Hidden Anchors in Multi-Agent LLM Deliberation

# Hidden Anchors in Multi-Agent LLM Deliberation
Source: [https://arxiv.org/html/2606.19494](https://arxiv.org/html/2606.19494)
Ram Dantu Department of CSE University of North Texas 76207, Denton, TX, USA apurba\.pokharel@unt\.eduram\.dantu@unt\.edu

###### Abstract

Multi\-agent LLM deliberation, where agents exchange and revise answers over several rounds, is increasingly used to improve reasoning and accuracy, yet how and why it works is rarely modelled\. Such deliberation mirrors how humans reach decisions\. As social animals we are pulled both by the group, the herd effect that classical opinion\-dynamics models such as DeGroot and Friedkin–Johnsen capture, and by our own internal belief, which they do not\. We model multi\-agent deliberation as a closed\-loop dynamical system in which each agent carries a hidden internal belief, itsanchor, that continually pulls its opinion regardless of its neighbours\. We show this anchor can be recovered from the deliberation alone, and that it explains a behaviour classical consensus rules forbid: an agent’s confidence in the correct answer can climb past where any agent started, escaping the space \(convex hull\) formed by the initial beliefs\. Checking whether the recovered anchor also predicts held\-out runs \(generalizes\) gives a simple test for when a model is truly driven by such an anchor\. Across three open\-weight model families this is a spectrum, not all\-or\-nothing\. All anchors’ influence are about equally strongly, but they differ in where the anchor sits, and only when it sits far from the initial opinions does deliberation escape the hull and need the full closed\-loop model\.

Hidden Anchors in Multi\-Agent LLM Deliberation

Apurba Pokharel and Ram DantuDepartment of CSEUniversity of North Texas76207, Denton, TX, USAapurba\.pokharel@unt\.eduandram\.dantu@unt\.edu

## 1Introduction

Multi\-Agent Systems \(MAS\):LLM\-based multi\-agent systems are showing remarkable performance, and as a result their adoption is growing rapidlyBecker \([2024](https://arxiv.org/html/2606.19494#bib.bib1)\)\. In particular, deliberation\-based multi\-agent systems, where agents exchange and revise answers over multiple rounds, have seen massive research effortDu et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib4)\); Liang et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib12)\); Chan et al\. \([2023](https://arxiv.org/html/2606.19494#bib.bib2)\)\. These works are overwhelmingly directed at improving performance by designing better deliberation frameworks, while how and why deliberation works remains largely unexamined\. Prior work treats deliberation as a black box that empirically improves accuracy, and to our knowledge nobody models the deliberation itself as a dynamical system\. This is the focus of our work, and we show that this helps explain the behavior of argentic deliberation\.

Modelling MAS as a dynamic system:The opinion\-dynamics literature offers classical linear consensus rules, DeGrootProskurnikov and Tempo \([2017](https://arxiv.org/html/2606.19494#bib.bib16)\), Hegselmann–KrauseHegselmann and Krause \([2002](https://arxiv.org/html/2606.19494#bib.bib9)\), and Friedkin–JohnsenFriedkin and Johnsen \([1999](https://arxiv.org/html/2606.19494#bib.bib6)\)\(§[2\.2](https://arxiv.org/html/2606.19494#S2.SS2)\), but these cannot reproduce LLM deliberation\. We frequently observe that across deliberation runs the probability of the class the group settles on rises while the others dip, so the correct trajectory escapes the convex hull/space formed by the group’s initial beliefs\. Classical consensus cannot model this, as every opinion at every round stays inside that initial hull\. We hypothesize a hidden driving force these classical models omit, and argue it is the LLM’s own internal belief\.

LLM deliberation is a closed\-loop system:Multi\-agent systems are commonly cast as either open\-loop or closed\-loopÅström and Murray \([2008](https://arxiv.org/html/2606.19494#bib.bib19)\)\. An open\-loop system evolves from its own state and inputs with no external control fed to steer it, whereas a closed\-loop system adds a feedback/control term that drives the state toward a reference, letting the trajectory settle away from the average of its inputs\. LLM deliberation’s implementation, in prior work cited above and in this work, uses no such feedback\. Each agent only exchanges opinions with its neighbours, so the update is a function of the agents’ states alone\. This is exactly why we apply classical open loop consensus dynamics, but find that it cannot reproduce the escape above\. We therefore model deliberation as a closed\-loop system in which each agent is assigned a hidden anchor, its internal belief\. This anchor is the control signal absent from the open\-loop view, and it is what lets the trajectory, leave the initial hull\.

Contributions:Our contributions are the following\. \(a\) A new closed\-loop interaction dynamics for multi\-agent LLM deliberation that includes a hidden per\-agent anchor\. \(b\) An empirical characterisation of where this dynamics converges, showing that deliberation settles within the convex hull defined by the agents’ recovered anchor beliefs \(§[7\.4](https://arxiv.org/html/2606.19494#S7.SS4), Figure[2](https://arxiv.org/html/2606.19494#S7.F2)\)\. \(c\) A system identification and held\-out validation procedure that recovers the anchor from trajectories and acts as a model selection test, showing that anchor strength is a spectrum across model families rather than a uniform property\.

## 2Related Work

### 2\.1Multi\-Agent LLM Debate

Multi\-agent deliberation among LLMs improves reasoning and accuracy\.Du et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib4)\)let model instances debate over multiple rounds, reporting gains on mathematical and strategic reasoning\.Liang et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib12)\)push agents into adversarial “tit\-for\-tat” exchanges refereed by a judge, countering degeneration of thoughts\. ChatEvalChan et al\. \([2023](https://arxiv.org/html/2606.19494#bib.bib2)\)turns role\-based multi\-agent debate into a stronger automatic evaluator\. In these works the round\-by\-round trajectory of belief is never modelled, and there is no account of why deliberation converges where it does\. Closing that gap is the contribution of this paper\.

### 2\.2Opinion Dynamics and Consensus

The trajectory of interacting beliefs is formalised in the opinion\-dynamics literature\. DeGroot learningProskurnikov and Tempo \([2017](https://arxiv.org/html/2606.19494#bib.bib16)\)replaces each opinion with a weighted average of its neighbours’, the Friedkin–Johnsen modelFriedkin and Johnsen \([1999](https://arxiv.org/html/2606.19494#bib.bib6)\)adds per\-agent anchoring to the initial opinion, and the Hegselmann–KrauseHegselmann and Krause \([2002](https://arxiv.org/html/2606.19494#bib.bib9)\)bounded\-confidence rule averages only over sufficiently close neighbours\. All three share a convex\-hull bound\. Every update is a convex combination of current \(and, for Friedkin–Johnsen, initial\) opinions, so no coordinate can leave the convex hull of the initial opinions\. We observe LLM deliberation that violates this bound for some model families, which a classical linear rule cannot reproduce and which motivates the closed\-loop model we develop\.

### 2\.3Opinion\-Dynamics Simulation via LLMs

A separate line of work uses LLM agents to simulate classical opinion dynamics and asks whether LLMs reproduce human social behaviour\. OpinioNetLiu et al\. \([2026](https://arxiv.org/html/2606.19494#bib.bib13)\)models ideological community agents updating through external\-event influence, network structure, and opinion inertia, outperforming Friedkin–Johnsen, Hegselmann–Krause, and Deffuant–Weisbuch on real social\-media trajectories\.He et al\. \([2026](https://arxiv.org/html/2606.19494#bib.bib8)\)run multi\-round LLM dialogues that retain each agent’s initial opinion and conclude that LLM opinion formation is “largely consistent with Friedkin–Johnsen”\.Chuang et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib3)\)report that networked LLM agents bias toward accurate consensus and fragment only when prompted with confirmation bias\. All three impose a prescribed classical rule and use the LLM for simulation\. None analyse the deliberation system itself or recover a per\-agent latent state from trajectories\. The contrast is sharpest againstHe et al\. \([2026](https://arxiv.org/html/2606.19494#bib.bib8)\): their Friedkin–Johnsen consistency predicts every coordinate stays inside the initial hull, yet we find this holds only for families whose recovered anchor coincides with the initial opinion and fails for those whose anchor lies elsewhere, where the gold\-class coordinate leaves the hull\.

