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Predicting the scale limits of social mechanisms in agent societies
Zengqing Wu, Chuan Xiao
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper introduces a prospective audit method to predict the scale limits of social mechanisms in societies of interacting language-model agents. The audit decomposes mechanism effects into structural exposure and agent response, allowing researchers to predict whether mechanisms like reciprocity, consensus, or gossip will survive as population size increases. Key findings include that a single structural term (e.g., message reach and lifetime) can determine the failure point of gossip, and that language-model agents' responses are sensitive to information formats (counts vs. percentages), leading to different scaling behaviors. The method was validated across different model families and code implementations.
Entities (10)
Relation Signals (7)
Zengqing Wu → affiliatedwith → University of Osaka
confidence 99% · Zengqing Wu University of Osaka Osaka, Japan
Chuan Xiao → affiliatedwith → University of Osaka
confidence 99% · Chuan Xiao ∗ University of Osaka Osaka, Japan
Audit → predicts → Scale Limits
confidence 95% · Here we introduce an audit that predicts a mechanism’s fate as a population grows.
Gossip → dependson → Message Reach and Lifetime
confidence 92% · For gossip, the population at which the mechanism fails is set by the reach and lifetime of its messages.
Reciprocity → isa → Social Mechanism
confidence 92% · whether reciprocity, consensus or punishment survives scaling.
Language-Model Agents → respondsto → Information Formats
confidence 90% · In language-model societies, agents respond not only to social information but to how it is expressed: counts and percentages led to different scale behaviour.
GPT-5.5 → usedin → experiments
confidence 85% · The panel shows the six primary probe cells on the three engines... gpt-5.5
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Abstract
Abstract:Societies of interacting language-model agents offer a controllable and repeatable way to study collective behaviour at scales that would be difficult to test with people. Their scientific value, however, depends on whether a social mechanism that works in a small group still operates when thousands of agents interact, and testing this directly requires costly large-scale runs. Here we introduce an audit that predicts a mechanism's fate as a population grows. It asks how often the mechanism can act, whether agents use the information it supplies, and whether the measurement itself creates apparent scale effects. Controlled experiments show that a single structural term can decide whether reciprocity, consensus or punishment survives scaling. For gossip, the population at which the mechanism fails is set by the reach and lifetime of its messages. In language-model societies, agents respond not only to social information but to how it is expressed: counts and percentages led to different scale behaviour. Predictions made before execution held on third-party code and a second model family, while a failed prediction exposed the boundary of the finding. The audit provides a prospective way to decide which social mechanisms can be interpreted across population scales.
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- Source: https://arxiv.org/abs/2608.22884v1
- Canonical: https://arxiv.org/abs/2608.22884v1
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PREDICTING THE SCALE LIMITS OF SOCIAL MECHANISMS IN AGENT SOCIETIES Zengqing Wu University of Osaka Osaka, Japan Chuan Xiao ∗ University of Osaka Osaka, Japan ABSTRACT Societies of interacting language-model agents offer a controllable and repeatable way to study collective behaviour at scales that would be difficult to test with people. Their scientific value, however, depends on whether a social mechanism that works in a small group still operates when thousands of agents interact, and testing this directly requires costly large-scale runs. Here we introduce an audit that predicts a mechanism’s fate as a population grows. It asks how often the mechanism can act, whether agents use the information it supplies, and whether the measurement itself creates apparent scale effects. Controlled experiments show that a single structural term can decide whether reciprocity, consensus or punishment survives scaling. For gossip, the population at which the mechanism fails is set by the reach and lifetime of its messages. In language-model societies, agents respond not only to social information but to how it is expressed: counts and percentages led to different scale behaviour. Predictions made before execution held on third-party code and a second model family, while a failed prediction exposed the boundary of the finding. The audit provides a prospective way to decide which social mechanisms can be interpreted across population scales. 1 Introduction Societies built from language-model agents offer a new experimental window on collective behaviour. Unlike traditional rule-based agents, language models can interpret open-ended descriptions, occupy different social roles and respond to the history of an interaction. Because the same population can be copied, controlled and rerun, these systems make it possible to explore social processes and counterfactual worlds that would be costly, slow or impractical to study with human participants. This possibility has prompted calls to study agents as behavioural entities that can be observed and experimentally perturbed, rather than only as models to be benchmarked [1]. The field has moved quickly. Early generative-agent work showed that a small town of language-model agents could produce believable daily social life [2], and generative agent-based modelling frameworks connected such agents to the questions of classical social simulation [3]. Individual language models can predict aggregate outcomes of survey experiments [4], play repeated social dilemmas [5], reproduce patterns of trust between partners [6], and generate recognizable features of citation networks [7]. General-purpose platforms now populate simulations of social media, cities and markets [8–12], in some cases assembling10 4 to10 6 interacting agents. The community is meanwhile debating what such simulations can support, including where their appropriate boundary lies [13], how a persistent validation gap should be closed [14], and how often current practice violates basic validity principles, with reported collective phenomena vanishing or reversing once those principles are enforced [15]. At this point, the validity of an artificial society depends on more than whether one agent resembles one person: it also depends on whether the processes connecting agents remain meaningful as the population grows. Scaling up is not a neutral change. In a small group, two individuals may meet repeatedly. In a large population, the same pair may never meet again. A piece of gossip sent to four people can cover most of a group of five but almost none of a group of five thousand. A public record can remove that bottleneck by making the same event visible to everyone. Language-model agents introduce a further complication because the same social fact can be expressed as a count, a percentage or a story about a named person, and the model need not treat these forms as equivalent. ∗ Corresponding author: chuanx@ist.osaka-u.ac.jp arXiv:2608.22884v1 [cs.MA] 24 Aug 2026 Scale limits of social mechanisms in agent societiesPreprint The realism of natural-language interaction can therefore work against explanation when it obscures the link between local behaviour and system-level emergence [16]. Consequently, a change in the output of a large simulation can have at least three origins: the social mechanism lost structural reach, the agents did not use the information supplied to them, or the observation protocol manufactured an apparent change. An aggregate curve alone cannot separate these explanations [17,18]. The ambiguity is expensive, since each large run of a language-model society consumes substantial computation, and resolving it by trial and error means paying that cost repeatedly. A prediction that precedes the run turns the question into a check. Decades of work show that cooperation in large populations can be sustained when information and institutions compensate for anonymity [19–21]. Experiments likewise show that group size often acts through monitoring, observability and critical mass rather than as a cause on its own [22–25]. Recent studies have begun to vary population size in language-model groups and to compare their collective dynamics with those of rule-based particles. They reveal changes in convention formation, consensus, segregation and collective performance [26–29]. Related theory shows why maintaining stable cooperation requires proportionally more rounds of gossip as the population grows [30]. Recent language-model studies further show that simulation results can be sensitive to seemingly minor design choices [31], while natural-language gossip can sustain cooperation in small groups of five or nine agents [32]. Increasing the size of the language model may improve simulation fidelity [33], but that is distinct from increasing the number of agents and the interactions among them. These studies establish that scale and interaction structure matter. What is still missing is a procedure that can predict, before a large run, which part of a mechanism will survive, which part will disappear and where the transition should occur. We address this gap by treating scale behaviour as a property of three objects that can be examined separately: the mechanism, the agents and the observation plan. For each route by which a mechanism can change behaviour, the audit asks three plain questions. How frequently does this route reach a decision? Does the agent respond when it does? And can the response be compared fairly across population sizes? Structural terms can often be read from code, whereas responses of language-model agents must be measured directly. The resulting path-level predictions are combined only when all active routes are known; otherwise the aggregate outcome is declared unidentified rather than guessed. The experiments reveal three general findings. First, the scale fate of a social mechanism can hinge on a single structural quantity, and the failure point of gossip follows a simple law set by message reach and lifetime. Second, language-model agents respond to the representation of social information: counts and explicit percentages can place an otherwise identical mechanism in different scale regimes. Third, directional responses are more reproducible across model versions than their absolute levels. Predictions made before execution were tested on unmodified external code and across model families. Successes establish the reach of the audit, while a failed prediction narrows its scope. Together, the results ask whether the fate of a mechanism at scale can be computed from parts that are measurable before the full society is run. 2 Results 2.1 An audit from mechanism to population The audit rests on a simple accounting idea. A social mechanism changes behaviour through one or more information pathways. The contribution of each pathway depends on how often it reaches a decision and how strongly an agent responds when it does. The first quantity can often be computed from the implementation before an experiment runs, and the second can be measured one decision at a time. We write the total effect as E(N ) = P j q j (N ) ∆ j (N ) + interactions,(1) whereE(N )is the mechanism’s effect relative to a matched baseline population without it,q j (N )is the exposure or coverage of pathwayj, and∆ j (N )is the behavioural change produced by one exposure. For gossip, a pathway is the route by which a report about the current partner reaches a decision:q j (N )is the fraction of relevant decisions at which the report is visible, whereas∆ j (N )is the change in giving conditional on seeing it. Their product is the expected contribution of that pathway to the aggregate effect. Equation(1)is a path-wise decomposition, conceptually related to, but not an estimator of, causal mediation effects [34, 35]. The decomposition turns an aggregate scaling curve into a set of testable parts (Fig. 1). Structural quantities such as re-encounter and message coverage are derived from code. The responses of language-model agents cannot be read from their weights and are therefore measured with controlled probes. An aggregate prediction is made only when the active pathways have been enumerated, their gains are known and stable over the target range, and important interactions have been specified or ruled out. A residual between the path sum and the observed aggregate signals an interaction or a missing pathway. A forecast must also exclude changes created by the experiment rather than by the mechanism. The audit therefore checks that an outcome has the same meaning at every population size, that the mechanism can activate, and that 2 Scale limits of social mechanisms in agent societiesPreprint Frozenauditchecklist Exampleentriesinapathtable Auditobjects Agentpopulation Mechanism implementation Observationplan Structuralaudit (derive푞 ! (푁)) Interfaceprobes (identify effective input and estimateΔ ! (푁)) Retunemeasurement orstop Aggregate identifiable? Level2 Aggregateprediction 퐸푁=∑푞 ! 