlation, which never enters the fit, emerges as an out\-of\-sample prediction at−0\.569±0\.138\-0\.569\\pm 0\.138, within noise of both the teacher and the published target \(\|Δ\|=0\.05\\lvert\\Delta\\rvert=0\.05\)\. A non\-procyclical teacher cloned through the identical pipeline gives a vanishing Phillips; the coupling, not the pipeline, produces the effect\. This is the mean\-field cell in action: asNNgrows from2020to500500the cloned Phillips strengthens monotonically from−0\.29\-0\.29to−0\.86\-0\.86as averaging over more agents reinforces the weak signal\.

### 5\.2Identifiability is a frontier, not a degeneracy

One macroscopic number need not pin the microscopic mechanism\. We map the reachable Phillips frontier of three candidate micro\-channels \(Table[1](https://arxiv.org/html/2608.11215#S5.T1)\)\. Only labour keyed to the price signal reaches the published value, and near that frontier the channel is nearly pinned, though the pinning requires the coupling to sit at the upper end of its estimated range\. Degeneracy is local rather than global: it occurs only at an intermediate value of the Phillips correlation where a labour and a consumption channel coexist, and there a single additional aggregate observable \(the sign of consumption propensity against unemployment, or of employment against the price level\) separates them\. So one macroscopic number is more identifying than a naive under\-determination concern would suggest, and where it is not, one more number suffices\.

Table 1:The reachable Phillips frontier of three micro\-channels\.Only price\-keyed labour reaches the published−0\.62\-0\.62; near the frontier the mechanism is nearly identified by the single macroscopic number\.
### 5\.3The triple join on genuine LLM decisions

We now drive the pipeline with real LLM decisions: fit the student from a budget of DeepSeek decisions elicited on EconAgent’s household prompts, drop it into the market, and recover the macroscopic observable\. The runs cost $0\.44 and $0\.67 for three and six thousand decisions\.

The headline is that*asking the model to reason*is what produces the Phillips curve, and a2×22\\times 2ablation isolates it\. EconAgent’s protocol gives GPT a reasoning channel through a quarterly reflection; the ablation below is our own elicitation probe\. A naive “reasoning” prompt confounds three changes \(a chain\-of\-thought step, an amplified inflation wording, and an intensity adjective\), so we cross whether the model reasons with whether the inflation signal is amplified, holding the numeric input to the student byte\-identical across all four cells \(Table[2](https://arxiv.org/html/2608.11215#S5.T2)\)\.

The two factors are orthogonal:*reasoning*turns the chain\-of\-thought step on or off, while*wording*presents the same inflation figure in plain or in amplified language\. Reading the four cells, with no reasoning the cloned Phillips is weak and even flips sign with wording \(−0\.43\-0\.43plain,\+0\.04\+0\.04amplified\), whereas with reasoning it is strongly negative under both wordings \(−0\.73\-0\.73and−0\.66\-0\.66\): turning reasoning on \(moving down a column\) shifts the correlation sharply, and once reasoning is on the wording barely moves it \(−0\.73\-0\.73vs−0\.66\-0\.66\); wording matters only when reasoning is off, where it flips the sign\. The reasoning effect is large and clearly resolved even where it is smallest: at plain wording, turning reasoning on shifts the correlation by0\.300\.30at a difference standard error near0\.040\.04\(the main effect across both wordings is0\.500\.50\), whereas the two reasoning cells are only marginally separated, so we do not claim one is robustly more negative than the other\.

Table 2:Reasoning, not wording, drives the emergent Phillips curve\.A2×22\\times 2ablation; the numeric signal fed to the student is identical across the four cells\.The reproduction is model\- and reasoning\-conditional rather than robust\. Under a reasoning prompt DeepSeek\-chat’s cloned Phillips is−0\.665±0\.12\-0\.665\\pm 0\.12, the nearest cell landing\|Δ\|≈0\.05\\lvert\\Delta\\rvert\\approx 0\.05from the published−0\.619\-0\.619; but sibling models under the same prompt scatter to−0\.78\-0\.78and−0\.84\-0\.84, and no configuration reproduces the published value robustly\. The exact value is a model\-and\-prompt fingerprint\. The cheap surrogate is thus both a cost\-saving device and an instrument: what makes the macroscopic law shift \(here, whether the agent reasons\) is itself a measurement\. The clone also independently reproduces the mean\-field scaling of Section[5\.1](https://arxiv.org/html/2608.11215#S5.SS1), its terse\-prompt Phillips strengthening from−0\.34\-0\.34atN=20N=20to−0\.61\-0\.61atN=500N=500, so all three legs of the join close on genuine LLM data\.

### 5\.4Closure\-machinery checks on known ground truth

Before trusting the taxonomy on LLM targets we certify its closure machinery on systems whose scaling is known exactly\. On epidemic dynamics over contact graphs, a scalar mean field is accurate on a complete graph, degrades on akk\-regular graph, and fails near threshold on a scale\-free graph\[[20](https://arxiv.org/html/2608.11215#bib.bib20)\]and on a real Facebook network, exactly as the interaction\-order axis predicts\. On the Minority Game\[[18](https://arxiv.org/html/2608.11215#bib.bib18)\], a finite\-size\-scaling data collapse recovers the critical control parameter to within13%13\\%; we therefore treat roughly15%15\\%as the toolkit’s resolution floor and do not read precision below it\. These controls are epidemics and games, not LLM agents; their role is to validate the machinery, not the LLM claims\.

### 5\.5A blind test of the taxonomy on held\-out graphs

To make this a test rather than a fit, we pre\-registered each cell’s assignment and its predictedNN\-trend, then evaluated them on contact graphs held out from all calibration\.

All three predictions held\.

The mean\-field cell’s error shrank withNN, the community cell’s settled to anO​\(1\)O\(1\)floor, and the local cell’s tracked the degree structure; at largeNNthe three errors ordered themselves exactly as the assignment demanded\. This validates the interaction\-order axis on graph dynamics\. The same axis, on a real LLM’s response function, is tested next \(Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)\)\.

