Beyond Consensus: Trace-Level Synthesis in Mixture of Agents

# Trace-Level Synthesis in Mixture of Agents
Source: [https://arxiv.org/html/2605.29116](https://arxiv.org/html/2605.29116)
## Beyond Consensus: Trace\-Level Synthesis in Mixture of Agents

Shreyas Fadnavis Bioscope AI shreyas\.fadnavis@bioscope\.ai &Praitayini Kanakaraj Bioscope AI praitayini\.kanakaraj@bioscope\.ai &Felix Wyss Bioscope AI felix\.wyss@bioscope\.ai

###### Abstract

When multiple LLM agents solve the same problem, standard practice compresses each agent’s reasoning into a majority vote or layered synthesis, treating agreement as the finish line\. We show this is unnecessarily lossy: an LLM aggregator that reads complete reasoning traces recovers correct solutions even when agents unanimously agree, with beneficial corrections consistently outweighing harmful ones—the*aggregation paradox*\. Majority voting has a ceiling that perturbation diversity does not raise \(error correlations are identical\); the aggregator’s gain comes from trace\-level complementarity, assembling correct intermediate steps from minority chains that voting discards\. These findings motivate Self\-Consistent Mixture of Agents which generates trace diversity through semantic\-preserving input perturbations, safeguards the majority via anchored refinement with provable non\-degradation guarantees, and always synthesizes—never gates on consensus\. A single model with perturbation\-induced trace variation outperforms heterogeneous model pools across structured reasoning, PhD\-level science, competition mathematics, and competitive programming\. The unit of aggregation should be the reasoning trace, not the answer\.

## 1Introduction

Inference\-time reasoning in large language models increasingly relies on generating multiple candidate solutions and combining them\. Self\-consistency\(Wanget al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib15)\)votes over chain\-of\-thought answers, Mixture of Agents\(Wanget al\.,[2025a](https://arxiv.org/html/2605.29116#bib.bib39)\)synthesizes outputs across model layers, multi\-agent debate\(Duet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib31)\)lets agents revise after reading each other’s reasoning, and verification\-guided selection\(Khalifaet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib29); Dorneret al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib30)\)picks the best candidate\. Despite their differences, all reduce a set of reasoning traces to a single answer—by vote, convergence, or selection \(see Appendix[Y](https://arxiv.org/html/2605.29116#A25)for an architectural comparison\)\.

This reduction discards most of the information that reasoning produces\. A chain\-of\-thought trace records proof strategies, intermediate calculations, dead ends, and implicit assumptions—yet at the point of aggregation, each trace is compressed to a single answer token \(in voting\) or an output passage \(in layered synthesis\)\. When agents converge, the standard heuristic is to accept the consensus and move on\(Huanget al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib24)\), setting aside minority chains that may contain correct intermediate steps the majority missed\.

We find that this information loss has a measurable cost\. When we replace majority voting with an LLM aggregator that reads the complete reasoning traces, it recovers correct solutions even when agents unanimously agree on the wrong answer\. Across benchmarks spanning structured reasoning, PhD\-level science, competition mathematics, and competitive programming, beneficial corrections consistently outweigh harmful ones \(§[3](https://arxiv.org/html/2605.29116#S3)\)\. We call this the*aggregation paradox*: synthesis over traces improves accuracy precisely where voting indicates no improvement is needed\. Figure[1](https://arxiv.org/html/2605.29116#S1.F1)illustrates the full pipeline on a GPQA\-Diamond problem where five perturbations produce five different proof strategies, yet trace\-level synthesis assembles evidence that no single trace contains \(Appendix[T\.1](https://arxiv.org/html/2605.29116#A20.SS1)\)\. The mechanism goes beyond what existing frameworks can explain\. The voting ceiling is not raised by perturbation diversity—error correlations are statistically identical \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px3)\)—and a single model with input perturbations matches or exceeds heterogeneous model pools \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px7)\)\. The improvement comes from trace\-level complementarity—different chains contain different correct intermediate steps, and the aggregator assembles solutions that no individual chain produced—with cheap surface variation sufficing to induce it \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4);Wuet al\.[2026](https://arxiv.org/html/2605.29116#bib.bib25); Kruszewskiet al\.[2026](https://arxiv.org/html/2605.29116#bib.bib54)\)\. These findings motivateSC\-MoA\(Self\-Consistent Mixture of Agents; §[4](https://arxiv.org/html/2605.29116#S4), Figure[2](https://arxiv.org/html/2605.29116#S4.F2)\): perturbation\-based trace diversity that sidesteps mode collapse, anchored refinement that converts the debate martingale\(Choiet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib34)\)into a submartingale with provable non\-degradation guarantees, and synthesis that never gates on consensus\.

#### Contributions\.

- •The*aggregation paradox*: trace\-level synthesis outperforms voting even at unanimous consensus, with beneficial flips outweighing harmful ones \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px2); Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1), Corollary[2](https://arxiv.org/html/2605.29116#Thmtheorem2)\)\.
- •Perturbation diversity does not decorrelate errors \(ρ¯\\bar\{\\rho\}is identical for diverse vs\. i\.i\.d\. proposals\), yet synthesis exceeds the voting ceiling via trace\-level complementarity \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px3)\); cheap surface perturbation suffices \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\)\.
- •SC\-MoA \(§[4](https://arxiv.org/html/2605.29116#S4), Figure[2](https://arxiv.org/html/2605.29116#S4.F2)\): perturbation diversity, anchored refinement with submartingale guarantees \(Proposition[3](https://arxiv.org/html/2605.29116#Thmtheorem3)\), and universal aggregation\. Highest accuracy on all five benchmarks under our evaluation protocol \(greedy decoding, matched compute; Table[1](https://arxiv.org/html/2605.29116#S5.T1)\); a single model outperforms heterogeneous pools \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px7)\)\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x1.png)Figure 1:SC\-MoA on a GPQA\-Diamond chemistry problem\.Five semantic perturbations of the same question produce structurally different reasoning traces: two identify the correct answer \(A\) via different proof strategies, while three reach wrong answers through plausible but incomplete arguments\. Anchored refinement freezes the majority answer and revises minorities, raising consensus from 2/5 to 5/5\. Trace\-level synthesis then reads*all*traces—including the now\-corrected minorities—and integrates complementary evidence \(e\.g\., the symmetric\-benzyne argument from one trace and the D\-retention isotopomer count from another\) into a single coherent explanation\. Majority voting would return the same answer \(A\) but discard all reasoning; synthesis recovers a richer, more reliable justification by treating the trace, not the answer, as the unit of aggregation\.

## 2Related Work

#### Self\-consistency and the voting ceiling\.

Self\-consistency\(Wanget al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib15)\)samples multiple chains and takes a plurality vote\. Subsequent work refines voting: ranked\(Wanget al\.,[2025b](https://arxiv.org/html/2605.29116#bib.bib20)\), confidence\-weighted\(Taubenfeldet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib17)\), inconsistency\-aware\(Huanget al\.,[2024b](https://arxiv.org/html/2605.29116#bib.bib18)\), adaptive budgets\(Fenget al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib19); Huanget al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib24)\), certified stopping\(Cordero\-Encinar and Duncan,[2025](https://arxiv.org/html/2605.29116#bib.bib21)\), and improved calibration\(Wang and Stengel\-Eskin,[2026](https://arxiv.org/html/2605.29116#bib.bib46)\)\. All improve*how*votes are counted, but the ballot remains an answer token\(Wuet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib25); Kruszewskiet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib54)\)\.Hamidiehet al\.\([2026](https://arxiv.org/html/2605.29116#bib.bib45)\)address diversity collapse across models; we suggest reading the traces voting discards\.

#### Multi\-agent debate and layered collaboration\.

Multi\-agent debate\(Duet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib31); Smitet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib33)\)and Mixture of Agents\(Wanget al\.,[2025a](https://arxiv.org/html/2605.29116#bib.bib39)\)let agents read and respond to full outputs, but interaction without structure can fail:Choiet al\.\([2025](https://arxiv.org/html/2605.29116#bib.bib34)\)proved unconstrained debate is a martingale,Khanet al\.\([2024](https://arxiv.org/html/2605.29116#bib.bib32)\)showed debate helps only under asymmetric knowledge, andRizvi\-Martelet al\.\([2026](https://arxiv.org/html/2605.29116#bib.bib38)\)characterize when communication helps at all\. GoA\(Yunet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib41)\)and AgentAuditor\(Yanget al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib55)\)address specific failure modes but require model diversity or additional training\. SC\-MoA’s anchored refinement \(§[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px2)\) constrains what can change rather than structuring the interaction \(Figure[13](https://arxiv.org/html/2605.29116#A25.F13)\)\.

#### Verification, selection, and the synthesis ceiling\.

Test\-time compute scaling\(Snellet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib22); Chenet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib23)\)improves accuracy through sampling and selection; ThinkPRM\(Khalifaet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib29)\)scores verification chains for best\-of\-NN, andDorneret al\.\([2026](https://arxiv.org/html/2605.29116#bib.bib30)\)characterize the best\-of\-NNceiling\. Critique–revise methods\(Madaanet al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib26); Shinnet al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib27); Yuksekgonulet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib49)\)iterate on a single solution but lack convergence guarantees\(Wanget al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib50); Huanget al\.,[2024a](https://arxiv.org/html/2605.29116#bib.bib28)\)\. All share a bound: accuracy cannot exceed the best candidate\. Trace\-level synthesis exceeds this by integrating evidence across chains \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px2)\)\.

#### Perturbation diversity and traces\.

DiVeRSe\(Liet al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib16)\)combines diverse prompts with step\-aware verification; SPUQ\(Gaoet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib42)\)introduces semantic\-preserving perturbations for uncertainty quantification; BayesPE\(Tonoliniet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib44)\)formalizes prompt ensembles in a Bayesian framework; boosted ensembles\(Pitiset al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib8); Freund and Schapire,[1997](https://arxiv.org/html/2605.29116#bib.bib6)\)adapt reweighting to prompt selection; Polyjuice\(Wuet al\.,[2021](https://arxiv.org/html/2605.29116#bib.bib47)\)and Tailor\(Rosset al\.,[2022](https://arxiv.org/html/2605.29116#bib.bib48)\)compose perturbation primitives with meaning\-*changing*edits\. All use perturbation to improve diversity or estimate uncertainty, reducing traces to votes or confidence intervals\. We extend perturbation to reasoning synthesis and find that perturbation*content*is not load\-bearing within the tested families \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\)\.Kambhampatiet al\.\([2025](https://arxiv.org/html/2605.29116#bib.bib56)\)argued that traces are computational scaffolding, andValmeekamet al\.\([2025](https://arxiv.org/html/2605.29116#bib.bib57)\)showed that semantically corrupted traces preserve accuracy, suggesting trace content serves as computational scaffolding\. Our findings are complementary inference\-time observations \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px3),[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\): trace\-level complementarity is a statistical property of the output distribution, not a claim about internal cognition\.

## 3The Aggregation Paradox

We now test the claim that trace\-level complementarity carries exploitable information that standard aggregation discards\. Three observations together constitute the*aggregation paradox*: synthesis beats voting even at consensus \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px2)\), the voting ceiling is binding yet synthesis exceeds it \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px3)\), and perturbation content is not load\-bearing—cheap surface variation suffices \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\)\.

#### Controlled setting\.

To isolate the aggregation mechanism from confounds—model heterogeneity, prompt engineering, temperature sampling—we fix a single controlled protocol throughout this section\.N=5N\{=\}5agents use the same model \(gpt\-oss\-120b\) with greedy decoding \(temperature 0\)\. Semantic\-preserving input perturbations \(SPUQ;Gaoet al\.[2024](https://arxiv.org/html/2605.29116#bib.bib42)\) generate structurally different reasoning traces from each agent\. We compare majority voting against LLM synthesis applied to the*same*set of proposals, so that the sole variable is the aggregation mechanism\. Evaluation spans five benchmarks \(BBH\-3, MMLU\-ML, GPQA\-Diamond, AIME, LCB\-Hard; details in §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px1)\)\.

#### Synthesis beats voting even at consensus\.

Standard practice gates aggregation on consensus: if agents agree, accept the majority answer and save compute\(Huanget al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib24)\)\. The logic is sound—why run an expensive synthesis call when the answer is already determined?

But this logic is wrong\.In a controlled pre\-refinement setting on GPQA\-Diamond \(n=198n\{=\}198\), trace\-level synthesis produces 12 beneficial flips versus 5 harmful \(2\.4×2\.4\{\\times\};\+3\.5\+3\.5pp; McNemarp=0\.09p\{=\}0\.09; Table[29](https://arxiv.org/html/2605.29116#A20.T29), Appendix[T](https://arxiv.org/html/2605.29116#A20)\)\. On BBH\-3 \(n=296n\{=\}296\), the same pre\-refinement analysis yields 16 beneficial versus 5 harmful \(3\.2×3\.2\{\\times\};\+3\.7\+3\.7pp;p=0\.016p\{=\}0\.016; Appendix[T](https://arxiv.org/html/2605.29116#A20)\)\. After anchored refinement, the harmful flips are eliminated: across all four QA benchmarks in the full pipeline \(Table[31](https://arxiv.org/html/2605.29116#A20.T31)\), BBH and AIME show zero harmful flips—refinement converts the five pre\-refinement harmful flips into correct answers, an important finding in its own right\. The following decomposition formalizes this asymmetry\.

###### Proposition 1\(Synthesis advantage decomposition\)\.

LetVoteandSynthbe two aggregation procedures applied to the sameNNproposals\. Define the*synthesis fidelity*Fs=ℙ​\(Synthcorrect∣Votecorrect\)F\_\{s\}=\\mathbb\{P\}\(\\textsc\{Synth\}\\text\{ correct\}\\mid\\textsc\{Vote\}\\text\{ correct\}\)and the*recovery rate*Rs=ℙ​\(Synthcorrect∣Votewrong\)R\_\{s\}=\\mathbb\{P\}\(\\textsc\{Synth\}\\text\{ correct\}\\mid\\textsc\{Vote\}\\text\{ wrong\}\)\. Then:

ℙ​\(Synthcorrect\)−ℙ​\(Votecorrect\)⏟synthesis advantage=Pw⋅Rs⏟beneficial flips−Pc⋅\(1−Fs\)⏟harmful flips\\underbrace\{\\mathbb\{P\}\(\\textsc\{Synth\}\\text\{ correct\}\)\-\\mathbb\{P\}\(\\textsc\{Vote\}\\text\{ correct\}\)\}\_\{\\text\{synthesis advantage\}\}\\;=\\;\\underbrace\{P\_\{w\}\\cdot R\_\{s\}\}\_\{\\text\{beneficial flips\}\}\\;\-\\;\\underbrace\{P\_\{c\}\\cdot\(1\-F\_\{s\}\)\}\_\{\\text\{harmful flips\}\}\(1\)wherePc=ℙ​\(Votecorrect\)P\_\{c\}=\\mathbb\{P\}\(\\textsc\{Vote\}\\text\{ correct\}\)andPw=1−PcP\_\{w\}=1\-P\_\{c\}\.

The decomposition follows from the law of total probability \(Appendix[\.5](https://arxiv.org/html/2605.29116#Ax1.SS5)\)\. Its value is not as a bound but as a diagnostic: it decomposes the net synthesis effect into two independently measurable quantities\.

###### Corollary 2\(Synthesis dominance condition\)\.

Synthesis dominates voting whenever the recovery\-to\-corruption ratio exceeds the correctness odds:

Rs1−Fs\>PcPw\.\\frac\{R\_\{s\}\}\{1\-F\_\{s\}\}\\;\>\\;\\frac\{P\_\{c\}\}\{P\_\{w\}\}\.\(2\)

The dominance condition is satisfied on both GPQA\-Diamond \(2\.4×2\.4\{\\times\}\) and BBH\-3 \(3\.2×3\.2\{\\times\}; Appendix[\.7](https://arxiv.org/html/2605.29116#Ax1.SS7)\)\. On the full BBH \(2,561 problems, Appendix[B](https://arxiv.org/html/2605.29116#A2)\), SC\-MoA wins 91 problems where SC fails and loses 48—a 65\.5% head\-to\-head win rate, concentrating on disambiguation, temporal reasoning, and state tracking\. At weak aggregators the dominance condition fails; the score\-based override provides the safety net \(Appendix[N](https://arxiv.org/html/2605.29116#A14)\)\. Consensus gating forfeits this recoverable accuracy—but what explains*why*synthesis succeeds where voting fails?

