The Contagion Tensor: A Framework for Measuring Output-Distribution Coupling in Multi-Agent LLM Systems -- and Auditing the Claims It Enables

# A Framework for Measuring Output-Distribution Coupling in Multi-Agent LLM Systems —and Auditing the Claims It Enables
Source: [https://arxiv.org/html/2606.28839](https://arxiv.org/html/2606.28839)
###### Abstract

We introduce the*Contagion Tensor*\(𝔗∈ℝM×N×T\\mathfrak\{T\}\\in\\mathbb\{R\}^\{M\\times N\\times T\}\), a measurement framework for quantifying how large language model \(LLM\) output distributions couple across modalities \(MM\), agents \(NN\), and time steps \(TT\)\. From the tensor we derive the*Coupling Amplification Factor*\(CAF\), a family of ratio\-based metrics that share the formCAF=𝔼​\[𝔗condition\]/𝔼​\[𝔗baseline\]\\text\{CAF\}=\\mathbb\{E\}\[\\mathfrak\{T\}\_\{\\text\{condition\}\}\]/\\mathbb\{E\}\[\\mathfrak\{T\}\_\{\\text\{baseline\}\}\]\. CAF provides a unitless, baseline\-referenced measurement with bootstrap confidence intervals, applicable to any multi\-agent configuration for which agent output distributions can be discretized\.

We instantiate CAF in four variants—network \(CAFnet\\text\{CAF\}\_\{\\text\{net\}\}\), cross\-modal \(CAFcross\\text\{CAF\}\_\{\\text\{cross\}\}\), temporal \(CAFtemp\\text\{CAF\}\_\{\\text\{temp\}\}\), and full\-axis reference \(CAFbase\\text\{CAF\}\_\{\\text\{base\}\}\)—each replacing a previously conflictingΓ\\Gammasymbol from prior work\. The strongest variant \(CAFbase\\text\{CAF\}\_\{\\text\{base\}\}\) is evaluated in a complete2×2×22\\times 2\\times 2block\-orthogonal simulation design with a modality\-specific ablation\. The ablation reveals that an apparent image\-condition super\-linear effect \(CAF=1\.40\\text\{CAF\}=1\.40\) collapses to sub\-linear \(CAF=0\.87\\text\{CAF\}=0\.87\) when the image perturbation module is disabled, a shift of−0\.53\-0\.53with zero effect on text conditions\.

We supplement the simulation with real\-API experiments across two model families: DeepSeek\-Chat \(R=30R=30, both uniform and diverse personas\) and GPT\-4o\-mini \(R=15R=15, within\-model, real vision\)\. Under uniform personas, text\-only BOUNDARY\_SYNC producesCAF≈1\.0\\text\{CAF\}\\approx 1\.0in both models\. Diverse personas drive convergence \(CAF = 0\.88\)\. A within\-model comparison on GPT\-4o\-mini reveals: C3 \(text\)CAF=1\.02\\text\{CAF\}=1\.02vs\. C5 \(real vision,R=30R=30\)CAF=1\.72\\text\{CAF\}=1\.72\[1\.700, 1\.733\],Δ=\+0\.70\\Delta=\+0\.70, validating the simulation’s super\-linear image\-condition prediction\. Of 11 conditions, 5 have been tested on real APIs and 6 remain unverified\.

Our contribution is two\-layered: \(1\) at the measurement\-instrument level, the CAF family—a baseline\-referenced, unitless ratio that makes output\-distribution coupling quantitatively falsifiable for the first time; and \(2\) at the methodology level, a transferable ablation protocol that any modular multi\-agent simulator can adopt to distinguish genuine coupling effects from design artifacts\. The framework, the protocol, and the audit together provide the tools and standards the field currently lacks\.

## 1Introduction

When multiple LLMs interact—as chatbot ensembles, retrieval\-augmented systems, or agent\-based social simulations—their output distributions evolve through repeated exchange\. A bias in one agent can propagate across the network, coupling outputs in ways that single\-model benchmarks cannot capture\[[1](https://arxiv.org/html/2606.28839#bib.bib1),[2](https://arxiv.org/html/2606.28839#bib.bib2)\]\. Yet the community lacks a reusable measurement framework for this phenomenon\. We have single\-model bias probes \(BBQ\[[5](https://arxiv.org/html/2606.28839#bib.bib5)\], StereoSet\[[6](https://arxiv.org/html/2606.28839#bib.bib6)\]\) and qualitative multi\-agent case studies\[[1](https://arxiv.org/html/2606.28839#bib.bib1)\], but no principled metric to answer questions like:*how much*does adding a second modality amplify coupling?*Does*memory increase temporal persistence?*Can*a calibration run on conditions A–D guarantee valid measurements on conditions E–H?

This paper provides the metric, the measurement protocol, and the audit\.

The framework\.We define the*Contagion Tensor*𝔗∈ℝM×N×T\\mathfrak\{T\}\\in\\mathbb\{R\}^\{M\\times N\\times T\}, where each cell𝔗​\[m,n,t\]=DJS⁡\(wn,tm∥w0\)\\mathfrak\{T\}\[m,n,t\]=\\operatorname\{D\}\_\{\\text\{JS\}\}\(w\_\{n,t\}^\{m\}\\\|w\_\{0\}\)is the Jensen\-Shannon divergence of an agent’s output distribution from a uniform reference\. From the tensor we derive the Coupling Amplification Factor \(CAF\), a family of ratio\-based metrics that all share the same form—condition over baseline—but differ in which axis they perturb \(Table[1](https://arxiv.org/html/2606.28839#S3.T1)\)\. This unification replaces four conflictingΓ\\Gammasymbols across our prior work with a single notation\.

The strongest case\.We instantiateCAFbase\\text\{CAF\}\_\{\\text\{base\}\}in a2×2×22\\times 2\\times 2full\-factorial simulation \(modality×\\timesagent\-count×\\timestimestep\-count\)\. Under default parameters, image conditions appear super\-linear \(CAFimage=1\.40\\text\{CAF\}\_\{\\text\{image\}\}=1\.40\) while text conditions are sub\-linear \(CAFtext=0\.84\\text\{CAF\}\_\{\\text\{text\}\}=0\.84\)—superficially, a modality\-driven bifurcation\. We then disable the single module that differentiates image from text in the simulator\. All four image conditions collapse from super\-linear to sub\-linear \(Δ=−0\.53\\Delta=\-0\.53\); text conditions are untouched\. The “bifurcation” is an artifact\.

The audit\.We compile a verification completeness table \(Table[11](https://arxiv.org/html/2606.28839#S5.T11)\), reporting the empirical status of every CAF variant\. Of 11 conditions, 5 have been tested on real APIs across two model families with functional BOUNDARY\_SYNC communication and controlled personas \(C3u and C8u atR=15R=15, C3d atR=30R=30, C5 atR=30R=30\)\. Under these controlled conditions, both C3 and C8u converge toCAF≈1\.0\\text\{CAF\}\\approx 1\.0, indicating that functional BOUNDARY\_SYNC communication alone—without persona diversity—does not measurably alter output\-distribution coupling relative to isolation\. The previously reported sign reversal \(C3 from 0\.877 to 3\.300 atR=5R=5\) was a small\-sample noise artefact\. We document this resolution not to minimize the earlier gap but to demonstrate that CAF, when measured at adequate repetitions with controlled confounds, converges to the theory\-expected value\.

What this paper is\.It is two things at different levels\. At the measurement\-instrument level, it is a framework that makes output\-distribution coupling quantitatively falsifiable for the first time\. At the methodology level, it is a transferable ablation protocol that any modular multi\-agent simulator can adopt to distinguish genuine coupling effects from design artifacts\. It is not a claim of emergent behavior\. It is built on the premise that making a phenomenon measurable—and providing the tools to audit those measurements—is a scientific contribution, even when the first measurements reveal more about the instruments than about the phenomenon\.

## 2Related Work

##### Multi\-agent LLM systems\.

Generative agents\[[1](https://arxiv.org/html/2606.28839#bib.bib1)\], multi\-agent debate\[[3](https://arxiv.org/html/2606.28839#bib.bib3)\], and negotiation\[[4](https://arxiv.org/html/2606.28839#bib.bib4)\]demonstrate emergent social dynamics but measure task performance, not output\-distribution coupling\. Frameworks including MetaGPT\[[23](https://arxiv.org/html/2606.28839#bib.bib23)\], CAMEL\[[24](https://arxiv.org/html/2606.28839#bib.bib24)\], and AutoGen\[[25](https://arxiv.org/html/2606.28839#bib.bib25)\]provide engineering infrastructure for multi\-agent workflows but offer no standardized coupling metric\. Agent\-based modeling textbooks provide simulation methodology\[[11](https://arxiv.org/html/2606.28839#bib.bib11)\]but no coupling metric\. Our CAF is orthogonal: it quantifies*how much*outputs couple, independent of task success\.

##### LLM bias evaluation\.

Single\-model bias probes—BBQ\[[5](https://arxiv.org/html/2606.28839#bib.bib5)\], StereoSet\[[6](https://arxiv.org/html/2606.28839#bib.bib6)\], Winogender\[[7](https://arxiv.org/html/2606.28839#bib.bib7)\], and broader safety benchmarks\[[2](https://arxiv.org/html/2606.28839#bib.bib2)\]—measure static distributional disparities\. Information\-theoretic diversity measures for text generation\[[8](https://arxiv.org/html/2606.28839#bib.bib8)\]quantify output variety but not inter\-agent coupling\. CAF fills the gap between single\-model bias measurement and multi\-agent dynamics\.

##### Simulation artifact detection\.

Verification and validation of simulation models has a long history\[[9](https://arxiv.org/html/2606.28839#bib.bib9),[10](https://arxiv.org/html/2606.28839#bib.bib10),[11](https://arxiv.org/html/2606.28839#bib.bib11)\]\. Standard protocols \(ODD\[[12](https://arxiv.org/html/2606.28839#bib.bib12)\]\) require design documentation but do not prescribe module\-specific ablation\. Our modality\-ablation protocol provides a concrete, transferable procedure applicable to any multi\-agent LLM simulator with independently toggleable modules\.

##### Network contagion and spectral methods\.

Threshold models\[[13](https://arxiv.org/html/2606.28839#bib.bib13)\]and spectral epidemiology\[[14](https://arxiv.org/html/2606.28839#bib.bib14)\]model propagation in networks\. JSD has been used as a divergence measure in multi\-agent settings\[[15](https://arxiv.org/html/2606.28839#bib.bib15)\]but not as the basis for a coupling ratio\.CAFnet\\text\{CAF\}\_\{\\text\{net\}\}bridges spectral radius to distributional coupling via a mean\-field mapping \(see §4\.2, Table[1](https://arxiv.org/html/2606.28839#S3.T1)\)\.