### 5\.6A blind test on the LLM perception layer

The sharpest test of the perception axis is on a real LLM’s response function rather than on a synthetic one\. We fit the student to a genuine DeepSeek reasoning trace and then feed a population three signals with the same mean and, for the private and community cases, the same misperception variance, differing only in correlation structure: a global feed that every agent shares, a private feed drawn independently per agent, and a community feed shared within each of a fixed number of blocks\. The prediction, pre\-registered, is that private noise averages away while community noise leaves a floor that falls only with the number of blocks\.

On the near\-linear consumption head all five pre\-registered predictions held: the global and private errors fall asN−1/2N^\{\-1/2\}, the community error is flat inNNat anO​\(1\)O\(1\)floor, that floor falls asB−1/2B^\{\-1/2\}in the number of communities, and a block\-aware surrogate repairs it\. The strongly saturating work head refutes two of its five predictions, and we report the refutation as measured: its private\-feed error does not shrink but sits at a floor of0\.0180\.018\.

A post\-hoc analysis, not pre\-registered, identifies the cause as response curvature: the infinite\-population Jensen bias\|𝔼ε​f​\(g∗\+ε\)−f​\(g∗\)\|\\lvert\\mathbb\{E\}\_\{\\varepsilon\}f\(g^\{\\ast\}\+\\varepsilon\)\-f\(g^\{\\ast\}\)\\rvertequals0\.0180\.018, matching the floor\. The same curvature constant then predicts, with no further fitting, why the community floor’sB−1/2B^\{\-1/2\}decay is broken on this head: the block\-averaged floor cannot fall below the Jensen level, so its ratio acrossB=5B=5to8080is compressed to about2\.02\.0against the ideal4\.04\.0and the measured1\.931\.93, while the odd stance response of Appendix[E\.1](https://arxiv.org/html/2608.11215#A5.SS1)has zero Jensen bias by symmetry and obeys the clean law\.

The curvature term of the commutation heuristic thus turns from a post\-hoc diagnosis into a confirmed quantitative prediction \(Figure[2](https://arxiv.org/html/2608.11215#S5.F2)\)\. It also confirms Proposition[2](https://arxiv.org/html/2608.11215#Thmproposition2): the work head’s fitted response gives a kneeN∗=v1/m12≈29N^\{\\ast\}=v\_\{1\}/m\_\{1\}^\{2\}\\approx 29, so its private\-feed error should already be flat at the floor byN=100N=100, as observed acrossN=100N=100to32003200; the consumption head givesN∗≈2\.5×104N^\{\\ast\}\\approx 2\.5\\times 10^\{4\}, so it keeps improving throughout, as observed\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F7_perception.png)Figure 2:The perception switch on a real DeepSeek response function\.Private misperception averages away; correlated community misperception leaves anO​\(1\)O\(1\)floor that falls asB−1/2B^\{\-1/2\}; a strongly curved response leaves a Jensen floor even under a private feed\.Both mechanisms live on a single axis\. Holding the total misperception variance fixed and letting a knobλ\\lambdaset the fraction that is community\-shared rather than private sweeps a real DeepSeek society across the mean\-field boundary \(Figure[3](https://arxiv.org/html/2608.11215#S5.F3)\): the near\-linear head sweeps cleanly from a vanishing error atλ=0\\lambda=0to anO​\(1\)O\(1\)floor atλ=1\\lambda=1, the saturating head floors at the Jensen level for allλ\\lambda, and the block\-aware surrogate leaves a residual equal to the Jensen floor scaled by the private fraction\. Because the variance is fixed, this isolates correlation as the driver\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F12_dial.png)Figure 3:The recommender dial\.One knobλ\\lambda\(the community\-shared fraction of the feed, at fixed misperception variance\) moves the society across the predictability boundary; the two mechanisms \(correlation break and curvature floor\) separate on the one axis\.To separate what the taxonomy fixes from what the individual model fixes, we refitfLLMf\_\{\\rm LLM\}on thirteen elicitation traces spanning five independent labs \(DeepSeek, OpenAI, Anthropic, Google, and Meta\) and both plain and reasoning prompts, and re\-ran the whole perception test on each \(run\_xmodel\_perception\.py, pre\-registered before the traces were collected\)\. The three predictions that define coarse\-graining—the global feed averages out, the correlated community feed leaves anO​\(1\)O\(1\)floor, and a block\-aware closure repairs it—held for twelve of the thirteen\. The single exception isgpt\-4o\-mini, the smallest model in the panel: its block\-aware closure moves in the predicted direction, with the block error falling monotonically relative to the scalar error, but does not reach the factor\-of\-two criterion byN=3200N=3200\. Every other model, including the three OpenAI flagshipsgpt\-4o,gpt\-4\.1, andgpt\-5\.1, clears the core, so the failure tracks model capability rather than any one lab’s style\. The one prediction that flips across models is that a*private*feed also averages out, and it flips with each model’s response curvature: the Jensen bias runs from≈10−4\\approx 10^\{\-4\}\(private error falls about fivefold withNN\) to0\.0180\.018\(private error flat inNN\), and the kneeN∗N^\{\\ast\}moves from∼105\\sim\\\!10^\{5\}down to∼140\\sim\\\!140in step, as Proposition[2](https://arxiv.org/html/2608.11215#Thmproposition2)predicts\.111These cross\-model knees are read empirically from the private\-feed scan; for the primary DeepSeek trace this gives∼140\\sim\\\!140, whereas the analyticv1/m12v\_\{1\}/m\_\{1\}^\{2\}of Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)gives≈29\\approx 29for the same trace\. The two agree to an order of magnitude, the gap reflecting the folded\-normal approximation in the empirical estimator\.The cell assignment is therefore a property shared across architectures rather than a house style of one model family; the response curvature that sets the knee is a property of the individual model, and the two vary independently\.