#### Beyond the voting ceiling\.

The natural hypothesis is that perturbation diversity helps by decorrelating errors, as predicted by the Condorcet jury theorem\(de Condorcet,[1785](https://arxiv.org/html/2605.29116#bib.bib3); Ladha,[1992](https://arxiv.org/html/2605.29116#bib.bib4)\)\(Appendix[\.1](https://arxiv.org/html/2605.29116#Ax1.SS1)\)\. We test this directly by measuring the mean pairwise error correlationρ¯\\bar\{\\rho\}\. Perturbation\-diverse and i\.i\.d\. proposals yield nearly identicalρ¯\\bar\{\\rho\}:0\.6330\.633vs\.0\.6030\.603on GPQA\-Diamond,0\.5710\.571vs\.0\.6070\.607on LCB\-Hard, with fully overlapping 95% confidence intervals \(Appendix[K](https://arxiv.org/html/2605.29116#A11)\)\. Error decorrelation is not the mechanism\. Yet on the same proposals, trace\-level synthesis outperforms majority voting by\+3\.5\+3\.5pp on GPQA; the full pipeline extends this to\+5\.3\+5\.3pp on LCB\-Hard \(Table[1](https://arxiv.org/html/2605.29116#S5.T1)\)\. The explanation is structural: any vote\-based aggregatorAV:V→y^A\_\{V\}\\\!:V\\to\\hat\{y\}satisfiesI​\(y∗;y^\)≤I​\(y∗;𝐕\)I\(y^\{\*\};\\hat\{y\}\)\\leq I\(y^\{\*\};\\mathbf\{V\}\)by the data processing inequality, so information present in traces but absent from votes is inaccessible by construction\. The trace ablation confirms this empirically: collapsing traces to answer labels reduces accuracy to the vote baseline \(Table[36](https://arxiv.org/html/2605.29116#A24.T36)\)\.

What predicts recovery?We measure trace diversityDtD\_\{t\}as mean pairwise cosine distance across reasoning traces in TF\-IDF space \(5000 features\)\. Beneficial flips exhibit significantly higherDtD\_\{t\}:0\.5850\.585vs\.0\.4600\.460on GPQA\-Diamond \(p=0\.015p\{=\}0\.015; Figure[5](https://arxiv.org/html/2605.29116#S5.F5)a; Appendix[U](https://arxiv.org/html/2605.29116#A21)\)\. The effect strengthens under dense pre\-trained embeddings \(1024\-dimensional;p=0\.0005p\{=\}0\.0005, Cohen’sd=0\.92d\{=\}0\.92\), confirming it reflects structural reasoning differences, not lexical surface variation \(Appendix[V](https://arxiv.org/html/2605.29116#A22)\)\. The nontrivial question is whether the aggregator can extract this surplus\. We decompose the recovery rate into two factors \(Proposition[7](https://arxiv.org/html/2605.29116#Thmtheorem7), Appendix[\.6](https://arxiv.org/html/2605.29116#Ax1.SS6)\): the probability that minority traces contain recoverable content \(psubp\_\{\\mathrm\{sub\}\}\) and the probability that the aggregator identifies it \(qq\)\. Synthesis dominates voting whenFsF\_\{s\}exceeds a threshold determined by these quantities \(Corollary[2](https://arxiv.org/html/2605.29116#Thmtheorem2)\)\. A targeted trace ablation tests each factor \(Table[36](https://arxiv.org/html/2605.29116#A24.T36), Appendix[X\.1](https://arxiv.org/html/2605.29116#A24.SS1)\): answer\-only and majority\-only inputs collapse to the vote baseline \(q≈0q\\approx 0when traces are absent\), while minority\-only proposals recover the full benefit \(73\.2% vs\. 72\.2% full\-trace;psubp\_\{\\mathrm\{sub\}\}is high\)\. Permuting steps within traces preserves accuracy—the aggregator treats traces as a bag of evidence\. The active ingredient is minority\-trace content\. One question remains:does the method by which trace diversity is induced matter?

#### Perturbation content is not load\-bearing\.

Within the family of meaning\-preserving perturbations, we find no evidence that perturbation content affects accuracy\. Hand\-crafted personas, SPUQ paraphrases, and GPT\-generated strategies are statistically indistinguishable \(McNemarp\>0\.25p\{\>\}0\.25; 72\.7%, 72\.7%, 70\.2% on GPQA\-Diamond; Appendices[J\.1](https://arxiv.org/html/2605.29116#A10.SS1),[J\.2](https://arxiv.org/html/2605.29116#A10.SS2)\)\. Controlled perturbations from a Polyjuice/Tailor taxonomy\(Wuet al\.,[2021](https://arxiv.org/html/2605.29116#bib.bib47); Rosset al\.,[2022](https://arxiv.org/html/2605.29116#bib.bib48)\)yield 76\.0% vs\. 74\.0% for free\-form SPUQ \(p=1\.0p\{=\}1\.0; Appendix[W](https://arxiv.org/html/2605.29116#A23)\)\.Valmeekamet al\.\([2025](https://arxiv.org/html/2605.29116#bib.bib57)\)showed that even semantically*corrupted*traces preserve accuracy, suggesting LLMs are robust to trace\-level noise\. Our finding is complementary but distinct: semantic\-*preserving*perturbations that change only surface form suffice to induce diverse reasoning strategies, consistent with the mode\-concentration hypothesis\(Wuet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib25); Kruszewskiet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib54)\)\. SC\-MoA therefore requires no prompt engineering or domain expertise within semantic\-preserving perturbation families\. We have not tested adversarial or meaning\-changing perturbations; the claim should not be extrapolated beyond surface variation that preserves semantics\.

## 4SC\-MoA: From Paradox to Algorithm

ProblemqqSemantic perturbationsx1x\_\{1\}x2x\_\{2\}⋯\\cdotsxNx\_\{N\}r1r\_\{1\}rkr\_\{k\}⋯\\cdots⋯\\cdotsr1r\_\{1\}rkr\_\{k\}⋯\\cdotsp1p\_\{1\}⋯\\cdotspNp\_\{N\}×N\{\\times\}\\,N×k\{\\times\}\\,kαi\\alpha\_\{i\}Phase 1: Proposal generationi\.i\.d\.MoA/GoARAG/toolsHumanAlt\. sourcesorCluster by answer equivalenceAnchored refinementfreezea∗a^\{\*\}; refine minorities w\.r\.t\.p∗p^\{\*\}Consensus\-evolution recordclusters, shifts, labels,αi\\alpha\_\{i\}scoresPhase 2: Clustering & refinementPhase 3: Universal aggregationTrace\-level aggregatorallNNtraces\+\+consensus recordSynthesized outputNo early exiteven at unanimousconsensusscore\-based overrideFinal outputa^\\hat\{a\}proposalstracesT1​…​TNT\_\{1\}\\\!\\ldots\\\!T\_\{N\}consensus record

Figure 2:SC\-MoA pipeline\.Phase 1 generatesNNproposals via semantic perturbations and inner self\-consistency\. Phase 2 clusters by answer equivalence and refines minority traces with the majority answer frozen\. Phase 3 synthesizes allNNtraces and the consensus record into a single output, with no early exit even at unanimous consensus\.Section[3](https://arxiv.org/html/2605.29116#S3)imposes four design constraints: cheap perturbation suffices for diversity \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\), the aggregator must read full traces since vote\-based methods cannot access trace\-level information by construction \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px3)\), aggregation must never gate on consensus \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px2)\), and refinement must protect the majority against the debate martingale\(Choiet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib34)\)\. SC\-MoA addresses these in three phases \(Algorithm[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px3), Figure[2](https://arxiv.org/html/2605.29116#S4.F2)\): perturbation\-based diversity, anchored refinement, and universal synthesis with verification\.

#### Phase 1: Diversity without degradation\.

RL\-trained language models concentrate on a small number of reasoning modes\(Wuet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib25); Kruszewskiet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib54)\); repeated sampling at a fixed prompt explores within a mode rather than across modes\. Since perturbation content is not load\-bearing \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\), SC\-MoA uses the cheapest mechanism: a single LLM call producesNNsemantic\-preserving rephrasingsx1,…,xNx\_\{1\},\\ldots,x\_\{N\}\(SPUQ;Gaoet al\.[2024](https://arxiv.org/html/2605.29116#bib.bib42)\), with a validation step ensuring protected tokens appear verbatim \(Appendix[L](https://arxiv.org/html/2605.29116#A12); controlled\-perturbation variant in Appendix[W](https://arxiv.org/html/2605.29116#A23)\)\. Eachxix\_\{i\}is fed to the base model with a generic CoT prompt\. Fork\>1k\>1, a neutral variant tag \(Appendix[J\.5](https://arxiv.org/html/2605.29116#A10.SS5)\) creates diversity at greedy decoding; the best\-of\-kkis selected by quality score, with intra\-agreementαi=\|\{j:ai​j=ai,best\}\|/k\\alpha\_\{i\}=\|\\\{j:a\_\{ij\}=a\_\{i,\\text\{best\}\}\\\}\|/kas confidence signal \(N⋅kN\{\\cdot\}kcalls total\)\.

#### Phase 2: Anchored refinement\.

To avoid the debate martingale\(Choiet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib34)\), SC\-MoA freezes the majority and revises only the minorities\.

Proposals are clustered by answer equivalence: string\-match for QA \(deterministic, faithful\); test\-pass signatures for code \(stochastic, potentially unfaithful; §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px4)\)\. The consensusC=kmax/NC=k\_\{\\max\}/Nis the majority cluster fraction\. All minorities are refined while the majority remains frozen \(Proposition[3](https://arxiv.org/html/2605.29116#Thmtheorem3)\): each minority receives a single call with the majority trace as reference andαi\\alpha\_\{i\}as confidence, yielding a revised proposal that either converges to the majority solution, integrates complementary reasoning from both traces, or retains its original answer with justification \(≤,N−kmax\{\\leq\},N\-k\_\{\\max\}calls\)\.

###### Proposition 3\(Anchoring invariant and refinement submartingale\)\.

Leta0∗a^\{\*\}\_\{0\}be the initial majority answer with supportk0k\_\{0\}\. SC\-MoA returnsa0∗a^\{\*\}\_\{0\}unless allN−k0N\-k\_\{0\}refined minority proposals converge to the samea′≠a0∗a^\{\\prime\}\\neq a^\{\*\}\_\{0\}that wins the post\-refinement vote or aggregation\. Fork0≥⌈N/2⌉k\_\{0\}\\geq\\lceil N/2\\rceil\(i\.e\., a strict majority\), override is impossible\. Moreover, the majority\-vote accuracy after anchored refinement is non\-decreasing:

ℙ​\(post\-refinement vote correct\)≥ℙ​\(pre\-refinement vote correct\)\.\\mathbb\{P\}\\bigl\(\\text\{post\-refinement vote correct\}\\bigr\)\\;\\geq\\;\\mathbb\{P\}\\bigl\(\\text\{pre\-refinement vote correct\}\\bigr\)\.\(3\)

On QA tasks the score\-based override does not fire \(quality score is always 1\.0\); non\-degradation instead follows from high synthesis fidelity alone\.

###### Corollary 4\(QA non\-degradation\)\.

On tasks where the override does not fire, synthesis is non\-degrading wheneverFs≥Pc/\(Pc\+Pw⋅Rs\)F\_\{s\}\\geq P\_\{c\}\\,/\\,\(P\_\{c\}\+P\_\{w\}\\cdot R\_\{s\}\)\.

EmpiricalFsF\_\{s\}values exceed the non\-degradation thresholds on both benchmarks \(Appendix[\.7](https://arxiv.org/html/2605.29116#Ax1.SS7)\)\.

#### Phase 3: Universal aggregation with verification\.

The aggregation paradox \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px2)\) rules out consensus gating\(Huanget al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib24)\); SC\-MoA always runs the aggregator\. A single call synthesizes across allNNpost\-refinement proposals with a consensus\-evolution narrative \(cluster shifts, action labels,αi\\alpha\_\{i\}scores; prompts in Appendices[J\.3](https://arxiv.org/html/2605.29116#A10.SS3),[J\.4](https://arxiv.org/html/2605.29116#A10.SS4)\)\. Any consensus thresholdθ<∞\\theta<\\inftyfor early exit*hurts*accuracy \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px4)\)\.

Algorithm 1Self\-Consistent Mixture of Agents

1:

qq, perturbation count

NN, samples/pert\.

kk
2:Final answer

a^\\hat\{a\}
3:Perturbation & Proposal\(§[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px1)\)

4:

\(x1,…,xN\)←Perturb​\(q\)\(x\_\{1\},\\ldots,x\_\{N\}\)\\leftarrow\\textsc\{Perturb\}\(q\)
5:for

i∈\[N\]i\\in\[N\]do

6:Draw

\{ri​j\}j=1k∼Generate​\(xi\)\\\{r\_\{ij\}\\\}\_\{j=1\}^\{k\}\\sim\\textsc\{Generate\}\(x\_\{i\}\)
7:

pi←arg​maxj⁡Score​\(ri​j\)p\_\{i\}\\leftarrow\\operatorname\*\{arg\\,max\}\_\{j\}\\textsc\{Score\}\(r\_\{ij\}\)
8:

αi←\|\{j:Ans​\(ri​j\)=Ans​\(pi\)\}\|/k\\alpha\_\{i\}\\leftarrow\|\\\{j:\\textsc\{Ans\}\(r\_\{ij\}\)\{=\}\\textsc\{Ans\}\(p\_\{i\}\)\\\}\|/k
9:Anchored Refinement\(§[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px2)\)

10:

𝒞,a∗,kmax←Cluster​\(\{pi\}\)\\mathcal\{C\},\\,a^\{\*\},\\,k\_\{\\max\}\\leftarrow\\textsc\{Cluster\}\(\\\{p\_\{i\}\\\}\)
11:for

i∈\[N\]i\\in\[N\]s\.t\.

Ans​\(pi\)≠a∗\\textsc\{Ans\}\(p\_\{i\}\)\\neq a^\{\*\}do

12:

pi←Refine​\(pi,p∗,αi\)p\_\{i\}\\leftarrow\\textsc\{Refine\}\(p\_\{i\},\\;p^\{\*\},\\;\\alpha\_\{i\}\)
13:Aggregation & Verification\(§[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px3)\)

14:

a^←Aggregate​\(\{p1,…,pN\},𝒞\)\\hat\{a\}\\leftarrow\\textsc\{Aggregate\}\(\\\{p\_\{1\},\\ldots,p\_\{N\}\\\},\\;\\mathcal\{C\}\)
15:if

Score​\(a^\)<maxi⁡Score​\(pi\)\\textsc\{Score\}\(\\hat\{a\}\)<\\max\_\{i\}\\textsc\{Score\}\(p\_\{i\}\)then

16:

a^←arg​maxi⁡Score​\(pi\)\\hat\{a\}\\leftarrow\\operatorname\*\{arg\\,max\}\_\{i\}\\textsc\{Score\}\(p\_\{i\}\)
17:return

a^\\hat\{a\}

A score\-based override reverts to the best proposal when the aggregator underperforms \(code: public test pass rate; QA: always 1\.0, so override never fires; Appendix[N](https://arxiv.org/html/2605.29116#A14)\)\. We call clusteringΦ\\Phi*faithful*if cluster agreement implies answer equivalence; for code, test\-pass clustering is*unfaithful*\(Appendix[I](https://arxiv.org/html/2605.29116#A9)\)\. This QA/code asymmetry pervades the pipeline \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px8), §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px4)\)\.

## 5Experiments

#### Setup\.

All methods usegpt\-oss\-120bwith greedy decoding \(temperature 0\)\. Ground truth never enters a prompt\. Fork\>1k\>1, a neutral variant tag creates distinct LLM calls; all baselines use the same variant\-tag protocol\. Sampling protocol and token\-budget comparisons \(Appendices[Q](https://arxiv.org/html/2605.29116#A17),[R](https://arxiv.org/html/2605.29116#A18)\) confirm greedy\+tag is representative and gains are not explained by token expenditure\. Full details in Appendix[J](https://arxiv.org/html/2605.29116#A10)\. SC\-MoA uses two operating points\. Main results \(Table[1](https://arxiv.org/html/2605.29116#S5.T1)\):N=5N\{=\}5,k=2k\{=\}2\(11 calls\)\. Mechanistic analyses \(Figures[5](https://arxiv.org/html/2605.29116#S5.F5)–[6](https://arxiv.org/html/2605.29116#S5.F6)\):N=4N\{=\}4,k=5k\{=\}5\(21 calls\), matching MM\-MoA’s peak operating point on LCB\-Hard and the intra\-perturbation saturation point \(Appendix[H\.1](https://arxiv.org/html/2605.29116#A8.SS1)\)\.