##### Multi\-agent coupling metrics\.

Several recent frameworks target dynamics and coupling in multi\-agent LLM systems from complementary angles\. CASPIAN monitors causal influence tensors and spectral propagation for cascade attack detection in multi\-agent systems\[[16](https://arxiv.org/html/2606.28839#bib.bib16)\]\. Riedl \(2026\) uses partial information decomposition with time\-delayed mutual information to separate synergistic, redundant, and unique information in multi\-agent coordination\[[17](https://arxiv.org/html/2606.28839#bib.bib17)\]\. Bridgeford & Helm \(2026\) employ temporal data kernel perspective space embedding for detecting behavioral shifts in black\-box multi\-agent systems\[[18](https://arxiv.org/html/2606.28839#bib.bib18)\]\. MultiAgentBench evaluates collaboration and competition quality in LLM agent teams\[[19](https://arxiv.org/html/2606.28839#bib.bib19)\]\. G\-Memory introduces hierarchical graph\-based memory for multi\-agent systems\[[20](https://arxiv.org/html/2606.28839#bib.bib20)\]\. EcoLANG addresses agent communication language induction for social simulation\[[21](https://arxiv.org/html/2606.28839#bib.bib21)\]\. A recent critical assessment questions whether LLMs reliably solve agent\-based modeling problems at all\[[22](https://arxiv.org/html/2606.28839#bib.bib22)\]\. CAF differs from these in being a simple, baseline\-referenced ratio that requires only discretized output distributions and a reference condition—no causal modeling, no information decomposition, no kernel embedding\. This simplicity is both a strength \(easy to adopt\) and a limitation \(captures degree but not structure of coupling\)\. Future work should compare CAF against these richer metrics on shared multi\-agent benchmarks\.

## 3The Contagion Tensor Framework

### 3\.1Contagion Tensor

Consider a system withMMmodalities,NNagents,TTdiscrete time steps, andKKoutput categories\. For each\(m,n,t\)\(m,n,t\), agentnnproduces an output whose distribution over categories iswn,tm∈ΔK−1w\_\{n,t\}^\{m\}\\in\\Delta^\{K\-1\}\.

###### Definition 1\(Contagion Tensor\)\.

𝔗∈ℝM×N×T\\mathfrak\{T\}\\in\\mathbb\{R\}^\{M\\times N\\times T\}where𝔗​\[m,n,t\]=DJS⁡\(wn,tm∥w0\)\\mathfrak\{T\}\[m,n,t\]=\\operatorname\{D\}\_\{\\text\{JS\}\}\(w\_\{n,t\}^\{m\}\\,\\\|\\,w\_\{0\}\), withw0w\_\{0\}the uniform reference andDJS\\operatorname\{D\}\_\{\\text\{JS\}\}the Jensen\-Shannon divergence \(base 2, bits\)\.DJS\\operatorname\{D\}\_\{\\text\{JS\}\}is symmetric, bounded in\[0,1\]\[0,1\], andDJS\\sqrt\{\\operatorname\{D\}\_\{\\text\{JS\}\}\}is a proper metric\[[15](https://arxiv.org/html/2606.28839#bib.bib15)\]\.

Each cell captures one agent’s distributional drift at one moment\. The full tensor enables comparisons along any axis: modality effects, agent\-level heterogeneity, and temporal trajectories\.

Discretization\.The tensor requires agent outputs to be discretized intoKKcategories\. For simulation experiments, agents produceKK\-way categorical distributions directly \(Ksim=10K\_\{\\text\{sim\}\}=10\)\. For real\-API experiments, free\-form text responses are mapped toKapi=5K\_\{\\text\{api\}\}=5categories via a two\-stage procedure: first, the response text is parsed for JSON\-format probability distributions; if parsing fails, a keyword\-frequency match against predefined category labels is used as a fallback\. The full discretization pipeline is documented in the supplementary code \(deepseek\_backend\.py, methodget\_strategy\_distribution\)\. The five real\-API categories are:neutral,biased\_female,biased\_male,stereotype\_avoidant, andstereotype\_reinforcing\. These capture both the bias direction \(female vs\. male\) and the framing \(avoidant vs\. reinforcing\)\. The keyword\-matching fallback uses category\-name substrings; formal human validation of this fallback has not been conducted, and production CAF deployments should use structured extraction or independently validated classifiers\. Sensitivity toKKand category design is examined in Appendix[K](https://arxiv.org/html/2606.28839#A11): the modality bifurcation emerges clearly atK≥10K\\geq 10and is attenuated atK≤5K\\leq 5, consistent with the expectation that finer\-grained category spaces provide more resolution for distributional comparisons\.

Choice ofw0w\_\{0\}\.We use the uniform distributionw0=\(1/K,…,1/K\)w\_\{0\}=\(1/K,\\dots,1/K\)as the reference for three reasons\. First, it is parameter\-free and universally applicable across tasks, models, and domains, ensuring reproducibility\. Second, under the null hypothesis of no systematic bias, an ideal agent would produce a uniform distribution over categories, makingw0w\_\{0\}the natural zero\-coupling reference\. Third, the uniform reference produces a bounded JSD \(DJS≤1\\operatorname\{D\}\_\{\\text\{JS\}\}\\leq 1for base\-2;DJS≤ln⁡2≈0\.693\\operatorname\{D\}\_\{\\text\{JS\}\}\\leq\\ln 2\\approx 0\.693for base\-ee\), which keeps CAF ratios interpretable\. We note that task\-specific or empirically grounded references \(e\.g\., per\-condition empirical marginals\) may be more appropriate in settings where a non\-uniform prior distribution is known a priori; CAF supports such references by substitution ofw0w\_\{0\}without modification to the ratio formula\.

![Refer to caption](https://arxiv.org/html/2606.28839v1/x1.png)Figure 1:The Contagion Tensor𝔗∈ℝM×N×T\\mathfrak\{T\}\\in\\mathbb\{R\}^\{M\\times N\\times T\}\. Each cell𝔗​\[m,n,t\]=DJS⁡\(wn,tm∥w0\)\\mathfrak\{T\}\[m,n,t\]=\\operatorname\{D\}\_\{\\text\{JS\}\}\(w\_\{n,t\}^\{m\}\\\|w\_\{0\}\)encodes one agent’s distributional divergence from the uniform reference at one time step\. Warmer colors indicate larger JSD \(greater drift from uniformity\)\. \(Conceptual illustration; cell colors are synthetic and not derived from experimental data\.\)
### 3\.2The CAF Family

###### Definition 2\(CAF family\)\.

Each variant is a ratio of tensor expectations:

CAFv=𝔼​\[𝔗condition\]𝔼​\[𝔗baseline\],𝔼​\[𝔗\]=1M​N​T​∑m,n,t𝔗​\[m,n,t\]\.\\text\{CAF\}\_\{\\text\{v\}\}=\\frac\{\\mathbb\{E\}\[\\mathfrak\{T\}\_\{\\text\{condition\}\}\]\}\{\\mathbb\{E\}\[\\mathfrak\{T\}\_\{\\text\{baseline\}\}\]\},\\qquad\\mathbb\{E\}\[\\mathfrak\{T\}\]=\\frac\{1\}\{MNT\}\\sum\_\{m,n,t\}\\mathfrak\{T\}\[m,n,t\]\.\(1\)CAF\>1\\text\{CAF\}\>1: amplified coupling\.CAF=1\\text\{CAF\}=1: independent superposition\.CAF<1\\text\{CAF\}<1: output convergence \(JSD decreases relative to baseline\)\. We use “homogenization” as a descriptive label for this pattern, noting that it is an empirical observation—agents tend toward similar output distributions under repeated text exchange—not a claim about the causal mechanism producing it\.

Table 1:The four CAF variants\. Each corresponds to one case study\. “Replaces” shows the previous, conflictingΓ\\Gammasymbols\.Table[1](https://arxiv.org/html/2606.28839#S3.T1)summarizes the four variants\. This unification resolves a symbol drift that previously made cross\-paper comparison impossible\. All four variants are implemented inunified\_caf\.pyand are mathematically well\-defined; the “Status” column reflects empirical verification completeness \(see §5\)\.

Agent–agent coupling \(CAFpair\\text\{CAF\}\_\{\\text\{pair\}\}\)\.A reviewer may ask: does divergence from a uniform reference truly measure*coupling*between agents, or merely individual drift? We define a direct pairwise variant:CAFpair=𝔼​\[Dpair\]/𝔼​\[Dpairiso\]\\text\{CAF\}\_\{\\text\{pair\}\}=\\mathbb\{E\}\[D\_\{\\text\{pair\}\}\]/\\mathbb\{E\}\[D\_\{\\text\{pair\}\}^\{\\text\{iso\}\}\], whereDpair=1N​\(N−1\)​∑i≠jDJS⁡\(wi∥wj\)D\_\{\\text\{pair\}\}=\\frac\{1\}\{N\(N\-1\)\}\\sum\_\{i\\neq j\}\\operatorname\{D\}\_\{\\text\{JS\}\}\(w\_\{i\}\\\|w\_\{j\}\)is the mean pairwise JSD between agent output distributions, and the baseline is the pairwise JSD under the ISOLATED communication mode\.CAFpair\>1\\text\{CAF\}\_\{\\text\{pair\}\}\>1means inter\-agent communication*increases*distributional divergence between agents relative to isolated drift;CAFpair<1\\text\{CAF\}\_\{\\text\{pair\}\}<1means communication causes agents to*converge*\(output homogenization\)\. The aggregation is:DpairD\_\{\\text\{pair\}\}is computed per time step across all agent pairs, then averaged over steps and repetitions to obtain𝔼​\[Dpaircond\]\\mathbb\{E\}\[D\_\{\\text\{pair\}\}^\{\\text\{cond\}\}\], with bootstrap CIs computed analogously toCAFbase\\text\{CAF\}\_\{\\text\{base\}\}\(B=2000B=2000, resampling over repetitions\)\. Section[4\.1\.6](https://arxiv.org/html/2606.28839#S4.SS1.SSS6)presents the full CAFpair\{\}\_\{\\text\{pair\}\}computation across all eight conditions, confirming that the pairwise variant tracks CAFbase\{\}\_\{\\text\{base\}\}directionally \(text: sub\-linear; image: super\-linear\) and validating that JSD\-to\-uniform captures inter\-agent coupling rather than isolated drift\.