## 3Problem Setting

We study a population ofnnLLM agents that deliberate over a fixed multiple\-choice question withddpossible answer classes\. The agents communicate over a directed graphG=\(V,E\)G=\(V,E\)withV=\{1,…,n\}V=\\\{1,\\dots,n\\\}\. We write𝒩i=\{j:\(j,i\)∈E\}\\mathcal\{N\}\_\{i\}=\\\{j:\(j,i\)\\in E\\\}for the neighbours whose outputs agentiiobserves, andA∈\{0,1\}n×nA\\in\\\{0,1\\\}^\{n\\times n\}for the adjacency matrix withAi​j=1A\_\{ij\}=1iffj∈𝒩ij\\in\\mathcal\{N\}\_\{i\}\. Deliberation proceeds in synchronous roundsk=0,1,…,Kk=0,1,\\dots,K\. At each round, agentii’s belief is a probability vector

𝐱i​\(k\)=\(xi,1​\(k\),…,xi,d​\(k\)\)∈Δd−1,\\mathbf\{x\}\_\{i\}\(k\)=\\big\(x\_\{i,1\}\(k\),\\dots,x\_\{i,d\}\(k\)\\big\)\\in\\Delta^\{d\-1\},\(1\)whereΔd−1=\{𝐩∈ℝ≥0d:∑cpc=1\}\\Delta^\{d\-1\}=\\\{\\mathbf\{p\}\\in\\mathbb\{R\}^\{d\}\_\{\\geq 0\}:\\sum\_\{c\}p\_\{c\}=1\\\}is the probability simplex over theddclasses\. We letg∈\{1,…,d\}g\\in\\\{1,\\dots,d\\\}denote the index of the gold \(correct\) class and refer toxi,g​\(k\)x\_\{i,g\}\(k\)as the gold\-class coordinate\.

### 3\.1Multi\-Agent Deliberation Protocol

![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/merged_dynamics_limits.png)Figure 1:Probability trajectories \(snippet\) for the gold class \(COVID\) and a competing class \(Common Cold\) under the protocol of §[3\.1](https://arxiv.org/html/2606.19494#S3.SS1)\(n=3n\{=\}3agents, ring topology\)\. Left and middle panels show the open\-loop baselines \(DeGroot and Friedkin–Johnsen\) initialised with the agents’ real first\-round beliefs: every class stays inside the band of its initial values, as Property[1](https://arxiv.org/html/2606.19494#Thmproperty1)guarantees\. Right panels \(highlighted\) show the real LLM round\-robin deliberation: the gold\-class probability grows past the maximum initial value across all agents, the empirical anomaly \([5](https://arxiv.org/html/2606.19494#S3.E5)\) that the rest of the paper explains\.We instantiateGGas a directed ring where agentiiobserves its ring\-predecessor,𝒩i=\{\(i−1\)modn\}\\mathcal\{N\}\_\{i\}=\\\{\(i\-1\)\\bmod n\\\}, so every agent has exactly one neighbour and influence propagates around the cycle\. At round0each agent answers the question independently, producing its initial belief𝐱i​\(0\)\\mathbf\{x\}\_\{i\}\(0\)\. At every subsequent roundk\>0k\>0agentiiis re\-prompted with \(i\) the original question, \(ii\) its own previous answer, and \(iii\) the previous\-round answer of its neighbour in𝒩i\\mathcal\{N\}\_\{i\}, and is asked to reconsider and re\-rank the candidate classes\. The model returns a ranked top\-55list with self\-reported probabilities\. We map each entry to the default class label and renormalise\. This yields, for every run, a fully observed discrete\-time trajectory which is the object the rest of the paper models\. Figure[1](https://arxiv.org/html/2606.19494#S3.F1)\(right\) shows one such trajectory\.

### 3\.2Open\-Loop Consensus Baselines

Under DeGroot dynamicsProskurnikov and Tempo \([2017](https://arxiv.org/html/2606.19494#bib.bib16)\)each agent moves a fixed stepε∈\(0,1\)\\varepsilon\\in\(0,1\)towards the average of its neighbours\.

𝐱i​\(k\+1\)=𝐱i​\(k\)\+ε​∑j∈𝒩iAi​j​\(𝐱j​\(k\)−𝐱i​\(k\)\),\\begin\{split\}\\mathbf\{x\}\_\{i\}\(k\{\+\}1\)=\\;&\\mathbf\{x\}\_\{i\}\(k\)\\\\ &\+\\varepsilon\\\!\\\!\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}\\\!\\\!A\_\{ij\}\\big\(\\mathbf\{x\}\_\{j\}\(k\)\-\\mathbf\{x\}\_\{i\}\(k\)\\big\),\\end\{split\}\(2\)The Friedkin–Johnsen modelFriedkin and Johnsen \([1999](https://arxiv.org/html/2606.19494#bib.bib6)\)adds per\-agent stubbornness/bias: with susceptibilityλ∈\[0,1\]\\lambda\\in\[0,1\],

𝐱i​\(k\+1\)=λ​\(𝐱i​\(k\)\+ε​∑j∈𝒩iAi​j​\(𝐱j​\(k\)−𝐱i​\(k\)\)\)\+\(1−λ\)​𝐱i​\(0\),\\begin\{split\}\\mathbf\{x\}\_\{i\}\(k\{\+\}1\)=\\;&\\lambda\\Big\(\\mathbf\{x\}\_\{i\}\(k\)\+\\varepsilon\\\!\\\!\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}\\\!\\\!A\_\{ij\}\\big\(\\mathbf\{x\}\_\{j\}\(k\)\-\\mathbf\{x\}\_\{i\}\(k\)\\big\)\\Big\)\\\\ &\+\(1\-\\lambda\)\\,\\mathbf\{x\}\_\{i\}\(0\),\\end\{split\}\(3\)so each agent is pulled back towards its own initial opinion𝐱i​\(0\)\\mathbf\{x\}\_\{i\}\(0\)\.

###### Property 1\(Convex\-hull bound\)\.

For the updates in \([2](https://arxiv.org/html/2606.19494#S3.E2)\) and \([3](https://arxiv.org/html/2606.19494#S3.E3)\) withε\\varepsilonsmall enough that every update is a convex combination, and for every classcc,

min1≤j≤n⁡xj,c​\(0\)≤xi,c​\(k\)≤max1≤j≤n⁡xj,c​\(0\)∀i,∀k\.\\min\_\{1\\leq j\\leq n\}x\_\{j,c\}\(0\)\\;\\leq\\;x\_\{i,c\}\(k\)\\;\\leq\\;\\max\_\{1\\leq j\\leq n\}x\_\{j,c\}\(0\)\\quad\\forall i,\\,\\forall k\.\(4\)That is, no coordinate of any opinion can ever leave the convex hull of the initial opinionsconv​\{𝐱j​\(0\)\}j\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}\_\{j\}\.

Figure[1](https://arxiv.org/html/2606.19494#S3.F1)\(left and middle\) illustrates this initialised with the agents’ real first\-round beliefs, both updates keep every class inside the band of its initial values\. These baselines therefore do not merely fit poorly\. They are*structurally incapable*of any trajectory that leaves the initial hull\.

#### Empirical anomaly: escape from the convex hull\.

The motivating observation of this paper is that real deliberation violates Property[1](https://arxiv.org/html/2606.19494#Thmproperty1)on the gold\-class coordinate\. Across runs we repeatedly observe

maxi,k⁡xi,g​\(k\)\>maxj⁡xj,g​\(0\),\\max\_\{i,\\,k\}\\;x\_\{i,g\}\(k\)\\;\>\\;\\max\_\{j\}\\;x\_\{j,g\}\(0\),\(5\)the gold\-class probability rising strictly above the largest value any agent held initially, so the trajectory escapesconv​\{𝐱j​\(0\)\}j\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}\_\{j\}\. The fraction of runs in which this happens depends strongly on the model family \(§[7\.5](https://arxiv.org/html/2606.19494#S7.SS5)\)\. A hidden, per\-agent driving force must therefore be added to the dynamics\.