#### Benchmarks\.

We evaluate on five benchmarks spanning question answering, mathematics, and code \(selection rationale in Appendix[A](https://arxiv.org/html/2605.29116#A1)\): BBH \(27 tasks, 2,400\+ problems;Suzgunet al\.[2023](https://arxiv.org/html/2605.29116#bib.bib14)\), MMLU\-ML \(112 questions;Hendryckset al\.[2021](https://arxiv.org/html/2605.29116#bib.bib11)\), GPQA\-Diamond \(198 PhD\-level science;Reinet al\.[2024](https://arxiv.org/html/2605.29116#bib.bib12)\), AIME \(90 competition\-math problems from AIME 2022–2024, integer answers 0–999;AI Mathematical Olympiad[2024](https://arxiv.org/html/2605.29116#bib.bib58)\), and LCB\-Hard \(171 hard competitive\-programming problems, post\-training cutoff, private\-test scored;Jainet al\.[2025](https://arxiv.org/html/2605.29116#bib.bib13)\)\. Contamination analysis in Appendix[S](https://arxiv.org/html/2605.29116#A19)\.

#### Baselines\.

All baselines are compute\-equalized at∼\{\\sim\}7–11 calls \(AppendixLABEL:app:compute\-budget\): Zero\-shot CoT \(1 call\), SCk=10k\{=\}10\(10 calls;Wanget al\.[2023](https://arxiv.org/html/2605.29116#bib.bib15)\), MoA \(9 calls;Wanget al\.[2025a](https://arxiv.org/html/2605.29116#bib.bib39)\), Self\-MoA \(7 calls\), and TextGrad \(∼\{\\sim\}10 calls;Yuksekgonulet al\.[2025](https://arxiv.org/html/2605.29116#bib.bib49)\)\. All share the same evaluation pipeline\. MoA usesN=4N\{=\}4SPUQ paraphrases \(2​N\+1=92N\{\+\}1\{=\}9calls\), isolating aggregation architecture\. Details in Appendix[J](https://arxiv.org/html/2605.29116#A10)\.

Table 1:Main results \(gpt\-oss\-120b,N=5N\{=\}5,k=2k\{=\}2, 11 calls\)\. Best inbold; second\-bestunderlined\. GoA reports the GoAMean\{\}\_\{\\text\{Mean\}\}variant \(3\-model pool; GoAMax\{\}\_\{\\text\{Max\}\}is 2\.5 pp lower on GPQA; Appendix[F](https://arxiv.org/html/2605.29116#A6)\); MoA uses 4 heterogeneous proposers; MoA \(para\) uses SPUQ paraphrases with a single model\.†\\daggerAnswer\-extraction artifact \(Appendix[B](https://arxiv.org/html/2605.29116#A2)\)\. Baseline details, gating ablation, and sampling protocol comparisons in Appendices[F](https://arxiv.org/html/2605.29116#A6),[B](https://arxiv.org/html/2605.29116#A2),[Q\.1](https://arxiv.org/html/2605.29116#A17.SS1)\.SC\-MoA holds the highest point estimate on all five benchmarks \(Table[1](https://arxiv.org/html/2605.29116#S5.T1); pairwise significance tests in Appendix[G](https://arxiv.org/html/2605.29116#A7); per\-task breakdowns in Appendices[B](https://arxiv.org/html/2605.29116#A2)–[D](https://arxiv.org/html/2605.29116#A4)\)\. Seven of twelve pairwise McNemar tests reachp<0\.05p\{<\}0\.05; the five non\-significant comparisons concentrate on GPQA \(n=198n\{=\}198\) and LCB\-Hard \(n=171n\{=\}171\), where statistical power is inherently limited by sample size \(Appendix[G](https://arxiv.org/html/2605.29116#A7)\)\.

On BBH\-3 \(296 problems\), SC\-MoA reaches 86\.5% \(\+\+6\.1 pp over SC,\+\+2\.7 pp over TextGrad,\+\+3\.7 pp over GoA\), with gains concentrating on structured\-reasoning tasks where perturbation diversity surfaces distinct parsing strategies \(full BBH results in Appendix[B](https://arxiv.org/html/2605.29116#A2)\)\. On GPQA\-Diamond, SC\-MoA reaches 73\.2% \(\+\+2\.5 pp over SC,\+\+3\.5 pp over TextGrad; Appendix[C](https://arxiv.org/html/2605.29116#A3)\)\. Self\-MoA collapses to 64\.1%—*below*zero\-shot \(66\.7%\)—illustrating the unanchored debate martingale\(Choiet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib34)\)\. On AIME \(90 competition\-math problems\), SC\-MoA reaches 91\.1% \(\+\+1\.1 pp over Self\-MoA,\+\+5\.5 pp over SC\)\. On LCB\-Hard \(code\), SC\-MoA reaches 62\.6% \(\+\+5\.3 pp over SC at greedy decoding,\+\+10\.6 pp over TextGrad; Appendix[D](https://arxiv.org/html/2605.29116#A4)\)\. Under temperature sampling \(T=0\.7T\{=\}0\.7\), SC reaches 67\.3% on LCB\-Hard \(Appendix[Q](https://arxiv.org/html/2605.29116#A17)\); SC\-MoA’s advantage on code is specific to the greedy\+tag protocol\. The override fires on 14\.6% of problems; a best\-of\-NNablation confirms it is essential at weak aggregators \(Appendix[N](https://arxiv.org/html/2605.29116#A14); oracle ceiling in Appendix[O](https://arxiv.org/html/2605.29116#A15)\)\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x2.png)Figure 3:Compute–accuracy Pareto frontier\.SC\-MoAk=1k\{=\}1\(N=4N\{=\}4,∼5\{\\sim\}5calls\) exceeds SC\(k=10k\{=\}10\) on GPQA \(74\.7% vs\. 70\.7%\) at roughly half the compute; on LCB\-Hard it slightly trails SC greedy\+tag \(55\.6% vs\. 57\.3%\)\.N=5N\{=\}5,k=2k\{=\}2\(11 calls\) reaches 73\.2% on GPQA and 62\.6% on LCB\-Hard, extending the frontier\.*Note:*k=1k\{=\}1andk=2k\{=\}2use differentNN; points are not directly comparable across the two operating points\.The aggregation paradox predicts that if the value lives in trace\-level synthesis rather than in the vote, then even a single synthesis pass—with no majority to fall back on—should already capture the complementary reasoning that voting discards\. SC\-MoA atk=1k\{=\}1\(N=4N\{=\}4,∼5\{\\sim\}5calls\) tests this directly: each perturbation produces a single trace, and the aggregator synthesizes once with no majority vote\. This minimal configuration reaches 74\.7% on GPQA \(\+4\.0\+4\.0pp over SC\) and 55\.6% on LCB\-Hard at roughly half the compute \(fullk=1k\{=\}1results in Table[35](https://arxiv.org/html/2605.29116#A23.T35), Appendix[W\.1](https://arxiv.org/html/2605.29116#A23.SS1)\)\. AtN=5N\{=\}5,k=2k\{=\}2\(11 calls\), accuracy reaches 73\.2% on GPQA \(Table[1](https://arxiv.org/html/2605.29116#S5.T1)\), further extending the Pareto frontier \(Figure[3](https://arxiv.org/html/2605.29116#S5.F3)\)\. Thek=1k\{=\}1vs\.k=2k\{=\}2comparison is confounded by differentNNand configuration choices; a direct significance test on LCB\-Hard at matchedNNyields McNemarp=0\.51p\{=\}0\.51, consistent with inner\-sampling saturation and the unreliability of public\-test selection under unfaithful clustering \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px4)\)\.

#### Aggregation safety and override analysis\.

Section[4](https://arxiv.org/html/2605.29116#S4)introduced two safety mechanisms: the anchoring invariant, which structurally prevents consensus degradation \(Proposition[3](https://arxiv.org/html/2605.29116#Thmtheorem3)\), and score\-based override, which reverts to the best individual proposal when the aggregator underperforms\. We now verify both empirically\. The consensus transition matrix pooled over 867 problems \(Figure[4](https://arxiv.org/html/2605.29116#S5.F4)a\) has*zero*entries below the diagonal, confirming the submartingale prediction\. Figure[4](https://arxiv.org/html/2605.29116#S5.F4)b stratifies expected consensus improvement by difficulty: low\-consensus problems exhibit the largest gain \(𝔼​\[Δ​C\]≈0\.3\\mathbb\{E\}\[\\Delta C\]\\approx 0\.3\), while high\-consensus problems are already near the ceiling, concentrating refinement where the anchoring constraint is most load\-bearing\. The net accuracy effect \(Figure[4](https://arxiv.org/html/2605.29116#S5.F4)c\) ranges from\+1\.8\+1\.8pp \(MMLU\-ML\) to\+6\.1\+6\.1pp \(BBH\), with the full pipeline rescuing problems across all benchmarks\. The score\-based override is essential on code \(\+5\.8\+5\.8pp on LCB\-Hard\) but contributes≤0\.3\{\\leq\}0\.3pp on BBH and MMLU\-ML \(Appendix[N](https://arxiv.org/html/2605.29116#A14)\)—the QA/code asymmetry predicted by the faithful/unfaithful clustering distinction \(§[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px3)\)\. On LCB\-Hard, 69% of problems reach unanimous consensus, yet always\-aggregate outperforms gating by5\.15\.1pp because test\-pass clustering is unfaithful—48\.3% of unanimous clusters mask hidden\-test divergence \(Table[14](https://arxiv.org/html/2605.29116#A8.T14); Appendices[H](https://arxiv.org/html/2605.29116#A8),[I](https://arxiv.org/html/2605.29116#A9),[P](https://arxiv.org/html/2605.29116#A16)\)\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x3.png)Figure 4:Anchored refinement never degrades consensus\(N=5N\{=\}5,k=2k\{=\}2, 867 problems\)\.\(a\)Transition matrix: zero degradations\.\(b\)Low\-consensus problems benefit most\.\(c\)Net accuracy gain per benchmark\.![Refer to caption](https://arxiv.org/html/2605.29116v1/x4.png)Figure 5:Beneficial flips occur when traces are diverse; recovery dominates corruption on all benchmarks\.Mechanistic config:N=4N\{=\}4,k=5k\{=\}5\(21 calls; §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px1)\)\.\(a\)Trace diversityD¯t\\bar\{D\}\_\{t\}stratified by outcome: beneficial flips show significantly higher diversity on GPQA, AIME, and LCB\-Hard \(p<0\.01p\{<\}0\.01\)\.\(b\)Synthesis decomposition \(Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1)\): recovery dominates corruption on all four benchmarks, significantly on three \(p<0\.05p\{<\}0\.05; BBHp=0\.30p\{=\}0\.30\)\.The mechanism: trace\-level synthesis\.Section[3](https://arxiv.org/html/2605.29116#S3)established trace\-level complementarity as the mechanism behind the synthesis advantage\. We now dissect it under the mechanistic configuration \(N=4N\{=\}4,k=5k\{=\}5, 21 calls; §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px1)\), which provides sufficient per\-perturbation samples for decomposition and ablation tests\. Beneficial flips exhibit significantly higher inter\-trace cosine distance on three of four benchmarks \(p<0\.01p\{<\}0\.01; Figure[5](https://arxiv.org/html/2605.29116#S5.F5)a\), with recovery consistently dominating corruption on all four \(Table[31](https://arxiv.org/html/2605.29116#A20.T31); Figure[5](https://arxiv.org/html/2605.29116#S5.F5)b; significantly on three,p<0\.05p\{<\}0\.05; BBHp=0\.30p\{=\}0\.30\)\. Synthesis fidelity is high \(Fs=96\.4%F\_\{s\}=96\.4\\%GPQA,97\.9%97\.9\\%BBH\): the aggregator rarely corrupts correct majorities\. Across the three QA benchmarks in Table[31](https://arxiv.org/html/2605.29116#A20.T31)\(BBH, GPQA, AIME;n=575n\{=\}575\), beneficial flips total 81 versus 4 harmful, concentrating corrections on problems where trace diversity is highest\.

#### Aggregation, not refinement, is the active ingredient\.

Removing minority refinement entirely changes accuracy by≤1\.8\{\\leq\}1\.8pp \(within noise\)\. On QA, where the override never fires, the aggregator alone accounts for all accuracy gains \(Figure[4](https://arxiv.org/html/2605.29116#S5.F4)c\)\. Inner sampling saturates atk=2k\{=\}2\(Appendix[H\.1](https://arxiv.org/html/2605.29116#A8.SS1); selection baselines in Appendix[X\.3](https://arxiv.org/html/2605.29116#A24.SS3)\)\. SweepingN∈\{1,…,10\}N\\in\\\{1,\\ldots,10\\\}on GPQA \(Figure[6](https://arxiv.org/html/2605.29116#S5.F6)c\), SC\-MoA consistently exceeds self\-consistency at everyNN, with the gap peaking at\+4\.1\+4\.1pp \(N=7N\{=\}7\)\. SC\-MoA atN=3N\{=\}3\(4 calls\) already surpasses SC atk=10k\{=\}10\(10 calls\)\. The two\-level procedure implements stratified sampling with provably lower variance than flat i\.i\.d\. sampling \(Proposition[5](https://arxiv.org/html/2605.29116#Thmtheorem5); Appendix[\.2](https://arxiv.org/html/2605.29116#Ax1.SS2)\):N=5N\{=\}5stratified samples outperform1010i\.i\.d\. samples by\+5\.0\+5\.0pp\.

#### Minority traces are the active ingredient\.

A targeted trace ablation \(Table[36](https://arxiv.org/html/2605.29116#A24.T36), Appendix[X\.1](https://arxiv.org/html/2605.29116#A24.SS1)\) identifies which trace content drives the gain\. Answer\-only extracts and majority\-only proposals both degrade to the majority\-vote baseline, confirming that the aggregator extracts load\-bearing information from reasoning traces and that majority agreement alone is insufficient\. Showing the aggregator*only*minority\-dissenting proposals recovers the full benefit \(73\.2%; flip ratio 2\.75\)\. Permuting steps within traces preserves accuracy, indicating the aggregator treats traces as a bag of evidence rather than ordered chains\. An information\-ladder ablation \(Appendix[X\.2](https://arxiv.org/html/2605.29116#A24.SS2)\) confirms the gain is not extra compute: without proposals the aggregator scores 55\.6% \(−14\.1\-14\.1pp;p<10−4p\{<\}10^\{\-4\}\), while providing traces restores recovery dominance \(5\.5×5\.5\{\\times\}\)\. The pipeline is also stable under perturbation choice: cross\-perturbation agreement is 82\.0% \(Fleiss’κ=0\.681\\kappa\{=\}0\.681; Appendix[X\.4](https://arxiv.org/html/2605.29116#A24.SS4)\); full ablation suite in Appendix[X](https://arxiv.org/html/2605.29116#A24)\)\. All propositions rest on idealized assumptions; they are design heuristics, not performance guarantees \(Appendix[Proofs](https://arxiv.org/html/2605.29116#Ax1)\)\.

#### Scaling and robustness\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x5.png)Figure 6:SC\-MoA generalizes across models and scales withNN\(N=4N\{=\}4,k=5k\{=\}5\)\.\(a\)Gains across 6 families \(8B–480B\)\.\(b\)MM\-MoA degrades with weak models; Multi\-SC\-MoA filters them\.\(c\)SC\-MoA exceeds SC at everyNN; SC plateaus\.SC\-MoA generalizes across 6 model families \(8B–480B parameters; panel a\), with gains appearing above a capability threshold\. Proposer quality is the binding constraint: dropping both proposer and aggregator from 120B to 8B reduces GPQA from 73% to 31%, whereas dropping only the aggregator \(prop=120B, agg=8B\) loses 2\.5 pp on QA \(Appendix[M](https://arxiv.org/html/2605.29116#A13)\)\. This asymmetry is consistent with the trace\-level complementarity mechanism: the quality of the reasoning traces determines the ceiling, not the sophistication of the reader that synthesizes them\. Multi\-model pools hurt flat aggregation but not SC\-MoA \(panel b\): MM\-MoA degrades by−6\.1\-6\.1pp as weak models enter the pool, while Multi\-SC\-MoA’s anchored pipeline filters weak proposals and keeps improving with pool size \(Appendix[E](https://arxiv.org/html/2605.29116#A5.SS0.SSS0.Px4);Hamidiehet al\.[2026](https://arxiv.org/html/2605.29116#bib.bib45)\)\. The pipeline is robust to poisoning: mixing three near\-zero\-accuracy agents degrades accuracy by only1\.21\.2pp \(Appendix[E](https://arxiv.org/html/2605.29116#A5.SS0.SSS0.Px2)\)\.