Why a linear ratio\.An earlier formulation used a product\-denominatorCAFprod=𝔼​\[𝔗\]/\(𝔼​\[𝔗M\]⋅𝔼​\[𝔗N\]⋅𝔼​\[𝔗T\]\)\\text\{CAF\}^\{\\text\{prod\}\}=\\mathbb\{E\}\[\\mathfrak\{T\}\]/\(\\mathbb\{E\}\[\\mathfrak\{T\}\_\{M\}\]\\cdot\\mathbb\{E\}\[\\mathfrak\{T\}\_\{N\}\]\\cdot\\mathbb\{E\}\[\\mathfrak\{T\}\_\{T\}\]\), which requires marginal projections and conflates dimension\-size artifacts with coupling\. We deprecate this form\. Under the independence null, the numerator and denominator of the linear ratio have the same expectation, so CAF converges to 1 regardless of tensor dimensions—removing the need for the dimension\-correction factors required by the product\-denominator form\. The linear ratio is

The C1 identity\.The baseline condition C1 \(isolated agent, no communication\) hasCAF​\(C1\)≡1\\text\{CAF\}\(C\_\{1\}\)\\equiv 1as an*identity*, not a measurement: the denominator and numerator of Eq\. \([1](https://arxiv.org/html/2606.28839#S3.E1)\) are the same tensor, so the ratio is exactly 1\. Its confidence interval of\[1\.0,1\.0\]\[1\.0,1\.0\]contains no statistical information\. C1 is a reference point, not a data point\.

### 3\.3Null Model and Bootstrap

We construct a null by shuffling𝔗\\mathfrak\{T\}entries independently along each axis, breaking correlational structure while preserving marginals\. All CIs are bootstrap percentile intervals \(B=2000B=2000,α=0\.05\\alpha=0\.05\)\. The convergence of the CAF estimator under independence \(a standard application of the weak law of large numbers and the continuous mapping theorem\) is proved in Appendix A\.

## 4Case Studies

We now instantiate the framework\. Section 4\.1 presents our strongest case: a complete simulation experiment with a modality ablation that demonstrates the framework’s ability to*detect*artifacts\. Section 4\.2 briefly defines three additional CAF variants, each corresponding to a prior experiment whose originalΓ\\Gammametric concealed a specific limitation that the CAF re\-expression exposes\.

### 4\.1Case Study: Modality Ablation \(CAFbase\\text\{CAF\}\_\{\\text\{base\}\}\)

This is the canonical instantiation of the framework\. We present the full experimental pipeline: design, baseline measurement, and ablation\.

#### 4\.1\.1Design:2×2×22\\times 2\\times 2Full Factorial

Three factors: modalityM∈\{text,image\}M\\in\\\{\\text\{text\},\\text\{image\}\\\}, agent countN∈\{3,5\}N\\in\\\{3,5\\\}, timestep countT∈\{10,20\}T\\in\\\{10,20\\\}\. The23=82^\{3\}=8conditions are listed in Table[2](https://arxiv.org/html/2606.28839#S4.T2)\. C1 is the isolated reference \(no inter\-agent communication\)\. C2–C8 use BOUNDARY\_SYNC with pre\-isolation communication and two cooldown steps\. All runs use 30 repetitions\. The master seed0xDEADBEEFis fixed for reproducibility; sub\-seeds for each \(condition, repetition, agent, time\-step\) are derived via MD5 hashing of the master seed concatenated with the condition ID, repetition index, agent index, and step index, ensuring independent stochastic streams across repetitions\. Bootstrap CIs resample at the repetition level, which is valid under the independence of sub\-seed\-derived runs\. Default parameters: bias strength=0\.3=0\.3,K=10K=10categories\.111Full configuration in Appendix B\.

Table 2:Condition definitions\.
#### 4\.1\.2Experiment 1: Full CAF Matrix

Table[3](https://arxiv.org/html/2606.28839#S4.T3)reports the CAF matrix under default parameters\. The pattern is clean: text conditions are sub\-linear \(CAF∈\[0\.80,0\.88\]\\text\{CAF\}\\in\[0\.80,0\.88\]\), image conditions super\-linear \(CAF∈\[1\.36,1\.43\]\\text\{CAF\}\\in\[1\.36,1\.43\]\)\. The 95% bootstrap CIs for image conditions lie entirely above 1\.0; for text conditions, entirely below 1\.0\. The text sub\-linearity is itself a finding: in text\-modality multi\-agent configurations, repeated interaction causes output*homogenization*\(agents converge to similar distributions\)\. CAF is the first quantitative measurement of this effect\.

Table 3:Full CAF matrix \(modal injection ON, 30 reps/condition\)\.CondModalityNNTTCAF95% CIC1text3101\.000\[1\.000, 1\.000\]C2text3200\.844\[0\.820, 0\.868\]C3text5100\.877\[0\.851, 0\.903\]C4text5200\.800\[0\.780, 0\.821\]C5image3101\.424\[1\.373, 1\.476\]C6image3201\.370\[1\.330, 1\.414\]C7image5101\.432\[1\.388, 1\.478\]C8image5201\.362\[1\.324, 1\.402\]Image mean=1\.397=1\.397, Text mean=0\.840=0\.840, Modality gap=0\.557=0\.557
#### 4\.1\.3Experiment 2: Modality Ablation

The simulator contains exactly one code path that differentiates image from text: whenmodality == "image", the agent’s output distribution receives noise∼𝒩​\(0,0\.03\)\\sim\\mathcal\{N\}\(0,0\.03\)and a structural shift∼Beta​\(2,5\)×0\.1×bias\_strength\\sim\\text\{Beta\}\(2,5\)\\times 0\.1\\times\\text\{bias\\\_strength\}\. We disable this module and re\-run all eight conditions with all seeds and parameters identical\.

Table 4:Modality ablation with 95% bootstrap CIs\. Text conditions \(C2–C4\) are identical in ON and OFF and not shown\. Text ON==OFF because the simulator’s image\-specific module is the only code path that differs between modalities; text\-condition executions are byte\-identical in both modes\.CondMMNNTTON \[CI\]OFF \[CI\]Δ\\DeltaC5image3101\.424 \[1\.37, 1\.48\]0\.898 \[0\.87, 0\.93\]−0\.526\-0\.526C6image3201\.370 \[1\.33, 1\.41\]0\.866 \[0\.84, 0\.89\]−0\.504\-0\.504C7image5101\.432 \[1\.39, 1\.48\]0\.884 \[0\.86, 0\.91\]−0\.548\-0\.548C8image5201\.362 \[1\.32, 1\.40\]0\.814 \[0\.79, 0\.84\]−0\.548\-0\.548Image mean1\.3970\.866−0\.532\\mathbf\{\-0\.532\}Text mean\(C2–C4\)0\.8400\.8400\.000\\mathbf\{\\phantom\{\-\}0\.000\}Result\.All four image conditions collapse from super\-linear to sub\-linear\. The mean shift is−0\.53\-0\.53with zero effect on text conditions\. Agent count \(NN\) showed negligible main effect under image conditions \(C5 vs\. C7: 1\.424 vs\. 1\.432,Δ=0\.008\\Delta=0\.008; C6 vs\. C8: 1\.370 vs\. 1\.362,Δ=−0\.008\\Delta=\-0\.008\), while timestep count \(TT\) showed a modest main effect under text conditions \(C3 vs\. C4: 0\.877 vs\. 0\.800,Δ=−0\.077\\Delta=\-0\.077\)\. Full factor\-effect decomposition is provided in Appendix D\.

To isolate whether the artifact originates in the modality perturbation or the communication logic, we ran an additional ISOLATED variant for each image condition \(C5i–C8i\): identicalM,N,TM,N,Tparameters but with no inter\-agent communication \(ISOLATED mode, as in C1\)\. Table[5](https://arxiv.org/html/2606.28839#S4.T5)shows the comparison\.

Table 5:ISOLATED vs\. BOUNDARY\_SYNC image conditions\. ISOLATED CAF≈1\\approx 1when injection is OFF, confirming the artifact is injection\-only and not communication\-dependent\.ModeCondONOFFΔ\\DeltaBOUNDARY\_SYNCC51\.4240\.898−0\.526\-0\.526C61\.3700\.866−0\.504\-0\.504C71\.4320\.884−0\.548\-0\.548C81\.3620\.814−0\.548\-0\.548ISOLATEDC5i1\.520\.99−0\.53\-0\.53C6i1\.510\.95−0\.56\-0\.56C7i1\.551\.03−0\.53\-0\.53C8i1\.480\.98−0\.51\-0\.51ISOLATED OFF values cluster near 1\.0, confirming no residual coupling artifact\.The ISOLATED OFF values cluster nearCAF≈1\.0\\text\{CAF\}\\approx 1\.0, confirming that disabling the modality perturbation restores independent superposition regardless of communication mode\. The artifact is entirely in the perturbation module\.

![Refer to caption](https://arxiv.org/html/2606.28839v1/x2.png)Figure 2:Modality\-ablation results for image conditions C5–C8\. Modal injection ON \(orange\) vs\. OFF \(blue\), with 95% bootstrap CIs\. The text\-condition mean \(dotted green line,CAF=0\.84\\text\{CAF\}=0\.84\) is unchanged\. All four image conditions collapse from super\-linear to sub\-linear when the injection module is disabled\.
#### 4\.1\.4Experiment 3: Real\-API with Functional BOUNDARY\_SYNC

We ran real\-API experiments across two model families with functional BOUNDARY\_SYNC communication \(30% population\-mean blending, 2\-step cooldown\) and the framework\-consistent JSD definition \(DJS⁡\(w∥w0\)\\operatorname\{D\}\_\{\\text\{JS\}\}\(w\\\|w\_\{0\}\)\)\. All CAF values are base\-invariant: the simulation uses base\-ee\(nats\) and real\-API uses base\-2 \(bits\), but the CAF ratio cancels the base, making values directly comparable\. At each synchronization stept≡0\(mod3\)t\\equiv 0\\pmod\{3\}, every agent’s output distribution is updated aswi\(t\)←\(1−λ\)​wi\(t\)\+λ​w¯\(t\)w\_\{i\}^\{\(t\)\}\\leftarrow\(1\-\\lambda\)w\_\{i\}^\{\(t\)\}\+\\lambda\\bar\{w\}^\{\(t\)\}, wherew¯\(t\)=1N​∑j=1Nwj\(t\)\\bar\{w\}^\{\(t\)\}=\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}w\_\{j\}^\{\(t\)\}andλ=0\.3\\lambda=0\.3\. Cooldown steps \(t≢0t\\not\\equiv 0\) proceed without communication\. Sensitivity toλ\\lambdais reported in Appendix[L](https://arxiv.org/html/2606.28839#A12)\(CV<<2\.1% acrossλ∈\[0,1\]\\lambda\\in\[0,1\]\)\. All CAF values are base\-invariant: the simulation uses base\-ee\(nats\) and real\-API uses base\-2 \(bits\), but the CAF ratio cancels the base, making values directly comparable\.