## 4Hidden\-Anchor Model

The missing ingredient behind the escape \([5](https://arxiv.org/html/2606.19494#S3.E5)\) is a force that does not depend on the observed opinions at all\. We therefore augment the consensus update with a hidden, per\-agent anchor𝐛i\\mathbf\{b\}\_\{i\}, the agent’s own internal belief, that continually pulls𝐱i\\mathbf\{x\}\_\{i\}towards𝐛i\\mathbf\{b\}\_\{i\}regardless of its neighbours\. We find empirically that this closed\-loop system settles intoconv​\{𝐛i\}i\\mathrm\{conv\}\\\{\\mathbf\{b\}\_\{i\}\\\}\_\{i\}rather thanconv​\{𝐱j​\(0\)\}j\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}\_\{j\}\(§[7\.4](https://arxiv.org/html/2606.19494#S7.SS4)\), reproducing the anomaly that linear consensus forbids\.

### 4\.1Hidden\-Anchor Update Rule

Each agent updates as

𝐱i​\(k\+1\)=𝐱i​\(k\)−α​∑j∈𝒩iAi​j​\(𝐱i​\(k\)−𝐱j​\(k\)\)−βi​\(𝐱i​\(k\)−𝐛i\),\\begin\{split\}\\mathbf\{x\}\_\{i\}\(k\{\+\}1\)=\\;&\\mathbf\{x\}\_\{i\}\(k\)\\\\ &\-\\alpha\\\!\\\!\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}\\\!\\\!A\_\{ij\}\\big\(\\mathbf\{x\}\_\{i\}\(k\)\-\\mathbf\{x\}\_\{j\}\(k\)\\big\)\\\\ &\-\\beta\_\{i\}\\big\(\\mathbf\{x\}\_\{i\}\(k\)\-\\mathbf\{b\}\_\{i\}\\big\),\\end\{split\}\(6\)with a single shared consensus gainα≥0\\alpha\\geq 0, a per\-agent anchor gainβi≥0\\beta\_\{i\}\\geq 0, and a hidden anchor𝐛i∈Δd−1\\mathbf\{b\}\_\{i\}\\in\\Delta^\{d\-1\}\. The first correction is the consensus pull: a scaled DeGroot step \(α\\alphaplays the role ofε\\varepsilonin Eq\. \([2](https://arxiv.org/html/2606.19494#S3.E2)\)\) that models the agent reacting to the neighbour opinion injected into its prompt\. The second is the anchor pull: a persistent attraction towards the agent’s own latent reasoning prior𝐛i\\mathbf\{b\}\_\{i\}that the prompt context does not override and that never appears in the observed opinions\. This prior is not arbitrary: an LLM carries inherent beliefs fixed by its pre\-training data and architecture, much as a person reasons from the background knowledge and predispositions they bring to a discussionTversky and Kahneman \([1974](https://arxiv.org/html/2606.19494#bib.bib17)\)\.

Equation \([6](https://arxiv.org/html/2606.19494#S4.E6)\) strictly generalises both baselines: settingβi=0\\beta\_\{i\}=0recovers DeGroot \([2](https://arxiv.org/html/2606.19494#S3.E2)\), and replacing the latent𝐛i\\mathbf\{b\}\_\{i\}by the observed initial opinion𝐱i​\(0\)\\mathbf\{x\}\_\{i\}\(0\)recovers Friedkin–Johnsen \([3](https://arxiv.org/html/2606.19494#S3.E3)\)\.

## 5System Identification

The hidden\-anchor model \([6](https://arxiv.org/html/2606.19494#S4.E6)\) is parameterised by\(α,βi,𝐛i\)\(\\alpha,\\beta\_\{i\},\\mathbf\{b\}\_\{i\}\)\. This section recovers those parameters from observed trajectories\.

Linear reparameterisation\.LetΔ​𝐱i​\(k\):=𝐱i​\(k\+1\)−𝐱i​\(k\)\\Delta\\mathbf\{x\}\_\{i\}\(k\):=\\mathbf\{x\}\_\{i\}\(k\{\+\}1\)\-\\mathbf\{x\}\_\{i\}\(k\)and𝜸i:=βi​𝐛i\\boldsymbol\{\\gamma\}\_\{i\}:=\\beta\_\{i\}\\mathbf\{b\}\_\{i\}\. With this reparameterisation, \([6](https://arxiv.org/html/2606.19494#S4.E6)\) becomes

Δ​𝐱i​\(k\)=−α​∑j∈𝒩iAi​j​\(𝐱i​\(k\)−𝐱j​\(k\)\)−βi​𝐱i​\(k\)\+𝜸i,\\Delta\\mathbf\{x\}\_\{i\}\(k\)=\-\\alpha\\\!\\\!\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}\\\!\\\!A\_\{ij\}\\big\(\\mathbf\{x\}\_\{i\}\(k\)\-\\mathbf\{x\}\_\{j\}\(k\)\\big\)\-\\beta\_\{i\}\\mathbf\{x\}\_\{i\}\(k\)\+\\boldsymbol\{\\gamma\}\_\{i\},\(7\)linear inθ=\(α,\{βi\},\{𝜸i\}\)\\theta=\(\\alpha,\\\{\\beta\_\{i\}\\\},\\\{\\boldsymbol\{\\gamma\}\_\{i\}\\\}\)\. Stacking \([7](https://arxiv.org/html/2606.19494#S5.E7)\) across all agents, rounds, class coordinates, and runs yields an overdetermined system𝐀​θ=𝐲\\mathbf\{A\}\\theta=\\mathbf\{y\}, solved by ordinary least\-squares\. Fit quality is reported as

R2​\(Δ​𝐱\)=1−∑‖Δ​𝐱i​\(k\)−Δ​𝐱^i​\(k\)‖2∑‖Δ​𝐱i​\(k\)−Δ​𝐱¯‖2,R^\{2\}\(\\Delta\\mathbf\{x\}\)=1\-\\frac\{\\sum\\\|\\Delta\\mathbf\{x\}\_\{i\}\(k\)\-\\widehat\{\\Delta\\mathbf\{x\}\}\_\{i\}\(k\)\\\|^\{2\}\}\{\\sum\\\|\\Delta\\mathbf\{x\}\_\{i\}\(k\)\-\\overline\{\\Delta\\mathbf\{x\}\}\\\|^\{2\}\},\(8\)evaluated on one\-step displacements so that the metric is not dominated by the trivial𝐱i​\(k\+1\)≈𝐱i​\(k\)\\mathbf\{x\}\_\{i\}\(k\{\+\}1\)\\approx\\mathbf\{x\}\_\{i\}\(k\)baseline\.

Identifiability\.Equation \([7](https://arxiv.org/html/2606.19494#S5.E7)\) is linear in\(α,βi,𝜸i\)\(\\alpha,\\beta\_\{i\},\\boldsymbol\{\\gamma\}\_\{i\}\)but not in\(α,βi,𝐛i\)\(\\alpha,\\beta\_\{i\},\\mathbf\{b\}\_\{i\}\), because the latter pair enters only through the product𝜸i=βi​𝐛i\\boldsymbol\{\\gamma\}\_\{i\}=\\beta\_\{i\}\\mathbf\{b\}\_\{i\}\.

Anchor recovery\.Anchors are recovered as𝐛~i=𝜸^i/β^i\\tilde\{\\mathbf\{b\}\}\_\{i\}=\\hat\{\\boldsymbol\{\\gamma\}\}\_\{i\}/\\hat\{\\beta\}\_\{i\}and projected onto the simplex,𝐛^i=ΠΔd−1​\(𝐛~i\)\\hat\{\\mathbf\{b\}\}\_\{i\}=\\Pi\_\{\\Delta^\{d\-1\}\}\(\\tilde\{\\mathbf\{b\}\}\_\{i\}\), by theO​\(d​log⁡d\)O\(d\\log d\)algorithm ofDuchi et al\. \([2008](https://arxiv.org/html/2606.19494#bib.bib5)\)\. Whenβ^i\\hat\{\\beta\}\_\{i\}is small the division is ill\-conditioned\. We reportβ^i\\hat\{\\beta\}\_\{i\}alongside every recovered anchor and flag agents below a threshold as unreliable\.