푁Δ " N+interactions Level1 Per-pathpredictions Aggregateunidentified Runtargetsystem acrosspopulationsizesN Compareregisteredquantities withobservedpathsand퐸푁, ifpredicted Nonzero composition residual? Interactionormissing path Adjudicatewithfrozenchecklist Out-of-checklistexplanation= failure PathEffectiveinput풒 풋 (푵)GainevidenceScaleclass Re- encounter partneridentityP(re-encounter) ~1/N code-derived; positive; stable decay(A1) Gossiptargetedreportcoveragec(N) ~min1,푓휏 $%% /N probe-estimated sanction; stable decay(A2/I1) Public board publicrecordvisibilityv(N) ≈constant probe-estimated response; stable flat(B3) Yes No YesNo Yes BEFORE SCALE OUTCOMES TEST AND ADJUDICATE Logpredictionbeforeexecution (curveclass,applicablerange,failureassumption) No Level2 Record success, failure, or unresolved outcome in ledger All active paths and gains identified? Gains stable; interactions specified or negligible. Level1 Mechanism implementation Protocolscreen passed? Figure 1: Prospective audit and adjudication workflow. Before any scale outcome exists, the mechanism implementation yields structural quantitiesq j (N ), interface probes identify the effective inputs and per-encounter gains∆ j , and protocol checks screen the observation plan. The completed path table supports per-path (Level 1) predictions, and an aggregate (Level 2) prediction is issued only when all active paths are enumerated, theirq j and∆ j identified, the gains stable, and interactions specified or negligible; otherwise the aggregate is declared unidentified. Predictions are logged with curve class, applicable range and failure assumption before execution. After execution, a nonzero composition residual diagnoses an interaction or missing path, and any explanation outside the frozen checklist is recorded as an audit failure. The path table shows example entries with their checklist codes. the observation window is long enough for the relevant dynamics. It also distinguishes information about a named individual, whose coverage commonly shrinks with population, from type-level or public information that may remain visible. These checks produce two outputs: path-level predictions whenever structural exposure is known, and an aggregate prediction only when the behavioural responses are known as well. The complete checklist and pseudocode are given in Supplementary Information. 2.2 Single structural terms decide survival We first tested the audit in rule-based societies, where every decision rule is visible and a prediction can be checked directly against the code. Four paired experiments kept the behavioural mechanism fixed while changing one structural feature (Methods, Fig. 2). Direct reciprocity collapsed betweenN = 20andN = 40, where random partners stopped meeting often enough to sustain cooperation. Network reciprocity on a fixed-degree lattice instead held a stable minority of cooperators, unchanged fromN = 10to10 5 , because clusters of neighbours continue to meet regardless of population. The pair differs in scale class, collapse against flatness, not in who cooperates at small sizes. Likewise, global sampling preserved consensus while local sampling on a ring froze into separate domains, and independent aggregation improved with population while sequential social influence limited the number of independent signals to about four [36]. In each case, changing the predicted bottleneck restored the mechanism: fixing partners restored cooperation, adding long-range links restored consensus, and adding independent information restored aggregation accuracy. A more direct intervention changed a single term in one update equation. When a reputation update was divided by population size, its effect fell from−0.43to zero as the population grew. Removing that factor while leaving the rest of 3 Scale limits of social mechanisms in agent societiesPreprint 10 2 10 4 N 0.0 0.2 0.4 0.6 0.8 1.0 cooperator share A direct reciprocity network reciprocity 10 2 10 4 N 0.0 0.2 0.4 0.6 0.8 1.0 consensus rate B local sampling global sampling 10 2 10 4 N 0.6 0.7 0.8 0.9 1.0 aggregation accuracy analytic plateau C sequential influence independent votes 10 1 10 2 10 3 N −0.4 −0.3 −0.2 −0.1 0.0 effect size D update step ∝ 1/N per-event update 10 1 10 2 10 3 N −0.5 −0.4 −0.3 −0.2 −0.1 0.0 effect size E organic transmission public record Figure 2: Single structural terms set the scale class in rule-based societies. (A) Direct reciprocity collapses where measured re-encounter falls below threshold, while network reciprocity on fixed degree stays flat. (B) Consensus under global sampling holds at every size, while the same update rule under local sampling on a ring freezes from roughlyN = 30upward. (C) Independent aggregation improves with population, while sequential social influence pins accuracy near the analytic plateau of0.692. (D) A reputation mechanism whose update step carries a1/Ndilution factor loses its effect as the population grows, while the per-event variant holds. (E) Replacing organic transmission with a public record flips the scale class of the same gossip mechanism with individual responses held fixed. Points are means over seeds (Methods). the mechanism unchanged held the effect near−0.45(Fig. 2D). Population size therefore did not weaken the agents themselves. It weakened the quantity delivered to them. The same distinction separates an institution from the people who use it. In an organic gossip system, a victim informed four randomly selected agents. As population increased by two orders of magnitude, the share of decisions carrying relevant gossip fell from0.75to0.013, and the sanction effect almost disappeared. Replacing private transmission with a public record kept visibility near0.93and the effect near−0.48(Fig. 2E). Individual responses were unchanged. The public record altered only the scale of information coverage. These experiments also expose two common misreadings of scale. A mechanism measured at a ceiling can appear ineffective at every size, and a fixed time horizon can make a slow system look as though it has failed. After the saturated punishment experiment was moved into its dynamic range, its effect was flat across populations. Finally, the direction of an effect and its scaling are different properties. Reversing how victims responded to punishment reversed the sign of the effect but left the same coverage decay underneath. The audit predicts this decay. The implementation determines whether the delivered effect is beneficial or harmful. 2.3 Message reach and lifetime set the failure scale The audit can predict not only whether a mechanism decays, but where the decay begins. In the gossip system, a report reachesfpeople and remains available for an effective lifetimeτ eff . Their product is the mechanism’s information budget. The predicted crossover therefore obeysN c ∝ f τ eff : once the population substantially exceeds this budget, too few decisions carry relevant gossip. 4 Scale limits of social mechanisms in agent societiesPreprint 10 1 10 2 fτ 10 1 10 2 crossover scale N c bounded memory slope = 0.245 A 10 1 10 2 fτ memory bound removed slope = 1.000 95% CI [0.988, 1.011] B Figure 3: Failure scale against the product of fan-out and mark lifetimefτ. (A) In the first design, which used memory bounded at twenty marks and a relative crossing threshold, the exponent fitted against the raw lifetime is0.245, outside the registered band. The bound can truncate the effective lifetimeτ eff belowτ. (B) In the amended memory-unbounded design, which uses fixed absolute crossing thresholds and hasτ eff = τ, the exponent is1.00with bootstrap intervals inside [0.97, 1.04] over ten seeds at three registered thresholds. The experiments recovered this relation quantitatively (Fig. 3). With unbounded memory, the fitted exponent of the crossover againstfτwas1.00, with bootstrap intervals inside[0.97, 1.04]. Two systems with different fan-out and lifetime but the same product failed at the same measured population of63.1. A finite memory design using a relative crossing threshold yielded a compressed exponent of0.245and failed its registered test. Report turnover provides a plausible account: new reports displace old ones and truncate the effective lifetime, approximated by τ eff = (1/τ + λ/M ) −1 . A separately registered amendment removed the memory bound and used a fixed informed- fraction threshold, and it yielded the exponent1.00. Because both features changed, this comparison does not by itself show that correcting the bounded data byτ eff recovers the law. It shows that the product law holds in the amended design where retained and nominal lifetimes coincide. 2.4 Language-model agents respond to information formats For a language-model society, code reveals what information is shown to an agent but not what the agent uses. We therefore isolated one decision at a time: each probe asks an agent to decide whether to give part of its endowment to a partner in a donation game (Methods). In the primary engine, an agent with no information gave to its partner, whereas a single report that the current partner had previously kept a donation produced complete sanctioning. The same report about a third party produced no detected effect (Fig. 4). The response was therefore tied to the identity of the current partner rather than to a general negative tone. Its direction survived five independent rewordings and appeared on five of six tested engines, although its magnitude differed. These probes measure the behavioural gain of a pathway without requiring the full social process to ignite. The representation of the evidence determined which quantity entered the decision. We crossed the number of negative reports with a stated population of40,200or1000(Fig. 5). When the prompt gave a count, for example “five participants reported this partner”, responses changed with the number of reports, but we detected no additional effect of the stated population size. This result was reproduced on further engines under criteria fixed in advance, with one engine responding only to the presence rather than the number of reports. The lattice wordings are deliberately mild so that responses stay away from ceilings, which is why report cells can sit above the uninformed baseline. The registered quantities are the dose and size slopes within each lattice, where the wording is constant, rather than the report-versus-baseline contrast. By contrast, when the report made the percentage explicit, both model families tested used