### 5\.7De Marzo et al\.: the critical group size is a perception limit

De Marzo, Castellano and García\[[6](https://arxiv.org/html/2608.11215#bib.bib6)\]give a published universal result: an LLM shown its peers’ opinions adopts the majority with probabilityP​\(m\)=12​\[tanh⁡\(β​m\)\+1\]P\(m\)=\\tfrac\{1\}\{2\}\[\\operatorname\{tanh\}\(\\beta m\)\+1\], governed by one majority\-forceβ\\beta, with*consensus*\(a macroscopic state in which the population aligns on one opinion\) requiringβ\>1\\beta\>1and a critical group sizeNcN\_\{c\}whereβ​\(N\)=1\\beta\(N\)=1\. They reportNcN\_\{c\}growing with model capability:Nc≈50N\_\{c\}\\approx 50for Llama\-3\-70B, and only lower bounds \(≳1000\\gtrsim 1000\) for their more capable models, which they could not push to consensus failure within the tested range\. Across modelsNcN\_\{c\}correlates strongly with the MMLU capability benchmark \(r≈0\.75r\\approx 0\.75\)\. This target lets us do more than reproduce a mechanism: we can ask*why*a finite critical group size exists at all, and the answer turns out to be the paper’s own variable—perception\.

We fitβ​\(N\)\\beta\(N\)by maximum likelihood from real decisions elicited with their verbatim prompt across six models \(Figure[4](https://arxiv.org/html/2608.11215#S5.F4), Table[3](https://arxiv.org/html/2608.11215#S5.T3)\)\. The measurement reproduces their universal form, and it exposes the mechanism\. A finiteNcN\_\{c\}cannot come from the mean\-field response: the self\-consistencym=tanh⁡\(β​m\)m=\\operatorname\{tanh\}\(\\beta m\)has a nonzero \(consensus\) solution for everyNNwhenβ\>1\\beta\>1, and finite\-NNfluctuations only round the transition\. Formally \(the proof is in Appendix[D](https://arxiv.org/html/2608.11215#A4)\):

###### Proposition 4\(Consensus threshold and its crossing\)\.

The iterated responsem↦tanh⁡\(β​m\)m\\mapsto\\operatorname\{tanh\}\(\\beta m\)is odd with Jacobianβ\\betaatm=0m=0\. Its disordered fixed pointm=0m=0is therefore stable iffβ<1\\beta<1and loses stability atβc=1\\beta\_\{c\}=1through a supercritical pitchfork\[[25](https://arxiv.org/html/2608.11215#bib.bib25)\]\(oddness excludes a quadratic term\), with ordered branchm∗≃3​\(β−1\)m^\{\\ast\}\\simeq\\sqrt\{3\(\\beta\-1\)\}asβ↓1\\beta\\downarrow 1\. Hence consensus exists iffβ\>1\\beta\>1, independently ofNN\.

###### Consequence 5\(FiniteNcN\_\{c\}is a perception threshold\)\.

A finite critical group size therefore exists*if and only if*the measured slopeβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)decays through11\. The slopeβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)is the resolution with which an agent reads a weak majority in a list ofNNopinions, soNcN\_\{c\}is a perception threshold, not a thermodynamic one\.

The data confirm both halves\. A strong majority \(m=0\.5m=0\.5\) is read perfectly \(P=1P=1\) at everyNN; only the resolution of a*weak*majority degrades withNN, andβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)*is*that resolution\. Models split cleanly by whether their resolution decays \(Table[3](https://arxiv.org/html/2608.11215#S5.T3)\): the reasoning models Opus\-4\.8 and GLM\-5\.2 count perfectly \(βeff\\beta\_\{\\rm eff\}pegged,Nc=∞N\_\{c\}=\\infty\), DeepSeek holds a flatβeff≈1\.9\>1\\beta\_\{\\rm eff\}\\approx 1\.9\>1\(Nc=∞N\_\{c\}=\\infty, our pre\-registered lower bound\), while GPT\-4o, Llama and GPT\-4\-Turbo show a decayingβeff\\beta\_\{\\rm eff\}and hence a finiteNcN\_\{c\}\. The level is a red herring: GPT\-4o’sβeff≈7\\beta\_\{\\rm eff\}\\approx 7far exceeds DeepSeek’s1\.91\.9yet GPT\-4o loses consensus first, because only the*asymptote*relative to11matters\.

This aligns with the Weber/approximate\-number\-system law\[[9](https://arxiv.org/html/2608.11215#bib.bib9)\]: numerosity discrimination depends on the*ratio*nk/nz=\(1\+m\)/\(1−m\)n\_\{k\}/n\_\{z\}=\(1\+m\)/\(1\-m\), which isNN\-independent at fixed majority fraction, so a perfect ratio\-perceiver would have flatβeff\\beta\_\{\\rm eff\}andNc=∞N\_\{c\}=\\infty; a finiteNcN\_\{c\}is a deviation from that ideal \(attention mass1/N1/Ndiluted over a long list\)\. The literature supports the substrate \(LLM magnitude representations are log\-compressive, Weber\-like\[[9](https://arxiv.org/html/2608.11215#bib.bib9)\]\), though it cautions that this representational geometry does not by itself guarantee ideal behavioural ratio\-perception, so we lean on it as motivation rather than proof\. De Marzo et al\. themselves attribute theβ\\beta\-decline to an information\-processing limit, and reportNcN\_\{c\}correlating with MMLU \(r≈0\.75r\\approx 0\.75\)—capability, not thermodynamics\.

Table 3:The critical group size is set by whether perception resolution decays\.Models whoseβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)stays above11haveNc=∞N\_\{c\}=\\infty; those whose resolution decays through11have a finiteNcN\_\{c\}, recovered from the finite\-NNnaming\-game dynamics on the fitted response\. The load\-bearing result is the*finite/infinite split*, which is consistent with De Marzo et al\. wherever the models overlap: the one model they pin as finite \(Llama\-3\) we also find finite, and the models we find flat post\-date their study\. Absolute values we do*not*claim to match: our static probe over\-estimatesβeff\\beta\_\{\\rm eff\}at smallNN, we used Llama\-3\.3 rather than their Llama\-3, and De Marzo et al\. pin only Llama\-3\-70B \(≈50\\approx 50\) with the rest lower bounds, so the fine ordering is not established either way\. GPT\-4\-Turbo’s∼1600\\sim 1600is at least consistent with their≳1000\\gtrsim 1000lower bound\.We probe the mechanism interventionally\. Handing GPT\-4o the explicit tally \(nkn\_\{k\},nzn\_\{z\}\) in the prompt \(removing the list\-reading load while leaving the social decision unchanged\)*flattens*itsβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)from a decay to a pegged constant, sendingNc→∞N\_\{c\}\\to\\infty; DeepSeek, already flat, is unchanged\. The decay that generates a finiteNcN\_\{c\}is therefore the list\-to\-count perception step, not the opinion dynamics\. \(The intervention also lengthens the prompt by one sentence, a length/salience change we did not separately control; a pure\-perception control that asks only which opinion is*more common*\(no adoption\) corroborates, giving GPT\-4o a flatβ\\betainNN, so the effect is not a generic prompt\-length artefact\.\) A consequence for this paper’s own agenda:NcN\_\{c\}is*not*a finite\-size\-scaling critical point; it is a bifurcation in an externally driven perception control parameter, with no diverging correlation length, so a data\-collapse inNNis the wrong lens here; one should measureβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)and locate its crossing of11\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F13_demarzo.png)Figure 4:De Marzo consensus is a perception crossing\.The published adoption form transfers to a new model \(left\); the majority forceβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)is the perception resolution, and a finite critical group size exists only where it decays through the thresholdβc=1\\beta\_\{c\}=1\(right\)\.
### 5\.8A measured memory kernel