## 6Experimental Setup

We run all experiments on three open\-weight instruction\-tuned LLMs, Llama\-3\.1\-70B\-InstructGrattafiori et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib7)\), Qwen3\-32BYang et al\. \([2025](https://arxiv.org/html/2606.19494#bib.bib18)\), and gpt\-oss\-20bOpenAI \([2025](https://arxiv.org/html/2606.19494#bib.bib14)\), on a symptom→\\rightarrowdisease diagnosis task, in which each agent ranks candidate diseases for a symptom set drawn from a4242\-class diagnosis benchmarkitachi9604 \([2020](https://arxiv.org/html/2606.19494#bib.bib10)\)\. We use1010cases, each a distinct target disease, to span a range of initial\-opinion geometries\. Each\(model,case\)\(\\text\{model\},\\text\{case\}\)cell uses the protocol of §[3\.1](https://arxiv.org/html/2606.19494#S3.SS1)withn=3n\{=\}3agents on a directed ring,K=5K\{=\}5rounds, and33random seeds, yielding9090deliberation trajectories \(3030per model\)\. We implement deliberation as a round\-robin message\-passing graph in LangGraphLangChain \([2024](https://arxiv.org/html/2606.19494#bib.bib11)\), where each agent is a node that reads its own and its neighbour’s previous response and emits an updated prediction\. The interaction protocol follows the multi\-agent debate paradigm ofDu et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib4)\), and most closely resembles the more recent deliberation frameworks built on the consensus protocol ofPokharel et al\. \([2025](https://arxiv.org/html/2606.19494#bib.bib15)\)\. Evaluation metrics \(R2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\), held\-out MSE, bootstrap CIs, hull\-containment rate\), full decoding hyperparameters, prompt templates, and checkpoint identifiers are deferred to Appendix[A](https://arxiv.org/html/2606.19494#A1)\. Code, data, and analysis scripts are available at the provided zip file\.

### 6\.1Analysis Procedures

Every stored trajectory is post\-processed by a fixed battery of analyses, referenced by: Experiment \+ letter throughout the paper:

A\. Open\-loop baseline\.Initialise the linear consensus rules \(§[3\.2](https://arxiv.org/html/2606.19494#S3.SS2)\) with the agents’ first\-round beliefs and simulate forward, confirming they stay insideconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}\(Property[1](https://arxiv.org/html/2606.19494#Thmproperty1)\)\.

B\. Observed vs\. linear\.Compare the real LLM trajectory against the round\-by\-round linear\-consensus prediction, exposing where the open\-loop model fails\.

C\. System identification\.Fit the hidden\-anchor update \(§[5](https://arxiv.org/html/2606.19494#S5)\) by least squares, recovering\(α^,β^i,𝐛^i\)\(\\hat\{\\alpha\},\\hat\{\\beta\}\_\{i\},\\hat\{\\mathbf\{b\}\}\_\{i\}\)and the in\-sampleR2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)\.

D\. Anchor drift\.Refit on an early and a late window of rounds and compare anchors, distinguishing compliance \(stable anchor\) from internalisation \(moving anchor\)\.

E\. Bootstrap CIs\.Block\-bootstrap the Experiment C fit per run for parameter confidence intervals \(§[7\.7](https://arxiv.org/html/2606.19494#S7.SS7)\)\.

F\. Held\-out cross\-run validation\.Fit on a subset of a problem’s seeds and predict a held\-out seed \(leave\-one\-out ensemble\), measuring whether recovered parameters generalise\.

We additionally fit nested restrictions of the anchor model with the same least\-squares machinery as Experiment C, DeGroot \(βi=0\\beta\_\{i\}\{=\}0\) and Friedkin–Johnsen \(𝐛i=𝐱i​\(0\)\\mathbf\{b\}\_\{i\}\{=\}\\mathbf\{x\}\_\{i\}\(0\)\), so that the three models are compared on an identical target both in\-sample and under the held\-out protocol of Experiment F\. This nested comparison is a structural ablation of the anchor mechanism\.

## 7Analysis

We analyse the symptom→\\rightarrowdisease deliberation benchmark: three open\-weight models \(Llama\-3\.1\-70B, Qwen3\-32B, gpt\-oss\-20b\),1010distinct target diseases, and33random seeds per problem, for3030independent deliberation runs per model \(9090total;n=3n\{=\}3agents,K=5K\{=\}5reflection rounds, ring topology\)\. Cross\-run quantities use the leave\-one\-seed\-out protocol of Experiment F\.

### 7\.1Trajectories Are Not Uniform Across Model Families

The motivating observation comes from inspecting the trajectories directly \(Experiments A and B; Figure[1](https://arxiv.org/html/2606.19494#S3.F1)\)\. The open\-loop baselines, initialised with real first\-round beliefs, never leave the band of initial opinions, as Property[1](https://arxiv.org/html/2606.19494#Thmproperty1)guarantees\. The real deliberation behaves very differently, but not uniformly across models\. Llama\-3\.1\-70B produces sharp, non\-monotone swings in the gold\-class probability that overshoot the initial band substantially\. Qwen3\-32B and gpt\-oss\-20b are markedly flatter: their gold\-class coordinate moves little and is, in many runs, well described by a linear convex update\. Quantitatively, taking the per\-run range to be the spread \(maximum minus minimum\) of the gold\-class probability across rounds0–KKwithin a single run, its mean over the3030runs is0\.260\.26for Llama versus0\.090\.09\(Qwen\) and0\.120\.12\(gpt\-oss\) \(Table[5](https://arxiv.org/html/2606.19494#A2.T5)\), and the total variation follows the same ordering\. This rules out a single universal claim and motivates a per\-family comparison of the linear baselines against the hidden\-anchor model, in\-sample and, crucially, under held\-out validation\.

Table 1:Nested model comparison on observed deliberation trajectories \(3030runs per model\)\. The three models are the hidden\-anchor update and its restrictions DeGroot \(βi=0\\beta\_\{i\}\{=\}0\) and Friedkin–Johnsen \(𝐛i=𝐱i​\(0\)\\mathbf\{b\}\_\{i\}\{=\}\\mathbf\{x\}\_\{i\}\(0\)\), all fit by the same least squares \(Experiment C\)\. Both blocks report the meanR2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\): in\-sample over the3030runs per model, held\-out over the1010disease groups\. Held\-outR2R^\{2\}is the leave\-one\-seed\-out reconstruction of the step changes of an unseen seed \(Experiment F\)\. A negative held\-out value means the fitted model predicts the step changes worse than their own mean, so the extra freedom does more harm than good \(gpt\-oss, see text\)\. “Anchor sel\. %” is the fraction of the1010disease groups in which the full anchor model attains the best held\-outR2R^\{2\}\. Best per block in bold\.
### 7\.2In\-Sample Fit Favours the Anchor Model, but Cannot Settle It

Fit in\-sample, the full hidden\-anchor model dominates both baselines for every family \(Table[1](https://arxiv.org/html/2606.19494#S7.T1), left block\):R2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)rises from0\.120\.12–0\.300\.30\(DeGroot\) and0\.300\.30–0\.370\.37\(Friedkin–Johnsen\) to0\.610\.61–0\.860\.86\(full\)\. DeGroot is insufficient everywhere, confirming that pure neighbour\-averaging does not describe deliberation\. But the full model also carries the most free parameters \(thennlatent anchors\), so an in\-sample advantage is expected by construction and cannot on its own establish that the anchors are real rather than fit to noise\. The decisive test is whether the recovered parameters generalise\.

### 7\.3Held\-Out Validation Reveals Fit

Under leave\-one\-seed\-out validation \(Table[1](https://arxiv.org/html/2606.19494#S7.T1), right block\) the three families separate sharply\. Full breakdown in Table[6](https://arxiv.org/html/2606.19494#A2.T6)\.