#### Calibration as a byproduct\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x6.png)Figure 7:Consensus as calibrated confidence\(N=5N\{=\}5,k=2k\{=\}2\)\.\(a\)Reliability: QA tracks the diagonal \(ECE==0\.064–0\.154\); LCB\-Hard is overconfident \(ECE==0\.288\)\.\(b\)Selective prediction: abstaining on low\-consensus AIME problems raises accuracy to 100% \(AUROC==0\.85\)\.SC\-MoA’s consensus structure provides uncertainty quantification as an emergent property, requiring no reward model or calibration data \(Appendix[\.7](https://arxiv.org/html/2605.29116#Ax1.SS7.SSS0.Px1)\)\. Under faithful clustering with per\-perturbation competencepmin\>12p\_\{\\min\}\>\\tfrac\{1\}\{2\}, consensus fidelity grows exponentially in the consensus count \(Proposition[6](https://arxiv.org/html/2605.29116#Thmtheorem6); Appendix[Proofs](https://arxiv.org/html/2605.29116#Ax1)\), predicting that consensus should track correctness probability\. Under faithfulΦ\\Phi\(QA string\-match\), calibration is moderate to strong \(ECE==0\.064–0\.154; Figure[7](https://arxiv.org/html/2605.29116#S5.F7)a\); under unfaithfulΦ\\Phi\(code test\-pass\), the fidelity bound breaks and consensus becomes overconfident \(LCB\-Hard ECE==0\.288\)—confirming the faithful/unfaithful distinction that Section[4](https://arxiv.org/html/2605.29116#S4.SS0.SSS0.Px3)predicted\. Selective prediction \(Figure[7](https://arxiv.org/html/2605.29116#S5.F7)b\) demonstrates practical value: on AIME, abstaining on the lowest\-consensus 46% raises accuracy from 91\.1% to 100% \(AUROC==0\.85\)\.

## 6Conclusion

We identified the aggregation paradox: trace\-level synthesis outperforms majority voting even at unanimous consensus, exceeding the voting ceiling through trace\-level complementarity\. The mechanism is precise: different reasoning chains contain different correct intermediate steps, and an aggregator that reads full traces can assemble solutions that no individual chain produced\. Cheap surface perturbation suffices to induce this complementarity without model heterogeneity or domain expertise, within the semantic\-preserving families tested\. SC\-MoA operationalizes these findings via cheap perturbation for diversity, anchored refinement with non\-degradation guarantees \(Proposition[3](https://arxiv.org/html/2605.29116#Thmtheorem3); Corollary[4](https://arxiv.org/html/2605.29116#Thmtheorem4)\), and universal synthesis that never gates on consensus, achieving the highest accuracy under our evaluation protocol on all five benchmarks with calibrated uncertainty as a free byproduct\.Societal impact\.By recovering correct reasoning steps that majority voting discards, trace\-level synthesis can improve the reliability of AI\-assisted decisions in domains where silent errors carry real consequences—scientific peer review, medical differential diagnosis, and safety\-critical code generation—while calibrated consensus lets practitioners abstain when confidence is low\.

## References

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## Proofs

#### Notation\.

Throughout the proofs we usepip\_\{i\}to denote per\-perturbation*competence*\(the probability that perturbationii’s proposal is correct\) andρi​j\\rho\_\{ij\}to denote the Pearson correlation between error indicators𝟏​\[Xi≠y∗\]\\mathbf\{1\}\[X\_\{i\}\\neq y^\{\*\}\]and𝟏​\[Xj≠y∗\]\\mathbf\{1\}\[X\_\{j\}\\neq y^\{\*\}\], wherey∗y^\{\*\}is the ground\-truth answer\. The mean pairwise error correlation isρ¯=2N​\(N−1\)​∑i<jρi​j\\bar\{\\rho\}=\\frac\{2\}\{N\(N\{\-\}1\)\}\\sum\_\{i<j\}\\rho\_\{ij\}\. The minimum competence across perturbations ispmin=mini⁡pip\_\{\\min\}=\\min\_\{i\}p\_\{i\}\.

### \.1Voting Ceiling Bound

For completeness, we state the correlated\-voter bound fromLadha \[[1992](https://arxiv.org/html/2605.29116#bib.bib4)\]in our notation\. At the paper’s operating point \(ρ¯≈0\.6\\bar\{\\rho\}\\approx 0\.6,ε≈0\.15\\varepsilon\\approx 0\.15,N=5N\{=\}5\), the bound givesPr⁡\(error\)≤0\.175\\Pr\(\\text\{error\}\)\\leq 0\.175vs\.0\.1700\.170for i\.i\.d\.—quantitatively indistinguishable, confirming that the main text’s empiricalρ¯\\bar\{\\rho\}measurement is the operative result\.

###### Proof\.

LetXi∈\{0,1\}X\_\{i\}\\in\\\{0,1\\\}indicate whether perturbationii’s proposal is correct, with𝔼​\[Xi\]=pi≥12\+ε\\mathbb\{E\}\[X\_\{i\}\]=p\_\{i\}\\geq\\frac\{1\}\{2\}\+\\varepsilon\. The majority vote is correct iffS=∑i=1NXi\>N/2S=\\sum\_\{i=1\}^\{N\}X\_\{i\}\>N/2\. We boundPr⁡\(S≤N/2\)\\Pr\(S\\leq N/2\)\.

DefineYi=Xi−piY\_\{i\}=X\_\{i\}\-p\_\{i\}\(zero\-mean\)\. ThenS−𝔼​\[S\]=∑iYiS\-\\mathbb\{E\}\[S\]=\\sum\_\{i\}Y\_\{i\}and𝔼​\[S\]≥N​\(12\+ε\)\\mathbb\{E\}\[S\]\\geq N\(\\frac\{1\}\{2\}\+\\varepsilon\)\.

Variance of correlated sum\.

Var​\(S\)=∑ipi​\(1−pi\)\+2​∑i<jρi​j​σi​σj\\mathrm\{Var\}\(S\)=\\sum\_\{i\}p\_\{i\}\(1\-p\_\{i\}\)\+2\\sum\_\{i<j\}\\rho\_\{ij\}\\,\\sigma\_\{i\}\\sigma\_\{j\}whereσi=pi​\(1−pi\)≤12\\sigma\_\{i\}=\\sqrt\{p\_\{i\}\(1\-p\_\{i\}\)\}\\leq\\frac\{1\}\{2\}\. ThereforeVar​\(S\)≤N4​\(1\+\(N−1\)​ρ¯\)\\mathrm\{Var\}\(S\)\\leq\\frac\{N\}\{4\}\\bigl\(1\+\(N\-1\)\\bar\{\\rho\}\\bigr\)\. Applying the correlated Hoeffding inequality\[Ladha,[1992](https://arxiv.org/html/2605.29116#bib.bib4)\]witht=N​εt=N\\varepsilonyields the stated bound\. Whenρ¯=0\\bar\{\\rho\}=0, this reduces to the classical Condorcet–Hoeffding bound1−exp⁡\(−2​N​ε2\)1\-\\exp\(\-2N\\varepsilon^\{2\}\); whenρ¯→1\\bar\{\\rho\}\\to 1, the bound becomes vacuous\. ∎

### \.2Hierarchical Marginalization \(Proposition[5](https://arxiv.org/html/2605.29116#Thmtheorem5)\)

This result instantiates classical stratified sampling\[Cochran,[1977](https://arxiv.org/html/2605.29116#bib.bib10)\]; no novelty is claimed\. We include it to clarify the variance\-reduction mechanism of the two\-level procedure\.

###### Proposition 5\(Hierarchical marginalization\)\.

Let\{x1,…,xN\}\\\{x\_\{1\},\\ldots,x\_\{N\}\\\}be semantic\-preserving perturbations of the input, each defining a conditional distributionP​\(a∣xi\)P\(a\\mid x\_\{i\}\)over answers\. SC\-MoA’s two\-level procedure computes:

P^​\(a∣q\)=1N​∑i=1N1k​∑j=1k𝟏​\[ai​j=a\]⏟inner SC:​P^​\(a∣xi\)\\hat\{P\}\(a\\mid q\)\\;=\\;\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\;\\underbrace\{\\frac\{1\}\{k\}\\sum\_\{j=1\}^\{k\}\\mathbf\{1\}\[a\_\{ij\}=a\]\}\_\{\\text\{inner SC: \}\\,\\hat\{P\}\(a\\mid x\_\{i\}\)\}\(4\)which is a consistent estimator of the mixture distributionP​\(a∣q\)=1N​∑iP​\(a∣xi\)P\(a\\mid q\)=\\frac\{1\}\{N\}\\sum\_\{i\}P\(a\\mid x\_\{i\}\)ask→∞k\\to\\infty\.

###### Proof\.

*Inner SC\.*For each strategysis\_\{i\}, drawkki\.i\.d\. samplesyi,1,…,yi,k∼p\(⋅∣x,si\)y\_\{i,1\},\\ldots,y\_\{i,k\}\\sim p\(\\cdot\\mid x,s\_\{i\}\)\. The inner SC selects the representativey^i=arg⁡maxy​∑j=1k𝟏​\[yi,j=y\]\\hat\{y\}\_\{i\}=\\arg\\max\_\{y\}\\sum\_\{j=1\}^\{k\}\\mathbf\{1\}\[y\_\{i,j\}=y\], which estimatesarg⁡maxy⁡p​\(y∣x,si\)\\arg\\max\_\{y\}\\,p\(y\\mid x,s\_\{i\}\)—the MAP answer under strategysis\_\{i\}\. By the consistency of the empirical mode for discrete distributions,y^i→arg⁡maxy⁡p​\(y∣x,si\)\\hat\{y\}\_\{i\}\\to\\arg\\max\_\{y\}\\,p\(y\\mid x,s\_\{i\}\)ask→∞k\\to\\infty\.

*Outer SC\.*Given representativesy^1,…,y^N\\hat\{y\}\_\{1\},\\ldots,\\hat\{y\}\_\{N\}\(one per strategy\), the majority vote selects:

y∗=arg⁡maxy​∑i=1N𝟏​\[y^i=y\]\.y^\{\*\}=\\arg\\max\_\{y\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\[\\hat\{y\}\_\{i\}=y\]\.Under uniform priorπ​\(si\)=1/N\\pi\(s\_\{i\}\)=1/N, this is the MAP estimate of the mixture1N​∑iδy^i​\(y\)\\frac\{1\}\{N\}\\sum\_\{i\}\\delta\_\{\\hat\{y\}\_\{i\}\}\(y\)\.

*Comparison to flat SC\.*Standard SC withN​kNki\.i\.d\. samples from a single prompt marginalizes over a single strategy:y^SC=arg⁡maxy⁡p​\(y∣x,s0\)\\hat\{y\}\_\{\\text\{SC\}\}=\\arg\\max\_\{y\}\\,p\(y\\mid x,s\_\{0\}\)\. SC\-MoA instead marginalizes over the strategy mixture, capturing the outer sum\. When strategies have non\-redundant errors \(lowρ¯\\bar\{\\rho\}\), the mixture’s mode is more robust than any single strategy’s mode\. ∎

### \.3Consensus Fidelity Bound \(Proposition[6](https://arxiv.org/html/2605.29116#Thmtheorem6)\)

###### Proposition 6\(Consensus fidelity bound\)\.

Suppose proposal errors are conditionally independent given the correct answer, each with competencepi≥pmin\>12p\_\{i\}\\geq p\_\{\\min\}\>\\tfrac\{1\}\{2\}\. Under a faithful clustering functionΦ\\Phi, the consensus fidelity at thresholdc=k0/Nc=k\_\{0\}/Nsatisfies:

F​\(c;Φ\)≥1−\(N⌈N​c⌉\)​\(1−pminpmin\)⌈N​c⌉\.F\(c;\\,\\Phi\)\\;\\geq\\;1\\,\-\\,\\binom\{N\}\{\\lceil Nc\\rceil\}\\\!\\left\(\\frac\{1\-p\_\{\\min\}\}\{p\_\{\\min\}\}\\right\)^\{\\\!\\lceil Nc\\rceil\}\\\!\.\(5\)Under an unfaithfulΦ\\Phi, the bound does not hold: observed consensus may be inflated by spurious agreement\.

When consensus gating is applied at thresholdθ\\theta, the accuracy loss relative to always\-aggregating is bounded by:

Δgate≤\(1−F​\(θ;Φ\)\)⋅ℙ​\(CΦ≥θ\)⋅ℙ​\(aggregation corrects\)\.\\Delta\_\{\\text\{gate\}\}\\;\\leq\\;\\bigl\(1\-F\(\\theta;\\,\\Phi\)\\bigr\)\\;\\cdot\\;\\mathbb\{P\}\\\!\\bigl\(C\_\{\\Phi\}\\geq\\theta\\bigr\)\\;\\cdot\\;\\mathbb\{P\}\\\!\\bigl\(\\text\{aggregation corrects\}\\bigr\)\.\(6\)
###### Proof\.

Under faithfulΦ\\Phi, consensusC=kmax/N≥cC=k\_\{\\max\}/N\\geq cmeans at leastm=⌈N​c⌉m=\\lceil Nc\\rceilproposals produced the*same*answer\. For the majority answer to be wrong, allmmagreeing proposals must be independently wrong\. The probability of any specific set ofmmproposals being simultaneously wrong is at most\(1−pmin\)m\(1\-p\_\{\\min\}\)^\{m\}\. The probability that there exists a*wrong*answer receivingmmvotes is bounded by the number of wrong\-answer coalitions times the per\-coalition probability\.

More precisely: the majority answera∗a^\{\*\}receivesk0≥mk\_\{0\}\\geq mvotes\. Conditional ona∗a^\{\*\}being wrong, each of thek0k\_\{0\}proposals voting fora∗a^\{\*\}must be wrong \(probability≤\(1−pmin\)k0\\leq\(1\-p\_\{\\min\}\)^\{k\_\{0\}\}\)\. But we also need the wrong answer to beat the correct answer, which has at mostN−k0N\-k\_\{0\}votes\. Sincek0≥mk\_\{0\}\\geq mand each correct voter is independent with probability≥pmin\\geq p\_\{\\min\}:

Pr⁡\(majority wrong∣C≥c\)\\displaystyle\\Pr\(\\text\{majority wrong\}\\mid C\\geq c\)≤Pr⁡\(wrong answer gets≥m​votes\)\\displaystyle\\leq\\Pr\(\\text\{wrong answer gets\}\\geq m\\text\{ votes\}\)≤\(Nm\)​\(1−pmin\)m⋅pmin−\(N−m\)⋅pminN−m\\displaystyle\\leq\\binom\{N\}\{m\}\(1\-p\_\{\\min\}\)^\{m\}\\cdot p\_\{\\min\}^\{\-\(N\-m\)\}\\cdot p\_\{\\min\}^\{N\-m\}=\(Nm\)​\(1−pminpmin\)m\.\\displaystyle=\\binom\{N\}\{m\}\\Bigl\(\\frac\{1\-p\_\{\\min\}\}\{p\_\{\\min\}\}\\Bigr\)^\{m\}\.The last step uses the likelihood ratio form\. Taking the complement gives the fidelity bound\.

Under unfaithfulΦ\\Phi:IfΦ\\Phigroups proposals by test\-pass signatures rather than true answer equivalence, two proposals may be in the same cluster despite having different underlying algorithms\. The “independence” assumption breaks: proposals that coincidentally pass the same public tests are grouped together even though their correctness on private tests is not correlated with their cluster membership\. Formally, unfaithfulΦ\\Phicreates a latent confounder \(public test structure\) that inflates observed consensusCΦC\_\{\\Phi\}above the “true” consensus computed under a faithfulΦ′\\Phi^\{\\prime\}\(e\.g\., private test agreement\)\. The bound does not hold becausemmproposals agreeing underΦ\\Phidoes not implymm*independently correct*proposals\. ∎

### \.4Proof of the Anchoring Invariant \(Proposition[3](https://arxiv.org/html/2605.29116#Thmtheorem3)\)

Statement\.Leta0∗a^\{\*\}\_\{0\}be the initial majority answer with supportk0k\_\{0\}\. SC\-MoA returnsa0∗a^\{\*\}\_\{0\}unless allN−k0N\-k\_\{0\}refined minorities converge to the samea′≠a0∗a^\{\\prime\}\\neq a^\{\*\}\_\{0\}that wins post\-refinement vote or aggregation\. Fork0≥⌈N/2⌉k\_\{0\}\\geq\\lceil N/2\\rceil\(a strict majority\), override is impossible\.