population size: one showed a sharp threshold and the other a graded response. The percentage sentence still contains the count, so the added dependence tracks the added percentage. Cells describing the same percentage then produced similar behaviour even when their counts and population sizes differed. The contrast is visible in baseline-adjusted models across all six lattices. The report-by-population interaction was indistinguishable from zero for every count-format lattice and negative for both percentage-format lattices (Fig. 5D, full coefficients in Supplementary Information). Because the uninformed baseline itself can drift with stated size, the count result is an absence of detected size-dependence rather than proof of equivalence. Within that scope, the finding 5 Scale limits of social mechanisms in agent societiesPreprint 0.000.250.500.751.00 give rate no information targeted note third-party placebo count, N= 1000 proportion, N= 8 proportion, N= 1000 claude-sonnet-4-6claude-opus-4-8gpt-5.5 Figure 4: Single-decision probe rates by cell and engine. Points are give rates withnbetween 24 and 48 decisions per cell, and horizontal bars are 95% Wilson intervals. A targeted report about the current partner produces sanctioning, an identically worded report about a third party does not, and count-format reports act without detected dependence on the stated population size. The panel shows the six primary probe cells on the three engines with per-cell reruns. The remaining engines and cells are tabulated in Supplementary Information. Engine labels and run dates are listed in the Supplementary Information model manifest. is consequential: two mathematically equivalent descriptions of social evidence can make the same mechanism appear scale-stable or scale-sensitive. We next asked whether gains measured in isolated decisions survive inside a society. A full mechanism combined language-model decisions with mechanical bookkeeping of message transmission and one scripted defector to initiate the process. Away from the known final round, the measured path contributions summed to the aggregate effect with zero residual at both tested population sizes. The social process nevertheless had very different reach. AtN = 8, it triggered ten sanctions and generated second-order cascades. AtN = 64, it did not reach a single decision in945 opportunities. Scale failure here was not a subtle change in attitude: the mechanism stopped touching behaviour at all. 2.5 Directions persist while absolute levels drift Commercial language models can change even when users call the same product name. We therefore separated directional findings, whether one condition raises or lowers a response, from absolute levels such as a baseline rate or threshold location. Across repeated tests, the targeted-report, third-party-placebo and count-format conclusions persisted across model snapshots. A report about the current partner continued to reduce giving, a third-party report remained a placebo, and the count-format response retained its lack of detected size-dependence. This stability was not universal. Absolute levels were less stable. The uninformed giving rate moved from roughly 0.6to0.93within days under an unchanged model string, and a strong response to percentage-framed evidence on one snapshot was absent on another. A confirmatory rerun preserved the targeted and placebo contrasts but failed its registered baseline threshold. We therefore attach model identity and run date to every reported level. For language- model societies, the more reproducible object can be the direction of a controlled contrast rather than a timeless behavioural constant. That stability must nevertheless be established for each interface. 2.6 Predictions generalize and failures narrow the claim We tested whether the audit could travel beyond the systems from which it was developed. Adversarial language-model agents (Methods) first proposed six counterexamples, including spreading cascades and selection effects that could compensate for declining local coverage. Each produced a measurable diagnostic within the audit, while the exercise added explicit checks for scale-comparable outcomes and insufficient time horizons. The audit was then applied to unmodified external systems. On a community cooperation library [37], it predicted a sign change betweenN = 20andN = 40. A new parameter grid was frozen before execution, and all twenty predicted values agreed with the deterministic outputs, a check that the audit’s derivation is correct rather than a statistical test (Fig. 6A). Result-masked readings of published models produced two valid cases and one voided case [38,39]. The 6 Scale limits of social mechanisms in agent societiesPreprint 402001000 stated population size N 0 (base) 1 5 25 reports k 0.710.500.58 0.960.921.00 0.880.620.79 0.750.710.79 A count, claude-sonnet 402001000 stated population size N 0 (base) 1 5 25 reports k 0.830.710.58 1.001.001.00 1.001.001.00 0.000.961.00 B percentage, claude-sonnet 402001000 stated population size N 0 (base) 1 5 25 reports k 0.830.460.58 0.670.960.62 0.300.750.79 0.040.330.54 C percentage, deepseek-reasoner −6−4−202 report ×logN interaction (95% CI) sonnet reasoner gemini (F) gpt-5.5 reasoner sonnet (F) D baseline-adjusted models count format percentage format 0.0 0.2 0.4 0.6 0.8 1.0 give rate Figure 5: Count and percentage formats lead to different stated-size dependence. (A) Dose lattice in the count format on the primary engine, give rates withn = 24decisions per cell. Response tracks the number of reports, while no stated-size dependence was detected in this count-format lattice. Anti-diagonal cells state identical proportions. (B, C) The same lattice with explicit percentages on two model families. The stated size now acts, through a sharp threshold on one engine and a graded curve on the other. (D) Baseline-adjusted report× logNinteractions with 95% confidence intervals across all six lattices, a post hoc analysis specified after the registered per-lattice fits (Methods). Count-format intervals contain zero on every engine, while both percentage-format interactions are negative and exclude zero. F marks Firth fits under separation. audit recovered the curve class and memory-to-population collapse of a bounded-memory model, although it placed the absolute crossover too early by a factor of two to two and a half. For a lattice of language-model agents, it correctly refused an aggregate forecast until coupling and bias were separately measured. These outcomes show both the reach of the structural classes and the lower precision of absolute crossover locations. A final prediction was filed before testing a second model family (Fig. 6B). The engine with a graded response followed the predicted monotonic curve and met both interior prediction bands. The companion prediction that a threshold-style engine would remain flat failed every registered criterion. Follow-up probes showed that this engine read the explicit percentage through a threshold, narrowing the earlier count-format finding rather than changing it after the fact. All failed and inconclusive outcomes are reported in Supplementary Information. In this framework, a failure is not discarded: it marks the boundary of the mechanism that can be predicted. 7 Scale limits of social mechanisms in agent societiesPreprint 10 1 10 2 N −1.5 −1.0 −0.5 0.0 0.5 payoff advantage per turn predicted flip window A pool I, per-capita budget pool I, fixed match length pool I, per-capita budget pool I, fixed match length 10 1 10 2 10 3 stated population size N 0.0 0.2 0.4 0.6 0.8 1.0 give rate sealed interior predictions ±0.20 B graded arm, observed threshold arm, observed Figure 6: Predictions filed in advance and their outcomes. (A) Third-party cooperation library on a frozen parameter grid. Lines are the filed predictions and open markers the observed values, which agree exactly on this deterministic system. The shaded band is the filed sign-flip window. (B) Cross-model-family test filed before execution, with the report count fixed at three. Curves are observed give rates for the graded and threshold arms, and the blue ticks with bands are the two interior predictions filed in advance with their±0.20bands, both met by the graded arm. The threshold arm failed its registered flatness criteria. 3 Discussion Artificial societies are attractive because they can be enlarged, copied and intervened on in ways that human societies cannot. The same flexibility creates a basic interpretive problem: adding agents changes who meets whom, what information remains visible and how an outcome is measured. Our results show that these changes can be separated before a large simulation is run. Re-encounter, information coverage and available time determine whether a pathway reaches decisions. Controlled probes determine what agents do with the information once it arrives. Their combination predicts the fate of the mechanism rather than merely describing the curve that appears afterwards. The most general discovery is that scale dependence need not originate in the decision-maker. The same agents and behavioural rule produced opposite scale outcomes when one structural term changed. A public record preserved gossip by maintaining visibility, whereas private transmission lost coverage. More broadly, institutions can be understood as devices that preserve information and coordination as populations grow [19,20]. This interpretation is consistent with human experiments in which reputation remains effective when scores are visible to all participants [22]. Scale invariance is therefore not necessarily an intrinsic property of a social mechanism. It can instead be an achievement of the surrounding information system. Language-model agents add another layer because natural language is part of the mechanism. Counts and percentages that encode the same underlying ratio led to different population-size responses. This result connects a design choice in the prompt to a population-level prediction. Human decision-makers also show ratio bias, denominator neglect and nonlinear responses to the number of sources [40–43], but such parallels do not make a language model a human participant. Instead, they show why the cognitive interface must be measured for each agent population. The observed drift across model snapshots strengthens this point: controlled directions were more stable than baseline levels, so behavioural rates from an agent society should be reported with model identity, date and information format. The same accounting can organize observations outside artificial societies without turning them into validation data. As an illustrative case, we applied the checklist to peer review at ICLR (the International Conference on Learning Representations), one of the largest machine-learning conferences, whose submissions grew roughly forty-fold in ten years as the field expanded. Its public records show deadline concentration, an expanding stock of coordination devices, and an apparent collapse of revision activity created by a change in the public record. The deadline concentration is largely absent at TMLR (Transactions on Machine Learning Research), a journal in the same field whose reviews run on per-paper rolling deadlines rather than one synchronized deadline, which makes it a natural contrast for deadline- driven behaviour. These descriptive results, reported in Supplementary Information, show that structural coverage, 8 Scale limits of social mechanisms in agent societiesPreprint compensation and measurement artifacts are recognizable in a real institution. They do not establish causal effects or test whether language-model agents reproduce human groups. The framework has clear limits. Absolute crossover locations should be treated as coarse forecasts: in the external bounded-memory case, the predicted crossover was a factor of two to two and a half lower than the observed one. The framework can therefore guide the choice of population sizes to test, but precise threshold estimates will require a more detailed model of memory turnover. The language-model findings concern one donation-game family, the tested wordings and the named model snapshots. Strong interactions between pathways prevent an aggregate prediction, and an unknown gain produces an unidentified verdict rather than a forecast. A nonzero composition residual is not merely a failure signal: it localises the interactions that a linear path sum cannot carry, and marks where studies of emergent coordination should look next. Predictions were logged internally before execution but were not independently preregistered. The framework is falsifiable on its own terms: a filed prediction that fails without an explanation inside the frozen checklist is an audit failure, a case requiring a new category rejects the checklist’s completeness, and unidentified verdicts are recorded rather than counted as successes. The field application concerns one institution observed without intervention. These restrictions define where the present evidence ends and where direct tests in human populations remain necessary. The audit changes the question asked of a large agent society. Instead of waiting to see whether an aggregate pattern changes with population, it asks which route carries the mechanism, how that route scales, and whether the agents use the information it delivers. Those quantities are measurable before the full run. This makes population scale a property to be predicted and designed for, rather than a robustness check performed after an expensive simulation has already produced a result. 