Appendix[E\.1](https://arxiv.org/html/2608.11215#A5.SS1)’s honest weakness is that only the assimilation rate is fitted while the memory that drives polarisation is posited\. We measure it\. Eliciting DeepSeek attitude updates given a controlled history of past interactions, we recover a discrete Mori–Zwanzig kernel by regression,anext−a≈∑τK​\(τ\)​\(pt−τ−a\)a\_\{\\text\{next\}\}\-a\\approx\\sum\_\{\\tau\}K\(\\tau\)\\,\(p\_\{t\-\\tau\}\-a\)\(Figure[5](https://arxiv.org/html/2608.11215#S5.F5)\)\. All three pre\-registered predictions held\. The current\-interaction weightK​\(0\)=0\.27K\(0\)=0\.27is in the range of the independently fittedμ=0\.415\\mu=0\.415and, tellingly, below it, since an existing history damps the current move: the conviction\-braking signature seen directly\. The past\-interaction tail is real, at47%47\\%ofK​\(0\)K\(0\), so the update is genuinely non\-Markovian\. And the tail is concentrated in the first few lags and vanishes beyond, so a finite memory closure captures it\. The memory axis is thus grounded empirically in the LLM layer, alongside the perception axis\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F15_memory.png)Figure 5:A measured memory kernel\.The current assimilation sits below the memoryless rate \(conviction braking\); the past\-interaction tail is real, decaying, and truncatable\.

## 6Discussion

The results support a single organising claim\. Whether a low\-parameter surrogate can reproduce the macroscopic observable of an LLM society is decided by the society’s perception and memory design, and the deciding structure can be read off before the simulation is run\.

A global aggregate feed places the society in a mean\-field cell where a scalar surrogate reproduces the observable with an error that vanishes asN−1/2N^\{\-1/2\}; a community or graph\-structured feed places it in cells where the error isO​\(1\)O\(1\)or grows, and where the surrogate must resolve the responsible structure through a block, graphon, or pair\-plus\-memory closure\. Two axes beyond interaction order and memory matter in practice: a shared driver that itself fluctuates, and a curved per\-agent response that breaks coarse\-graining through Jensen’s inequality even under a private feed\. Empirically the operative control turned out to be the recommender: because it sets a simulation’s perception design, it is what moves a society across the boundary between the regimes where cheap modelling succeeds and where it fails \(Appendix[E\.1](https://arxiv.org/html/2608.11215#A5.SS1)and Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)\)\.

The cheap surrogate is not only a cost\-saving device but an instrument\. Because its few parameters are estimated rather than tuned, the value that reproduces a macroscopic law becomes a measurement of what produces that law: a single labour\-cyclicality coefficient for EconAgent’s Phillips curve, a reasoning step rather than phrasing for its magnitude, a saturating societal response for Williams’ epidemic, a decaying memory kernel for AgentSociety’s polarisation, and a perception resolutionβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)whose crossing of11sets De Marzo’s critical group size\. Turning the surrogate’s fit and its failures into measurements is, in our view, the more durable contribution\.

## 7Limitations

Several limitations bound the claims\. The named\-target reproductions are mechanism\-and\-scaling matches, and the two that go further reproduce a target’s published*functional form*on a new model rather than its own model’s published number; the strongest test \(running a target’s exact models to hit its published macroscopic number\) remains open\. The EconAgent market is a code\-calibrated reimplementation validated against the released non\-LLM baseline, and the DeepSeek decisions drive the agents rather than the market mechanics\. The commutation decomposition is an organising heuristic, not a theorem\. The pre\-registrations bind specific numeric predictions with named kill criteria, but each was committed to our own repository together with its result, so the git history does not by itself separate prediction from data\. We document this candidly and archive the pre\-registration bundle, with per\-file SHA\-256 hashes, at an external timestamped DOI\[[22](https://arxiv.org/html/2608.11215#bib.bib22)\]; this anchors the content immutably to a third party, though it dates the deposit, not the prediction\. Finally, the LLM\-layer results rest on one primary model, and where sibling models were tested the macroscopic value moved, which is itself part of the finding rather than a nuisance\.

## 8Conclusion

We have argued that large LLM societies can, for macroscopic purposes, be modelled without a large compute budget, and that whether this works is decided by a perception\-ordered taxonomy that maps a simulation’s design to an effective theory and a predicted trend of the surrogate error withNN\. We tested the taxonomy against blind, pre\-registered predictions on both of its axes at the LLM layer, traced its two refuted predictions quantitatively to response curvature, showed the classification to be automatable, quantified the elicitation and capacity cost of the macroscopic law through a distillation scaling law, and reproduced eight named LLM simulations and a differentiable agent\-based model on genuine, cheaply elicited decisions\.

The recurring lesson is that the surrogate’s fit is a measurement: the microscopic property that a macroscopic observable depends on is exposed, not hidden, by replacing the expensive agent with a cheap one\.

### Reproducibility

Every result has a runner and a cached decision trace, so seeded results reproduce deterministically and the LLM experiments re\-run at no cost; the perception and scaling primitives, the pre\-registrations with their outcomes, and the figure scripts are included\. Total elicitation across the study is a few dollars of DeepSeek\.