###### Proof\.

Phase 3 modifies only minority proposals \(those with answer≠a0∗\\neq a^\{\*\}\_\{0\}\)\. Thek0k\_\{0\}majority proposals are frozen; they retaina0∗a^\{\*\}\_\{0\}throughout\.

*SC exit path \(Phase 4, consensus gate\):*If post\-refinement consensus≥θ\\geq\\theta, SC\-MoA returns the majority’s best proposal\. Sincek0k\_\{0\}frozen proposals holda0∗a^\{\*\}\_\{0\}, the majority answer isa0∗a^\{\*\}\_\{0\}unless the minorities create a larger cluster\. Fork0≥⌈N/2⌉\+1k\_\{0\}\\geq\\lceil N/2\\rceil\+1, theN−k0≤⌊N/2⌋−1N\-k\_\{0\}\\leq\\lfloor N/2\\rfloor\-1minorities cannot form a majority even if they all converge to the samea′a^\{\\prime\}\. ForN=5N=5:k0≥3k\_\{0\}\\geq 3means≤2\\leq 2minorities, which cannot outvote 3 majorities\.

*Aggregation path:*If the aggregator producesa′≠a0∗a^\{\\prime\}\\neq a^\{\*\}\_\{0\}, the override protection \(Phase 5\) checks whether the aggregated answer scores at least as well as the best individual\. If the majority answer is correct, its proposal likely has the highest quality score, and the override protection triggers\.

*Submartingale property\.*On strict\-majority instances \(k0≥3k\_\{0\}\\geq 3forN=5N=5\), the output is locked ata0∗a^\{\*\}\_\{0\}regardless of minority refinement\. Refinement can only change minority proposals, potentially improving them \(e\.g\., a minority adopting the majority approach, which might help on a future meta\-evaluation step\) without degrading the output\. Therefore:𝔼​\[correct after refinement∣k0≥3\]≥𝔼​\[correct before refinement∣k0≥3\]\\mathbb\{E\}\[\\text\{correct after refinement\}\\mid k\_\{0\}\\geq 3\]\\geq\\mathbb\{E\}\[\\text\{correct before refinement\}\\mid k\_\{0\}\\geq 3\], making the process a submartingale\. This contrasts with unconstrained debate, where the martingale property permits downward moves\.

Edge case:k0=⌈N/2⌉k\_\{0\}=\\lceil N/2\\rceil\.When the majority holds exactly a bare majority \(k0=3k\_\{0\}=3forN=5N\{=\}5\), theN−k0=2N\-k\_\{0\}=2minorities cannot form a majority but the aggregator could still producea′≠a0∗a^\{\\prime\}\\neq a^\{\*\}\_\{0\}\. In this case Phase 5 \(override\) provides the safety net: ifa′a^\{\\prime\}scores worse, the system reverts\. Empirically, the bare\-majority edge case accounts for<5%\{<\}5\\%of problems; the override mechanism handles it correctly in all observed instances \(Appendix[N](https://arxiv.org/html/2605.29116#A14)\)\. ∎

### \.5Synthesis Advantage Decomposition \(Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1)\)

Statement\.LetVoteandSynthbe two aggregation procedures applied to the sameNNproposals\. DefineFs=Pr⁡\(Synthcorrect∣Votecorrect\)F\_\{s\}=\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\\mid\\textsc\{Vote\}\\text\{ correct\}\),Rs=Pr⁡\(Synthcorrect∣Votewrong\)R\_\{s\}=\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\\mid\\textsc\{Vote\}\\text\{ wrong\}\),Pc=Pr⁡\(Votecorrect\)P\_\{c\}=\\Pr\(\\textsc\{Vote\}\\text\{ correct\}\), andPw=1−PcP\_\{w\}=1\-P\_\{c\}\. Then:

Pr⁡\(Synthcorrect\)−Pc=Pw⋅Rs−Pc⋅\(1−Fs\)\.\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\)\-P\_\{c\}\\;=\\;P\_\{w\}\\cdot R\_\{s\}\\;\-\\;P\_\{c\}\\cdot\(1\-F\_\{s\}\)\.
###### Proof\.

By the law of total probability:

Pr⁡\(Synthcorrect\)\\displaystyle\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\)=Pr⁡\(Synthcorrect∣Votecorrect\)⋅Pc\\displaystyle=\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\\mid\\textsc\{Vote\}\\text\{ correct\}\)\\cdot P\_\{c\}\+Pr⁡\(Synthcorrect∣Votewrong\)⋅Pw\\displaystyle\\quad\+\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\\mid\\textsc\{Vote\}\\text\{ wrong\}\)\\cdot P\_\{w\}=Fs⋅Pc\+Rs⋅Pw\.\\displaystyle=F\_\{s\}\\cdot P\_\{c\}\+R\_\{s\}\\cdot P\_\{w\}\.SubtractingPcP\_\{c\}:

Pr⁡\(Synthcorrect\)−Pc\\displaystyle\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\)\-P\_\{c\}=Fs⋅Pc\+Rs⋅Pw−Pc\\displaystyle=F\_\{s\}\\cdot P\_\{c\}\+R\_\{s\}\\cdot P\_\{w\}\-P\_\{c\}=Pc⋅\(Fs−1\)\+Rs⋅Pw\\displaystyle=P\_\{c\}\\cdot\(F\_\{s\}\-1\)\+R\_\{s\}\\cdot P\_\{w\}=Pw⋅Rs−Pc⋅\(1−Fs\)\.∎\\displaystyle=P\_\{w\}\\cdot R\_\{s\}\-P\_\{c\}\\cdot\(1\-F\_\{s\}\)\.\\qed

### \.6Extraction Decomposition \(Proposition[7](https://arxiv.org/html/2605.29116#Thmtheorem7)\)

Since𝐕=ϕ​\(𝐓\)\\mathbf\{V\}=\\phi\(\\mathbf\{T\}\)is a deterministic function of the trace tuple, the data processing inequality givesI​\(y∗;𝐓\)≥I​\(y∗;𝐕\)I\(y^\{\*\};\\mathbf\{T\}\)\\geq I\(y^\{\*\};\\mathbf\{V\}\)\. This is trivially true; the nontrivial question is when an aggregatorAAcan extract the surplus in practice\. The following proposition decomposes the recovery rateRsR\_\{s\}into two independently measurable factors, making the synthesis advantage \(Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1)\) actionable\.

###### Proposition 7\(Extraction decomposition\)\.

Condition on the majority vote being wrong\. Letpsubp\_\{\\mathrm\{sub\}\}be the probability that at least one minority trace contains reasoning content that, if identified, suffices to recover the correct answer; and letqqbe the probability that the aggregator identifies and acts on such content when it exists\. ThenRs≥q⋅psubR\_\{s\}\\geq q\\cdot p\_\{\\mathrm\{sub\}\}, and by Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1):

Pr⁡\(A​\(𝐓\)=y∗\)−Pc≥Pw⋅q⋅psub−Pc⋅\(1−Fs\)\.\\Pr\\bigl\(A\(\\mathbf\{T\}\)=y^\{\*\}\\bigr\)\-P\_\{c\}\\;\\geq\\;P\_\{w\}\\cdot q\\cdot p\_\{\\mathrm\{sub\}\}\\;\-\\;P\_\{c\}\\cdot\(1\-F\_\{s\}\)\.The bound is non\-negative \(synthesis dominates voting\) wheneverFs≥Pc/\(Pc\+Pw⋅q⋅psub\)F\_\{s\}\\geq P\_\{c\}\\,/\\,\(P\_\{c\}\+P\_\{w\}\\cdot q\\cdot p\_\{\\mathrm\{sub\}\}\)\.

###### Proof\.

Immediate\. When the majority vote is wrong, a recoverable sub\-argument exists with probability≥psub\\geq p\_\{\\mathrm\{sub\}\}; the aggregator acts on it with probability≥q\\geq q\. By independence of existence and identification,Rs≥q⋅psubR\_\{s\}\\geq q\\cdot p\_\{\\mathrm\{sub\}\}\. Substituting into Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1)yields the bound\. ∎

The value of the decomposition is diagnostic: it isolates three independently testable failure modes—absence of recoverable content \(psub≈0p\_\{\\mathrm\{sub\}\}\\approx 0\), aggregator blindness \(q≈0q\\approx 0\), and fidelity collapse \(FsF\_\{s\}below threshold\)\. The trace ablation \(Table[36](https://arxiv.org/html/2605.29116#A24.T36)\) tests each:

- •Recoverable content \(psubp\_\{\\mathrm\{sub\}\}\):minority\-only traces recover the full benefit \(73\.2% vs\. 72\.2% full\-trace\), confirming useful sub\-arguments exist in minorities\.
- •Aggregator competence \(qq\):answer\-only inputs collapse to the vote baseline, confirming the aggregator uses trace content, not merely answer labels\.
- •Fidelity \(FsF\_\{s\}\):96\.4%96\.4\\%on GPQA\-Diamond \(\>92\.0%\>92\.0\\%threshold\) and97\.9%97\.9\\%on BBH\-3 \(\>93\.9%\>93\.9\\%threshold\)\.

### \.7QA Non\-Degradation \(Corollary[4](https://arxiv.org/html/2605.29116#Thmtheorem4)\)

On QA tasks, the score\-based override \(Phase 5\) does not fire because all LLM\-generated answers receive a quality score of 1\.0\. Non\-degradation therefore depends entirely on synthesis fidelity\.

###### Proof of Corollary[4](https://arxiv.org/html/2605.29116#Thmtheorem4)\.

From Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1), synthesis is non\-degrading \(i\.e\.,Pr⁡\(Synthcorrect\)≥Pc\\Pr\(\\textsc\{Synth\}\\text\{ correct\}\)\\geq P\_\{c\}\) iff:

Pw⋅Rs−Pc⋅\(1−Fs\)≥0⟺Fs≥PcPc\+Pw⋅Rs\.P\_\{w\}\\cdot R\_\{s\}\-P\_\{c\}\\cdot\(1\-F\_\{s\}\)\\;\\geq\\;0\\quad\\Longleftrightarrow\\quad F\_\{s\}\\;\\geq\\;\\frac\{P\_\{c\}\}\{P\_\{c\}\+P\_\{w\}\\cdot R\_\{s\}\}\.This is the dominance condition \(Corollary[2](https://arxiv.org/html/2605.29116#Thmtheorem2)\) restated as a threshold onFsF\_\{s\}\. ∎

#### Empirical verification\.

We compute the fidelity threshold for each QA benchmark using the contingency\-table values from the aggregation\-value analysis \(Table[29](https://arxiv.org/html/2605.29116#A20.T29)and surrounding text; mechanistic configurationN=4N\{=\}4,k=5k\{=\}5\):

In both cases the empiricalFsF\_\{s\}exceeds the threshold, confirming non\-degradation without the override\. The margin is\+4\.4\+4\.4pp on GPQA\-Diamond and\+4\.0\+4\.0pp on BBH\-3\.

#### Note on agreement\-based confidence estimation\.

The use of inter\-classifier agreement as a confidence proxy has formal precedent:Madaniet al\.\[[2004](https://arxiv.org/html/2605.29116#bib.bib51)\]showed that disagreement rates between independent classifiers suffice to estimate accuracy;Platanioset al\.\[[2017](https://arxiv.org/html/2605.29116#bib.bib52)\]extended this to multiple classifiers via probabilistic logic constraints\. SC\-MoA’s consensusC=kmax/NC=k\_\{\\max\}/Ninstantiates this principle—the consensus fidelity bound \(Proposition[6](https://arxiv.org/html/2605.29116#Thmtheorem6)\) formalizes the connection\.

## Appendix ABenchmark Selection Rationale

The five benchmarks were chosen to span three axes that stress\-test trace\-level synthesis:*domain*\(language, science, mathematics, code\),*difficulty*\(graduate\-level to competition\-grade\), and*verification modality*\(string\-match vs\. executable tests\), which directly probes the faithful/unfaithful clustering distinction central to the aggregation safety analysis \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px4)\)\. Each benchmark targets a distinct failure mode of answer\-level aggregation and a corresponding strength of trace\-level complementarity\.

#### BBH \(BIG\-Bench Hard, 27 tasks, 2,400\+ problems\)\.

BBH\[Suzgunet al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib14)\]comprises precisely those BIG\-Bench tasks where prior language models failed to surpass the average human rater under standard prompting—the “Hard” designation is an empirical performance criterion, not a subjective label\. Tasks span multi\-step symbolic manipulation \(Boolean expressions, Dyck languages, tracking shuffled objects\), natural language inference \(disambiguation QA, formal fallacies\), and commonsense reasoning \(date understanding, penguins in a table\)\. Performance improves dramatically with chain\-of\-thought\[wei2022chain\], confirming that the reasoning process—not just the final answer—is load\-bearing\. For SC\-MoA, BBH tests*process\-level complementarity*: each task requires a different reasoning modality, and perturbation\-induced diversity can surface distinct parsing or tracking strategies within a single task\. Because base\-model accuracy is already high under self\-consistency \(∼80%\{\\sim\}80\\%\), the voting ceiling is tight, making BBH a sensitive detector of marginal gains from trace\-level synthesis\.

#### MMLU\-ML \(112 questions\)\.

The machine\-learning subset of MMLU\[Hendryckset al\.,[2021](https://arxiv.org/html/2605.29116#bib.bib11)\]contains graduate\-level questions on optimization theory, statistical learning, information theory, and generalization bounds—topics where correctness hinges on domain\-specific facts rather than multi\-step procedural reasoning\. MMLU\-ML tests*knowledge\-level complementarity*: when one agent’s trace recalls a theorem that another agent confuses with a similar result, the aggregator can identify the more precise derivation by reading the full reasoning chain\. The small sample size \(n=112n\{=\}112\) also creates a regime where per\-question trace information matters most, since answer\-level statistics have limited smoothing power\.

#### GPQA\-Diamond \(198 PhD\-level science questions\)\.

GPQA\-Diamond\[Reinet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib12)\]is the highest\-purity tier of the GPQA benchmark: 198 questions where both expert annotators answered correctly but non\-expert validators with unrestricted web access achieved only∼34%\{\\sim\}34\\%—hence “Google\-Proof\.” Questions span biology, physics, and chemistry at post\-graduate difficulty; domain experts themselves score∼65%\{\\sim\}65\\%\. GPQA\-Diamond is the hardest QA benchmark in our suite: base\-model accuracy is near chance on the most difficult items, maximizing the room for synthesis to operate\. It tests*domain complementarity*: different agents may correctly apply domain knowledge in different scientific subfields, and an aggregator reading full traces can identify which agent’s reasoning is grounded in the relevant domain\. Critically, GPQA also exposes a fundamental*limit*of perturbation diversity: when the base model systematically lacks knowledge for a subfield, all perturbations share the same blind spot \(10 such failures; §[C](https://arxiv.org/html/2605.29116#A3)\)\.

#### AIME 2022–2024 \(90 competition\-mathematics problems\)\.

AIME\[AI Mathematical Olympiad,[2024](https://arxiv.org/html/2605.29116#bib.bib58)\]is the qualifying exam for the USA Mathematical Olympiad, testing advanced algebra, combinatorics, geometry, and number theory at difficulty levels that challenge even talented competition students \(median score among qualifiers:∼5/15\{\\sim\}5/15\)\. Each answer is an integer in\[0,999\]\[0,999\]—no multiple choice, no partial credit—so random performance is0\.1%0\.1\\%and majority voting over wrong answers always fails\. AIME problems characteristically admit multiple valid solution paths \(constructive counting, generating functions, coordinate geometry, algebraic manipulation\), making them an ideal test of*strategy complementarity*: one agent may correctly set up a recurrence while another correctly computes a binomial sum, and a trace\-level aggregator can assemble the correct derivation from complementary partial chains that no individual agent completed\.

#### LCB\-Hard \(171 competitive\-programming problems, post\-training cutoff\)\.

LiveCodeBench\-Hard\[Jainet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib13)\]contains competitive programming problems from AtCoder and LeetCode that were released*after*the base model’s training cutoff, eliminating contamination by construction\. The “Hard” tier is where meaningful model differentiation occurs: frontier models achieve3535–58%58\\%pass@1 on Hard versus near\-saturation on Easy\. Solutions are scored by executing held\-out private test cases with no partial credit—code either passes all tests or fails\. LCB\-Hard is the only benchmark in our suite with*unfaithful*clustering: solutions are grouped by test\-pass signatures rather than string\-match, so two programs that pass identical public tests may implement fundamentally different algorithms\. This directly probes the QA/code asymmetry \(Corollary[4](https://arxiv.org/html/2605.29116#Thmtheorem4)\) and the score\-based override mechanism\. It tests*implementation complementarity*: one agent may choose the correct algorithm but introduce an off\-by\-one error, while another may handle edge cases correctly within a suboptimal framework—a trace\-level aggregator can combine the correct algorithmic choice with the correct edge\-case handling\.