4 Methods 4.1 Audit procedure and checklist The audit uses a frozen, closed checklist fixed before the external test batch and proceeds in four steps. First, it enumerates the routes through which a mechanism can affect a decision. Second, it derives how often each route reaches a decision as population size changes. Third, it measures the behavioural response delivered by that route. Fourth, it combines routes only when their gains and interactions are identified; otherwise it retains the path-level predictions and marks the aggregate as unidentified. Before a prediction is filed, the checklist also screens for changes in outcome meaning, mechanism activation, observation time, capacity, saturation and compensating mechanisms. Predictions cite checklist codes and state a curve class and an applicable range. Explanations outside the frozen checklist count as audit failures. Compensation claims require quantitative diagnostics, with parallel paths identified and their ablations registered in advance. The adversarial attacks and the result-masked audits were performed by clean-session language-model agent auditors isolated from the result data, with access limited to methods-only briefs and, for the red team, to the frozen checklist. They are not an independent research team, the authors adjudicated every outcome against the checklist, and auditor recognition of the research lineage voided one case. Retrodictions of published results are performed under result masking, with contamination self-reports treated as a weak filter, since training-data exposure of published results cannot be excluded. The full checklist, the probe materials with their registered criteria and adjudications, and the twelve-entry failure ledger are in Supplementary Information. 4.2 Rule-based experiments All rule-based simulations use seeded NumPy code with explicit seeds and at least three seeds per configuration, with populations fromN = 10to10 5 as stated per experiment. The paired designs hold the update rule fixed and vary one term, the sampling source for consensus, the presence of1/Nin an update step for reputation dilution, the information routing for the gossip pair, and the coverage generator for the institutional comparison (five seeds, fifty rounds,N ∈10, 100, 1000). In the causal interventions the restored arms sit at the ceiling, which is harmless since the verdicts require only recovery while the degraded baselines remain interior. The failure scale law uses ten seeds per configuration, three registered coverage thresholds (0.2,0.3,0.4), and least squares fits oflogN c onlog(f τ ) with bootstrap intervals over seeds, run on the amended memory-unbounded design in whichτ eff = τ, using a fixed absolute informed-fraction threshold. The bounded first design used a relative half-plateau threshold. Its registered failure at exponent 0.245 and the truncation formula τ eff = (1/τ + λ/M ) −1 are reported in Results. The transfer and prospective tests run on the unmodified Axelrod-Python library [37] with frozen parameter grids and predictions filed before execution. 9 Scale limits of social mechanisms in agent societiesPreprint 4.3 Language-model probes Probes present a donation game vignette and elicit a single give or keep decision, withn = 24decisions per cell unless stated, frozen wordings, and resume-safe logging. Later runs record the full provider envelope including the resolved model identity per call. The earliest probes predate envelope logging and carry the most specific engine label available, as detailed in the Supplementary model manifest. Engines span five model families (Claude Sonnet 4.6 and Sonnet 5, Claude Opus 4.8, GPT-5.5, Gemini 3.5 Flash, and DeepSeek v4-flash accessed with its reasoning mode on and off over fixed weights). Three of the six engines were accessed through vendor agent CLIs and three through direct APIs, and the directional and format results replicate across both access routes. Every reported comparison is made within a single model snapshot, and replication across snapshots is itself reported in Results. The count lattice crossesk ∈1, 5, 25with statedN ∈40, 200, 1000plusk = 0baselines, with per-engine wording calibration to place the anchor cell in the interior. The percentage lattice states the arithmetic-consistent percentage inside a directly worded report. The calibrated count wordings are milder than the percentage wording, so the lattices are not matched sentence pairs, and the format contrast rests on within-lattice slopes, in which the wording is constant, together with the matched core probe cells whose count and percentage sentences differ only by the inserted percentage. Calibration used a separate pilot batch of twelve decisions per candidate wording, pilot calls enter no reported analysis, and every lattice cell, including the calibration point, was collected fresh in the confirmatory runs. Per-engine calibration supports interface measurement within each engine rather than same-stimulus comparison across engines. The cross-model-family test fixed the report count at three, varied stated population over8, 40, 200, 1000, and filed monotonicity, two anchor values and two interior values from a logistic interpolation with bands of±0.20before any data existed. The composed system uses one scripted defector and mechanical note transmission, with registered verdicts adjudicated inside the operating window and final-round cells reported as failures. Temperature is provider-fixed on the primary engine, and a sweep at t∈0, 0.7, 1.0 on another engine reproduced all cells identically. 4.4 Field study Data are public OpenReview records retrieved with the official client under authenticated access, at most two requests per second, with resume-safe provenance logs. Submissions are enumerated by invitation as unique root forums with a seven-category status breakdown, reviews are official reviews on those forums, and revisions are public edit records classified by signature, with reviewer revisions defined as edits signed by the review’s own reviewer pseudonym more than120seconds after review creation. Trusted timestamps (tcdate, tmdate) are used throughout, since user-editable dates are not relied on. Enumerated counts reconcile against official totals, including an exact match of11,672root forums for 2025, with remaining gaps documented in Supplementary Information. The 2025 area split used a fixed seed (20260719), development-half analyses preceded a single validation run with three criteria set in advance, and the 2026 incident windows were fixed from the official timeline. Device coding used only public policy documents across 2017 through 2026, with fifty devices in five categories, per-device sources, and a persistence rule stated in Supplementary Information. Coding was performed by one coder, and the released per-device source list supports independent recoding. 4.5 Ethics and data use This study recruited no participants and performed no interventions. The field component is a secondary analysis of publicly accessible, human-generated peer review records retrieved from the platform’s public API in accordance with its terms of use (https://openreview.net/legal/terms). The authors had no interaction with reviewers and accessed no private information. Review comments on the platform are published under a C BY 4.0 licence and metadata under C0. No review text is quoted or redistributed, anonymous identities are not linked across papers or to natural persons, and only aggregate counts and timing distributions are reported. The randomized feedback intervention described in Supplementary Information was run by the conference itself under its own institutional approval, and this study neither participated in it nor analyses it causally. 4.6 Statistics and reproducibility The statistical unit of every probe analysis is a single API call in an isolated session with no shared context. Calls were treated as conditionally independent given the engine, wording and sampling configuration, and the randomness is the provider’s sampling randomness rather than sampling from a population of models. All inference is therefore scoped to the tested snapshot and prompt distribution, which is the same scope as the directional-versus-level distinction reported in Results, and the five-rewording robustness runs bound sensitivity to any single wording. Per-cell samples ofn = 24 10 Scale limits of social mechanisms in agent societiesPreprint are powered for the large contrasts the probes target, rates near zero against rates near one, and lattice-level inference pools 216 decisions per engine in the registered models rather than testing single cells. All probe rates are proportions of valid parsed decisions with exactnreported per cell. Infrastructure failures were retried to completion and the superseded records retained in the raw logs. Dose lattice analyses are binomial logistic regressions (generalized linear models, GLMs) of the keep decision onlogkandlogN. These regressions are the test of which quantity enters the decision: a nonzero coefficient onlogkmeans the reported count is used, and a nonzero coefficient onlogNmeans the stated population size is used (n = 216decisions per engine per lattice, 215on one engine), two-sided Wald tests, with Firth’s penalised likelihood under separation, and the ratio-reading contrast testsH 0 : β logk = −β logN (two-sided). For the primary count lattice the exact values areβ logk = +0.48 (z = 3.17,p = 0.0015) andβ logN = 0.00(z = 0.00,p = 1.00). For the percentage lattice they are+0.66(z = 5.29, p = 1.2× 10 −7 ) and−0.48(z =−3.95,p = 7.8× 10 −5 ) on one engine with contrastz = 1.15(p = 0.25), and Firth +4.8and−4.1on the other with contrastz = 0.68(p = 0.50). For the count lattice on further engines theβ logk dose tests givez = 7.49(p < 10 −10 ),z = 3.05(p = 0.0023), andz = 1.10(p = 0.27, the gated engine), withβ logN intervals [−0.17, +0.33], [−0.36, +0.10] and [−0.19, +0.26]. The baseline-adjusted models reported in Results add thek = 0no-information cells and fitkeep∼ 1 + R + R logk + logN + R logNper lattice and engine, whereRindicates report presence andR logkcarries the dose among report cells. The reported quantity is theR× logNinteraction with two-sided Wald 95% intervals, Firth’s penalised likelihood where the plain fit separates, and no multiplicity correction