- •Llama\-3\.1\-70B:the full model’s mean held\-outR2R^\{2\}is0\.440\.44against≈0\.05\\approx 0\.05for both baselines, and it is selected in8/108/10groups\. The latent anchor is a genuine, transferable property: parameters fit on two seeds predict an unseen seed where linear consensus cannot\.
- •Qwen3\-32B:the full model is still selected in7/107/10groups, but its mean held\-outR2R^\{2\}\(0\.080\.08\) sits at or just below the baselines \(≈0\.10\\approx 0\.10\): a weak anchor signal on near\-linear dynamics, where the few overfit folds already pull the mean down\. The transfer here lives in the selection count rather than in the average fit\.
- •gpt\-oss\-20b:the baselines win \(mean≈0\.13\\approx 0\.13vs\.−0\.94\-0\.94, full selected in only2/102/10groups\)\. The full model’s mean held\-outR2R^\{2\}is sharply negative: on an unseen seed it predicts the step changes worse than simply using their mean, so adding the hidden anchor does more harm than good\. Its deliberation is essentially linear consensus, with no transferable latent anchor\.

We therefore read held\-out validation as amodel\-selectioncriterion: it determines when a hidden anchor governs deliberation rather than imposing one\. That the procedure selects the linear baselines for gpt\-oss, and penalises the over\-parameterised anchor model exactly where no anchor is present, is evidence that the recovered anchors elsewhere are not an artefact of model capacity\.

Table 2:Recovered\-anchor geometry \(Experiment C,3030runs per model\)\.β^¯\\bar\{\\hat\{\\beta\}\}is the mean per\-agent anchor gain\. “𝐛^\\hat\{\\mathbf\{b\}\}margin” is the median overshoot of the recovered anchors past the initial bandconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}on the gold coordinate\. “% out” is the fraction of runs whose anchors lie more than0\.100\.10outside that band\. “SS cont\.” is the steady\-state \(final\-round\) containment rate inconv​\{𝐛^j\}\\mathrm\{conv\}\\\{\\hat\{\\mathbf\{b\}\}\_\{j\}\\\}at tolerance0\.050\.05\.
### 7\.4Recovered Anchor Geometry Explains the Gradient

Why does the same procedure certify an anchor for Llama but reduce to a linear baseline for gpt\-oss? The recovered parameters answer this directly \(Table[2](https://arxiv.org/html/2606.19494#S7.T2)\)\. The anchor gain does not discriminate: the mean per\-agent gainβ^¯\\bar\{\\hat\{\\beta\}\}is essentially flat across families \(0\.340\.34–0\.360\.36\)\. What differs is the anchor location\. For Llama the recovered anchors𝐛^i\\hat\{\\mathbf\{b\}\}\_\{i\}lie far outside the band of initial opinions on the gold coordinate \(median margin0\.330\.33, outside in92%92\\%of runs\)\. For Qwen and gpt\-oss they sit essentially at the initial band \(median0\.100\.10\)\. An anchor that coincides with the initial opinion,𝐛^i≈𝐱i​\(0\)\\hat\{\\mathbf\{b\}\}\_\{i\}\\\!\\approx\\\!\\mathbf\{x\}\_\{i\}\(0\), is exactly the Friedkin–Johnsen special case of \([6](https://arxiv.org/html/2606.19494#S4.E6)\) \(cf\. \([3](https://arxiv.org/html/2606.19494#S3.E3)\)\), which explains why the held\-out criterion selects FJ for gpt\-oss: that model is anchored, to its own starting belief, and so cannot leave the initial hull\.

This geometry also fixes where deliberation converges\. Empirically the final\-round opinions settle inside the recovered anchor hullconv​\{𝐛^j\}\\mathrm\{conv\}\\\{\\hat\{\\mathbf\{b\}\}\_\{j\}\\\}rather than the initial one, in74%74\\%\(Llama\),74%74\\%\(Qwen\), and60%60\\%\(gpt\-oss\) of runs, far above the2929–48%48\\%whole\-trajectory rate, since intermediate iterates overshoot before settling\. The trajectory therefore escapesconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}precisely when some recovered anchor lies outside it\. This containment is the empirical content of contribution \(b\)\. Figure[2](https://arxiv.org/html/2606.19494#S7.F2)shows a representative case \(GERD, seed 2\): Llama’s opinions converge into the anchor hull while Qwen’s never enter it\. The gpt\-oss case is indistinguishable from Qwen on this run \(no point inside the hull,75%75\\%PCA variance captured\)\.

![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/hull_llama_gerd_seed2.png)\(a\)Llama\-3\.1\-70B: opinions settle insideconv​\{b^i\}\\mathrm\{conv\}\\\{\\hat\{b\}\_\{i\}\\\}\.
![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/hull_qwen_gerd_seed2.png)\(b\)Qwen3\-32B: opinions never enter the anchor hull \(0/180/18\)\.

Figure 2:Recovered anchor hullconv​\{b^i\}\\mathrm\{conv\}\\\{\\hat\{b\}\_\{i\}\\\}\(shaded\) and the deliberation trajectory \(PCA\-projected; rounds0–55\) for one case \(GERD, seed 2\)\. Llama’s later\-round opinions converge into the hull \(steady state contained\), whereas Qwen’s stay outside throughout\. The gpt\-oss case matches Qwen \(omitted,0/180/18inside,75%75\\%variance captured\)\. This is the geometric counterpart of the steady\-state containment rates in Table[2](https://arxiv.org/html/2606.19494#S7.T2)\.
### 7\.5Hull Escape Is Family\-Dependent

The geometric anomaly of \([5](https://arxiv.org/html/2606.19494#S3.E5)\), the gold\-class probability leavingconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}, which Property[1](https://arxiv.org/html/2606.19494#Thmproperty1)forbids for the linear baselines, tracks the same ordering \(Table[3](https://arxiv.org/html/2606.19494#S7.T3)\)\. We call a run an escape at toleranceτ\\tauwhen its escape margin, the largest overshoot of any agent’s gold\-class probability past the round\-0bandconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}in any later round \(Table[3](https://arxiv.org/html/2606.19494#S7.T3)\), exceedsτ\\tau, withτ\\taumeasured in probability units\. The verdict depends on the tolerance required: at a negligibleτ=0\.02\\tau\{=\}0\.02all three models escape in≈75%\\approx 75\\%of runs, but at a meaningfulτ=0\.10\\tau\{=\}0\.10only Llama escapes substantially \(77%77\\%of runs; mean overshoot0\.220\.22\), while Qwen and gpt\-oss escape in≈25%\\approx 25\\%of runs with mean overshoot≈0\.07\\approx 0\.07\. The hull\-violating dynamics are thus concentrated in the family for which held\-out validation certifies a latent anchor, and follow directly from its anchor geometry \(§[7\.4](https://arxiv.org/html/2606.19494#S7.SS4)\)\. Only Llama’s recovered anchors lie far enough outside the initial hull to drive the trajectory past it\.

Table 3:Gold\-coordinate escape margin beyondconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}\(the band of initial opinions\), over3030runs per model\. The margin is the largest overshoot of any agent’s gold\-class probability past the round\-0band in any later round\.
### 7\.6Anchor Behaviour is a Spectrum

Hidden\-anchor behaviour is not all\-or\-nothing but a spectrum across model families: Llama\-3\.1\-70B≫\\ggQwen3\-32B\>\>gpt\-oss\-20b, with gpt\-oss at the linear\-consensus \(Friedkin–Johnsen\) end of the scale\. The mechanism is uniform, every family anchors to a latent prior with comparable strength, but only when that prior sits far from the initial opinions does deliberation escape the convex hull and demand the full closed\-loop model\. Our held\-out criterion recovers that boundary without supervision\.