#### Design coverage\.

Table[2](https://arxiv.org/html/2605.29116#A1.T2)summarizes the evaluation design\. The five benchmarks create a stress test across the axes that matter most for SC\-MoA: difficulty \(high base accuracy on BBH/MMLU\-ML vs\. near\-chance on GPQA/AIME\), verification modality \(faithful string\-match on all QA benchmarks vs\. unfaithful test\-pass on LCB\-Hard\), and complementarity type \(process, knowledge, domain, strategy, implementation\)\. Any method that improves across all five benchmarks cannot be exploiting a single favorable regime\.

Table 2:Benchmark selection axes\. Each benchmark targets a distinct complementarity type and occupies a different region of the difficulty×\\timesverification\-fidelity space\.

## Appendix BBBH Per\-Task Breakdown

Table[3](https://arxiv.org/html/2605.29116#A2.T3)reports per\-task accuracy on the BBH\-3 subset \(296 problems\) used in the main paper, including GoA and TextGrad\. Table[4](https://arxiv.org/html/2605.29116#A2.T4)reports results on the full BBH benchmark \(2,561 problems\) for the subset of methods that were evaluated on all 27 tasks\.

#### Answer\-extraction artifact \(†\\dagger\)\.

Zero\-shot CoT scores 2% on*date\_understanding*due to an answer\-extraction regex failure: the model outputs dates in free\-form prose that the extraction pipeline cannot parse into the expected format\. This affects only the ZS\-CoT baseline; all other methods use structured answer extraction that handles this format correctly\. The†\\daggersymbol in Tables[1](https://arxiv.org/html/2605.29116#S5.T1)and[3](https://arxiv.org/html/2605.29116#A2.T3)marks BBH\-3 averages that include this artifact\.

Table 3:BBH\-3 per\-task accuracy \(%\)\. All methods usegpt\-oss\-120b; GoA uses a 3\-model pool \(120B, 70B, 8B\)\. Best inbold; second\-bestunderlined\.Table 4:Full BBH per\-task accuracy \(%, 2,561 problems, persona perturbationsN=4N\{=\}4\)\.Note:this table uses a different configuration from Table[1](https://arxiv.org/html/2605.29116#S5.T1)\(which reports SPUQN=5N\{=\}5,k=2k\{=\}2on the BBH\-3 subset\); numbers are not directly comparable across the two tables\. Tasks where SC\-MoA differs from SC\(k=10k\{=\}10\) by≥5\{\\geq\}5pp are highlighted\. 14 tasks are solved perfectly by both SC and SC\-MoA and are omitted for space\.
#### Head\-to\-head analysis\.

The wins concentrate on five tasks:*disambiguation\_qa*\(\+\+20 problems\),*dyck\_languages*\(\+\+14\),*word\_sorting*\(\+\+14\),*penguins\_in\_a\_table*\(\+\+13\), and*date\_understanding*\(\+\+9\)\. These tasks share a common structure: they require coordinating multiple parsing or tracking steps where different reasoning strategies decompose the problem differently\. Perturbation diversity surfaces these alternative decompositions; i\.i\.d\. sampling with a single strategy explores locally around one approach\.

#### Where SC\-MoA loses\.

The losses concentrate on*web\_of\_lies*\(−\-9 pp\) and*word\_sorting*\(−\-6 pp\)\. On*web\_of\_lies*, the task reduces to a parity computation; a single correct chain\-of\-thought suffices, and aggregation occasionally introduces ambiguity by combining redundant reasoning chains\.*Word\_sorting*is dominated by Self\-MoA \(66%\), whose layered aggregation suits the straightforward comparison\-based structure\. These failure modes suggest that SC\-MoA’s overhead is unjustified when the problem has a single natural algorithmic approach\.

#### Consensus diagnostics\.

75\.2% of BBH problems reach unanimous pre\-refinement consensus \(C=1\.0C\{=\}1\.0\), with 97\.1% accuracy among these\. Accuracy degrades monotonically with consensus:C=0\.8→80\.5%C\{=\}0\.8\\to 80\.5\\%,C=0\.6→57\.3%C\{=\}0\.6\\to 57\.3\\%,C=0\.4→30\.0%C\{=\}0\.4\\to 30\.0\\%\. Refinement is rare: only 370 minority proposals are generated across all 2,561 problems, of which 190 adopt the majority answer, 21 improve their own approach, and 15 defend their minority position\.

#### Why MoA and Self\-MoA fail\.

MoA \(86\.4%\) and Self\-MoA \(86\.3%\) both underperform SC \(88\.1%\) on BBH\. The per\-task comparison reveals the mechanism: on*date\_understanding*, MoA scores 60\.0% vs\. SC\-MoA’s 95\.0% \(−\-35 pp\); on*penguins\_in\_a\_table*, Self\-MoA scores 71\.9% vs\. SC\-MoA’s 100\.0% \(−\-28\.1 pp\)\. These tasks require precise state tracking \(calendar arithmetic, table lookups\) where debate\-style aggregation causes proposals to converge on the majority’s parsing strategy—losing the structural diversity that makes alternative approaches visible\. SC\-MoA avoids this because only minorities are refined, and the frozen majority anchor prevents convergence collapse\.

## Appendix CGPQA\-Diamond Analysis

GPQA\-Diamond\[Reinet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib12)\]contains 198 PhD\-level multiple\-choice science questions where domain experts achieve∼\{\\sim\}65% accuracy\. Table[5](https://arxiv.org/html/2605.29116#A3.T5)presents full results\.

Table 5:GPQA\-Diamond results \(gpt\-oss\-120b, mechanistic config:N=4N\{=\}4,k=5k\{=\}5unless noted\)\.MethodAccuracy \(%\)Avg\. CallsZero\-shot CoT66\.71Self\-MoA64\.17SC \(k=10k\{=\}10\)70\.710TextGrad69\.7∼\{\\sim\}10MoA \(para,N=4N\{=\}4\)65\.29SC\-MoA \(N=4N\{=\}4,k=5k\{=\}5, ungated\)72\.711\.5SC\-MoA \(N=4N\{=\}4,k=1k\{=\}1\)74\.75#### Key observations\.

\(1\) SC\-MoA \(k=5k\{=\}5, ungated\) achieves 72\.7%, a\+\+2\.0 pp gain over SC and\+\+3\.0 pp over TextGrad\. \(2\) Thek=1k\{=\}1variant \(74\.7%\) outperforms the headlinek=5k\{=\}5configuration at lower compute \(McNemarp=0\.63p\{=\}0\.63, not significant atn=198n\{=\}198\), suggesting that intra\-perturbation SC may dilute the perturbation signal on hard problems where each rephrasing captures a unique angle, though the difference is within sampling noise\.

#### Consensus diagnostics\.

Among the 127/198 problems reaching unanimous consensus \(C=1\.0C\{=\}1\.0\), accuracy is 89\.8%\. The remaining∼\{\\sim\}10% of consensus\-1\.0 problems are*confidently wrong*— all five perturbations converge on the same incorrect answer\. At lower consensus levels, accuracy drops sharply:C=0\.8→48\.5%C\{=\}0\.8\\to 48\.5\\%\(n=33n\{=\}33\),C=0\.6→42\.3%C\{=\}0\.6\\to 42\.3\\%\(n=26n\{=\}26\),C=0\.4→46\.2%C\{=\}0\.4\\to 46\.2\\%\(n=13n\{=\}13\)\. On hard science questions, partial consensus is barely better than chance among four choices \(25%\), confirming that GPQA stresses the limits of ensemble methods\.

#### Head\-to\-head\.

SC\-MoA \(ungated\) vs\. SC\(k=10k\{=\}10\): 16 wins, 5 losses \(76\.2% win rate, net\+\+11 problems\)\. The ungated\-vs\.\-gated comparison yields 5 wins vs\. 3 losses—a smaller margin, reflecting the fewer problems where the aggregation step flips the answer\.

#### The difficulty effect\.

GPQA\-Diamond represents a natural experiment separating the two axes of the gate model\. Unlike LCB\-Hard \(low fidelity, high difficulty\), GPQA has high fidelity but high difficulty\. The−\-1\.5 pp gate cost on GPQA demonstrates that difficulty alone is sufficient to make gating harmful, independent of clustering fidelity\.

#### Failure modes\.

10 problems are solved by zero\-shot CoT but not by SC\-MoA \(ungated\)\. These problems tend to have moderate\-to\-high consensus \(C≥0\.6C\\geq 0\.6\), indicating that all five perturbations converge on the same plausible but incorrect answer\. When the underlying model systematically lacks domain knowledge for a specific subfield, perturbation diversity cannot compensate—all reasoning approaches share the same factual blind spot\. This represents a fundamental limitation: SC\-MoA amplifies*reasoning diversity*but cannot create*knowledge*that the base model does not possess\.

## Appendix DLCB\-Hard Extended Analysis

LiveCodeBench\-Hard contains 171 hard competitive\-programming problems \(129 AtCoder, 42 LeetCode\), all post\-training\-cutoff\. Solutions are scored by executing held\-out private test cases\.

### D\.1Per\-platform breakdown

Table 6:LiveCodeBench\-Hard per\-platform results \(n=171n\{=\}171: 129 AtCoder, 42 LeetCode; mechanistic config:N=4N\{=\}4,k=5k\{=\}5\)\. SC\-MoA uses the ungated variant\.MethodOverallAtCoder \(129\)LeetCode \(42\)CallsZero\-shot CoT42\.139\.550\.01TextGrad52\.049\.659\.5∼\{\\sim\}6\.5MoA \(para,N=4N\{=\}4\)50\.945\.766\.79SC \(k=10k\{=\}10\)57\.352\.771\.410Self\-MoA57\.352\.771\.47SC\-MoA \(gated\)55\.6——10\.1SC\-MoA \(ungated\)59\.656\.669\.011\.5SC\-MoA leads overall \(59\.6%\) and on AtCoder \(56\.6%\)\. On LeetCode, SC \(71\.4%\) and Self\-MoA \(71\.4%\) lead SC\-MoA \(69\.0%\); however, all methods achieve higher accuracy on LeetCode than AtCoder, consistent with LeetCode’s more structured specifications and stronger public test suites\.

### D\.2Head\-to\-head comparisons

Per\-problem win/loss counts for SC\-MoA \(ungated\) vs\. each baseline:

Table 7:Per\-problem head\-to\-head on LCB\-Hard \(171 problems\)\.SC\-MoA wins 3–4×\\timesmore problems than it loses against every baseline\. Against SC\(k=10k\{=\}10\), 34 problems are rescued by SC\-MoA that SC cannot solve, while only 9 go the other direction\.

### D\.3Unique solves

9 problems are solved*only*by SC\-MoA \(ungated\) and no other method\. For comparison: SC\(k=10k\{=\}10\) uniquely solves 3, SC\-MoA \(gated\) uniquely solves 2, MoA uniquely solves 1, and TextGrad uniquely solves 1\. Zero\-shot CoT and Self\-MoA have no unique solves\. This demonstrates that universal aggregation surfaces solutions that neither voting, debate, nor iterative refinement can reach independently\.

### D\.4Consensus diagnostics and the fidelity problem

The gated SC\-MoA routes 98% of LCB\-Hard problems through the SC exit \(consensus≥0\.6\\geq 0\.6\) because test\-pass\-signature clustering is coarse\. Among the 118/171 problems reaching unanimous consensus \(C=1\.0C\{=\}1\.0\), gated accuracy is 67\.8%—far below the 97\.1% observed atC=1\.0C\{=\}1\.0on BBH\. This gap directly quantifies the clustering fidelity difference between QA \(string\-match, faithful\) and code \(test\-pass\-signature, unfaithful\)\.

The ungated variant aggregates*all*171 problems regardless of consensus, reaching 73\.9% among theC=1\.0C\{=\}1\.0subset \(\+\+6\.1 pp over gated\)\. The aggregator recovers information that noisy consensus discards: solutions that “agree” on public tests may implement fundamentally different algorithms, and the aggregator can synthesize the strongest elements from each\.

At lower consensus levels, accuracy drops sharply for both variants:C=0\.8→36\.7%C\{=\}0\.8\\to 36\.7\\%\(gated\) vs\. 46\.4% \(ungated\);C=0\.6→38\.9%C\{=\}0\.6\\to 38\.9\\%\(gated\) vs\. 36\.4% \(ungated\)\. The ungated variant’s advantage is largest at high consensus, precisely where the fidelity problem is worst\.

### D\.5Gated vs\. ungated: problem\-level analysis

Comparing gated and ungated SC\-MoA problem\-by\-problem: the ungated variant wins on 16 problems and loses on 6 \(net\+\+10\)\. Among the 16 ungated wins, the average pre\-refinement consensus is 0\.89—these are problems where all five perturbations appeared to agree \(under the noisy clustering\), yet the aggregator found a better solution\. 9 of the 16 wins occur atC=1\.0C\{=\}1\.0, confirming that the fidelity problem is the dominant mechanism\.

### D\.6Override protection

The override fires on 20/171 problems \(11\.7%\) in the ungated variant\. See Appendix[N](https://arxiv.org/html/2605.29116#A14)for the full override analysis across aggregator strengths\.

### D\.7Refinement actions

In the ungated variant, 78 minority proposals adopt the majority approach\. The adopt action has a 37\.2% success rate \(29 correct out of 78\), indicating that majority adoption is not a guaranteed path to correctness on code—the majority itself may be wrong\. This further justifies the universal aggregation strategy: rather than relying on minority→\\tomajority convergence, the aggregator synthesizes across all proposals including those that defended their minority position\.

### D\.8Unsolvable problems

45 of 171 problems \(26\.3%\) remain unsolved by every method tested, indicating a capability ceiling for the underlying model on the hardest competitive\-programming problems\. The remaining 126 solvable problems represent the space where method design matters: SC\-MoA \(ungated\) solves 109/126 \(86\.5%\) of solvable problems, vs\. SC’s 84/126 \(66\.7%\) and Self\-MoA’s 86/126 \(68\.3%\)\.

## Appendix EPersona Diversity vs\. Model Diversity: Extended Results

This section reports full pipeline metrics for all five conditions of the ablation described in §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px7)\. All runs use LCB\-Hard \(n=171n\{=\}171\), temperature 0\.7, seed 42\.4persona@120B,@70B,@8B, and@20Buse four code personas \(analytical, intuitive, adversarial, proof\-sketch\) on the respective single model;4model \(mixed\)uses a generic chain\-of\-thought prompt across four models withgpt\-oss\-120bas refiner and aggregator\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x7.png)Figure 8:Pipeline heatmap for all five conditions\(n=171n\{=\}171each\)\. Cells are colored per\-column by normalised rank \(green = best, red = worst\)\. Thick borders highlight the two competitive conditions\. 4persona@70B and @8B collapse on accuracy despite reasonable consensus\. 4persona@20B†ran under severe TPM throttling \(0\.8 LLM calls/problem\); its SC\-self\-exit and aggregation\-free metrics reflect incomplete pipeline execution rather than algorithmic behaviour\.Table 8:Full pipeline metrics for the persona\-vs\.\-model diversity ablation \(n=171n\{=\}171problems each\)\.*Draft stability*=100−%100\-\\%minority drafts adopted;*Refinement independence*=100×\(1−Δ​C/Δ​Cmax\)100\\times\(1\-\\Delta C/\\Delta C\_\{\\max\}\)whereΔ​C\\Delta Cis post−\-pre consensus lift\. Higher is better on every metric except*Consensus lift*\.†\\dagger4persona@20B ran under TPM throttling; pipeline metrics are unreliable\.#### Model capacity is the binding constraint\.

4persona@70B and @8B achieve near\-zero accuracy \(8\.8% and 1\.8%\) despite reasonable pipeline mechanics \(compile rates 100% and 98%, pre\-exit rates 71% and 77%\)\. The failure is not in consensus formation—both reachCpre\>0\.78C\_\{\\text\{pre\}\}\>0\.78—but in the quality of the agreed\-upon solution: the models converge confidently on wrong answers\.

#### Why 4model \(mixed\) matches 4persona@120B despite weak agents\.

The 120B backbone anchors the 4model condition at two levels: \(i\) it contributes the majority proposal in≈\\approx60% of problems \(the strongest single agent\), and \(ii\) it rewrites every minority draft via the adoption action \(96\.5% adoption rate\)\. The 70B and 8B agents increase compute and pipeline complexity without contributing net signal\. McNemar’s test on 171 paired problems confirms accuracy parity with 4persona@120B \(p=0\.855p\{=\}0\.855\)\.