1. 1\.DeepSeek\-Chat\(R=30R=30, both uniform and diverse personas\): C1, C3u, C3d, and C8u, all text\-only\. Uniform\-persona conditions use identical system prompts; diverse\-persona C3d uses five distinct agent profiles to isolate the persona\-diversity confound\.∼\\sim5400 API calls\.
2. 2\.GPT\-4o\-mini\(R=30R=30for C5,R=15R=15for C3\): C3 \(text/5/10\) and C5 \(image/3/10, real vision\)\. C5 uses PIL\-generated synthetic scene JPEGs via GPT\-4o\-mini’s vision API\.∼\\sim1650 API calls\.

Table 6:Real\-API experiments with functional BOUNDARY\_SYNC\. R values: DeepSeek\-Chat C3dR=30R=30, C3u/C8uR=15R=15; GPT\-4o\-mini C3R=15R=15, C5R=30R=30\. All conditions use uniform personas unless marked “diverse\.”CondModelModalitySim\. CAFReal CAF \[95% CI\]Class\.C3uDeepSeek\-Chattext0\.8770\.998aINDEPC3dDeepSeek\-Chattext \(diverse\)0\.8770\.880SUBC8uDeepSeek\-Chattext1\.3621\.025aINDEPC3GPT\-4o\-minitext0\.8771\.017INDEPC5GPT\-4o\-minireal image1\.4241\.717 \[1\.700, 1\.733\]SUPER
DeepSeek\-Chat C3d and GPT\-4o\-mini C5 atR=30R=30; all other conditions atR=15R=15\. C3d uses five distinct agent personas\.aC3u and C8u values are from theR=15R=15uniform\-persona control run\. C5realtransmits PIL\-generated synthetic scene JPEGs via GPT\-4o\-mini’s vision API \(R=30R=30\)\.

DeepSeek\-Chat uniform conditions\.Both C3u \(CAF = 0\.998\) and C8u \(CAF = 1\.025\) are near\-independent atR=15R=15, demonstrating that functional BOUNDARY\_SYNC communication alone—without persona diversity— does not measurably alter output\-distribution coupling relative to the isolated baseline\.

DeepSeek\-Chat C3d: persona\-diversity effect\.CAF = 0\.880\. When agents are assigned five intentionally distinct personas \(C3d\) rather than identical prompts \(C3u\), CAF drops from 0\.998 to 0\.880—a*sub\-linear*shift indicating output*convergence*\. This is the opposite of the original diverse\-persona C8 finding \(CAF = 11\.720\), which used both persona diversity*and*text\-proxied “image” prompts\. Isolating the persona factor in a clean text\-only condition reveals that persona diversity causes agent outputs to*homogenize*\(CAF<1<1\), not diverge\. The BOUNDARY\_SYNC population\-mean blending pulls diverse viewpoints toward the center\.

GPT\-4o\-mini within\-model C3 vs\. C5\.We ran the first within\-model text\-vs\-image CAF comparison on a single vision\-capable model \(GPT\-4o\-mini,R=15R=15for C3,R=30R=30for C5, 2550 total calls\)\. C3 atR=15R=15was already stable \(CAF = 1\.017, within 0\.02 of the DeepSeek C3u value of 0\.998 atR=15R=15\); C5 was prioritized for the larger sample to obtain publication\-grade CIs on the key image\-modality result\. Results:

- •C3 \(text/5/10/BOUNDARY\_SYNC\):CAF = 1\.017 — near unity, consistent with the DeepSeek\-Chat C3u finding that text\-only BOUNDARY\_SYNC produces no measurable coupling\.
- •C5 \(image/3/10/BOUNDARY\_SYNC, real vision,R=30R=30\):CAF = 1\.717 \(95% CI \[1\.700, 1\.733\]\)\. The CI excludes 1\.0 and excludes the C3 point estimate \(1\.017\), confirming a statistically significant super\-linear image\-condition effect within the same model\.
- •Δ\\Delta\(image−\-text\) =\+0\.700\+0\.700\. This is a clean within\-model estimate of the image\-modality effect, free of cross\-model confounds\.

Interpretation\.Three conclusions follow from the complete dataset \(R=30R=30DeepSeek\-Chat text conditions \+R=15R=15GPT\-4o\-mini within\-model\): \(1\) text\-only BOUNDARY\_SYNC with uniform personas does not measurably alter coupling atR≥15R\\geq 15\(C3u CAF≈\\approx1\.0 across both models\); \(2\) persona diversity drives*convergence*\(C3d CAF = 0\.88\), not the divergence previously attributed to image\-modality effects; \(3\) real vision inputs through a vision\-capable model produce*super\-linear*coupling \(C5 CAF = 1\.717 atR=30R=30\), consistent in direction with the simulation’s prediction—suggesting that the simulation’s image perturbation module, while artificially inflating the effect \(Δ=−0\.53\\Delta=\-0\.53under ablation\), captures a qualitative pattern that real vision models exhibit\. TheΔ\\Deltaof\+0\.70\+0\.70for the within\-model text\-vs\-image comparison is directionally consistent with the simulation’s ablation effect \(−0\.53\-0\.53\)—both indicate that image modality drives coupling while text does not—indicating that the relationship between modality and coupling strength is not a simulation artifact\.

Known limitations:\(1\) only 4 of 8 blueprint conditions tested on real APIs \(C3u, C3d, C8u, C5\); C2, C4, C6, C7 remain unmeasured; \(2\) C5 usesR=30R=30with synthetic scene stimuli generated via PIL; real\-world photographs may produce different CAF values\. The synthetic scenes \(e\.g\., “boardroom,” “construction site”\) use abstract geometric shapes and neutral color palettes, which may attenuate visual bias signals relative to real photographs containing human faces, clothing, and contextual cues\. If real\-world images carry stronger visual bias signals, the CAF amplification effect \(CAF\>1\\text\{CAF\}\>1\) observed in the simulation and partially validated in GPT\-4o\-mini could be*larger*in magnitude with natural stimuli—making the synthetic\-scene estimate a conservative lower bound\. Replication with real\-world photographs atR≥30R\\geq 30is needed for publication\-grade CIs; \(3\) the within\-model comparison is limited to GPT\-4o\-mini; replication on other vision\-capable models \(Gemini, Claude\) is needed for generalization; \(4\) two model families tested; generalization to other architectures is unknown; \(5\) C1 baseline JSD differs markedly across backends: simulation∼0\.018\\sim 0\.018, GPT\-4o\-mini∼0\.21\\sim 0\.21–0\.220\.22, DeepSeek\-Chat∼0\.31\\sim 0\.31nats—a 17×\\timesrange\. This gap indicates that simulation and real\-model output distributions operate on fundamentally different absolute scales\. CAF’s ratio formulation cancels this scale difference—a deliberate design choice enabling cross\-backend comparisons—but the absolute JSD gap implies that the simulation does not capture the raw distributional statistics of real LLM outputs\. Within\-backend CAF comparisons are unaffected; cross\-backend comparisons should treat simulation CAF magnitudes as qualitative rather than quantitative estimates of real\-model coupling strength; \(6\) C1’s CI is degenerate\[1\.0,1\.0\]\[1\.0,1\.0\]by construction \(CAF​\(C1\)≡1\\text\{CAF\}\(\\text\{C1\}\)\\equiv 1\)\. \(7\) the text\-to\-category discretization pipeline \(JSON parsing with keyword fallback\) has not been validated against human annotations; the keyword\-fallback rate has not been systematically measured, and systematic misclassification could bias CAF estimates\. Formal validation with human\-annotated responses is deferred to future work\.

#### 4\.1\.5What This Case Demonstrates

CAFbase\\text\{CAF\}\_\{\\text\{base\}\}provided quantitative answers that qualitative inspection could not: the exact magnitude of the artifact \(−0\.53\-0\.53\), the per\-condition decomposition \(C5–C8 behave near\-identically\), and the ISOLATED check that isolated the cause to the perturbation module\. The framework*enabled*the ablation: before CAF, the image\-text bifurcation was a qualitative impression; after CAF, it was a measurable quantity that could be decomposed\.

This case also demonstrates a general principle: calibration on a subset of conditions does not guarantee validity in unseen cells\. If we had calibrated the simulator to match desired CAF values on C1, C3, C4, and C7, the artifact in C5, C6, and C8 would have remained undetected\. This directly answers the third question posed in the Introduction: calibration on conditions A–D does*not*guarantee valid measurements on conditions E–H—and CAF, paired with the ablation protocol, provides the tool to detect when it fails\.

#### 4\.1\.6Experiment 4: Direct Pairwise Coupling \(CAFpair\\text\{CAF\}\_\{\\text\{pair\}\}\)

A reviewer may ask whether JSD\-to\-uniform genuinely captures*coupling*between agents, as opposed to individual drift from uniformity\. We address this directly by computingCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}, the pairwise variant defined in §3\.2:CAFpair=𝔼​\[Dpaircond\]/𝔼​\[DpairC1\]\\text\{CAF\}\_\{\\text\{pair\}\}=\\mathbb\{E\}\[D\_\{\\text\{pair\}\}^\{\\text\{cond\}\}\]/\\mathbb\{E\}\[D\_\{\\text\{pair\}\}^\{\\text\{C1\}\}\], whereDpairD\_\{\\text\{pair\}\}is the mean pairwise JSD between agent output distributions at each time step\.

Table 7:CAFpair\{\}\_\{\\text\{pair\}\}\(pairwise JSD ratio\) with bootstrap 95% CIs \(R=30R=30,B=10,000B=10\{,\}000\)\.Image meanCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}= 1\.160; Text meanCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}= 0\.719\.CAFpair\\text\{CAF\}\_\{\\text\{pair\}\}tracksCAFbase\\text\{CAF\}\_\{\\text\{base\}\}direction in all 7 non\-baseline conditions\.

Result\(Table[7](https://arxiv.org/html/2606.28839#S4.T7)\)\. CAFpair\{\}\_\{\\text\{pair\}\}tracks CAFbase\{\}\_\{\\text\{base\}\}directionally for image conditions \(C5–C8 all super\-linear, CAF∈pair\[1\.45,1\.53\]\{\}\_\{\\text\{pair\}\}\\in\[1\.45,1\.53\]\) and shows near\-unit values for text conditions \(C2–C4 CAF∈pair\[0\.94,1\.02\]\{\}\_\{\\text\{pair\}\}\\in\[0\.94,1\.02\]\)\. The CIs confirm that the image super\-linearity is statistically robust \(all CIs exclude 1\.0\), while text conditions show mixed evidence: C3’s pairwise CI \[0\.98, 1\.06\] contains 1\.0, meaning we cannot reject the null of independent pairwise evolution for this condition; C2 and C4 CIs exclude 1\.0, confirming homogenization there\. CAF makes this distinction explicit rather than collapsing all text conditions into a single claim\.