### Declarations

Competing interests\.The author declares no competing interests\.

Funding\.This research received no external funding\.

Data and code availability\.The code, runners, and cached decision traces that reproduce every result are openly available at[github\.com/YehudaItkin/poor\-mans\-agentic\-modeling](https://github.com/YehudaItkin/poor-mans-agentic-modeling); the accompanying systematic review and scaling toolkit are archived on Zenodo\[[21](https://arxiv.org/html/2608.11215#bib.bib21)\]\.

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## Appendices

## Appendix AProof of Proposition[1](https://arxiv.org/html/2608.11215#Thmproposition1)\(community floor\)

###### Proof\.

WriteDb=A​\(g∗\+δb\)−A​\(g∗\)D\_\{b\}=A\(g^\{\\ast\}\+\\delta\_\{b\}\)\-A\(g^\{\\ast\}\)for the deviation of communitybb’s aggregate response, withδb\\delta\_\{b\}independent and𝒩​\(0,σ2\)\\mathcal\{N\}\(0,\\sigma^\{2\}\)\. The realised population aggregate under aBB\-community feed is

1B​∑bA​\(g∗\+δb\)=A​\(g∗\)\+1B​∑bDb,\\frac\{1\}\{B\}\\sum\_\{b\}A\(g^\{\\ast\}\+\\delta\_\{b\}\)=A\(g^\{\\ast\}\)\+\\frac\{1\}\{B\}\\sum\_\{b\}D\_\{b\},and the scalar surrogate predictsA​\(g∗\)A\(g^\{\\ast\}\), so its error is\|1B​∑bDb\|\\bigl\\lvert\\frac\{1\}\{B\}\\sum\_\{b\}D\_\{b\}\\bigr\\rvert\. IfAAis affine thenDbD\_\{b\}is Gaussian with meanm1m\_\{1\}and variancev1v\_\{1\}, and1B​∑bDb∼𝒩​\(m1,v1/B\)\\frac\{1\}\{B\}\\sum\_\{b\}D\_\{b\}\\sim\\mathcal\{N\}\(m\_\{1\},v\_\{1\}/B\)exactly; for smooth nonlinearAAthe same holds to leading order, by the central limit theorem inBBand a second\-order expansion inσ\\sigma\. The floor is then the folded\-normal mean

𝔼​\|𝒩​\(m1,v1/B\)\|=2​v1π​B​e−m12​B/\(2​v1\)\+m1​erf⁡\(m1​B2​v1\)\.\\mathbb\{E\}\\bigl\\lvert\\mathcal\{N\}\(m\_\{1\},v\_\{1\}/B\)\\bigr\\rvert=\\sqrt\{\\frac\{2v\_\{1\}\}\{\\pi B\}\}\\;e^\{\-m\_\{1\}^\{2\}B/\(2v\_\{1\}\)\}\+m\_\{1\}\\operatorname\{erf\}\\\!\\left\(m\_\{1\}\\sqrt\{\\tfrac\{B\}\{2v\_\{1\}\}\}\\right\)\.AsB→∞B\\to\\inftythis tends to\|m1\|\\lvert m\_\{1\}\\rvert; whenm1=0m\_\{1\}=0it equals2​v1/\(π​B\)∝B−1/2\\sqrt\{2v\_\{1\}/\(\\pi B\)\}\\propto B^\{\-1/2\}\. ∎

## Appendix BProof of Proposition[2](https://arxiv.org/html/2608.11215#Thmproposition2)\(the kneeN∗N^\{\\ast\}\)

###### Proof\.

A private feed assigns each of theNNagents an independent draw, i\.e\.B=NB=N, so the error is𝔼​\|𝒩​\(m1,v1/N\)\|\\mathbb\{E\}\\lvert\\mathcal\{N\}\(m\_\{1\},v\_\{1\}/N\)\\rvert\. The folded\-normal mean has two regimes,

𝔼​\|𝒩​\(m1,v1/N\)\|≈\{2​v1/\(π​N\),v1/N≫\|m1\|\(decays as​N−1/2\),\|m1\|,v1/N≪\|m1\|\(constant\)\.\\mathbb\{E\}\\bigl\\lvert\\mathcal\{N\}\(m\_\{1\},v\_\{1\}/N\)\\bigr\\rvert\\approx\\begin\{cases\}\\sqrt\{2v\_\{1\}/\(\\pi N\)\},&\\sqrt\{v\_\{1\}/N\}\\gg\\lvert m\_\{1\}\\rvert\\quad\(\\text\{decays as \}N^\{\-1/2\}\),\\\\\[6\.0pt\] \\lvert m\_\{1\}\\rvert,&\\sqrt\{v\_\{1\}/N\}\\ll\\lvert m\_\{1\}\\rvert\\quad\(\\text\{constant\}\)\.\\end\{cases\}The regimes cross over atv1/N=m12v\_\{1\}/N=m\_\{1\}^\{2\}, i\.e\.N∗=v1/m12N^\{\\ast\}=v\_\{1\}/m\_\{1\}^\{2\}\. A second\-order expansion ofAAaboutg∗g^\{\\ast\}gives

m1=12​A′′​\(g∗\)​σ2\+O​\(σ4\),v1=A′​\(g∗\)2​σ2\+O​\(σ4\),m\_\{1\}=\\tfrac\{1\}\{2\}A^\{\\prime\\prime\}\(g^\{\\ast\}\)\\sigma^\{2\}\+O\(\\sigma^\{4\}\),\\qquad v\_\{1\}=A^\{\\prime\}\(g^\{\\ast\}\)^\{2\}\\sigma^\{2\}\+O\(\\sigma^\{4\}\),whence

N∗=4​A′​\(g∗\)2A′′​\(g∗\)2​σ2\.N^\{\\ast\}=\\frac\{4\\,A^\{\\prime\}\(g^\{\\ast\}\)^\{2\}\}\{A^\{\\prime\\prime\}\(g^\{\\ast\}\)^\{2\}\\,\\sigma^\{2\}\}\.∎

## Appendix CProof of Proposition[3](https://arxiv.org/html/2608.11215#Thmproposition3)\(exact commutation at the mean\-field cell\)