across the six lattices since the six intervals are reported in full. This analysis was specified after the registered per-lattice fits and is reported as post hoc. Full coefficient tables are in Supplementary Information. The failure scale law reports least squares fits with bootstrap intervals from1,000resamples over ten seeds at three thresholds. Field clustering shares are proportions with exact denominators (29,045development and17,703validation reviews), and area correlations are two-sided Spearman rank correlations with tie handling,ρ = −0.49(p = 0.15, n = 10areas) andρ = +0.18(p = 0.60,n = 11areas). Language-model calls total roughly6,600. Every level-type quantity carries the most specific available engine label and a run date in the project ledger, and fully resolved provider identities are available for the later envelope-logged runs. Data availability All raw experimental outputs, including JSON and JSONL files containing one record per language-model decision with its engine label and, where logged, the resolved provider envelope, together with the field-study reconciliation counts and provenance logs, device coding matrix, and prediction ledger, are available athttps://github.com/ wuzengqing001225/scale_limits_agent_societiesand archived at Zenodo (https://doi.org/10.5281/ zenodo.21532739). The field study also analysed publicly accessible OpenReview records. The raw OpenReview records are not redistributed in the repository; they can be retrieved fromhttps://openreview.netthrough the OpenReview API, subject to the platform’s Terms of Use, using the extraction code provided in the repository. Code availability All simulation, probe, extraction and analysis code, including the scripts that regenerate every figure and ev- ery Supplementary table from the released data, are available in the same repository (https://github.com/ wuzengqing001225/scale_limits_agent_societies) and archived at Zenodo (https://doi.org/10.5281/ zenodo.21532739). Acknowledgements C.X. discloses support for the research of this work from JSPS KAKENHI (JP23K17456, JP23K28096, JP25H01117 and JP26K03246) and JST CREST (JPMJCR22M2). Z.W. discloses support for the research of this work from JST BOOST (JPMJBS2402). 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The paragraphs that follow state the standards governing their application. Table S1: The closed checklist, part one: protocol checks (C1–C5) and input identity rules (E1–E4). Every prediction cites codes from this list, and an explanation outside it is an audit failure. CodeNameRule and diagnosticUse in predictions C1Scale-comparable observable The observable must have anN-comparable interpretation.Extensive counts and whole-population conjunctions,such as P (all agree) = p N , carry mechanical size dependence and require normalisation or a separate null model prediction undefined if vio- lated C2ActivatabilityMechanism must activate at the reference scale or under exogenous seeding. Zero ignition only at largeNis a positive A2-class result, zero ignition everywhere is an invalid test validity gate C3Causalneighbour- hood Neighbourhood within the measurement hori- zon must clear the boundary and stay locally isomorphic acrossN. On rings this reduces toN≳ 2rT. Globally coupled systems need separate argument validity gate for small N C4Endgame isolationFinal rounds of known horizons excluded or reported separately measurement window C5Dynamic rangeNo ceilings or floors. Saturated cells are re- tuned and excluded from adjudication measurement window E1Identity-indexed in- put Decision function indexed by the identity of the object routes to A2 coverage audit E2Type or density inputType frequency does not dilute mechanically with the denominator, but its generating dy- namics remain subject to A4/A5 waives A2 only E3Count versus propor- tion The agent’s actual weighting of numerator, de- nominator and format is measured by probes fixes the effective input and supplies ∆ j E4 Public record versus private transmission A public record is only a candidate for B3. Measured visibility must decouple from N precondition for B3 Two-level output. Level 1 predicts structural quantitiesq j per causal path without using the target system’s scale results, from code, mechanism descriptions, and independent interface probes. Level 2 predicts the aggregateE(N ) only when every active path’s gain is derived or independently measured and shown stable over the target range, or measured at several sizes. Otherwise the verdict is aggregate unidentified. Each prediction carries a curve class among flat, monotone decay, monotone growth, non-monotone or transition, and conditional, together with an applicable range, a turning region when computable, cited checklist codes, and the assumption most likely to fail. Protocol checks (C1–C5).Observables must be scale-comparable. Extensive counts and whole-population conjunc- tions carry mechanical size dependence and require normalisation or a separate null model. Mechanism activatable at the reference scale or under exogenous seeding, where non-ignition at largeNalone is a result and non-ignition everywhere is an invalid test. Causal neighbourhoods must clear the boundary and remain locally isomorphic across sizes, which on rings reduces to a lower bound onNof order twice the interaction radius times the horizon, while globally coupled systems need separate argument. Final rounds of known horizons are excluded or reported separately. Measurements must sit away from ceilings and floors, and saturated cells are retuned and excluded. Input identity rules (E1–E4).Identity-indexed information enters the coverage audit. Type and density information does not dilute mechanically, though its generating dynamics still require audit. Count versus proportion consumption 15 Scale limits of social mechanisms in agent societiesPreprint Table S2: The closed checklist, part two: degradation classes (A1–A5), correction terms (I1–I3), and compensation channels (B1–B4). Correction terms shift the absolute position of a turning scale and never justify a decay prediction on their own. Compensation channels require the stated quantitative diagnostic at the time of invocation. CodeNameRule and diagnosticUse in predictions A1Re-encounter proba- bility Pair re-interaction probability, under random matching∼ 1/(N−1). Diagnostic: count the pair generator decay or transition of q j A2Targeted coverageProbability that a decision point holds infor- mation about the current object,∼ λτ eff /N. Diagnostic: measure the informed fraction decay or transition of q j A3Monitoring reach Probability that behaviour is observed by a potential sanctioner. Diagnostic: count obser- vation edges decay or transition of q j A4Time budget Convergence timeτ (N )against a fixed hori- zonT. Diagnostic: measureτ (N ), discrimi- nate by extending T decay or transition of q j A5Update-step dilution A1/Nfactor inside the update step of a global aggregate. Diagnostic: read the update rule decay or transition of q j I1Memory-fluxlife- time τ eff = (1/τ + λ/M ) −1 , harmonic mean-field approximation under independent hazards shifts turning scale only I2 Capacityand throughput Finite processing or storage limitsshifts turning scale only I3Behavioural satura- tion Saturation of the behavioural mapping, distinct from measurement saturation (C5) shifts turning scale only B1Supercritical spread- ing Compute or measureR 0 . Coverage capped rather than∼ 1/N flat, with diagnostic B2Selection amplifica- tion Stationary share converges to anN- independent value flat, with diagnostic B3 Institutional cover- age restoration Measured visibility or query rate decoupled from N (E4 precondition) flat, with diagnostic B4 ParallelN- independent path Path identified in the path table in advance. Ablation registered in advance, execution may follow conditional, with registered ablation is settled by probes. Public records are candidates for coverage restoration only after measured visibility decouples from N . Path decomposition.A mandatory table lists each path with activation status, effective input, degradation class, gain source among code, probe, and unknown, and the gain’s sign and magnitude or the response function from which the gain is derived. Any unknown gain halts the audit at Level 1. Degradation classes (A1–A5) and corrections (I1–I3). Re-encounter probability, targeted coverage, monitoring reach, time budget against convergence time, and update-step dilution. Corrections cover the memory-flux effective lifetime, capacity and throughput limits, and behavioural saturation, and none of them justifies a decay prediction on its own. Compensation channels (B1–B4).Supercritical spreading, selection amplification, institutionalized coverage restora- tion, and parallelN-independent paths. Every invocation requires a quantitative diagnostic, and parallel paths must be identified in advance with ablations registered in advance. Opposite-signed paths yield conditional verdicts with explicit switching conditions. Non-monotone standards. A non-monotone or transition verdict requires structure specified in advance, either opposite-signed paths with a predicted crossing region or an explicit single-path non-monotone equation with a turning region. Observed non-monotonicity without an advance basis counts as an audit failure. Retrodictions. Audits of published results are performed under result masking, with contamination self-reports treated as a weak filter. 16 Scale limits of social mechanisms in agent societiesPreprint S1.1 Audit procedure as pseudocode Input: mechanism implementation M , agent population A, observation plan O, frozen checklist C. 1. ScreenOagainst the protocol checks ofC; retune or exclude any measurement at a ceiling, on a non-scale- comparable observable, under a known final round, inside a too-small causal neighbourhood, or under a time budget shorter than convergence. 2. Enumerate the causal pathsjofMfrom its implementation; for each, identify the carrying quantity and classify its input under the input identity rules of C. 3. For each path, deriveq j (N )from the implementation, and measure∆ j with single-decision probes ofAin the formats that M actually uses, with frozen wordings and interior calibration. 4. Complete the path table; if any gain is unknown, stop at per-path predictions and return the verdict aggregate unidentified. 5. Otherwise predictE(N )from Eq. (1) of the main text; check compensation channels ofCwith quantitative diagnostics before predicting decay. 