The CAFpair\{\}\_\{\\text\{pair\}\}values are more conservative than CAFbase\{\}\_\{\\text\{base\}\}\(image mean 1\.16 vs\. 1\.40; text mean 0\.72 vs\. 0\.84\), consistent with the fact that pairwise divergence is a stricter measure than individual divergence from a uniform reference\. Both metrics converge on the same substantive conclusion: multi\-agent text communication causes output homogenization, and the image perturbation module drives artificial divergence\. As shown in Appendix[I](https://arxiv.org/html/2606.28839#A9), simpler alternatives—entropy and variance ratios—cannot distinguish text from image conditions \(both span<0\.03<0\.03across all conditions\), confirming that CAF captures category\-level structure that scalar dispersion measures miss\.

##### Comparison with pairwise mutual information\.

We computed a pairwise mutual information \(MI\) ratio—defined analogously to CAFpair\{\}\_\{\\text\{pair\}\}as the per\-condition mean pairwise MI over the C1 baseline—on the same simulation runs\. Across all seven non\-baseline conditions, the MI ratio spans a narrow range \[0\.95, 0\.99\] and cannot separate text from image modalities\. In contrast, CAFpair\{\}\_\{\\text\{pair\}\}separates text \(0\.67–0\.76\) from image \(1\.06–1\.15\) \(Table[8](https://arxiv.org/html/2606.28839#S4.T8)\)\. JSD captures category\-level structure that mutual information compresses: MI primarily reflects how much information one agent’s output carries about another’s—a quantity that remains nearly constant across modalities under the simulation’s noise model—while JSD is sensitive to*which*categories receive probability mass\. Both entropy/variance \(Appendix[I](https://arxiv.org/html/2606.28839#A9)\) and pairwise MI fail the discrimination test that CAF passes, converging on CAF’s unique value\.

Table 8:CAFpair\{\}\_\{\\text\{pair\}\}vs\. pairwise mutual information ratio\.CondModalityCAFpair\{\}\_\{\\text\{pair\}\}MI RatioC2text0\.6830\.955C3text0\.7590\.969C4text0\.6700\.992C5image1\.1500\.997C6image1\.0900\.951C7image1\.1500\.967C8image1\.0590\.967MI ratio∈\\in\[0\.95, 0\.99\]; CAFpair\{\}\_\{\\text\{pair\}\}separates text \[0\.67,0\.76\] from image \[1\.06,1\.15\]\.
##### Sensitivity toKK\(number of categories\)\.

Table[9](https://arxiv.org/html/2606.28839#S4.T9)reports CAFbase\{\}\_\{\\text\{base\}\}atK∈\{3,10,20\}K\\in\\\{3,10,20\\\}\(full table in Appendix[K](https://arxiv.org/html/2606.28839#A11)\)\. The modality bifurcation emerges atK≥10K\\geq 10; atK=3K=3, image conditions become sub\-linear, consistent with coarser categorizations masking distributional structure\. CAF values are stable acrossK≥10K\\geq 10for text \(CV<<5%\), while image CAF increases withKK, suggesting finer spaces amplify measurable coupling\. All main\-text experiments useK=10K=10\.

Table 9:CAFbase\{\}\_\{\\text\{base\}\}at selectedKK\(full: Appendix[K](https://arxiv.org/html/2606.28839#A11)\)\.

### 4\.2Other CAF Variants: Definitions and Diagnostic Audit

The remaining three variants correspond to prior experiments that each suffered from a specific measurement limitation\. Table[10](https://arxiv.org/html/2606.28839#S4.T10)summarizes their definitions, data status, and the diagnostic insight that the CAF re\-expression provides\. Each variant exposes a different failure mode of measurement without an explicit baseline: sub\-critical detection with no reference condition \(CAFnet\\text\{CAF\}\_\{\\text\{net\}\}\), baseline treated as optional \(CAFcross\\text\{CAF\}\_\{\\text\{cross\}\}\), and baseline version drift producing contradictory published values \(CAFtemp\\text\{CAF\}\_\{\\text\{temp\}\}\)\. These failure modes are generic risks in any multi\-agent measurement study\. Documenting them under a unified notation makes the risks visible and the remedies explicit\. Detailed descriptions of each variant and the underlying experiments appear in Appendix[N](https://arxiv.org/html/2606.28839#A14)\.

Table 10:Additional CAF variants: definition, data status, and diagnostic insight\.

## 5Empirical Status Audit

Table[11](https://arxiv.org/html/2606.28839#S5.T11)is the central honest assessment of the paper\. It reports, for every CAF variant, what has been measured against a real API and what remains unverified\. We encourage future work adopting CAF to maintain a similar table\.

Table 11:Empirical verification status across all four CAF variants\.✓\\checkmark= directionally verified;∼\\sim= partially measured under incompatible conditions;×\\times= unverified\.VariantClaimStatusNoteCAFbase\\text\{CAF\}\_\{\\text\{base\}\}Image artifactΔ=−0\.53\\Delta\{=\}\{\-0\.53\}✓\\checkmark\(sim\+API\)5 conds on 2 models, C5CAF=1\.72\\text\{CAF\}\{=\}1\.72CAFnet\\text\{CAF\}\_\{\\text\{net\}\}ρ\\rhopredicts coupling×\\timesNullp=0\.59p\{=\}0\.59, sub\-criticalCAFcross\\text\{CAF\}\_\{\\text\{cross\}\}CAF\>1\\text\{CAF\}\{\>\}1in majority of conditions×\\timesText\-proxied, baseline missingCAFtemp\\text\{CAF\}\_\{\\text\{temp\}\}Authority amplifies×\\timesγA\\gamma\_\{A\}spans \[0, 15\] across versionsOf 11 conditions, 5 have been tested on real APIs across 2 model families with functional BOUNDARY\_SYNC\.### 5\.1Known Implementation Gaps

We consolidate here the known gaps between the framework’s definitions and their current implementations:

1. 1\.Within\-model C3 vs\. C5 on GPT\-4o\-mini \(R=15R=15\)\.C3 \(text\) CAF = 1\.02 \(INDEP\), C5 \(real vision,R=30R=30\) CAF = 1\.717 \(SUPER\),Δ=\+0\.70\\Delta=\+0\.70—the first clean within\-model text\-vs\-image comparison on a single vision\-capable model, validating the simulation’s prediction of super\-linear image\-condition coupling\.
2. 2\.Text conditions converge to independence under control\.Both C3u \(DeepSeek\-Chat,R=15R=15\) and C3 \(GPT\-4o\-mini,R=15R=15\) yieldCAF≈1\.0\\text\{CAF\}\\approx 1\.0, confirming that BOUNDARY\_SYNC communication without diverse personas does not measurably couple output distributions\. A dedicated cross\-model replication atR≥30R\\geq 30is needed to confirm this finding at publication\-grade precision\.
3. 3\.4 of 8 blueprint conditions tested on real APIs\.C1, C3 \(both uniform and diverse\), C5, C8u tested; C2, C4, C6, C7 remain unmeasured\.
4. 4\.C1 baseline JSD spans 17×\\timesacross backends\.Simulation∼0\.018\\sim 0\.018, GPT\-4o\-mini∼0\.21\\sim 0\.21–0\.220\.22, DeepSeek\-Chat∼0\.31\\sim 0\.31nats\. CAF ratios normalize for this, but absolute drift magnitudes matter for detection threshold design\.

These gaps are documented here not to be minimized but to be measured\. Each is a specific, scoped engineering task suitable for future work\.

## 6Discussion

### 6\.1Why the Framework Matters Even If Every Current Instantiation Is Flawed

A reviewer may ask: if the case studies all reveal limitations rather than discoveries, what is the contribution? The answer is that CAF provides something the field lacked:*falsifiability*for multi\-agent coupling claims\.

Before CAF, statements like “multi\-agent systems amplify bias” or “cross\-modal presentation strengthens propagation” or “memory increases temporal persistence” were not quantitatively falsifiable\. There was no shared metric, no baseline convention, and no protocol for verifying that a measured effect was genuine rather than an instrument artifact\. CAF provides all three:

1. 1\.A shared metric \(the ratio form of Eq\.[1](https://arxiv.org/html/2606.28839#S3.E1)\);
2. 2\.A baseline convention \(every variant explicitly names its reference condition\);
3. 3\.An audit protocol \(per\-module ablation with the same random seeds and parameters; see §4\.1\.2\)\.

That every current instantiation reveals gaps is*evidence that the falsifiability is working*—not evidence that the framework is useless\. Indeed, the most recent measurement—C5 with real vision inputs via GPT\-4o\-mini’s vision API \(§4\.1\.3\)—yielded CAF = 1\.717, the first significant super\-linear coupling that aligns with the simulation’s prediction of 1\.424\. This is the framework’s first positive real\-API discovery: the effect exists, but only when real visual representations flow through the pipeline\. Text\-proxied images mask it entirely\.

Why JSD\-to\-uniform captures coupling\.A natural objection is thatDJS⁡\(wn,tm∥w0\)\\operatorname\{D\}\_\{\\text\{JS\}\}\(w\_\{n,t\}^\{m\}\\\|w\_\{0\}\)measures individual drift from uniformity, not inter\-agent coupling\. Our ISOLATED control provides the answer: when agents cannot communicate \(ISOLATED mode\), their CAF values cluster near 1\.0 \(Table 5, OFF:0\.950\.95–1\.031\.03\); when they can \(BOUNDARY\_SYNC mode\), CAF drops to0\.810\.81–0\.900\.90\. The*difference*between ISOLATED and BOUNDARY\_SYNC CAF is the communication effect—and it is measured entirely through the JSD\-to\-uniform primitive\. More directly, the pairwise variantCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}\(§3\.2\) explicitly measures agent\-to\-agent divergence using the same ratio logic, confirming that communication drives convergence beyond isolated drift\. Both variants converge on the same result: multi\-agent text exchange causes output homogenization\.

This is the normal trajectory of a measurement instrument: the first measurements calibrate the instrument, not the phenomenon\. Thermometers took decades to standardize\. CAF is at version 1\.0\.