The refinement dynamics reveal an asymmetric information flow that explains this robustness\. Across 171 problems, weak agents \(8B, 70B, 20B\) adopt the strong agent’s answer 147 times, while the 120B adopts from a weaker agent only 18 times—an8\.2×8\.2\{\\times\}ratio\. When weak agents adopt, accuracy is 65–67%; when the 120B adopts, accuracy drops to 56%, but these events are rare\. Anchored refinement thus acts as a directional filter: weak proposals converge toward the strong majority, but the strong majority is shielded from weak\-agent noise\.

#### Persona diversity is more compute\-efficient\.

Across the four metrics that differ between 4persona@120B and 4model \(mixed\)—CpreC\_\{\\text\{pre\}\}\(0\.91 vs\. 0\.65\), pre\-exit rate \(92% vs\. 53%\), draft stability \(85% vs\. 4%\), consensus lift \(0\.022 vs\. 0\.184\)—persona diversity dominates in every direction\. The 4model condition burns 0\.9 more LLM calls per problem and 7×\\timesmore aggregation budget \(8\.2% vs\. 1\.2%\) to achieve the same result\. For practitioners choosing between persona diversity \(fixed capable model, different prompts\) and model diversity \(multiple model weights, generic prompt\), persona prompting is the strictly preferred design under a fixed accuracy target\.

#### Multi\-SC\-MoA: pipeline contribution on heterogeneous pools\.

The preceding ablation fixes the proposer count at four\. A complementary question is whether the SC\-MoA pipeline \(cluster\+\+refine\+\+aggregate\+\+override\) helps when proposal diversity comes from*model heterogeneity*rather than persona or SPUQ perturbations\. To test this, we apply the full pipeline to the same eight\-model pool used in the MM\-MoA baseline \(Table[9](https://arxiv.org/html/2605.29116#A5.T9)\), sweepingN=3,…,8N\{=\}3\{,\}\\ldots\{,\}8and drawingk=3k\{=\}3i\.i\.d\. samples per proposer\. All other settings match MM\-MoA exactly: aggregatorgpt\-oss\-120b, deep CoT system prompt\.

Table 9:Proposer pool for the Multi\-SC\-MoA and MM\-MoA N\-sweeps, listed in the order they enter asNNincreases\.Table[10](https://arxiv.org/html/2605.29116#A5.T10)reports accuracy at eachNN\. On GPQA\-Diamond, Multi\-SC\-MoA outperforms MM\-MoA at everyNN, averaging\+10\.6\+10\.6pp\. On LCB\-Hard, the benefit isNN\-dependent: Multi\-SC\-MoA matches MM\-MoA at smallNN\(33–55, strong models only\) but gains\+8\.5\+8\.5pp on average at largeNN\(66–88, weak models added\)\.

Table 10:Multi\-SC\-MoA vs\. MM\-MoA N\-sweep on GPQA\-Diamond \(n=198n\{=\}198\) and LCB\-Hard \(n=171n\{=\}171\)\. Both use the same eight\-model pool and aggregator; Multi\-SC\-MoA adds clustering, refinement, aggregation, and override \(Phases 2–5\)\. Single\-model SC\-MoA reference: 72\.7% GPQA, 59\.6% LCB\-Hard\.TheNN\-dependent pattern on code isolates Phase 5 override as the mechanism\. At smallNN\(strong proposers only\), proposals converge on similar test\-pass signatures and flat aggregation captures most of the signal; the SC\-MoA pipeline’s extra phases add little\. At largeNN, weak\-model proposals introduce noise that the aggregator propagates into its synthesis\. The score\-based override reverts to the best\-scoring individual proposal, catching these aggregation errors\. MM\-MoA, which lacks override, sees accuracy drop from 58\.5% \(N=4N\{=\}4\) to 51\.5% \(N=8N\{=\}8\) as weak models are added; Multi\-SC\-MoA*climbs*from 59\.3% to 63\.9%\.

Despite these gains, single\-model SC\-MoA ongpt\-oss\-120b\(72\.7% GPQA, 59\.6% LCB\-Hard\) outperforms the best Multi\-SC\-MoA configuration at everyNNon GPQA and matches it on LCB\-Hard\. Paraphrase diversity on a capable model remains the preferred design\.

Per\-cell error counts range from 2–21 on GPQA and 18–31 on LCB\-Hard \(weak\-model timeouts and transient inference errors\)\. Withn=171n\{=\}171problems and∼60%\{\\sim\}60\\%accuracy, individual\-cell 95% CIs are≈±7\{\\approx\}\\pm 7pp; the mean\-level comparisons \(\+10\.6\+10\.6pp GPQA,\+3\.6\+3\.6pp LCB\-Hard\) are robust because they average over sixNNvalues\.

## Appendix FComparison with GoA \(Graph\-of\-Agents\)

GoA\[Yunet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib41)\]is a graph\-based multi\-agent collaboration framework that frames LLM collaboration as a graph problem\. A meta\-LLM reads model cards to select domain\-specialist agents \(node sampling\), agents score each other’s initial responses to build a weighted graph \(edge sampling\), messages pass bidirectionally along edges, and a final answer is produced via graph pooling \(max: take the highest\-scored agent’s answer; mean: meta\-LLM synthesizes all responses weighted by score\)\.

We run GoA on GPQA\-Diamond with a 3\-model pool spanning three capability tiers:gpt\-oss\-120b\(120B MoE\),llama\-3\.3\-70b\(70B dense\), andllama\-3\.1\-8b\(8B dense\)\. The meta\-LLM for node sampling and mean pooling isgpt\-oss\-120b\. We setk=3k\{=\}3\(select all three models\), temperature 0\.7, 1 round of message passing, and edge\-pruning threshold 0\.05—matching GoA’s recommended defaults\.

To isolate the contribution of the collaboration protocol from the model pool, we run two additional SC\-MoA controls at matched compute \(∼10\{\\sim\}10LLM calls per problem\):

- •SC\-MoAN=3,k=3: the full SC\-MoA pipeline \(SPUQ perturbation→\\topropose→\\tocluster→\\torefine→\\toaggregate→\\tooverride\) on a singlegpt\-oss\-120b, withN=3N\{=\}3paraphrases andk=3k\{=\}3samples per paraphrase\.
- •SC\-MoAN=4,k=5: the headline configuration reported in Table[1](https://arxiv.org/html/2605.29116#S5.T1), included for reference\.

Table 11:GoA vs\. SC\-MoA across all benchmarks\. GoA uses a heterogeneous 3\-model pool \(120B, 70B, 8B\); SC\-MoA uses a singlegpt\-oss\-120b\. SC\-MoA \(N=3N\{=\}3,k=3k\{=\}3\) is a matched\-compute variant \(∼10\{\\sim\}10calls\)\. Best inbold\.#### Analysis\.

SC\-MoA’s headline configuration \(N=4N\{=\}4,k=5k\{=\}5\) outperforms GoAMax\{\}\_\{\\text\{Max\}\}on GPQA \(\+\+2\.5 pp\) and LCB\-Hard \(\+\+2\.3 pp\), though at higher compute \(∼22\{\\sim\}22vs\.∼11\{\\sim\}11calls\)\. On MMLU\-ML, GoAMean\{\}\_\{\\text\{Mean\}\}reaches 92\.0%—slightly above SC\-MoA’s 91\.1%—likely because the heterogeneous pool gives GoA access to models with complementary ML domain knowledge\. GoAMean\{\}\_\{\\text\{Mean\}\}consistently outperforms GoAMax\{\}\_\{\\text\{Max\}\}by 2–5 pp across all benchmarks\. The result is notable because GoA has access to three distinct model weights while SC\-MoA uses a single model throughout\.

#### Matched\-compute comparison\.

To isolate the collaboration protocol from any compute advantage, we compare SC\-MoAN=3,k=3\(the full SC\-MoA pipeline on a singlegpt\-oss\-120bwithN=3N\{=\}3paraphrases andk=3k\{=\}3samples per paraphrase\) against GoA at matched compute \(∼10\{\\sim\}10LLM calls per problem\)\. On GPQA\-Diamond, SC\-MoAN=3,k=3reaches 73\.7%, outperforming GoAMax\{\}\_\{\\text\{Max\}\}by\+3\.5\+3\.5pp and GoAMean\{\}\_\{\\text\{Mean\}\}by\+1\.0\+1\.0pp—even at matched compute and with a single model\.

The gap reflects the perturbation\-vs\.\-heterogeneity distinction \(§[3](https://arxiv.org/html/2605.29116#S3.SS0.SSS0.Px4)\): on hard science questions, input perturbation on a strong model extracts more signal than routing across a heterogeneous pool\.

GoA’s graph\-based routing does provide a clear advantage in one setting: when the model pool is heterogeneous and includes weak models, GoA’s confidence\-weighted edge sampling and max pooling effectively ignore low\-quality agents \(GoAMax\{\}\_\{\\text\{Max\}\}: 70\.2%\), while SC\-MoA’s majority\-based aggregation cannot filter them \(Multi\-SC\-MoA with the same 3\-model pool: 58\.1%; Appendix[E](https://arxiv.org/html/2605.29116#A5.SS0.SSS0.Px4)\)\. This suggests the two approaches are complementary: GoA excels at*selecting which models to use*; SC\-MoA excels at*extracting more from each model once selected*\.

#### SC\-MoA as a protocol\-agnostic module\.

SC\-MoA’s post\-proposal pipeline \(cluster→\\torefine→\\toaggregate→\\tooverride\) operates on any set of candidate proposals without assuming how they were generated\. The lift over flat aggregation is independent of the diversity source:\+\+7\.5 pp on single\-model GPQA,\+\+10\.6 pp on an eight\-model pool \(§[E](https://arxiv.org/html/2605.29116#A5.SS0.SSS0.Px4)\)\. SinceYunet al\.\[[2026](https://arxiv.org/html/2605.29116#bib.bib41)\]show that MoA is a special case of GoA \(GoA Proposition 1\), SC\-MoA’s phases can replace graph pooling in any GoA\-family pipeline\. Head\-to\-head on GPQA\-Diamond, GoAMax\{\}\_\{\\text\{Max\}\}and SC\-MoA solve largely non\-overlapping problem subsets \(union ceiling: 79\.8%\), confirming their complementarity\.

#### SC\-GoA: full composition experiment\.

To validate this composability claim empirically, we build SC\-GoA: GoA’s node sampling selects three models from our model pool, SPUQ generatesPPparaphrases per model, each \(model, paraphrase\) slot generateskkself\-consistency samples \(best\-of\-kk\), and SC\-MoA’s Phases 2–6 replace graph pooling for final aggregation\. We optionally include GoA’s full edge sampling and message passing between the proposal stage and SC\-MoA’s clustering\. Table[12](https://arxiv.org/html/2605.29116#A6.T12)reports results on all 198 GPQA\-Diamond problems\.

Table 12:SC\-GoA on GPQA\-Diamond \(n=198n\{=\}198\)\. SC\-GoA composes GoA’s node sampling with SC\-MoA’s SPUQ, self\-consistency, and Phases 2–6\. Variants include \(\+\+MP\) or exclude \(−\-MP\) GoA’s edge sampling and message passing\.PP: paraphrases per model;kk: SC samples per slot\.Three findings emerge\.*First*, SC\-MoA’s phases compose cleanly with GoA’s routing: all SC\-GoA variants that include Phases 2–6 outperform GoAMax\{\}\_\{\\text\{Max\}\}\(70\.2%\), and SC\-GoA−MP\{\}\_\{\-\\text\{MP\}\}\(74\.7%\) exceeds standalone SC\-MoA \(72\.7%\) by\+\+2\.0 pp\.*Second*, GoA’s message passing is slightly net\-negative \(−\-1\.0 pp: 73\.7% vs\. 74\.7%\): bidirectional peer refinement without an anchoring invariant lets weaker models \(8B, 70B\) dilute the strongest agent’s proposals, whereas SC\-MoA’s Phase 3 refines*only*minorities against a frozen majority\.*Third*, ablating Phase 3 produces the largest accuracy drop \(−\-3\.5 pp: 73\.7%→\\to70\.2%\), confirming that majority\-anchored refinement—not just clustering or aggregation—is the primary contributor even atop GoA’s graph pipeline\. SC\-GoA−MP\{\}\_\{\-\\text\{MP\}\}\(74\.7%\) exceeds standalone SC\-MoA \(72\.7%\), suggesting that GoA’s heterogeneous routing and SC\-MoA’s refinement pipeline are complementary—composing them captures both model diversity and perturbation diversity\.

## Appendix GStatistical Significance

All experiments use greedy decoding with content\-hash caching, so re\-running produces identical results—there is no stochastic variance to measure\. We instead quantify*problem\-level*significance: for each SC\-MoA vs\. baseline pair, we compute McNemar’s test on the2×22\\times 2table of per\-problem correctness \(wins vs\. losses\) and bootstrap 95% confidence intervals on the accuracy difference \(10,000 resamples\)\.

Table 13:Statistical significance of SC\-MoA \(ungated, SPUQ paraphrases,N=5N\{=\}5,k=2k\{=\}2\) vs\. each baseline\. McNemar’s test on per\-problem correctness; bootstrap 95% CIs on accuracy difference \(10,000 resamples\)\. BBH\-3 uses the 296\-problem subset from Table[1](https://arxiv.org/html/2605.29116#S5.T1)\. Computed bycompute\_significance\.py\.7 of 12 comparisons are significant atp<0\.05p<0\.05\. All three BBH\-3 comparisons against MoA, Self\-MoA, and SC reach significance; the BBH\-3 vs\. TextGrad comparison \(p=0\.131p=0\.131\) has a positive but wide CI\. The five non\-significant comparisons concentrate on the two smaller benchmarks \(GPQA,n=198n\{=\}198; LCB\-Hard,n=171n\{=\}171\), where statistical power is inherently limited\. All 12 bootstrap CIs are consistent with a positive true effect \(upper bound\>0\>0in every case\), suggesting the gains are real but some individual comparisons are underpowered\.

Clopper–Pearson 95% CIs on SC\-MoA accuracy: BBH\-3\[82\.1%,90\.2%\]\[82\.1\\%,90\.2\\%\], GPQA\[66\.5%,79\.3%\]\[66\.5\\%,79\.3\\%\], LCB\-Hard\[54\.9%,69\.8%\]\[54\.9\\%,69\.8\\%\]\.

## Appendix HGating Threshold Sensitivity

We sweep the consensus\-gating thresholdθ∈\{0\.4,0\.5,0\.6,0\.7,0\.8,0\.9,1\.0,∞\}\\theta\\in\\\{0\.4,0\.5,0\.6,0\.7,0\.8,0\.9,1\.0,\\infty\\\}by simulating each threshold on existing results: for each problem, if the pre\-refinement consensusC≥θC\\geq\\theta, we use the gated \(SC\-exit\) result; otherwise we use the ungated \(always\-aggregate\) result\. Table[14](https://arxiv.org/html/2605.29116#A8.T14)reports the results\.

Table 14:Accuracy \(%\) under different consensus\-gating thresholds \(N=4N\{=\}4,k=5k\{=\}5,gpt\-oss\-120b\)\.θ=∞\\theta\{=\}\\inftyis always\-aggregate \(no gating\)\. Always\-aggregate dominates or matches gating on all benchmarks except MMLU\-ML, whereθ=0\.6\\theta\{=\}0\.6slightly exceeds always\-aggregate\.On all benchmarks except MMLU\-ML, always\-aggregate \(θ=∞\\theta\{=\}\\infty\) dominates or matches the gated variant\. On MMLU\-ML,θ=0\.6\\theta\{=\}0\.6achieves 92\.0% vs\. 91\.1% atθ=∞\\theta\{=\}\\infty, suggesting the aggregator occasionally introduces errors on easy problems where the SC majority is already correct\. The cost of gating is largest on AIME \(−24\.5\-24\.5pp atθ=0\.4\\theta\{=\}0\.4\), where frequent near\-tie consensus scores cause premature SC\-exit, and on BBH\-3 \(−6\.7\-6\.7pp\)\. On LCB\-Hard the cost is−4\.0\-4\.0pp, driven by unfaithful test\-pass clustering\. The monotonic improvement fromθ=0\.4\\theta\{=\}0\.4toθ=∞\\theta\{=\}\\inftyon most benchmarks supports the universal aggregation design\.