6. Log every prediction with curve class, applicable range, cited checklist codes, and the assumption most likely to fail, before results exist; adjudicate outcomes against the logged text; report failures. Diagnostics: a nonzero composition residual signals an interaction or a missing path; out-of-checklist explanation of any outcome is an audit failure. S2 Completed path tables Tables S3 and S4 consolidate the path tables of the main systems as completed during the audits. Structural quantities, gains and curve classes were recorded before the corresponding scale outcomes. The outcome column reports the adjudicated result and was added after. The anchor pairs A to C formed the inducing set from which the checklist was frozen and are marked accordingly. Their codes are the retrospective classification under the frozen list. Table S3: Path tables, rule-based systems. Gains derive from code. Sources in parentheses. Classes are flat, decay, or transition (decay crossing a registered threshold). SystemPathInput (E)q j (N ) (source)∆ j (source)CodesFiled classOutcome Pair A, direct (an- chor) rememberedre- encounter partner identity (E1) ∼ 1/Nunder random matching (code) full conditional defec- tion (code) A1transitioncollapseat N =20–40at threshold Pair A, networkneighbour clusteringneighbour pay- offs N -independent local struc- ture (code) imitation of best neigh- bour (code) –flat share stable near 0.1 to 10 5 Pair B, local (an- chor) convergence within budget neighbour opin- ions τ (N ) outruns fixedT (measured) majorityadoption (code) A4transition frozenfrom N ≈ 30 Pair B, globalmean-field samplingsampledopin- ions N -independent (code)same (code)–flatflat to 10 5 Pair C, sequential (anchor) informativedeci- sions public recordsaturates near four (mea- sured) cascadeadoption (code) –flatbelow analytic anchor pinnednear 0.689vs 0.692 Pair C, independent private signalsown signalgrows with N (code)majority count (code) –growth0.75→ 1.00 R4, dilutedreputationupdate step aggregated reports step∝ 1/N (code) defectionbelow threshold (code) A5decay −0.43 → 0.00 R4, per-eventreputationupdate step reportsN -independent (code)same (code)–flat holdsnear −0.45 R6, organictargeted coverageidentity marks (E1) ∼1/N(measured 0.75→ 0.013) multiplicative sanction (code) A2decayeffect −0.387 → −0.007 R6, boardinstitutional cover- age publicrecord (E4) visibility≡ 1by construc- tion same (code)B3flatholds at−0.48 R3 ′ gossiptargeted coverageidentity marks (E1) ≈min1,λτ eff /N (code) p-response to mark (code) A2, I1transition, N c ∝ fτ eff exponent1.00 [0.97, 1.04] 17 Scale limits of social mechanisms in agent societiesPreprint Table S4: Path tables, language model and external systems. Gains are probe-measured on the stated engines. Give rates with n per cell in the result tables. SystemPathInput (E)q j (N ) (source)∆ j (source)CodesFiled classOutcome Count channeltargeted reportreport count (E3, probed) statedNnot consumed, no dilution path (probes) give1.00 → 0.00on targeted report.Dose +0.48perlogk(probes) E3flat in stated N nodetected stated-size dependence, four engines Percentage channelstated ratio explicitpercent- age (E3, probed) report always delivered in probes; its stated ratio k/N falls with N threshold(Firth +4.8/− 4.1) or graded (+0.66/ −0.48) response to the ratio (probes) E3 decayin stated N negative inter- actions,both model families Composedmecha- nism ignition,transmis- sion, sanction identitymarks (E1) fired atN =8. Zero igni- tion atN =64over945 decisions probe values aboveA2, C2, C4 window- qualified composi- tion residualzero inwindow. Whole-run verdicts failed on final round Third-party library within-matchre- encounter partneridentity (E1) repeat turns per opponent fall under per-capita bud- get (code) strategy payoff structure (code) A1, A4sign flip in N =24–48 20/20filed quantities exact S3 Rule-based model specifications Complete implementations with all constants are in the released code (Code availability). This section specifies each model’s state, update order, and the single manipulated term of its pair. Pair A, direct versus network reciprocity (coop_abm.py).Nagents hold strategies TFT or ALLD (initialised uniformly at random), benefit3and cost1, and a bounded per-agent memory ofM = 20(partner identity, last move toward the agent) written circularly. Each generation plays50rounds of random perfect matching followed by payoff-based strategy updating, for30generations. TFT cooperates unless the remembered last move of the current partner was a defection. The measured structural quantity is the remembered re-encounter rate. The network arm is the classical spatial game: cooperators and defectors on a ring with two neighbours, payoffs summed over neighbours, and each agent imitating its best-performing neighbour, run for fifty rounds. The registered intervention in the direct arm fixes partners (A1 restored). The observable in both arms is the cooperator share. Pair B, consensus versus segregation (consensus_abm.py).Nagents hold opinions±1and adopt the majority of themselves plus four sampled agents, for a fixed budget of200sweeps. The manipulated term is the sample source only, fixed ring neighbours at offsets±1,±2against fresh uniform global samples. Consensus is unanimity. The discordant-edge fraction is recorded. The registered intervention samples globally with probabilityεper update. The surviving twin runs Schelling segregation on the same ring. Pair C, information aggregation (agg_abm.py). Binary ground truth, private signal accuracyq = 0.6,2,000 replicate runs evaluated at checkpointsNfrom11to10 5 (odd, no ties). The independent arm counts a majority of private signals. The sequential arm makes each agent adopt the sign of the running public action difference whenever its magnitude is at least two, ignoring its own signal (a cascade), and follow its signal otherwise. The analytic plateau is q 2 /(q 2 + (1− q) 2 ) = 0.692. The measured structural quantity is the number of informative, non-cascaded decisions. The intervention gives each agent probability λ of using its own signal regardless of the record. R4, update-step dilution (r4_dilution.py).Nagents with fixed cooperation propensities drawn fromU (0.4, 0.9) and a reputation score in[0, 1]initialised at1. Partners are drawn from ring offsets within distance two. Agents defect toward partners with reputation below0.5with probability0.8. After each round, keepers’ reputations fall by the update step and all reputations recover toward1at rate0.02. The manipulated term is the step alone,δk/Nwithδ = 0.15, k = 4(ak-report average diluted into anN-agent consensus) against the per-incident stepδ. Fifty rounds, thirty generations, five seeds, N ∈10, 100, 1000. The observable is the give-rate lift over a reputation-free baseline. R6, organic transmission versus public record (r6_board.py). Fixed cooperation propensities from U (0.4, 0.9), no evolution. A boolean tag matrix records who holds a mark about whom. Informed agents give with probability coop× 0.2(multiplicative sanction). The manipulated term is the coverage generator alone, a victim sending marks to four random others against every defection becoming visible to all. Marks decay at rate0.1per round.1,500rounds, five seeds,N ∈10, 100, 1000. Observables are the give-rate lift over a no-gossip arm and the measured coverage at decision points. 18 Scale limits of social mechanisms in agent societiesPreprint R3 and its amendment, failure scale law (r3_nc_scaling.py,r3_prime.py). Random perfect matching. A victim of a defection sends a mark about the keeper tofothers excluding the pair. Marked partners are defected against with probabilityp. Marks decay at rate1/τper round. The observable is the informed fraction at decision points over 100post-burn-in rounds (150total). The original design bounds each agent’s memory atM = 20marks and uses a relative threshold, and its registered exponent test failed at0.245(Results). The amended design registered before its data removes the bound (dense mark matrix, theM → ∞limit, mechanics otherwise identical) and definesN c as the log-interpolated crossing of an absolute informed-fraction threshold. Configurations crossf ∈1, 2, 3, 6, 10at τ = 10with(f =3,τ =20),(f =3,τ =5)and a secondaryp = 0.4arm. The reinforced run uses ten seeds, thresholds 0.2, 0.3, 0.4, and1,000bootstrap resamples of the least-squares slope oflogN c onlog(fτ ). In the truncation formula τ eff = (1/τ + λ/M ) −1 , λ is the per-agent mark arrival rate, proportional to the keep rate times f . R1 and R5 (r1_d2_midrange.py,r5_minimal_pair.py). R1 retunes the punishment mechanism of the ceiling demonstration into mid-range (cooperation near0.80) and measures the effect spread across populations. R5 holds a sanction mechanism fixed and varies information routing (global versus local), recording coverage, the aggregate lift, and its components, with the refusal variant reversing the sign of the victim response. Full parameterisation in the scripts. S4 Supporting arguments This section provides analytical derivations and scoped argument sketches rather than formal theorems. The composition identity is exact under its stated assumptions. The remaining arguments motivate the checklist diagnostics and explicitly state their scope conditions. S4.1 Composition Identity Suppose the mechanism acts only through a live indicator, so that the probability of the focal behaviour equalsaon encounters where the path is live andbotherwise, withaandbindependent ofN, and letqdenote the probability that the path is live at a decision point. The behaviour rate under the mechanism is thenq a + (1− q)bby the law of total expectation, the matched baseline rate is b, and subtracting gives E(N ) = q (a− b). The identity follows directly from the law of total expectation, and its value here is operational. Oncea,b, andqare measured separately, the aggregate follows with no free parameters. It held to numerical precision in the rule layer, where the residual across six population sizes stays below 0.003, and in the composed language model system, where the residual is 0.000 at both population sizes inside the operating window. S4.2 Survival Under Finite-Range Local Dynamics Claim (informal). Consider agents on a fixed graph of bounded degree in which each round every agent reads states, is paired, and interacts only within graph distancer, following rules that do not referenceN. Fix a measurement horizon ofTrounds. If for a given agent the neighbourhoods of radiusrTin two systems of different sizes are isomorphic and carry the same initial distribution, then the agent’s state distribution afterTrounds is the same in both systems. On a ring this holds onceNexceeds a threshold of order2rT, so intensive rates above that threshold should not depend on N beyond sampling noise. Argument sketch. Aftertrounds the state of agentidepends only on initial conditions and random draws inside a ball whose radius grows by at mostrper round. Coupling the random draws of the two systems on isomorphic balls propagates identical states round by round. The subtlety we have not formalized is the pairing process, whose randomness must be coupled consistently across overlapping neighbourhoods, and the constant in the threshold absorbs this. Three consequences matter for the paper. Flatness of local punishment is expected rather than discovered. Small populations below the threshold form a degenerate zone where paired comparisons lose meaning, which is where our one flatness failure occurred, and a registered retest above the threshold passed. The premise excludes global coupling, so roulette selection, public boards, mean field sampling, and dynamic rewiring void the guarantee. S4.3 Coverage and the Effective Lifetime Assume random matching that is well mixed, notes about specific individuals produced at ratek r fper agent per round for keep ratek r and fan outf, independent note lifetimes, and subcritical response cascades. Balancing production against loss at stationarity gives coverageq ≈ k r fτ eff /(N − 1). With unbounded memoryτ eff equals the nominal lifetimeτ. With bounded memory ofMslots an arriving note can evict an existing one, so a note faces decay at rate 19 Scale limits of social mechanisms in agent societiesPreprint 1/τand overwriting at rate nearλ/Mfor arrival rateλ = k r f, and treating the two as independent hazards gives τ eff ≈ (1/τ + λ/M ) −1 , whose limits recoverτandM/λ. These are mean field approximations with no exactness claim. A coverage thresholdθthen puts the crossover atN c ∝ f τ eff . The amended design, which removed the memory bound and replaced the relative crossing with a fixed absolute threshold, produced a fitted exponent of1.00. Because both features changed, the contrast with the bounded design is not a quantitative test of the truncation correction; the bounded exponent of0.245is only consistent with that account. A branching estimate gives a reproduction number R 0 ≈ f pτ eff /(N − 1)for response probabilityp, locating the small population regime where cascades amplify coverage. S4.4 Conditional Decay Bound Claim (informal). Suppose the mechanism differs from baseline only at live encounters, responses are bounded, no compensation channel of the checklist is present, and response cascades are subcritical. Then the effect is bounded by a constant multiple of coverage, so decay of coverage forces decay of the effect. Argument sketch. Couple a mechanism run and a baseline