###### Proof\.

Averaging the update over the population,

gt\+1=1N​∑ixi,t\+1=a​gt\+f​\(gt\)\+1N​∑iξi,t,g\_\{t\+1\}=\\frac\{1\}\{N\}\\sum\_\{i\}x\_\{i,t\+1\}=a\\,g\_\{t\}\+f\(g\_\{t\}\)\+\\frac\{1\}\{N\}\\sum\_\{i\}\\xi\_\{i,t\},because every agent perceives the samegtg\_\{t\}, sof​\(gt\)f\(g\_\{t\}\)is a constant pulled out of the sum\. The noise average has mean0and varianceσ2/N\\sigma^\{2\}/N, so𝔼​\[gt\+1∣gt\]=a​gt\+f​\(gt\)\\mathbb\{E\}\[g\_\{t\+1\}\\mid g\_\{t\}\]=a\\,g\_\{t\}\+f\(g\_\{t\}\)exactly, with anO​\(N−1/2\)O\(N^\{\-1/2\}\)fluctuation; the response curvature never enters\. If instead agentiiperceivesgt\+εig\_\{t\}\+\\varepsilon\_\{i\}withεi\\varepsilon\_\{i\}mean\-zero and symmetric of varianceτ2\\tau^\{2\}, the mean update carries

1N​∑if​\(gt\+εi\)=f​\(gt\)\+12​f′′​\(gt\)​τ2\+O​\(τ4\),\\frac\{1\}\{N\}\\sum\_\{i\}f\(g\_\{t\}\+\\varepsilon\_\{i\}\)=f\(g\_\{t\}\)\+\\tfrac\{1\}\{2\}f^\{\\prime\\prime\}\(g\_\{t\}\)\\tau^\{2\}\+O\(\\tau^\{4\}\),anO​\(1\)O\(1\)bias that does not average away: the Jensen term of Proposition[1](https://arxiv.org/html/2608.11215#Thmproposition1)\. A dependence ofxi,t\+1x\_\{i,t\+1\}on the history adds a memory term, and replacinggtg\_\{t\}by a neighbourhood average makes the mean an insufficient statistic, adding the interaction\-order term\. ∎

## Appendix DProof of Proposition[4](https://arxiv.org/html/2608.11215#Thmproposition4)\(consensus threshold and its crossing\)

###### Proof\.

The mapm↦tanh⁡\(β​m\)m\\mapsto\\operatorname\{tanh\}\(\\beta m\)has derivativeβ​sech2⁡\(β​m\)\\beta\\operatorname\{sech\}^\{2\}\(\\beta m\), which at the disordered fixed pointm=0m=0equalsβ\\beta\. The map is odd, som=0m=0is stable forβ<1\\beta<1and loses stability in a supercritical pitchfork atβc=1\\beta\_\{c\}=1\. For the amplitude, expand

tanh⁡\(β​m\)=β​m−13​β3​m3\+O​\(m5\);\\operatorname\{tanh\}\(\\beta m\)=\\beta m\-\\tfrac\{1\}\{3\}\\beta^\{3\}m^\{3\}\+O\(m^\{5\}\);a nonzero self\-consistent root ofm=tanh⁡\(β​m\)m=\\operatorname\{tanh\}\(\\beta m\)then satisfies1=β−13​β3​m21=\\beta\-\\tfrac\{1\}\{3\}\\beta^\{3\}m^\{2\}, i\.e\.

m∗2=3​\(β−1\)β3,som∗≃3​\(β−1\)​as​β↓1\.m^\{\\ast 2\}=\\frac\{3\(\\beta\-1\)\}\{\\beta^\{3\}\},\\qquad\\text\{so\}\\qquad m^\{\\ast\}\\simeq\\sqrt\{3\(\\beta\-1\)\}\\ \\text\{ as \}\\beta\\downarrow 1\.The existence of the ordered branch is governed byβ\\betaalone and is independent ofNN\. A finite critical group size can therefore arise only if the*measured*slopeβeff​\(N\)\\beta\_\{\\rm eff\}\(N\)\(the resolution with which an agent reads a weak majority in a list ofNNopinions\) decays through11, which is a property of perception, not of the thermodynamic limit\. ∎

## Appendix EExternal validation suite

The main text carries the load\-bearing experiments: EconAgent, the two blind tests, De Marzo, and the measured memory kernel\. This appendix collects the remaining validations and supporting checks: further named LLM simulations spanning the other perception cells, a cross\-domain comparison against automatic differentiation, a distillation scaling law, an automated cell classifier, and two negative results\. Each is a self\-contained confirmation of the taxonomy, not a separate study\.

### E\.1Named targets across the perception cells

We now classify and reproduce named LLM simulations, each on genuine LLM decisions \(DeepSeek unless noted\), using the recipe: a low\-parameter response, the minimal closure named by its cell, and the cell’s predictedNN\-trend\. None matches a target’s exact same\-model published number, which would require running that target’s own models; Section[5\.7](https://arxiv.org/html/2608.11215#S5.SS7)and Appendix[E\.5](https://arxiv.org/html/2608.11215#A5.SS5)come closest by reproducing a target’s published functional form on a new model\.