Table 4:Per\-run bootstrap uncertainty \(Experiment E,B=1000B\{=\}1000; all values % except the last\)\. “β^¯≤0\\bar\{\\hat\{\\beta\}\}\\\!\\leq\\\!0” is runs with non\-positive mean anchor gain\. “β^\\hat\{\\beta\}CI∋0\\ni 0” is agent\-runs whose anchor CI contains zero\. “α^\\hat\{\\alpha\}sig\.” is runs whose consensus\-gain CI excludes zero\. The last column is the median width of the per\-agentβ^\\hat\{\\beta\}CI\.
### 7\.7Per\-Run Parameter Uncertainty

Block\-bootstrapping each fit \(Experiment EB=1000B\{=\}1000resamples of the transition pool\) quantifies how well\-determined the recovered parameters are within a single run \(Table[4](https://arxiv.org/html/2606.19494#S7.T4)\)\. Two things follow\. First, the model is not a perfect fit: the mean anchor gain is non\-positive in3%3\\%\(Qwen\) and7%7\\%\(gpt\-oss\) of runs, and individual anchor confidence intervals \(CIs\) contain zero in46%46\\%,73%73\\%, and87%87\\%of agent\-runs, the per\-agent anchor is often not significant on its own\. Second, and consistently with every other diagnostic, the uncertainty itself tracks the gradient: the median anchor\-CI width is0\.610\.61for Llama against1\.061\.06and1\.151\.15for Qwen and gpt\-oss, so Llama’s anchors are roughly twice as well\-determined\. The consensus\-gain CI width, by contrast, is essentially constant across families \(≈0\.32\\approx 0\.32\)\. It is specifically the anchor that is sharper where held\-out validation certifies it\. These per\-run intervals are wide because a single run contributes onlyK=5K\{=\}5transitions, which is why we base the main claims on the population\-level and held\-out evidence above rather than on per\-run significance\. The bootstrap is reported here for the ordering it reveals, not as a per\-agent test\.

#### Anchor drift\.

We also tested whether the recovered anchor is stable within a deliberation \(Experiment D\): refitting on an early and a late window of rounds and comparing anchors\. This diagnostic did not generalise, each window has too few transitions to recover a stable anchor, so the early/late comparison is dominated by fit noise rather than a consistent drift signal, and we therefore draw no conclusions from it\.

#### Anchor diagnostic accuracy\.

Anchor diagnostic accuracy is discussed in Appendix[A\.5](https://arxiv.org/html/2606.19494#A1.SS5)\.

## 8Conclusion

We modelled multi\-agent LLM deliberation as a closed\-loop dynamical system in which each agent carries a hidden per\-agent anchor that pulls its opinion toward a latent prior independent of its neighbours\. The model nests DeGroot \(βi=0\\beta\_\{i\}\{=\}0\) and Friedkin–Johnsen \(𝐛i=𝐱i​\(0\)\\mathbf\{b\}\_\{i\}\{=\}\\mathbf\{x\}\_\{i\}\(0\)\) as special cases, and deliberation empirically settles withinconv​\{𝐛i\}\\mathrm\{conv\}\\\{\\mathbf\{b\}\_\{i\}\\\}\(§[7\.4](https://arxiv.org/html/2606.19494#S7.SS4)\), reproducing the gold\-class probability leaving the hull of initial opinions that linear consensus forbids \(Property[1](https://arxiv.org/html/2606.19494#Thmproperty1), Eq\. \([5](https://arxiv.org/html/2606.19494#S3.E5)\)\)\. Fitting the model by least squares and validating it on held\-out seeds turns parameter recovery into an unsupervised model\-selection test\. On a symptom→\\rightarrowdisease benchmark it certifies a transferable latent anchor for Llama\-3\.1\-70B, reduces to the linear baselines for gpt\-oss\-20b, and places Qwen3\-32B between them\. Hidden\-anchor behaviour is thus a spectrum across model families rather than a uniform property: the anchor gain is comparable across families, but only when the anchor sits far from the initial opinions does deliberation escape the hull and demand the full closed\-loop model\. Grounding the inferred anchor in model internals, and turning the open\-loop contention schedule into a predictive controller of the dynamics, are the natural next steps\.

## Limitations

We deliberately state the weaknesses of this work plainly\.

The positive result rests on one model\.Of the three families tested, held\-out validation certifies a transferable latent anchor only for Llama\-3\.1\-70B\. For Qwen3\-32B the margin over the linear baselines is within noise, and for gpt\-oss\-20b the baselines win outright\. On this benchmark the “hidden anchor” is therefore close to a single\-model phenomenon\. Three open\-weight models, on one English symptom→\\rightarrowdisease task with1010cases,n=3n\{=\}3agents andK=5K\{=\}5rounds, might not license claims about model families in general\.

The anchor is weakly identified\.A single run identifies only the product𝜸i=βi​𝐛i\\boldsymbol\{\\gamma\}\_\{i\}=\\beta\_\{i\}\\mathbf\{b\}\_\{i\}\. The anchor𝐛i=𝜸i/βi\\mathbf\{b\}\_\{i\}=\\boldsymbol\{\\gamma\}\_\{i\}/\\beta\_\{i\}is recovered only by stacking multiple seeds, is ill\-conditioned wheneverβ^i\\hat\{\\beta\}\_\{i\}is small, and is further biased by the simplex projection\. With onlyK=5K\{=\}5transitions per run, per\-agent anchor confidence intervals contain zero in4646–87%87\\%of agent\-runs and the mean anchor gain is non\-positive in up to7%7\\%\. The model is not significant at the per\-run level\. Every headline claim rests on population\-level aggregation, and even there the held\-out fit is modest: a meanR2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)of0\.440\.44for the best family \(Llama\) while the other two families sit at or below the linear baselines on held\-out data\.

The anchor is inferred, not measured\.The anchor is a latent quantity fit to output\-probability trajectories, not read from model internals\. We do not show it corresponds to any actual internal representation, so reading𝐛i\\mathbf\{b\}\_\{i\}as the model’s “internal belief” is an interpretation, not a verified claim\.

Escape is threshold\-sensitive and orthogonal to accuracy\.Whether a trajectory escapes the initial hull depends on the margin required: at a negligible tolerance all three models escape in≈75%\\approx 75\\%of runs, and the family separation appears only at a hand\-chosen0\.100\.10margin\. The dynamics we model are also orthogonal to correctness\. The most dynamic family \(Llama\) is the least accurate \(43%43\\%\), so the phenomenon we explain does not by itself improve deliberation\.

## Writing Assistance

We used Claude, an AI assistant, to help with writing in ways consistent with the ACL policy on AI assistance\. Its use was limited to surface\-level support: improving clarity, grammar, and phrasing of text we authored, and minor formatting and editing assistance\. All research ideas, the model and its analysis, the experiments, and the claims are entirely our own, and we verified the correctness of all content\. The assistant was not used to generate scientific content, results, or citations\.

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## Appendix AExperimental Setup Details

### A\.1Models

We use open\-weight instruction\-tuned LLMs only, for reproducibility\. The primary models are Llama\-3\.1\-70B\-InstructGrattafiori et al\. \([2024](https://arxiv.org/html/2606.19494#bib.bib7)\)and Qwen3\-32BYang et al\. \([2025](https://arxiv.org/html/2606.19494#bib.bib18)\), and we add one non\-Llama, non\-Qwen family \(gpt\-oss\-20bOpenAI \([2025](https://arxiv.org/html/2606.19494#bib.bib14)\)\) so that the generalisation claim is not specific to a single training lineage\. Llama and Qwen run at 4\-bit quantisation viabitsandbytes\(NF4 with double quantisation\)\. gpt\-oss\-20b runs at its native MXFP4 precision\. Decoding uses temperature0\.70\.7, top\-pp0\.90\.9, and a512512\-token cap per turn\.

### A\.2Task and Dataset

The task is symptom→\\rightarrowdisease diagnosis: each agent is presented with a symptom set and asked for a ranked top\-5 list of candidate diseases with self\-reported probabilities, drawn from a4242\-class symptom–disease benchmarkitachi9604 \([2020](https://arxiv.org/html/2606.19494#bib.bib10)\)\. We use1010cases, each with a distinct gold disease, run with33random seeds per case \(3030trajectories per model,9090in total\)\. A single deliberation domain is sufficient for the present claim, which concerns the per\-family dynamics of deliberation rather than cross\-domain generalisation\. Extending the study to further domains \(e\.g\. sentiment or legal\-judgement classification\) is left to future work\.

### A\.3Prompt Templates

All agents share the system message

> Respond only in the requested format\. Do not write your reasoning aloud\.

Round0elicits each agent’s initial opinion from the symptom set alone, with no neighbour context\. Placeholders in braces are filled per case:\{SYMPTOMS\}is the case symptom string and\{DISEASES\}is the fixed list of the4242class labels\.