run on the same random primitives. Behavioural divergences originate only at live encounters, and each divergence spawns further divergences with subcritical expectation, so the expected number of divergent decisions per round is controlled by the number of live encounters times a finite amplification factor. The amplification factor is heuristic in our treatment, and the scope conditions cannot be certified in advance for an arbitrary system. The red team results reported in the main text exhibit both failure modes, a supercritical channel that rebuilds coverage and a selection channel that amplifies an order1/Nseed to a constant. The bound therefore functions as the core of a procedure whose applicability is checked case by case. S5 Model manifest Engine (as reported)Resolved identityAccessNotes and run dates Primaryengine, early probes claude-sonnet-4-6vendor CLItargeted, placebo, count and propor- tion probes and wording robustness (2026-07-12). Envelopes not logged in this early batch Primary engine, later runs claude-sonnet-5vendor CLIconfirmatory rerun, count lattice, fac- tor separations, cross-model-family test, percentage lattice (2026-07-13 to 07-20). Full envelopes. Provider fixes sampling temperature Opus familyclaude-opus-4-8vendor CLI cross-model probes (2026-07-12) and factor-separation reruns (2026- 07-19) GPT-5.5gpt-5.5 via the codex agent CLI vendor CLIcross-model probes (2026-07-12) and count lattice (2026-07-22). Reasoning-capable configuration Geminigemini-3.5-flashpublic APIcross-family probes,count lat- tice,temperaturesweepat t ∈ 0, 0.7, 1.0(2026-07-19 to 07-22) DeepSeekdeepseek-v4-flashpublic APIaccessed with reasoning mode on (reasoner) and off (chat) over fixed weights. Percentage and count lat- tices, cross-model-family test (2026- 07-19 to 07-22) Auxiliaryclaude-haiku-4-5vendor CLIappears in CLI usage envelopes as an auxiliary process, never the sole responder. Behavioural attribution to the primary engine verified by token signatures Table S5: Engines, engine labels with resolved identities where logged, access routes, and run dates. Every level-type quantity in the paper carries the most specific of these labels and a run date in the released ledger. Fully resolved provider identities are available for the envelope-logged runs. 20 Scale limits of social mechanisms in agent societiesPreprint S6 Probe materials All probes present the following vignette, with braces filled per cell, and elicit a single decision. The full skeleton is identical across engines and experiments. You are participant Pagent in a multi-round exchange study with N participants. Each round you are paired with another participant. You may transfer 1 credit from your account; if you do, your partner receives 3 credits. Your partner independently faces the same choice toward you. Your goal: finish with as many credits as possible. After each round, a participant may write a short note about their partner’s choice, and the note may be shown to a few other participants. Round 3 of 5. Your credits: credits. Your partner: Ppartner. context "action":"give" or "action":"keep". Respond with only JSON. Core cell contexts (wording V1). Targeted report: “Note from PA: PS chose keep against me in an earlier round.” No information: “You have no information about your current partner.” Third-party placebo: the targeted note about an agent who is not the current partner, followed by the no-information line about the partner. Count report: “3 participants have reported that PP chose keep against them in earlier rounds.” Percentage reports: “3 of the 8 participants (37.5%) have reported that PP chose keep against them in earlier rounds.” and “3 of the 1000 participants (0.3%) have reported that PP chose keep against them in earlier rounds.” Robustness rewordings.Each core cell has four frozen rewordings in addition to V1 (five in total), varying reporting verb, voice, and sentence order while preserving content. All are archived in the released probe scripts verbatim. Directional verdicts held in at least four of five rewordings per cell in the primary run. Dose lattice wordings. Weak wordings used for interior calibration, with K a count phrase and X the partner label. W1: “K mentioned being unsure whether PX transferred in an earlier round.” W2: “K said PX seemed hesitant to transfer in an earlier round.” W3: “K reported that PX sometimes chose keep in earlier rounds.” The percentage lattice states the arithmetic-consistent percentage inside the report, “k of the N participants (pct%) has/have reported that PX chose keep against them in earlier rounds.” Per-engine calibration selects the wording whose give rate at (k = 5,N = 200) falls in [0.3, 0.7] nearest 0.5. Cross-model-family test, filed before execution.Report count fixed at three, stated population in8, 40, 200, 1000, so the stated percentage runs37.5,7.5,1.5,0.3. Filed before data: strict monotonicity, both anchors (≤ 0.2atN = 8, ≥ 0.8atN = 1000), interior values0.396and0.682from a two-anchor logistic in log proportion with bands of±0.20, and a rise of at least0.4. For the companion arm, flatness with spread< 0.15in the report cells and in their differences from same-run baselines. S7 Supplementary result tables: probes and lattices Tables S6 through S14 report every lattice cell with exact n, the cross-model-family arms, and the temperature sweep. Engine labels, and resolved identities where envelopes were logged, are listed in the model manifest. Table S6: Count-format dose lattice, primary engine (claude-sonnet-5, wording W3). Give rates withnin parentheses. N = 40 N = 200 N = 1000 k = 00.708 (24)0.500 (24)0.583 (24) k = 10.958 (24)0.917 (24)1.000 (24) k = 50.875 (24)0.625 (24)0.792 (24) k = 250.750 (24)0.708 (24)0.792 (24) 21 Scale limits of social mechanisms in agent societiesPreprint Table S7: Count-format lattice, deepseek-reasoner (calibrated wording W3). Give rates with n in parentheses. N = 40 N = 200 N = 1000 k = 00.667 (24)0.750 (24)0.875 (24) k = 10.333 (24)0.500 (24)0.292 (24) k = 50.167 (24)0.250 (24)0.174 (23) k = 250.208 (24)0.500 (24)0.167 (24) Table S8: Count-format lattice, gemini-3.5-flash (calibrated wording W3). Give rates with n in parentheses. N = 40 N = 200 N = 1000 k = 01.000 (24)1.000 (24)1.000 (24) k = 10.917 (24)1.000 (24)0.708 (24) k = 50.292 (24)0.458 (24)0.250 (24) k = 250.083 (24)0.208 (24)0.208 (24) Table S9: Count-format lattice, GPT-5.5 (calibrated wording W1). Give rates with n in parentheses. N = 40 N = 200 N = 1000 k = 00.958 (24)1.000 (24)0.917 (24) k = 10.833 (24)0.792 (24)0.833 (24) k = 50.667 (24)0.625 (24)0.792 (24) k = 250.583 (24)0.458 (24)0.708 (24) Table S10: Percentage-format lattice, claude-sonnet-5. Give rates with n in parentheses. N = 40 N = 200 N = 1000 k = 00.833 (24)0.708 (24)0.583 (24) k = 11.000 (24)1.000 (24)1.000 (24) k = 51.000 (24)1.000 (24)1.000 (24) k = 250.000 (24)0.958 (24)1.000 (24) Table S11: Percentage-format lattice, deepseek-reasoner. Give rates with n in parentheses. N = 40 N = 200 N = 1000 k = 00.833 (24)0.458 (24)0.583 (24) k = 10.667 (24)0.958 (24)0.625 (24) k = 50.304 (23)0.750 (24)0.792 (24) k = 250.042 (24)0.333 (24)0.542 (24) Table S12: Cross-model-family test filed before execution, deepseek-reasoner (graded arm). Give rates by stated population. N = 8 N = 40 N = 200 N = 1000 No information0.833 (24)0.833 (24)0.792 (24)0.750 (24) Proportion-format report (k = 3)0.000 (24)0.500 (24)0.708 (24)0.875 (24) Table S13: Cross-model-family test filed before execution, claude-sonnet-5 (threshold arm). Give rates by stated population. N = 8 N = 40 N = 200 N = 1000 No information1.000 (24)0.958 (24)0.792 (24)0.792 (24) Proportion-format report (k = 3)0.583 (24)1.000 (24)1.000 (24)1.000 (24) 22 Scale limits of social mechanisms in agent societiesPreprint Table S14: Temperature sweep, gemini-3.5-flash. Give rates at three sampling temperatures. t = 0t = 0.7 t = 1.0 No information1.000 (24)1.000 (24)1.000 (24) Targeted report0.000 (24)0.000 (24)0.000 (24) Count report, N = 10000.000 (24)0.000 (24)0.000 (24) 23 Scale limits of social mechanisms in agent societiesPreprint S8 Illustrative field application: peer review at forty-fold scale The audit was developed for agent societies, but its categories describe properties of interaction and observation rather than properties of silicon. We therefore asked whether they were recognizable in a human institution that had expanded rapidly. ICLR, the International Conference on Learning Representations and one of the largest machine-learning conferences, grew from roughly490submissions in 2017 to19,525in 2026 as the field expanded. The analysis uses public review records from 2024–2026, ten years of policy documents, and the full public corpus of TMLR (Transactions on Machine Learning Research), a journal in the same field whose reviews follow per-paper rolling deadlines rather than one synchronized deadline, as a design contrast. ICLR 2025 also randomized an LLM feedback intervention whose assignment labels are not public. Feedback was successfully delivered to18,946of44,831reviews (42.3percent), so no quantity here is read causally with respect to it, and the post-release revision rates below differ in kind from the published feedback-update figure [44]. Every claim in this section is descriptive. The application illustrates the checklist and does not validate its predictions for agent societies. The time-budget class appears as deadline concentration (Fig. S1A). In the development half of ICLR 2025,59.3 percent of on-time reviews arrived in the final72hours of a22-day window. The held-out half reproduced the pattern at60.8percent, against a threshold of27percent set before validation. TMLR provides a design contrast because it uses assignment-relative rather than one synchronized deadline. The sharpest three-day period contained35.4percent of ICLR reviews but6.8percent of TMLR reviews under a per-paper construction, and a single seven-day calendar window separates the venues even more sharply, at72.9against1.24percent (Fig. S1B). The contrast is cross-sectional and carries confounds beyond deadline design. Compensation appears in the changing stock of formal coordination devices (Fig. S1C). Under persistence coding, active devices rose from5to39and shifted from restoring information coverage toward admission control and sanctions as submissions passed roughly five to seven thousand. This co-movement is not a causal estimate. More broadly, machine-learning peer review remained comparably inconsistent when a NeurIPS consistency experiment was repeated after submissions had grown more than five-fold: in both experiments, roughly half of the accepted-paper list would have changed under an independent rerun [45,46]. Reviews of peer-review research describe the same difficulty of maintaining reliable decisions at scale [47]. These cross-venue observations motivate, but do not demonstrate, a compensating role for the growing apparatus. The public record also demonstrates a protocol artifact. A 2026 security response froze editing and rolled reviews back to their pre-discussion state. The resulting public data contain four reviewer-signed revisions where neighbouring years contain thousands. Reading this literally would mistake a change in the observation channel for a collapse of reviewer behaviour. Post-release revision behaviour otherwise reproduces across the two 2025 halves: revision rates were0.270 and0.282, median first-revision delays were24.8and24.9days (n = 7,845and4,995revised reviews), and roughly60 percent of revision events fell inside the discussion period. Revision magnitude is measurable only between consecutive public edits. Its median signed change was zero words across 1,212 intervals in 902 multiply revised reviews. 