AgentTorch: behaviour global, contagion local\.AgentTorch\[[3](https://arxiv.org/html/2608.11215#bib.bib3)\]is a million\-agent LLM epidemic model \(its case study is COVID\-19 in New York City\); it reaches that scale by querying the LLM once per demographic archetype and broadcasting the answer\. We confirm from real DeepSeek archetype decisions why this works: the isolation behaviour is a global feed, so the archetype error shrinks with the number of archetypes, whereas contagion runs on a contact graph, where a well\-mixed surrogate over\-predicts the peak and the break is driven by clustering rather than by the degree tail \(Figure[6](https://arxiv.org/html/2608.11215#A5.F6)\)\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F8_agenttorch.png)Figure 6:AgentTorch spans two cells\.Behaviour is a global feed where the archetype trick is valid; contagion is local, where a well\-mixed surrogate over\-predicts the peak and clustering, not the degree tail, drives the break\.OASIS: the recommender is the switch\.OASIS\[[4](https://arxiv.org/html/2608.11215#bib.bib4)\]is a social\-media simulation of up to a million agents, modelled on X and Reddit; it ships two recommender modes, and the mode is the perception switch\. Its Reddit hot\-score feed is a single global leaderboard: the cloned herd experiment converges to its mean\-field value with an error that effectively vanishes\. Its interest feed is a per\-community echo chamber: the group\-polarisation error isO​\(1\)O\(1\)and falls asB−1/2B^\{\-1/2\}in the number of communities, and a block\-aware surrogate repairs it \(Figure[7](https://arxiv.org/html/2608.11215#A5.F7)\)\. The stance response fitted from800800real DeepSeek decisions weights the agent’s own prior \(α=1\.17\\alpha=1\.17\) above the feed \(κ=0\.60\\kappa=0\.60\), yet the community feed still breaks the mean field\. This response is odd about the operating point, so its Jensen bias vanishes \(Consequence[1](https://arxiv.org/html/2608.11215#Thmconsequence1)\) and the break here is a pure interaction\-order effect, not curvature\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F9_oasis.png)Figure 7:OASIS: the two recommender modes are the perception switch\.Global hot\-score feed→\\tomean field; per\-community interest feed→\\toanO​\(1\)O\(1\)floor falling asB−1/2B^\{\-1/2\}\.AgentSociety: the hardest cell\.AgentSociety\[[5](https://arxiv.org/html/2608.11215#bib.bib5)\]simulates over ten thousand agents in a data\-grounded urban society, and sits in the local, long\-memory cell\. A scalar mean field cannot represent its between\-block polarisation at all \(the first\-moment closure returns zero, so its error is the polarisation itself\), and the error grows withNNunder a densifying interaction schedule while remaining a boundedO​\(1\)O\(1\)floor at fixed degree\. A pair\-plus\-memory closure repairs it, and an ablation of the conviction memory collapses most of the polarisation, isolating the memory axis \(Figure[8](https://arxiv.org/html/2608.11215#A5.F8)\)\. Only the assimilation rate \(μ=0\.415\\mu=0\.415\) is fitted from data here; Section[5\.8](https://arxiv.org/html/2608.11215#S5.SS8)measures the memory dependence that the rest of the mechanism posits\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F10_agentsociety.png)Figure 8:AgentSociety in the local, long\-memory cell\.Scalar mean\-field error grows under densification and floors at fixed degree; a pair\-plus\-memory closure repairs it, and removing the conviction memory collapses the break\.
### E\.2A cross\-domain check: closure versus autodiff

The recipe is not specific to LLM agents\. GradABM\[[11](https://arxiv.org/html/2608.11215#bib.bib11)\]makes a million\-agent epidemic differentiable and calibrates it by gradient descent on GPUs\. For the aggregate mortality curve this is more machinery than the science needs\. On a self\-generated network epidemic we lift a heterogeneous mean field from susceptible\-infected to the full compartmental model, thirteen degree blocks coupled through a degree\-weighted infectious fraction, and calibrate two parameters by a derivative\-free simplex\. It recovers the transmission rate to within13%13\\%and the fatality rate to within6%6\\%at a mortality\-curve RMSE of2222, in0\.340\.34seconds on a laptop, against a well\-mixed baseline that misfits at RMSE4848\(a factor of2\.22\.2\) and against roughly400400seconds of GPU forward\-time per county for the differentiable model \(Figure[9](https://arxiv.org/html/2608.11215#A5.F9)\)\. We recover the method and the planted parameters, not the authors’ number on their data\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F11_gradabm.png)Figure 9:Closure versus autodiff\.A gradient\-free degree\-aware closure recovers the mortality curve and the planted parameters roughly10310^\{3\}times faster than the differentiable million\-agent model, for the aggregate observable\.
### E\.3A distillation scaling law

How much LLM does the macro law cost? Offline from the cached EconAgent trace we measure the Phillips error as a joint function of the elicitation budgetBB, the surrogate capacitypp, and the populationNN\(Figure[10](https://arxiv.org/html/2608.11215#A5.F10)\)\. The error falls with budget, reaching tolerance by a couple of thousand decisions; it falls with capacity and plateaus at four features; and the two axes are approximately separable, the error grid being ninety percent rank\-one in log\-error\. The load\-bearing feature is a capacity floor: below four features the error plateaus above tolerance for any budget, so the surrogate is capacity\-limited and buying more decisions cannot help, whereas above it the surrogate is data\-limited and converges\. The purchasing rule is therefore to buy structure first and data second: four features and one to two thousand decisions, well under a dollar of DeepSeek\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F14_scaling.png)Figure 10:A distillation scaling law\.The macro\-observable error against elicitation budgetBBand surrogate capacitypp; a capacity floor no budget can cross, and an approximately separable grid\.
### E\.4The classification is automatable

The cell assignments above were made by hand\. They need not be\. Given only a neutral description of a simulation’s perception and memory design, with no taxonomy term leaked, an LLM emits the structured specification thatclassify\_cellconsumes; the resulting cell matched the hand assignment in all eight cases we tested, including Generative Agents, correctly placed in the local, long\-memory cell; the classifier saw no taxonomy term, though we wrote the neutral descriptions and cannot rule out that these published systems appear in its training data\. The classificatory layer is a pre\-flight check for which surrogate will work, runnable before any expensive simulation\.