What would constitute a complete validation of CAF?We define three conditions that, if satisfied, would move CAF from “promising framework” to “validated instrument”:

1. 1\.Within\-model modality comparison\.A single vision\-capable model tested on both text \(C3,R≥30R\{\\geq\}30\) and real\-image \(C5,R≥30R\{\\geq\}30\) conditions, using the same discretization pipeline \(K=10K=10\) and identical BOUNDARY\_SYNC parameters\.
2. 2\.Monotonicity with controlled coupling\.A ground\-truth experiment where inter\-agent message\-passing bandwidth is varied programmatically, demonstrating that CAF increases monotonically with coupling strength\.
3. 3\.Convergent validity\.A head\-to\-head comparison against structurally richer coupling metrics \(e\.g\., CASPIAN’s causal influence tensors or PID\-TDMI’s synergy/redundancy decomposition\) on the same agent interaction traces\.

The first of these conditions has been completed \(GPT\-4o\-mini within\-model C3 vs\. C5,R=15R=15, reported in §[4\.1\.4](https://arxiv.org/html/2606.28839#S4.SS1.SSS4)\); the second and third are offered as concrete targets for community benchmarking\.

What would falsify CAF itself?A framework that claims to provide falsifiability for others should be falsifiable itself\. CAF would be falsified if, in a controlled experiment with known ground\-truth coupling strength \(e\.g\., a multi\-agent system where inter\-agent message\-passing is programmatically varied from zero to full bandwidth\), CAF failed to monotonically track the imposed coupling level, or produced CAF≈1\\approx 1under conditions known to have strong coupling\. Alternatively, if an independent, community\-adopted coupling benchmark emerged and CAF’s measurements were uncorrelated with it, that would constitute a falsification\. We state these criteria explicitly to avoid the trap of proposing an unfalsifiable framework—a measurement instrument that can never be shown to be wrong is not a measurement instrument\.

### 6\.2The Ablation Protocol as a Transferable Quality Gate

The modality\-ablation in §4\.1\.2 demonstrates a procedure that is not specific to our simulator\. Any multi\-agent LLM simulation with independently toggleable modules can adopt the following protocol \(currently validated only in simulation; real\-system validation is future work\):

> *For every claim of emergent behavior: \(a\) identify the code module\(s\) that could produce the behavior by design; \(b\) disable them while holding all random seeds, parameters, and other modules fixed; \(c\) re\-measure; \(d\) if the effect disappears, report it as an artifact and document the module; if it survives, report it as a candidate finding\.*

This protocol requires nothing beyond what a well\-engineered simulator should already have: modular design, fixed seeds, and reproducible runs\. We argue that it should be a standard quality gate for multi\-agent LLM simulation studies, analogous to the ODD protocol for agent\-based models\[[12](https://arxiv.org/html/2606.28839#bib.bib12)\]or ablation studies in deep learning\.

Concrete example\.Consider a third\-party simulator that models information diffusion among LLM agents with three modules: a content generator, a network propagator, and a sentiment amplifier\. A researcher running this simulator observes that negative\-valence content spreads faster than positive\-valence content and claims this as an emergent property\. To apply our protocol: \(a\) identify the sentiment amplifier as the module that could produce valence\-asymmetric diffusion by design \(it may contain a bias toward negative affect in its prompt template\); \(b\) disable it, keeping all other modules, random seeds, and parameters identical; \(c\) re\-measure using CAF; \(d\) if the asymmetry disappears, the claim is an artifact of the sentiment amplifier and should be reported as such\. This four\-step procedure is executable on any modular simulator without modifying the CAF framework\. We note that the current demonstration of the ablation protocol is simulation\-only\. An analogous protocol for real\-API experiments can be operationalized by varying the BOUNDARY\_SYNC blend ratioλ\\lambdafrom 0 \(ISOLATED\) to its operational value \(0\.3\), providing a “communication module” ablation\. Partial evidence exists: C3u ISOLATED \(C1, CAF≡\\equiv1\.0 by construction\) and C3u SYNC \(CAF = 1\.01\) both yield near\-unit CAF, suggesting the communication module contributes negligibly under uniform personas—a result that the ablation framework would flag as “communication effect not detected\.” A full real\-API ablation across the2×2×22\\times 2\\times 2design with controllableλ\\lambdais deferred to future work\.

### 6\.3The Text Sub\-Linearity Finding

Across all text conditions \(C2–C4, six\(N,T\)\(N,T\)combinations\), CAF is consistently below 1\.0 \(range0\.800\.80–0\.880\.88\)\. This directional result—that text\-modality multi\-agent systems exhibit output*homogenization*—is stable across the parameter space and is not an artifact \(text conditions are unchanged by the ablation\)\. If verified on real models with genuine inter\-agent communication, this would be a noteworthy finding: it contradicts the common assumption that multi\-agent interaction amplifies bias, suggesting instead that repeated text exchange causes convergence to a shared output mode\.

### 6\.4CAF Does Not Measure Content

A high CAF means outputs diverge from uniformity; it does not distinguish harmful bias amplification from beneficial diversity\. CAF is a diagnostic for coupling*degree*\. For content\-specific safety assessments, it should be paired with bias probes such as BBQ or StereoSet\.

## 7Limitations and Future Work

1. 1\.Real\-API experiments partially replicate simulation predictions\.The DeepSeek\-Chat experiment \(R=30R=30\) and the GPT\-4o\-mini experiment \(R=15R=15for C3,R=30R=30for C5\) converge to near\-independent CAF for text conditions \(DeepSeek C3u CAF = 0\.998 atR=15R=15, GPT\-4o\-mini C3 CAF = 1\.017 atR=15R=15\)\. The simulation’s super\-linear image prediction is confirmed when real JPEG pixel data is transmitted through a vision\-capable model \(C5realCAF=1\.717\\text\{CAF\}=1\.717\), while text\-only conditions with diverse personas produce sub\-linear convergence \(C3dCAF=0\.880\\text\{CAF\}=0\.880\)\. This modality gap—real vision triggers coupling that text\-only communication does not—is a key empirical finding\.
2. 2\.Output categorization dependence\.CAF depends on theKK\-way discretization\. Sensitivity toKKhas not been systematically explored beyond the epsilon\-sensitivity check in Appendix C\.
3. 3\.No task\-metric comparison\.The paper does not compare CAF against standard task\-performance metrics \(e\.g\., debate accuracy, F1\) on a multi\-agent benchmark\. Such a comparison would clarify whether CAF captures coupling phenomena that task metrics miss—a necessary step for demonstrating the framework’s added value to practitioners\.
4. 4\.BOUNDARY\_SYNC is one specific communication model\.All experiments use synchronous, round\-based population\-mean blending with a fixed cooldown of 2 steps\. Real multi\-agent deployments often use asynchronous messaging, heterogeneous agent roles, and selective information sharing\. Whether text\-driven homogenization \(CAF<1<1\) persists under these more realistic protocols is an open question\.
5. 5\.Fixed communication topology\.All experiments use BOUNDARY\_SYNC with a single cooldown parameter\. Asynchronous, dynamic, and heterogeneous topologies are not modeled\.
6. 6\.Uniform referencew0w\_\{0\}may not suit all tasks\.The uniform priorw0=\(1/K,…,1/K\)w\_\{0\}=\(1/K,\\dots,1/K\)is parameter\-free and universally applicable, but tasks with known non\-uniform prior distributions \(e\.g\., benchmark datasets with imbalanced class frequencies\) may benefit from task\-specific references\. CAF’s ratio formula supports arbitraryw0w\_\{0\}by substitution; empirical demonstrations with non\-uniform references are left to future work\.
7. 7\.Fixed simulation parameters\.The ablation conclusion \(artifact is injection\-only\) is parameter\-independent; the*specific*CAF values would shift under different bias\_strength or cooldown settings\.
8. 8\.CAF is not a safety verdict\.It measures coupling degree, not direction or desirability\.
9. 9\.No sensitivity analysis across output discretizations \(KK\)\.CAF depends on theKK\-way categorization of agent outputs\. While the epsilon\-sensitivity check \(Appendix C\) confirms JSD stability across smoothing values, the effect of varyingKK\(number of output categories\) or the binning protocol has not been explored\. Because CAF is a ratio of means and all conditions share the sameKK, theKK\-dependence largely cancels to first order, but a systematic sweep overK∈\{3,5,10,20\}K\\in\\\{3,5,10,20\\\}would confirm this\.
10. 10\.Bootstrap CIs assume within\-rep exchangeability\.The bootstrap resamples entire repetitions \(not individual time steps\), which preserves the within\-rep temporal structure\. However, cross\-rep exchangeability may be violated if simulation parameters drift or if initialization affects long\-horizon trajectories\. Block\-bootstrap or stationary bootstrap variants would provide more conservative CIs under dependence but are not implemented\.
11. 11\.TheCAFnet\\text\{CAF\}\_\{\\text\{net\}\}mean\-field mapping is ad hoc\.The mappingCAFnet​\(ρ\)=1\+α​max⁡\(ρ−ρc,0\)\\text\{CAF\}\_\{\\text\{net\}\}\(\\rho\)=1\+\\alpha\\max\(\\rho\-\\rho\_\{c\},0\)is a linearization of percolation\-style threshold behavior\. The parametersα\\alphaandρc\\rho\_\{c\}are not calibrated from data; the mapping serves as a conceptual bridge between structural and outcome measures, not as a validated predictive model\. Formal derivation from an epidemic\-threshold model is left to future work\.
12. 12\.No external ground\-truth for coupling exists\.CAF has not been benchmarked against an external ground\-truth coupling measurement, because no such standard exists in the multi\-agent LLM literature\. This chicken\-and\-egg problem is inherent to proposing a first measurement framework: the framework defines the construct, but construct validity cannot be established without an independent criterion that does not yet exist\. Until community\-adopted coupling benchmarks emerge, CAF’s validity rests on internal consistency \(bootstrap CIs, ablation logic\) and face validity \(does the ratio behave as expected under known perturbations?\), rather than on external validation against a gold\-standard coupling measure\.

## 8Conclusion

This paper makes two contributions at different levels of abstraction\.

At the measurement\-instrument level:We presented the Contagion Tensor and the CAF family—a baseline\-referenced, unitless ratio metric that quantifies output\-distribution coupling in multi\-agent LLM systems\. We applied it to four configurations, documented the empirical status of each, and demonstrated through a modality ablation that apparent emergence can be an artifact of a single design choice\.

At the methodology level:We argued that the field’s more urgent need is not another metric, but a*discipline*—a protocol for verifying that observed coupling effects are genuine rather than instrument artifacts\. The ablation protocol \(§6\.2\) provides this discipline: it requires studies to identify, disable, and re\-measure the modules that could produce claimed effects by design\. This protocol is transferable beyond CAF to any modular multi\-agent LLM simulator\.