> You are given the following symptoms: \{SYMPTOMS\}\. You must choose ONLY from these diseases: \{DISEASES\}\. Provide your top 5 disease predictions with confidence probabilities \(must sum to 1\.0\)\. Format each line exactly as: RANK\. DISEASE\_NAME: PROBABILITY Example: 1\. Diabetes: 0\.40 2\. Hypertension: 0\.25 3\. Hypothyroidism: 0\.15 4\. Heart attack: 0\.12 5\. Malaria: 0\.08 Your top 5 predictions: Then after the predictions explain in 2\-3 sentences strictly why your top prediction is the most likely given the symptoms, and why the other candidates are less likely\. Explanation:

Each reflection round \(1≤k≤K1\\leq k\\leq K\) gives the agent its own and its neighbour’s previous response, in round\-robin order\.\{PROBLEM\}restates the diagnosis task,\{OWN\_PREV\}and\{NEIGHBOUR\_PREV\}are the two prior\-round answers \(predictions plus explanations\)\.

> Deliberation Topic: \{PROBLEM\} You previously predicted: \{OWN\_PREV\} Another agent predicted: \{NEIGHBOUR\_PREV\} You must choose ONLY from these diseases: \{DISEASES\}\. Considering both predictions and their explanations, provide your updated top 5 disease predictions with confidence probabilities \(must sum to 1\.0\)\. Format each line exactly as: RANK\. DISEASE\_NAME: PROBABILITY Your updated top 5 predictions: Then explain in 2\-3 sentences why your top prediction is the most likely given the symptoms, and why the other candidates are less likely\. Explanation:

The fixedRANK\. DISEASE\_NAME: PROBABILITYformat is what the parser maps to the per\-round opinion vector𝐱i​\(k\)∈Δd−1\\mathbf\{x\}\_\{i\}\(k\)\\in\\Delta^\{d\-1\}over thed=42d\{=\}42classes\.

### A\.4Metrics

We report four metrics so that the results section can stay terse\.

- •R2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)\([8](https://arxiv.org/html/2606.19494#S5.E8)\): one\-step prediction quality on stacked transitions\.
- •Held\-out MSE: mean\-squared one\-step prediction error under leave\-one\-run\-out, capturing transfer of recovered parameters across runs\.
- •Bootstrap CIs:B=1000B=1000resamples of the transition pool yielding95%95\\%confidence intervals onβ^i\\hat\{\\beta\}\_\{i\}and𝐛^i\\hat\{\\mathbf\{b\}\}\_\{i\}\.
- •Hull\-containment rate: fraction of runs whose final\-round opinions all lie inconv​\{𝐛^j\}j\\mathrm\{conv\}\\\{\\hat\{\\mathbf\{b\}\}\_\{j\}\\\}\_\{j\}on every coordinate, measuring whether deliberation settles within the recovered anchor hull \(§[7\.4](https://arxiv.org/html/2606.19494#S7.SS4)\)\.

### A\.5Anchor Diagnostic Accuracy

For completeness we report whether the deliberated consensus lands on the gold diagnosis: the round\-KKmean opinion has its argmax on the gold class in43%43\\%\(Llama\-3\.1\-70B\),57%57\\%\(Qwen3\-32B\), and57%57\\%\(gpt\-oss\-20b\) of runs\. Notably the most dynamic family \(Llama\) is the least accurate, so the rich, hull\-escaping dynamics are not driven by movement toward the correct answer\. This is consistent with the scope of this work: we model and explain the behaviour of multi\-agent deliberation, the latent\-anchor dynamics and when they govern a model, not its task accuracy\. Diagnostic correctness is orthogonal to the dynamical claim: a hidden anchor pulls an agent toward its own prior whether or not that prior is correct\. Improving accuracy is a separate problem addressable by a richer deliberation framework \(e\.g\. a judge agent or retrieval augmentation\)\.

## Appendix BSupplementary Tables

This appendix collects quantities that are cited in the body but do not warrant a table there\. We add to it as further such numbers accumulate\. Floats may be typeset away from this text\. The tables in this section are, by clickable reference:

- •Table[5](https://arxiv.org/html/2606.19494#A2.T5): gold\-coordinate per\-run dispersion \(§[7\.1](https://arxiv.org/html/2606.19494#S7.SS1)\)\.
- •Table[6](https://arxiv.org/html/2606.19494#A2.T6): per\-disease held\-outR2R^\{2\}behind the model\-selection counts \(§[7\.3](https://arxiv.org/html/2606.19494#S7.SS3)\)\.

Table 5:Gold\-coordinate dispersion within a deliberation \(§[7\.1](https://arxiv.org/html/2606.19494#S7.SS1)\)\. For each run the per\-run range is the spread \(maximum minus minimum\) of the gold\-class probability across rounds0–KK\. The table reports the mean over the3030runs per model\. Llama’s gold coordinate moves several times more than the other families, the quantitative form of the non\-uniform trajectories in Figure[1](https://arxiv.org/html/2606.19494#S3.F1)\.DiseaseDeGrootFJFullBestLlama\-3\.1\-70B\(Full best in8/108/10\)Fungal infection0\.020\.020\.020\.020\.73\\mathbf\{0\.73\}FullAllergy0\.050\.050\.020\.020\.66\\mathbf\{0\.66\}FullGERD0\.080\.080\.070\.070\.24\\mathbf\{0\.24\}FullChronic chol\.0\.07\\mathbf\{0\.07\}0\.060\.06−0\.03\-0\.03DeGrootDrug reaction0\.130\.130\.140\.140\.55\\mathbf\{0\.55\}FullCOVID0\.000\.000\.000\.000\.81\\mathbf\{0\.81\}FullPeptic ulcer0\.050\.050\.050\.050\.74\\mathbf\{0\.74\}FullAIDS0\.120\.120\.080\.080\.49\\mathbf\{0\.49\}FullDiabetes0\.000\.000\.000\.000\.62\\mathbf\{0\.62\}FullGastroenteritis−0\.02\\mathbf\{\-0\.02\}−0\.05\-0\.05−0\.40\-0\.40DeGrootQwen3\-32B\(Full best in7/107/10\)Fungal infection0\.260\.260\.260\.260\.34\\mathbf\{0\.34\}FullAllergy0\.240\.240\.24\\mathbf\{0\.24\}−0\.06\-0\.06FJGERD0\.030\.030\.040\.040\.08\\mathbf\{0\.08\}FullChronic chol\.0\.130\.130\.130\.130\.24\\mathbf\{0\.24\}FullDrug reaction0\.070\.070\.060\.060\.25\\mathbf\{0\.25\}FullCOVID0\.00\\mathbf\{0\.00\}−0\.01\-0\.01−0\.19\-0\.19DeGrootPeptic ulcer0\.040\.040\.040\.040\.10\\mathbf\{0\.10\}FullAIDS0\.08\\mathbf\{0\.08\}0\.070\.07−0\.25\-0\.25DeGrootDiabetes0\.090\.090\.090\.090\.13\\mathbf\{0\.13\}FullGastroenteritis0\.100\.100\.080\.080\.17\\mathbf\{0\.17\}Fullgpt\-oss\-20b\(Full best in2/102/10\)Fungal infection0\.18\\mathbf\{0\.18\}0\.170\.17−0\.12\-0\.12DeGrootAllergy0\.160\.160\.16\\mathbf\{0\.16\}0\.080\.08FJGERD0\.160\.160\.160\.160\.22\\mathbf\{0\.22\}FullChronic chol\.0\.13\\mathbf\{0\.13\}0\.130\.130\.080\.08DeGrootDrug reaction0\.10\\mathbf\{0\.10\}0\.090\.090\.060\.06DeGrootCOVID0\.02\\mathbf\{0\.02\}−0\.01\-0\.01−9\.43\-9\.43DeGrootPeptic ulcer0\.220\.220\.180\.180\.35\\mathbf\{0\.35\}FullAIDS0\.06\\mathbf\{0\.06\}0\.050\.05−0\.79\-0\.79DeGrootDiabetes0\.21\\mathbf\{0\.21\}0\.200\.200\.130\.13DeGrootGastroenteritis0\.04\\mathbf\{0\.04\}0\.020\.020\.010\.01DeGrootTable 6:Per\-disease held\-outR2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)behind the model\-selection counts of §[7\.3](https://arxiv.org/html/2606.19494#S7.SS3)and the “Full sel\. %” column of Table[1](https://arxiv.org/html/2606.19494#S7.T1)\. For each of the1010disease groups we fit the three nested models \(DeGroot, Friedkin–Johnsen, Full\) on two seeds and score one\-step prediction on the held\-out seed, averaged over the three leave\-one\-seed\-out folds\. The best model per row is in bold, and the “Best” column names it\. Counting the Full\-best rows gives the selection counts \(8/108/10,7/107/10,2/102/10\)\. The single extreme gpt\-oss COVID entry \(−9\.43\-9\.43\) is the catastrophic overfit that drags the gpt\-oss mean held\-outR2R^\{2\}to−0\.94\-0\.94in Table[1](https://arxiv.org/html/2606.19494#S7.T1)\. Values are read fromheldout\_nested\.csvproduced byheldout\_nested\.py\.
## Appendix CReproducibility

All code, the stored deliberation trajectories, and the analysis scripts that regenerate every table and figure are included in the supplementary material\.