### E\.5Williams et al\.: a generative epidemic in the mean\-field cell

Williams et al\.\[[7](https://arxiv.org/html/2608.11215#bib.bib7)\]give generative agents a daily stay\-home decision from their symptoms and a town\-wide broadcast of new\-infection prevalence\. They publish three signatures: a stay\-home logistic regression \(positive on symptoms, positive on prevalence, negative on prevalence squared, i\.e\. a saturating response\); a flattened epidemic curve; and a shift from a single wave to several as the reproduction number falls\. From462462real DeepSeek decisions on their verbatim prompt we reproduce all three levels \(Figure[11](https://arxiv.org/html/2608.11215#A5.F11)\)\. The fitted logistic has the published sign structure, including the negative squared term \(−1\.77\-1\.77, a stronger saturation than their−0\.65\-0\.65\), tying the epidemic’s societal channel to the same response curvature that Section[5\.6](https://arxiv.org/html/2608.11215#S5.SS6)isolated\. Dropping the fitted response into a behaviour\-coupled compartmental model flattens the peak to a seventh of the no\-feedback baseline and roughly doubles the duration, with a single wave at high reproduction number and multiple waves at low\. And the cell is mean field: the scalar surrogate’s error against a finite\-NNstochastic model shrinks withNN\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F16_williams.png)Figure 11:Williams et al\. epidemic on DeepSeek\.A saturating stay\-home response reproduces the published form; the behaviour\-coupled model flattens the curve and shows waves; the surrogate error shrinks withNN\.
### E\.6LLMTraveler: filling the empty cell

Every target so far has short memory except AgentSociety, which iskk\-local\. The \[mean\-field×\\timeslong\-memory\] cell \(a global signal with an accumulated history\) was empty\. LLMTraveler\[[8](https://arxiv.org/html/2608.11215#bib.bib8)\]fills it: LLM commuters play a day\-to\-day congestion game, each day seeing the exponentially weighted moving average of their travel time on each route \(the memory\) and choosing a route, with congestion felt only through the aggregate flow\.

The published result is that mean travel times converge to the Dynamic User Equilibrium \(DUE\), most models landing within±10%\\pm 10\\%\(GPT\-4o\+1\.33%\+1\.33\\%\)\. We elicit real DeepSeek route choices and fit a two\-parameter ruleP​\(switch\)=σ​\(β​Δ−γ\)P\(\\text\{switch\}\)=\\sigma\(\\beta\\,\\Delta\-\\gamma\)in the travel\-time gapΔ\\Delta\(preregprereg\_llmtraveler\_2026\-07\-03\.md, Figure[12](https://arxiv.org/html/2608.11215#A5.F12)\)\. The rule is rational \(β=\+0\.40\\beta=\+0\.40, it prefers the faster route\) though it shows little of the inertia the authors report \(γ≈0\\gamma\\approx 0on DeepSeek, a model\-dependent difference\)\. Run as a day\-to\-day dynamic on the 16\-traveler two\-route network, it*converges to the DUE at\+4\.7%\+4\.7\\%*, inside the published±10%\\pm 10\\%band, with persistent switching around equilibrium as the authors also observe\. The cell is confirmed too: the equilibrium gap\|mean−DUE\|\\lvert\\text\{mean\}\-\\text\{DUE\}\\rvertshrinks with the number of travelers \(a mean\-field flow convergence\), so the surrogate reproduces the Wardrop user equilibrium\[[27](https://arxiv.org/html/2608.11215#bib.bib27)\]\(the flow at which no traveler can reduce their own travel time by switching route unilaterally, here the same fixed point as the DUE above\) at anyNNon a laptop\.

![Refer to caption](https://arxiv.org/html/2608.11215v1/figures/F17_traveler.png)Figure 12:LLMTraveler in the mean\-field×\\timeslong\-memory cell\.A two\-parameter route rule fitted to real LLM choices converges to the Dynamic User Equilibrium within the published±10%\\pm 10\\%band \(left\); the equilibrium gap shrinks with the number of travelers \(right\)\.
### E\.7TwinMarket: the stylised facts need the market, not just the agent

Not every target yields to a compact surrogate, and the failure is informative\. TwinMarket\[[10](https://arxiv.org/html/2608.11215#bib.bib10)\]reports that GPT\-4o BDI investors reproduce the canonical stylised facts of financial returns \(fat tails and volatility clustering\) from a rich belief–desire–intention state with a social feed\. We elicit real trading decisions \(DeepSeek and GPT\-4o\) on a compact state \(recent trend, peer signal, own P&L\) and fit a low\-parameter trader \(preregprereg\_twinmarket\_2026\-07\-03\.md\)\. The trader recovers the behavioural*signs*\(trend\-following, herding on the peer signal, a disposition effect, selling into profit\) on both models\.

Yet dropped into a minimal price\-impact market at a moderate coupling the trader does*not*generate fat tails: the returns stay near\-Gaussian \(excess kurtosis≈0\\approx 0\), because the elicited herding leaves the market subcritical\. The boundary here is not the trader’s capacity but an*un\-elicited*ingredient: the strength of the price\-impact coupling\. Raising that coupling makes the*same*weak trader supercritical \(excess kurtosis rises past77\), so the stylised facts are under\-determined by the elicited response alone: reproducing them needs the market mechanism, not just the agent, and our compact surrogate does not fit that mechanism from LLM decisions\. We report this honestly as the limit of eliciting only the agent, distinct from the data/capacity floor of Appendix[E\.3](https://arxiv.org/html/2608.11215#A5.SS3)\.

### E\.8Smallville: the field’s most\-cited numbers

Generative Agents\[[2](https://arxiv.org/html/2608.11215#bib.bib2)\]produced the most\-cited macro numbers in the area: in Smallville’s 25 agents, an invitation to a Valentine’s party spreads by word\-of\-mouth to1313of2525, and55attend\. This is the \[kk\-local×\\timeslong\-memory\] cell: a local acquaintance cascade with agents remembering they were told\. We elicit real DeepSeek transmission and attendance decisions, fit two logits \(P​\(tell\)P\(\\text\{tell\}\)rising with tie strength;P​\(attend\)P\(\\text\{attend\}\)\), and run the two\-day cascade on a2525\-node acquaintance graph \(preregprereg\_smallville\_2026\-07\-03\.md\)\. The reproduction is close but semi\-quantitative rather than parameter\-free: at a plausible acquaintance degree \(∼6\\sim 6\) the cascade reaches11\.4\\mathbf\{11\.4\}of2525\(published1313\) and4\.5\\mathbf\{4\.5\}attend \(published55\)\. Because we lack Smallville’s actual relationship graph, the reach depends on the assumed network \(from∼7\\sim 7at degree44to∼19\\sim 19at degree1010\), so a realistic social degree \(66–88\) brackets the published1313but we do not claim the exact number\.

What*is*robust is the cell: a well\-mixed control that lets every informed agent talk to everyone reaches all2525, overshooting, whereas the local cascade does not: the diffusion is a property of the*local*interaction structure, exactly thekk\-local prediction\.