20242025 dev 2025 val 2026 0.0 0.2 0.4 0.6 0.8 last-72 h share of on-time reviews registered threshold uniform A calendar 7-day window per-paper 3-day window 0.0 0.2 0.4 0.6 0.8 sharpest-window share B ICLR 2025 TMLR 2017202020232026 year 0 10 20 30 40 active devices C feedback monitoring admission control sanction automation Figure S1: Peer review at forty-fold scale. All series are descriptive. (A) Share of on-time reviews arriving in the final 72hours of the review window, against the registered threshold of twice the uniform-rate benchmark. (B) Share of reviews in the sharpest calendar and assignment-relative windows for ICLR 2025 and TMLR:72.9versus1.24percent in the calendar window and35.4versus6.8percent in the per-paper window. (C) Active formal coordination devices coded from ten years of policy documents. Their composition shifts from feedback and monitoring toward admission control and sanctions as submissions grow. 24 Scale limits of social mechanisms in agent societiesPreprint A prediction filed before the data expected stronger clustering in higher-pressure areas. The data showed near uniformity instead, with the72-hour share inside0.57to0.63across all21areas and a rank correlation small and unstable in sign across the two halves, so the uniformity is the finding and the failed prediction is recorded in the ledger. Table S15 reports the per-area clustering and Table S16 the extraction reconciliation. The 2026 incident windows are, in UTC, clean before 2025-11-11 00:00, ambiguous from 2025-11-11 00:00 to 2025-11-28 00:00, and post-incident afterwards, fixed from the official response timeline. The device coding matrix behind the accumulation count, fifty devices with categories, first years, persistence and per-device sources, is provided as Supplementary Data 1. Persistence coding treats a policy as active from its first documented year until a documented discontinuation. A conservative documented-years-only variant is used as a lower bound. Table S15: Deadline clustering by primary area, ICLR 2025. D = development half, V = validation half. Clust. = share of on-time reviews arriving in the final 72 hours. HalfAreaSubmissionsReviewsClust. Dalignment, fairness, safety, privacy, and societal co...104842250.586 Dapplications to computer vision, audio, language, and...137355280.594 Dapplications to physical sciences (physics, chemistry...54322040.592 Dapplications to robotics, autonomy, planning2449730.624 Dgenerative models106342780.603 Dlearning theory36514600.609 Doptimization52120290.600 Dother topics in machine learning (i.e., none of the a...56222290.611 Dreinforcement learning67626610.572 Dunsupervised, self-supervised, semi-supervised, and s...86834580.577 Vapplications to neuroscience & cognitive science2148550.609 Vcausal reasoning1194710.626 Vdatasets and benchmarks76030510.626 Vfoundation or frontier models, including LLMs129751990.627 Vinfrastructure, software libraries, hardware, systems...893450.607 Vinterpretability and explainable AI51620590.610 Vlearning on graphs and other geometries & topologies42917590.574 Vlearning on time series and dynamical systems27010850.575 Vneurosymbolic & hybrid AI systems (physics-informed,...1054250.612 Vprobabilistic methods (Bayesian methods, variational ...2369610.581 Vtransfer learning, meta learning, and lifelong learning37414930.582 Table S16: Extraction reconciliation. Roots are unique submission forums enumerated by invitation; reviews are public official reviews with trusted timestamps. VenueRootsReviewsOfficial figureNote ICLR 20247,40428,0287,262 submissions reportedreviewed forums equal official count ex- actly ICLR 202511,67246,74811,672 / 11,603roots minus desk = 11,602, Fact Sheet 11,603 ICLR 202619,81475,85919,525 valid; 779 deskAPI desk count 908 at extraction; enforce- ment continued TMLR7,57621,454about 4,026 searchableinvitation enumeration more complete S9 Baseline-adjusted models and prospectively logged tests Table S21 reports the core probe cells for all six engines, the basis of the five-of-six statement in the main text. Table S17 reports full coefficients for the baseline-adjusted logistic models defined in Methods, one per lattice and engine, including thek = 0no-information cells. Table S18 lists the twenty registered quantities of the prospective third-party code test with their filed values. Table S19 reports the per-criterion adjudication of the cross-model-family test, and Table S20 the registered criteria for the count-format lattice on further engines. In every case the criteria were filed in the time-ordered ledger before the corresponding data existed, and the ledger digest with dates is archived with the released materials. 25 Scale limits of social mechanisms in agent societiesPreprint Table S17: Baseline-adjusted logistic modelskeep ∼ 1 + R + R logk + logN + R logNover all lattice decisions includingk = 0baselines. Estimates with Wald 95% intervals. The fit column reports a generalized linear model (GLM) or, under separation, Firth penalised likelihood. TheR× logNcolumn is the quantity reported in the main text. Format EngineFit n R logklogNR× logN countsonnet (primary) GLM 288 −1.18 [−3.75, 1.39]+0.48 [0.18, 0.77]+0.16 [−0.20, 0.52] −0.16 [−0.62, 0.29] countreasonerGLM 287 −0.22 [−2.80, 2.35]+0.13 [−0.10, 0.35] −0.37 [−0.81, 0.07] +0.41 [−0.08, 0.90] countgeminiFirth288 +2.38 [−6.06, 10.81] +1.09 [0.80, 1.38]+0.00 [−1.52, 1.52] +0.08 [−1.47, 1.62] countgpt-5.5GLM 288 +4.23 [−1.50, 9.96]+0.36 [0.13, 0.59]+0.34 [−0.60, 1.28] −0.47 [−1.44, 0.50] pctreasonerGLM 287 +3.54 [1.08, 6.01]+0.66 [0.41, 0.90]+0.34 [−0.04, 0.72] −0.82 [−1.27,−0.38] pctsonnetFirth288 +6.43 [−1.70, 14.56] +4.83 [2.33, 7.33]+0.37 [−0.03, 0.78] −4.50 [−6.58,−2.41] Table S18: The twenty registered quantities of the prospective third-party code test. Predictions for all twenty were filed in the time-ordered ledger before the experiment ran and all twenty observed values agreed exactly with the filed values on this deterministic system, so filed and observed columns coincide and are shown once. PoolArmNPayoff advantage per turn I Grudgerper-capita budget6+0.550 I Grudgerper-capita budget12+0.364 I Grudgerper-capita budget24+0.087 I Grudgerper-capita budget48 −1.575 I Grudgerper-capita budget96 −1.537 I Grudgerfixed match length6+0.300 I Grudgerfixed match length12+0.455 I Grudgerfixed match length24+0.522 I Grudgerfixed match length48+0.553 I Grudgerfixed match length96+0.568 I WSLSper-capita budget6 −0.700 I WSLSper-capita budget12 −0.727 I WSLSper-capita budget24 −0.783 I WSLSper-capita budget48 −1.575 I WSLSper-capita budget96 −1.537 I WSLSfixed match length6 −0.700 I WSLSfixed match length12 −0.455 I WSLSfixed match length24 −0.348 I WSLSfixed match length48 −0.298 I WSLSfixed match length96 −0.274 Table S19: Per-criterion adjudication of the cross-model-family test filed before execution. The graded arm ran on deepseek-reasoner and the companion flatness arm on claude-sonnet; criteria are quoted from the ledger entry. Criteria not applicable to an arm are marked –. Registered criterionGraded armFlatness arm Give rate strictly monotone in Npass– Anchor: give≤ 0.2 at N = 8pass– Anchor: give≥ 0.8 at N = 1000pass– Interior: 0.396± 0.20 at N = 40pass– Interior: 0.682± 0.20 at N = 200pass– Give-rate rise of at least 0.4 across the tested rangepass– Flatness: report-cell give-rate spread < 0.15–fail Flatness: spread of differences from same-run baselines < 0.15–fail Sanction direction: baseline-minus-report difference positive at every N–fail Flatness: end-to-end give-rate change≤ 0.15–fail Table S20: Registered criteria for the count-format lattice on further engines, filed before data collection. Criterion (i) requires the anti-diagonal cells, which state identical proportions, to differ, ruling out ratio reading. The engine failing (i) responded to the presence of reports rather than their number and is recorded as consistent rather than confirmed in the main text. Registered criteriongemini-3.5-flashgpt-5.5deepseek-reasoner (i) dose effect β log k > 0,|z| > 1.96passpassfail (i) 95% CI for β log N contains 0passpasspass (i) anti-diagonal give-rate spread > 0.15passpasspass 26 Scale limits of social mechanisms in agent societiesPreprint Table S21: Cross-engine give rates for the core probe cells (wording V1,n = 24decisions per cell). B, no information. T, targeted report about the current partner. X, identical report about a third party. Count, count-format report at stated N =1000. Pct 8 and Pct 1000 , percentage-format reports at statedN =8andN =1000. The directional structure, sanction on a targeted report with a placebo third-party report, appears on five of six engines, with a partial response on one and no response on deepseek-chat, which shares weights with deepseek-reasoner and differs only in whether the reasoning mode is active. Cells not run on an engine are marked –. EngineBTXCountPct 8 Pct 1000 claude-sonnet-4-61.000.001.000.000.96– claude-opus-4-81.000.501.000.750.96– gpt-5.50.920.000.960.000.000.17 gemini-3.5-flash1.000.001.000.000.000.04 deepseek-chat1.001.000.960.250.830.96 deepseek-reasoner0.920.000.750.000.170.88 27 Scale limits of social mechanisms in agent societiesPreprint S10 Failed and inconclusive registered outcomes Twelve registered outcomes failed, returned inconclusive verdicts, or exposed their own criteria as inadequate. Each row lists the outcome and what followed from it. Table S22: Failed, inconclusive, or criterion-inadequate registered outcomes. OutcomeWhat followed Exponent test failed under bounded memory (slope 0.245) memory turnover identified as a candidate violated as- sumption. A separately registered amendment using un- bounded memory and an absolute threshold passed with slope1.00, without isolating which change accounted for the difference Aggregate verdict inconclusive in the routing pair mandatory component metrics and the path decomposition step Flatness failed at the smallest populationcausal neighbourhood condition. Registered retest above threshold passed with spread 0.014 Registered sign prediction failed in the early gossip design sign treated as implementation property, demonstrated by the refusal variant Two composed-system verdicts failed on the final round endgame isolation rule for live runs A registered interpretation was refuted by its own control cell the format account of proportion framing Cross-model run failed three of five cells on one engine per-encounter gains treated as model parameters An exploratory observable reversed direction in the transfer test observable specification made part of the audit Confirmatory baseline fell short of threshold on a newer generation unconditional baselines version-scoped. Conditional struc- ture replicated Registered flatness prediction failed on all four criteria on a later snapshot count-format scoping of the stated-size result. Threshold response identified by the percentage lattice Dose gate fired on a count lattice whose in- terval favoured the tested claim engine recorded as consistent rather than confirmed. Word- ing strength added to the format dependence Field area-gradient prediction was not sup- ported: its sign criterion passed in one half only, while the effect was near zero and changed sign across halves criterion recorded as too coarse. Uniformity reported as the finding S11 Behavioural collapse in full games Full-game experiments with language model agents failed for instrument reasons that recur across game types. In donation games with social vocabulary, giving collapses to near unanimity in all non-final rounds and is insensitive to payoff structure across twelve prompt variants, including changed cost-benefit ratios, competitive framing, system- prompt role changes, and explicit payoff matrices. In coordination games with labelled options, choices collapse onto lexical or positional focal points and reach full coordination without any social process. Under a known horizon the final round collapses toward defection, with a defection rate that grows with the stated population size. Across all three, presentation dominates incentive structure, mechanisms that operate on behavioural deviations receive no input under such collapse, and conditional probes must precede any full-game study. 28