#### Environment\.

Experiments run under Python 3 with NumPy and SciPy for the analysis and PyTorch with Hugging Face Transformers for deliberation\. The three models, their quantisation settings, and the decoding hyperparameters are listed in Appendix[A](https://arxiv.org/html/2606.19494#A1)\. A pinned dependency list is included in the repository\.

#### Deliberation\.

Each\(model,case\)\(\\text\{model\},\\text\{case\}\)cell is run with the protocol of §[3\.1](https://arxiv.org/html/2606.19494#S3.SS1):n=3n\{=\}3agents on a directed ring,K=5K\{=\}5reflection rounds, and the33fixed random seeds\. The deliberation step writes one JSON trajectory per run, recording every agent’s per\-round probability vector over the4242classes\. These stored trajectories are the sole input to the analysis\. No model has to be re\-run to reproduce any number in the paper\.

#### Analysis\.

The analysis is deterministic: it reads the stored trajectories and refits the nested models by ordinary least squares \(§[5](https://arxiv.org/html/2606.19494#S5)\)\. The only stochastic step, the Experiment E block bootstrap, is seeded \(B=1000B\{=\}1000, fixed seed\), so its confidence intervals reproduce exactly\. Each table and figure maps to a single script:

- •Table[1](https://arxiv.org/html/2606.19494#S7.T1): in\-sample fits and the nested\-model comparison, with the leave\-one\-seed\-out held\-out block and the per\-disease selection counts \(the latter detailed in Table[6](https://arxiv.org/html/2606.19494#A2.T6)\)\.
- •Table[2](https://arxiv.org/html/2606.19494#S7.T2): recovered\-anchor strength and overshoot past the initial band, with the steady\-state containment rate\.
- •Table[3](https://arxiv.org/html/2606.19494#S7.T3): the gold\-coordinate escape margin beyondconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}\.
- •Table[4](https://arxiv.org/html/2606.19494#S7.T4): the per\-run bootstrap confidence intervals of Experiment E\.
- •Table[5](https://arxiv.org/html/2606.19494#A2.T5): the per\-run gold\-coordinate dispersion\.
- •Table[7](https://arxiv.org/html/2606.19494#A4.T7)and the figures of Appendix[D](https://arxiv.org/html/2606.19494#A4): the per\-run held\-out validation and the native per\-run plots for the showcased Llama COVID case\.

Every reported value was regenerated from the released trajectories with these scripts, and the script names and exact commands are given in the repository README\.

## Appendix DPer\-Run Figure Gallery

For concreteness we walk one representative run through the analysis battery of §[6\.1](https://arxiv.org/html/2606.19494#S6.SS1)\. We use Llama\-3\.1\-70B on the COVID case \(seed 2\), the family with the strongest hidden\-anchor signal\. These are the native per\-run outputs of Experiments B, C, E and F\. We omit Experiment A here, since this run starts from near\-identical agent beliefs, so the open\-loop baseline is static and uninformative as a picture\. The qualitative behaviour shown is representative of the Llama runs that the aggregate tables summarise\. This section is placed last so that its figures do not interrupt the surrounding appendix text\. Floats may be typeset away from this text\. The figures and table in this section are, by clickable reference:

- •Figure[3](https://arxiv.org/html/2606.19494#A4.F3): natural deliberation trajectory \(Experiment B\)\.
- •Figure[4](https://arxiv.org/html/2606.19494#A4.F4): hidden\-anchor system\-identification fit \(Experiment C\)\.
- •Figure[5](https://arxiv.org/html/2606.19494#A4.F5): per\-run bootstrap distributions \(Experiment E\)\.
- •Figure[6](https://arxiv.org/html/2606.19494#A4.F6): per\-run parameter confidence intervals \(Experiment E\)\.
- •Table[7](https://arxiv.org/html/2606.19494#A4.T7): held\-out leave\-one\-seed\-out validation \(Experiment F\)\.

![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/app_b_prob_llama_covid_seed2.png)Figure 3:Natural LLM deliberation \(Experiment B\), per\-class probability over rounds for all three agents\. The gold class \(COVID\) climbs from0\.600\.60to0\.930\.93, above every agent’s round\-0value, so the trajectory leavesconv​\{𝐱j​\(0\)\}\\mathrm\{conv\}\\\{\\mathbf\{x\}\_\{j\}\(0\)\\\}on that coordinate\. This is the escape that the open\-loop rules forbid \(Property[1](https://arxiv.org/html/2606.19494#Thmproperty1)\)\.![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/app_c_fit_llama_covid_seed2.png)Figure 4:Hidden\-anchor system identification \(Experiment C, §[5](https://arxiv.org/html/2606.19494#S5)\) for the same run\. Observed \(solid\) against the one\-step model prediction \(dashed\) per class, with in\-sampleR2​\(Δ​𝐱\)=0\.91R^\{2\}\(\\Delta\\mathbf\{x\}\)=0\.91\.![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/app_e_bootstrap_llama_covid.png)Figure 5:Per\-run bootstrap distributions \(Experiment E, §[7\.7](https://arxiv.org/html/2606.19494#S7.SS7);B=1000B\{=\}1000, five transitions\) of the recovered parameters for the same run\. Companion to Figure[6](https://arxiv.org/html/2606.19494#A4.F6)\.![Refer to caption](https://arxiv.org/html/2606.19494v1/images_final/app_e_paramci_llama_covid.png)Figure 6:Per\-run95%95\\%confidence intervals \(Experiment E, §[7\.7](https://arxiv.org/html/2606.19494#S7.SS7)\) from the bootstrap of Figure[5](https://arxiv.org/html/2606.19494#A4.F5)\. The consensus gainα^\\hat\{\\alpha\}excludes zero, while everyβ^i\\hat\{\\beta\}\_\{i\}interval crosses zero: with only five transitions per run the individual anchor strengths are not separately identifiable, which is why the anchor signal is read at the population level rather than per run\.Table 7:Held\-out leave\-one\-seed\-out validation \(Experiment F, §[7\.3](https://arxiv.org/html/2606.19494#S7.SS3)\) for the Llama COVID case\. The three deliberation seeds are split into two training seeds and one held\-out seed\. We fit the hidden\-anchor model \(§[5](https://arxiv.org/html/2606.19494#S5)\) on the pooled one\-step transitions of the two training seeds \(the ensemble fit\), forward\-simulate from the held\-out seed’s round\-0opinions, and score the prediction against that seed’s observed trajectory\.R2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)is the one\-step prediction quality on stacked transitions\.R2​\(state\)R^\{2\}\(\\text\{state\}\)is the trajectory\-level fit of the simulated states\. MSE is the mean squared per\-coordinate error\. The trajectory\-level fit is high and stable \(R2​\(state\)≥0\.98R^\{2\}\(\\text\{state\}\)\\geq 0\.98\) while the one\-stepR2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)is noisy and turns negative on seed 3, an artefact of having only five transitions per seed\. For reference the in\-sampleR2​\(Δ​𝐱\)R^\{2\}\(\\Delta\\mathbf\{x\}\)averages0\.820\.82\. Numbers are read fromheld\_out\_results\.jsonproduced byrun\_experiment\_f\.