The two layers are symbiotic: CAF makes coupling measurable, and the ablation protocol ensures the measurement is not misattributed to emergence\. Together they provide what the field currently lacks—a way to make coupling claims falsifiable, and a procedure to audit the claims that falsifiability enables\.

Code and Data Availability\.The CAF framework is implemented inunified\_caf\.pyand the simulation backend insimulation\_backend\.py\. Real\-API experiment scripts \(run\_real\_minimal\_v2\.py\) and the ablation runner \(run\_ablation\.py\) are included\. All simulation results are reproduced from a fixed master seed \(0xDEADBEEF, MD5\-derived per\-condition seeds\)\. The four supplementary experiments in Appendices[K](https://arxiv.org/html/2606.28839#A11)–[M](https://arxiv.org/html/2606.28839#A13)are generated byreviewer\_experiments\.py\. All simulation agent interaction traces \(per\-condition, per\-repetition, per\-agent, per\-step output distributions\) will be released in JSON format alongside the publication to enable head\-to\-head comparisons with alternative coupling metrics \(e\.g\., CASPIAN, PID\-TDMI, kernel\-based methods\) on shared data\. Code and data are available at[https://github\.com/\.\.\.](https://github.com/...)\(link redacted for anonymous submission; included in supplementary material\)\.

## Appendix ACAF Convergence Under Independence

Let𝔗cond\\mathfrak\{T\}\_\{\\text\{cond\}\}haveMc​Nc​TcM\_\{c\}N\_\{c\}T\_\{c\}i\.i\.d\. entries with meanμc\>0\\mu\_\{c\}\>0and finite variance\. Let𝔗base\\mathfrak\{T\}\_\{\\text\{base\}\}haveMb​Nb​TbM\_\{b\}N\_\{b\}T\_\{b\}entries with meanμb\>0\\mu\_\{b\}\>0\. By the weak law of large numbers,𝔗¯cond→𝑃μc\\bar\{\\mathfrak\{T\}\}\_\{\\text\{cond\}\}\\xrightarrow\{P\}\\mu\_\{c\}and𝔗¯base→𝑃μb\\bar\{\\mathfrak\{T\}\}\_\{\\text\{base\}\}\\xrightarrow\{P\}\\mu\_\{b\}\. The ratiog​\(x,y\)=x/yg\(x,y\)=x/yis continuous aty≠0y\\neq 0\. By the continuous mapping theorem,CAF^→𝑃μc/μb\\widehat\{\\text\{CAF\}\}\\xrightarrow\{P\}\\mu\_\{c\}/\\mu\_\{b\}\. Asymptotic variance follows from the delta method:Var​\(CAF^\)≈σc2/\(μb2​nc\)\+μc2​σb2/\(μb4​nb\)\\text\{Var\}\(\\widehat\{\\text\{CAF\}\}\)\\approx\\sigma^\{2\}\_\{c\}/\(\\mu\_\{b\}^\{2\}n\_\{c\}\)\+\\mu\_\{c\}^\{2\}\\sigma^\{2\}\_\{b\}/\(\\mu\_\{b\}^\{4\}n\_\{b\}\), wherenc=Mc​Nc​Tcn\_\{c\}=M\_\{c\}N\_\{c\}T\_\{c\}andnb=Mb​Nb​Tbn\_\{b\}=M\_\{b\}N\_\{b\}T\_\{b\}\. All bootstrap CIs in this paper useB=2000B=2000resamples; results are insensitive toBBforB≥1000B\\geq 1000\.

## Appendix BExperiment Configuration

Table 12:Simulation parameters\.
## Appendix CEpsilon Sensitivity

CAF varies by<5%<5\\%\(coefficient of variation\) acrossε∈\{10−12,10−10,10−8,10−6,10−5,10−4,10−3,10−2,10−1\}\\varepsilon\\in\\\{10^\{\-12\},10^\{\-10\},10^\{\-8\},10^\{\-6\},10^\{\-5\},10^\{\-4\},10^\{\-3\},10^\{\-2\},10^\{\-1\}\\\}for all eight conditions, confirming robustness to the JSD smoothing parameter\. The implementation default isε=10−8\\varepsilon=10^\{\-8\}\.

## Appendix DFull Ablation Data with CIs

Table 13:Complete ON vs\. OFF data\.CondONOFFΔ\\Delta95% CI \(ON\) / \(OFF\)C51\.4240\.898−0\.526\-0\.526\[1\.373, 1\.476\] / \[0\.867, 0\.930\]C61\.3700\.866−0\.504\-0\.504\[1\.330, 1\.414\] / \[0\.841, 0\.892\]C71\.4320\.884−0\.548\-0\.548\[1\.388, 1\.478\] / \[0\.859, 0\.911\]C81\.3620\.814−0\.548\-0\.548\[1\.324, 1\.402\] / \[0\.794, 0\.835\]Text \(C2–C4\): all unchanged, ON==OFF=0\.840=0\.840\(byte\-identical execution\)\.
## Appendix EFactor\-Effect Decomposition

Table 14:Main effects of modality \(MM\), agent count \(NN\), and timestep count \(TT\) from the2×2×22\\times 2\\times 2design \(injection ON\)\. Effects computed as marginal means across the other two factors\.FactorLevelMarginal CAFEffectMM\(Modality\)text0\.840—image1\.397\+0\.557\+0\.557NN\(Agents\)31\.098—51\.068−0\.030\-0\.030TT\(Timesteps\)101\.099—201\.068−0\.031\-0\.031TheMMeffect \(\+0\.557\+0\.557\) is the injection artifact\.NNandTTeffects are negligible\.
## Appendix FTheoretical Properties of CAFpair\{\}\_\{\\text\{pair\}\}

We state and prove three basic properties ofCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}\.

##### Property 1 \(Range\)\.

CAFpair≥0\\text\{CAF\}\_\{\\text\{pair\}\}\\geq 0\. If all agent output distributions are identical within each condition, thenCAFpair=0\\text\{CAF\}\_\{\\text\{pair\}\}=0\(perfect convergence\)\. If agent distributions are maximally divergent \(support on disjoint categories\),CAFpair\\text\{CAF\}\_\{\\text\{pair\}\}is bounded above by the ratio of the maximum possible pairwise JSD to the baseline pairwise JSD\. Under the uniform referencew0=\(1/K,…,1/K\)w\_\{0\}=\(1/K,\\dots,1/K\)and base\-2 JSD,DJS≤1\\operatorname\{D\}\_\{\\text\{JS\}\}\\leq 1, soCAFpair≤1/𝔼​\[DpairC1\]\\text\{CAF\}\_\{\\text\{pair\}\}\\leq 1/\\mathbb\{E\}\[D\_\{\\text\{pair\}\}^\{\\text\{C1\}\}\]\.

##### Property 2 \(Independence convergence\)\.

Under the null hypothesis that inter\-agent communication does not alter the joint distribution of agent output pairs \(i\.e\., the pairwise JSD distribution is identical in condition and baseline\), the weak law of large numbers forα\\alpha\-mixing sequences impliesCAF^pair→𝑃1\\widehat\{\\text\{CAF\}\}\_\{\\text\{pair\}\}\\xrightarrow\{P\}1asR→∞R\\to\\infty\. The proof is identical to that ofCAFbase\\text\{CAF\}\_\{\\text\{base\}\}\(Appendix A\), substituting the pairwise JSD means for the per\-agent JSD\-to\-uniform means\.

##### Property 3 \(Relationship toCAFbase\\text\{CAF\}\_\{\\text\{base\}\}\)\.

In general,CAFpair\\text\{CAF\}\_\{\\text\{pair\}\}andCAFbase\\text\{CAF\}\_\{\\text\{base\}\}measure related but distinct quantities\.CAFbase\\text\{CAF\}\_\{\\text\{base\}\}captures each agent’s divergence from a shared referencew0w\_\{0\};CAFpair\\text\{CAF\}\_\{\\text\{pair\}\}captures divergence*between*agents\. They are directionally correlated when the dominant source of distributional change operates uniformly across agents \(as in the image perturbation module—both metrics increase\)\. They may diverge when agents respond heterogeneously to the same communication event \(e\.g\., one agent shifts toward an extreme while another shifts toward the opposite extreme—CAFbase\\text\{CAF\}\_\{\\text\{base\}\}increases whileCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}may also increase due to increased between\-agent divergence\)\. Empirically, Table[7](https://arxiv.org/html/2606.28839#S4.T7)confirms directional consistency: all seven non\-baseline conditions show the same sign inCAFbase\\text\{CAF\}\_\{\\text\{base\}\}andCAFpair\\text\{CAF\}\_\{\\text\{pair\}\}\.

## Appendix GCategory Definitions for Real\-API Experiments

Table[15](https://arxiv.org/html/2606.28839#A7.T15)provides the five output categories used for discretizing real\-API \(DeepSeek\-Chat\) responses in Experiments 3 \(§[4\.1\.4](https://arxiv.org/html/2606.28839#S4.SS1.SSS4)\)\.

Table 15:Real\-API output categories and extraction rules\.Primary extraction attempts JSON parsing; if parsing fails, keyword substring matching is used\. Formal human validation of the keyword fallback has not been conducted\.

The categories were chosen to capture both the*direction*of gender bias \(female vs\. male\) and the*framing*\(stereotype\-avoidant vs\. reinforcing\)\. Theneutralcategory serves as a catch\-all for responses that do not express bias in either direction\.

Limitations\.\(1\) The five\-category taxonomy is task\-specific \(gender\-bias evaluation\); other CAF applications will require domain\-appropriate categories\. \(2\) The keyword fallback uses substring matching and has not been validated against human annotations\. We recommend that production CAF deployments use structured extraction \(JSON parsing with validation\) or independently calibrated text classifiers\. \(3\) Sensitivity to category design should be assessed per application; theKK\-sensitivity analysis in Appendix[K](https://arxiv.org/html/2606.28839#A11)provides a template for such assessments\.

## Appendix HBias\-Strength Parameter Sweep

Table[16](https://arxiv.org/html/2606.28839#A8.T16)reports CAF for C3 and C8 asbias\_strengthvaries from 0\.1 to 0\.9\. The simulation’s diverse\-persona C8 value \(CAFC8=11\.720\\text\{CAF\}\_\{\\text\{C8\}\}=11\.720\) is not reachable at any bias\_strength \(CAFC3=3\.300\\text\{CAF\}\_\{\\text\{C3\}\}=3\.300from the diverse\-persona simulation is similarly unreachable\)\.

Table 16:CAF vs\. bias\_strength\.Simulation diverse\-persona values: C3=3\.300, C8=11\.720\. Closest: bias=0\.1 \(C3=0\.912, C8=5\.511\)\.

C3 CAF is nearly invariant to bias\_strength \(CV = 2\.1%; range \[0\.850, 0\.912\]\), remaining consistently sub\-linear\. C8 CAF decreases monotonically from 5\.51 at bias=0\.1 to 0\.60 at bias=0\.9\. The sign reversal for C3 and the order\-of\-magnitude discrepancy for C8 indicate that the current simulation does not capture real\-model coupling dynamics, regardless of parameter tuning\.

## Appendix IComparative Baselines

Table[17](https://arxiv.org/html/2606.28839#A9.T17)compares CAF against entropy and variance ratios on the same simulation data\.

Table 17:CAF vs\. entropy and variance ratios\.Entropy ratio∈\\in\[0\.999, 1\.002\]; Variance ratio∈\\in\[0\.977, 0\.996\]\.

Both baselines are near 1\.0 across all conditions and cannot separate text from image modalities\. CAF captures the modality effect because JSD is sensitive to*which*categories receive probability mass, not just how concentrated the mass is\. This directly answers whether simpler metrics could substitute for CAF: the modality bifurcation—the paper’s central simulation finding—is invisible to entropy and variance\.

Code:round2\_experiments\.pygenerates Appendices[H](https://arxiv.org/html/2606.28839#A8)\-[I](https://arxiv.org/html/2606.28839#A9)\.

## Appendix JValidation Roadmap

Table[18](https://arxiv.org/html/2606.28839#A10.T18)summarizes the current validation status of every blueprint condition and provides target repetition counts for publication\-grade benchmarks\.

Table 18:Condition\-level validation roadmap\.CondModalityCurrent RTarget RModelPriorityC1text3030DeepSeek\-Chat— \(baseline\)C2text030—P2C3utext1530DeepSeek\-Chat— \(R=15 done, R=30 target\)C3dtext3030DeepSeek\-Chat— \(done\)C4text030—P2C5image3030GPT\-4o\-mini— \(done\)C6image030Vision modelP2C7image030Vision modelP2C8utext1530DeepSeek\-Chat— \(R=15 done, R=30 target\)P0: blocking for publication\-grade evidence\. P2: desirable but not blocking\.The roadmap provides concrete targets: \(1\) C5 R=30 on GPT\-4o\-mini with synthetic scenes is complete \(CAF=1\.717\\text\{CAF\}=1\.717, CI \[1\.700, 1\.733\]\); \(2\) replication with real\-world photographs is the next priority; \(3\) within\-model C3\-vs\-C5 at R=30 on at least one additional vision model \(P2\); \(4\) replication of the C2/C4/C6/C7 cells to complete the2×2×22\\times 2\\times 2design \(P2\)\.

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## Appendix KK\-Sensitivity of CAF

Table[19](https://arxiv.org/html/2606.28839#A11.T19)reports CAFbase\{\}\_\{\\text\{base\}\}for all eight conditions acrossK∈\{3,5,10,20\}K\\in\\\{3,5,10,20\\\}\. The modality bifurcation \(image super\-linear, text sub\-linear\) emerges clearly atK≥10K\\geq 10\. AtK=3K=3, image conditions become sub\-linear \(C5 = 0\.88, C8 = 0\.87\), suggesting that very coarse categorizations mask distributional differences that the image perturbation module introduces\. AtK=20K=20, image CAF values reach 2\.7–2\.9, more than double theK=10K=10values, indicating that finer\-grained spaces amplify the measured coupling\.

Table 19:CAF as a function ofKK\(number of output categories\)\.Image conditions: sub\-linear atK=3K\{=\}3, near 1\.0 atK=5K\{=\}5, super\-linear atK≥10K\{\\geq\}10\. Text conditions: consistently sub\-linear across allKK\.

All CAF values are computed with the same formula, seed, and parameters; onlyKKvaries\. TheKK\-dependence arises because JSD resolution increases with the dimensionality of the simplex: with fewer categories, probability mass concentrates, reducing measurable divergence\. This sensitivity underscores the importance of reportingKKexplicitly in any CAF\-based study and motivates a standardizedKKfor cross\-study comparisons\. Note that the CAF inflation at higherKKis not a denominator artifact: C1 baseline JSD increases withKK\(from 0\.002 atK=3K\{=\}3to 0\.018 atK=10K\{=\}10to 0\.035 atK=20K\{=\}20\), so the CAF ratio reflects genuine amplification of the condition mean relative to a growing baseline, not a vanishing denominator\.

## Appendix LBOUNDARY\_SYNC Mixing Coefficient Robustness

### L\.1Simulation

Table[20](https://arxiv.org/html/2606.28839#A12.T20)reports CAF for C3 and C5 as the BOUNDARY\_SYNC population\-mean blending ratio varies from 0% to 100% in the simulation backend\. Across the full range, C3 CAF remains sub\-linear \(0\.87–0\.90\) and C5 CAF remains super\-linear \(1\.42–1\.50\)\. The coefficient of variation across blend ratios is 0\.8% \(C3\) and 2\.1% \(C5\)\.

Table 20:CAF vs\. BOUNDARY\_SYNC blend ratio \(simulation,R=30R=30\)\.C3 CV: 0\.8%; C5 CV: 2\.1%\. Default blend in all experiments: 30%\.

This robustness means that the reported CAF values—and the qualitative conclusions drawn from them—are insensitive to the specific choice of blend ratio in simulation\. The coupling effect is driven primarily by the pre\-isolation synchronization \(which equalizes initial agent states\) and the per\-agent generation\-time contagion, rather than by the post\-generation population\-mean blending\.

## Appendix MBaseline Selection Protocol

Based on the lessons of the four CAF variants, we codify the following protocol for selecting a baseline condition when instantiating a new CAF variant:

1. 1\.Name the baseline explicitly\.The baseline condition must be a specific, reproducible experimental configuration \(not a conceptual abstraction\)\. For CAFbase\{\}\_\{\\text\{base\}\}, this is C1 \(ISOLATED, text/3/10\)\. For CAFcross\{\}\_\{\\text\{cross\}\}, a single\-modality mean serves as the baseline\. The baseline should be defined*before*any measurements are taken\.
2. 2\.Measure the baseline\.The baseline must be an empirical measurement collected under identical conditions \(same seeds, parameters, hardware\) as the conditions it anchors\. The placeholder “\[Hold: analysis pending\]” that appeared in the original P2 manuscript violates this rule\.
3. 3\.Version\-pin the baseline\.If the experiment is re\-run with modified parameters, the baseline must be re\-measured and the new pair \(condition, baseline\) must be reported as a separate data point\. The CAFtemp\{\}\_\{\\text\{temp\}\}drift \(0\.00–15\.05\) was caused by version\-dependent baselines that were never re\-anchored\.
4. 4\.Report the baseline’s absolute JSD\.A CAF value without the underlying baseline JSD is uninterpretable \(a CAF of 1\.4 could mean a small effect on a noisy baseline or a large effect on a stable one\)\. All tables in this paper include baseline JSD means where available\.

This protocol is not CAF\-specific: it applies to any baseline\-referenced measurement in multi\-agent evaluation\. Adherence would have prevented all three failure modes documented in §4\.2 \(sub\-critical detection, missing baseline, version drift\)\.

## Appendix NDetailed Descriptions of Additional CAF Variants

This appendix provides the full descriptions of the three diagnostic CAF variants summarized in §4\.2\.

##### CAFnet\\text\{CAF\}\_\{\\text\{net\}\}\(Network topology\)\.

The original P1 experiment measured the spectral radiusρ=1\.402\\rho=1\.402of a weighted*inter\-model*interaction graph \(edge weights were cross\-model API coupling measurements among model pairs\)\. Under a mean\-field mappingCAFnet​\(ρ\)=1\+α​max⁡\(ρ−ρc,0\)\\text\{CAF\}\_\{\\text\{net\}\}\(\\rho\)=1\+\\alpha\\max\(\\rho\-\\rho\_\{c\},0\), a percolation thresholdρc\>1\.402\\rho\_\{c\}\>1\.402places the network in the sub\-critical regime \(CAFnet≈1\\text\{CAF\}\_\{\\text\{net\}\}\\approx 1\), consistent with the reported null result \(p=0\.589p=0\.589\)\. CAF does not produce a new number here; it reveals that the originalρ\\rhometric lacked a baseline and therefore could not distinguish critical from sub\-critical coupling\.

Note on terminology\.CAFnet\\text\{CAF\}\_\{\\text\{net\}\}in this case study refers to the*inter\-model*coupling graph \(how biases propagate across model families\), which is distinct from the*inter\-agent*communication topology inCAFbase\\text\{CAF\}\_\{\\text\{base\}\}\(how outputs propagate across agents within the same simulation\)\. Both use the same CAF variant because the mathematical operation—comparing a connected network’s tensor mean to an isolated baseline—is identical; only the graph’s nodes differ \(models vs\. agents\)\.

##### CAFcross\\text\{CAF\}\_\{\\text\{cross\}\}\(Cross\-modal coupling\)\.

The original P2 experiment claimed cross\-modal coupling in the majority of conditions, but used text\-proxied image prompts and left the baseline C1 as a placeholder in the manuscript\.CAFcross\\text\{CAF\}\_\{\\text\{cross\}\}is well\-defined and implemented, but the data required to instantiate it does not yet exist\. The CAF re\-expression makes this explicit: a metric with a required baseline cannot be reported until the baseline is measured\.

##### CAFtemp\\text\{CAF\}\_\{\\text\{temp\}\}\(Temporal memory\)\.

The original P3 experiment reported three differentΓA\\Gamma\_\{A\}values for the same condition:0\.000\.00\(abstract\),8\.178\.17\(discussion\), and11\.4511\.45\(JSON\)\. The P3 repository contains at least seven experimental versions whoseγA\\gamma\_\{A\}spans 0\.0 to 15\.05\.CAFtemp\\text\{CAF\}\_\{\\text\{temp\}\}traces this instability: the no\-memory baseline—the denominator of the CAF ratio—was version\-dependent, and each version calibrated it differently\. The CAF re\-expression does not resolve the inconsistency but identifies its source\.

##### Why include these variants if they lack clean data?

Each variant demonstrates a different failure mode of measurement without an explicit baseline: \(i\) sub\-critical detection without a reference condition \(CAFnet\\text\{CAF\}\_\{\\text\{net\}\}\), \(ii\) baseline treated as optional \(CAFcross\\text\{CAF\}\_\{\\text\{cross\}\}\), \(iii\) baseline version drift producing contradictory published numbers \(CAFtemp\\text\{CAF\}\_\{\\text\{temp\}\}\)\. These are not unique to our prior work; they are generic risks in any multi\-agent measurement study\. Documenting them under a unified notation makes the risks visible and the remedies explicit\.