### H\.1Inner Sampling Budget \(kk\-Sweep\)

We sweep the per\-paraphrase sample budgetk∈\{3,4,5,6,7\}k\\in\\\{3,4,5,6,7\\\}with a fixed pool ofN=4N\{=\}4SPUQ paraphrases, drawn from a frozen cache to ensure strict apples\-to\-apples comparison across cells\. The proposer isgpt\-oss\-120b, the aggregator isllama\-3\.1\-8b\-instant, sampling temperature is 0\.7, and all other SC\-MoA stages \(cluster→\\torefine→\\toaggregate→\\tooverride\) are held constant\.Caveat:this sweep uses a weaker aggregator \(llama\-3\.1\-8b\-instant\) than the main results \(gpt\-oss\-120b\)\. The saturation\-at\-k=5k\{=\}5finding should therefore be interpreted as a lower bound; the main aggregator may benefit from additional samples\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/fig_ksweep.png)Figure 9:Inner sampling budget sweep\.Accuracy vs\.kk\(samples per paraphrase\) on GPQA\-Diamond \(n=198n\{=\}198\) and LCB\-Hard \(n=171n\{=\}171\)\. Both benchmarks peak atk=5k\{=\}5; beyond this point, post\-peak variation is within noise on GPQA and significantly regresses on LCB\-Hard \(Δ=−4\.7\\Delta\{=\}\-4\.7pp,p=0\.008p\{=\}0\.008\)\. Error bars: bootstrap 95% CIs\.Across 198 GPQA\-Diamond questions \(Figure[9](https://arxiv.org/html/2605.29116#A8.F9)\), accuracy climbs monotonically from 69\.2% \(k=3k\{=\}3\) to a peak of 71\.7% atk=5k\{=\}5, then oscillates within noise \(70\.2% atk=6k\{=\}6, 71\.2% atk=7k\{=\}7\); a paired bootstrap onk=5k\{=\}5vs\.k=6k\{=\}6yieldsΔ=−1\.5\\Delta=\-1\.5pp \(p=0\.48p=0\.48\), so the post\-peak variation is not statistically distinguishable\. On 171 LCB\-Hard problems, accuracy also peaks atk=5k\{=\}5\(64\.3%\) after a\+3\.5\+3\.5pp climb fromk=3k\{=\}3, but regresses significantly atk=6k\{=\}6to 59\.6% \(pairedΔ=−4\.68\\Delta=\-4\.68pp, 95% CI\[−8\.19,−1\.75\]\[\-8\.19,\-1\.75\],p=0\.008p=0\.008\)\. Both benchmarks therefore exhibit the same empirical optimum, while compute cost grows linearly inkk\(each step adds≈25%\{\\approx\}25\\%more LLM calls and, on LCB\-Hard,≈25%\{\\approx\}25\\%more sandbox executions\)\.

We adoptk=5k\{=\}5as the operating point for thekk\-sweep analysis on the basis that it maximizes accuracy on both benchmarks at the lowest budget where these gains are realized\. The main\-table results \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px3)\) usek=2k\{=\}2with the strongergpt\-oss\-120baggregator, where saturation occurs earlier \(§[4](https://arxiv.org/html/2605.29116#S5.F4)\)\.

## Appendix IPublic vs\. Private Test Fidelity

On LCB\-Hard, clustering uses test\-pass signatures from public tests, but scoring uses private \(hidden\) tests\. This creates a fidelity gap: proposals that pass all public tests may fail private tests\. We quantify this gap for all methods\.

Table 15:Public\-to\-private test fidelity on LCB\-Hard \(n=171n\{=\}171\)\.P​\(priv∣pub\)P\(\\text\{priv\}\\mid\\text\{pub\}\)is the probability that a solution passing all public tests also passes all private tests\.SC\-MoA achieves the highest public\-to\-private fidelity \(74\.0%\) among all methods—solutions that pass public tests are substantially more likely to also pass private tests\. SC has the worst fidelity \(53\.9%\) despite the highest public pass rate \(90\.1%\), indicating that i\.i\.d\. sampling with a single prompt overfits to public test structure\. SC\-MoA’s perturbation diversity produces solutions that are more robust to distribution shift between public and private tests\.

AtC=1\.0C\{=\}1\.0\(unanimous consensus\), 119 problems reach this level and 88/119 \(73\.9%\) are correct\. The 26\.1% false\-positive rate at unanimous consensus directly quantifies why gating is harmful on code: even perfect observed agreement does not guarantee correctness when the clustering metric \(public tests\) is only a proxy for the true metric \(private tests\)\.

## Appendix JImplementation Details

### J\.1QA personas \(5 strategies\)

#### Analytical\.

“You are a careful analytical thinker\. Break the problem into components and reason through each step explicitly\. Show your work\. Check each intermediate conclusion before proceeding\.”

#### Intuitive\.

“You are an intuitive problem solver\. Start with your best guess, then verify it against the constraints\. If verification fails, try the next most likely answer\.”

#### Adversarial\.

“You are a critical thinker\. First identify what answer most people would give, then check if that obvious answer is correct or a common misconception\. Look for traps and edge cases\.”

#### Chain\-tracer\.

“You are a meticulous state\-tracker\. Maintain an explicit ledger of all variables and their values\. Update the ledger at each reasoning step\. Your answer must be consistent with the final state of the ledger\.”

#### Counterfactual\.

“You are a hypothesis tester\. For each candidate answer, ask: if this were correct, what else would have to be true? Check each implication\. Eliminate answers whose implications contradict the problem setup\.”

### J\.2Code personas \(5 strategies\)

#### Analytical\.

“Analyze the problem’s constraints and complexity requirements first\. Select the tightest algorithm that meets the constraints\. Implement cleanly with explicit edge\-case handling\.”

#### Intuitive\.

“Write the shortest pragmatic solution using standard library helpers\. Favor readability and correctness over cleverness\.”

#### Adversarial\.

“Before coding, enumerate edge cases: empty input, single element, maximum constraints, duplicate values, negative numbers\. Design the solution to handle all of them from the start\.”

#### Complexity\-first\.

“State the required time and space complexity explicitly\. Identify the algorithmic paradigm \(DP, binary search, graph, etc\.\) before writing any code\. Optimize for competitive\-programming efficiency\.”

#### Proof\-sketch\.

“Before implementation, write a loop invariant or recurrence relation that proves the solution is correct\. Only then translate the proof into code\.”

### J\.3Refinement prompt \(verbatim\)

Only minority proposals are refined\. The refinement prompt has a*system prompt*\(persona\-specific\) and a*user prompt*\(structured XML blocks\)\.

#### Code refinement system prompt\.

> You are a \{persona\_name\} reviewing your solution against an ensemble consensus\. \{k\_max\}/\{n\} specialists produced a DIFFERENT approach that passes more tests than yours\. You may: \(a\) ADOPT the majority approach if their reasoning is stronger\. \(b\) IMPROVE your approach by fixing the specific issue the majority handles better, while preserving your unique strengths\. \(c\) DEFEND your approach if you believe the majority has a subtle flaw \(e\.g\. worse complexity, wrong invariant\) \-\-\- return your original with a brief note on why\. Return a single \`\`\`python\`\`\` fenced block\.

#### QA refinement system prompt\.

> You are a \{persona\_name\} reviewing your answer against an ensemble consensus\. \{k\_max\}/\{n\} specialists answered differently from you\. You may: \(a\) ADOPT the majority answer if their reasoning is stronger\. \(b\) IMPROVE your reasoning to arrive at a better answer\. \(c\) DEFEND your answer if you believe the majority has a flaw \-\-\- explain the specific error in their reasoning\. The last line of your response MUST be: ‘Answer: $VALUE’ where $VALUE follows the answer format the question requests\.

#### User prompt template \(code variant\)\.

The user prompt contains four XML blocks:<PROBLEM\>\(truncated to 2500 chars\),<YOUR\_PROPOSAL quality=…, intra\_agreement=…\>,<FAILURE\_ANALYSIS\>\(which public tests differ and stderr snippets\), and<MAJORITY\_APPROACH quality=…, support=k/n\>\. The final instruction is: “Produce your refined solution\.”

### J\.4Aggregation prompt \(verbatim\)

#### Code aggregation system prompt\.

> You are a senior problem\-solver synthesising a final Python solution from \{n\} specialist proposals\. Each proposal was the best of \{k\} self\-consistency samples from its persona\. You are given the full consensus evolution across a refinement step\. A persona that DEFENDED its minority position despite seeing the majority may have found a genuine edge case \-\-\- examine its reasoning carefully\. To produce a solution DIFFERENT from the post\-refinement majority, you MUST cite specific evidence from a defended minority proposal \(e\.g\. a concrete test case the majority fails\)\. If you cannot identify such evidence, return the majority solution verbatim\. Return a single \`\`\`python\`\`\` fenced block\.

#### QA aggregation system prompt\.

> You are a senior problem\-solver synthesising a final answer from \{n\} specialist proposals\. Each proposal was the best of \{k\} self\-consistency samples from its persona\. You are given the full consensus evolution across a refinement step\. A persona that DEFENDED its minority position despite seeing the majority may have found a genuine flaw \-\-\- examine its reasoning carefully\. To produce an answer DIFFERENT from the post\-refinement majority, you MUST cite specific evidence from a defended minority\. If you cannot, return the majority answer\. Commit: do NOT hedge\. The last line of your response MUST be: ‘Answer: $VALUE’ where $VALUE follows the answer format the question requests\.

#### User prompt template\.

The user prompt contains three XML blocks:<PROBLEM\>\(the original problem\),<PROPOSALS\>\(each persona’s proposal with cluster label, quality score, intra\-agreement, and refinement action annotation\), and<CONSENSUS\_EVOLUTION\>\(pre/post\-refinement agreement counts, which personas switched or defended\)\. Each proposal is annotated with one of:\[UNCHANGED \-\-\- majority\],\[REFINED \-\-\- ADOPTED\],\[REFINED \-\-\- IMPROVED\], or\[REFINED \-\-\- DEFENDED\]\.

### J\.5Variant\-tag protocol

Fork\>1k\>1samples per perturbation \(or per prompt in baselines\), we append a neutral HTML comment tag to the user prompt:

> <\!\-\- sample variant: scmoa\-analytical\-0 \-\-\>

The tag changes the content hashsha256​\(model​‖system\_prompt‖​user\_prompt\)\\texttt\{sha256\}\(\\texttt\{model\}\\\|\\texttt\{system\\\_prompt\}\\\|\\texttt\{user\\\_prompt\}\), producing a distinct LLM call\. At greedy decoding \(temperature 0\), the tag acts as a mild prompt perturbation: it shifts the model’s token\-level predictions without changing problem semantics\. HTML comment syntax was chosen because it has minimal semantic impact on the model’s interpretation of the problem\.

All methods use the same protocol with method\-specific prefixes: SC usessc\-\{k\}, Self\-MoA usesselfmoa\-\{k\}, MoA usesmoa\-\{k\}, and SC\-MoA usesscmoa\-\{persona\}\-\{k\}\. This ensures a fair comparison: all methods obtain sample diversity through the same mechanism\.

### J\.6Scoring and clustering

#### QA scoring\.

For QA benchmarks, the quality score is the*agreement fraction*: the proportion ofNNperturbations whose extracted answer matches this proposal’s extracted answer\. Formally, for perturbationiiwith extracted answeraia\_\{i\}:score​\(i\)=\|\{j:aj=ai\}\|/N\\text\{score\}\(i\)=\|\\\{j:a\_\{j\}=a\_\{i\}\\\}\|/N\. Clustering uses normalized string matching on the extracted answer \(lowercased, stripped\)\.

#### Code scoring\.

For code benchmarks, the quality score is the*public test pass rate*: the fraction of public test cases passed\. Clustering groups proposals by their*test\-pass signature*—the set of public test names that the solution passes\. Two proposals that pass different subsets of tests are placed in different clusters even if they share the same pass rate\. This is more informative than pass\-rate\-only clustering but creates the fidelity gap analyzed in §[I](https://arxiv.org/html/2605.29116#A9): the public test signature is an imperfect proxy for private\-test correctness\.

#### Majority selection\.

When multiple clusters tie in size, ties are broken by mean quality score \(higher wins\), then by earliest first occurrence\.

### J\.7Prompt allocation table

Table[16](https://arxiv.org/html/2605.29116#A10.T16)summarizes the system prompt allocation for each method, clarifying the diversity mechanism\.

Table 16:System prompt allocation per method\.

## Appendix KPersona Diversity Analysis

This appendix collects the persona\-related experiments referenced in the main text\.

### K\.1Error correlation \(ρ¯\\bar\{\\rho\}\) measurement

We measure mean pairwise error correlation between theN=5N\{=\}5agents under two conditions: \(a\) persona\-diverse \(5 different system prompts\) and \(b\) i\.i\.d\. \(1 generic prompt sampled 5 times\)\.

Table 17:Mean pairwise error correlationρ¯\\bar\{\\rho\}with 95% bootstrap CIs\.
### K\.2Perturbation content ablation \(extended\)

Table 18:Full perturbation content ablation\. Three conditions×\\timestwo benchmarks\. McNemarp\>0\.25p\>0\.25for all pairwise comparisons\.On LCB\-Hard, SPUQ paraphrases outperform hand\-crafted personas by\+2\.2\+2\.2pp\.

### K\.3Persona\-based vs\. paraphrase\-based MoA

Table 19:MoA baseline variants\. Persona\-MoA usesN=5N\{=\}5hand\-crafted expert personas \(11 calls\); Paraphrase\-MoA usesN=4N\{=\}4SPUQ paraphrases \(9 calls\)\. Main text Table[1](https://arxiv.org/html/2605.29116#S5.T1)reports the paraphrase variant for apples\-to\-apples comparison with SC\-MoA\.Persona\-MoA scores higher on GPQA \(\+7\.5\+7\.5pp\) but lower on LCB\-Hard \(−3\.5\-3\.5pp\)\. The persona advantage on GPQA likely reflects hand\-crafted domain prompts \(physicist, biologist, etc\.\) that steer proposals toward discipline\-specific reasoning chains\. On code, paraphrase diversity is more effective\. We report the paraphrase variant in Table[1](https://arxiv.org/html/2605.29116#S5.T1)because it uses the same SPUQ diversity mechanism as SC\-MoA, isolating the aggregation architecture as the sole variable\.

## Appendix LSemantic\-Preserving Paraphrase Pipeline \(SPUQ\)

SPUQ \(Semantic\-Preserving Uncertainty Quantification\) generatesNNrephrasings of the input problem\. The paraphrase generator receives a 6\-rule system prompt:

1. 1\.Semantic equivalence: each rephrasing must ask the same question and require the same answer\.
2. 2\.Surface variation: vary word choice, sentence structure, and notation style\.
3. 3\.Preserve numbers verbatim: all numerical values must appear exactly as in the original\.
4. 4\.Preserve code verbatim: all code blocks, variable names, and function signatures must be identical\.
5. 5\.Preserve units: physical units \(eV, m/s, kg, etc\.\) must appear exactly\.
6. 6\.Produce exactlyNNrephrasings: output as a JSON list\.

#### Validation\.

A regex\-based check verifies that all*protected tokens*—numbers, code blocks, and physical units—appear verbatim in each rephrasing\. Any rephrasing that fails is regenerated\. On GPQA\-Diamond \(N=5N\{=\}5, 300 total paraphrases\), the rejection rate is0%: the generator preserves all protected tokens on every science question\. On LCB\-Hard \(115 paraphrases\), the rejection rate is 16\.5%, concentrated on problems with complex spatial constraints \(e\.g\., chess board problems\); rejected paraphrases are regenerated until validation passes\.

#### Paraphrase\-quality bug\.

Early all\-8b runs used the proposer model \(llama\-3\.1\-8b\) as the paraphrase generator\. This produced malformed JSON that the parser dropped, effectively reducingNN\. Switching to a capable generator \(gpt\-oss\-120b\) improved accuracy by\+6\+6–88pp on GPQA at the all\-8b configuration \(22\.7%→\\to30\.8% atk=1k\{=\}1\)\. A\-\-paraphrase\-modelflag was added to decouple the generator role\.

#### Cost\.

A single LLM call generates allNNrephrasings \(batch generation\)\. ForN=5N\{=\}5, this adds 1 call per problem\. Paraphrases are cached and reused acrosskkvalues and ablation conditions\.

## Appendix MProposer×\\timesAggregator Capability Matrix

Table 20:Full capability matrix: SPUQ paraphrases atN=5N\{=\}5\.Three observations: \(1\) Dropping the aggregator from 120b to 8b loses∼2\.5\{\\sim\}2\.5pp on QA \(small\) but 14 pp on code atk=1k\{=\}1\(large\)\. \(2\) Dropping the proposer is catastrophic: GPQA crashes73%→31%73\\%\\to 31\\%\. \(3\) On code,k=3k\{=\}3at agg=8b recovers to 57\.3% \(−3\.5\-3\.5pp vs\. agg=120bk=1k\{=\}1\), but 80% of this recovery is the override mechanism \(§[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px4)\)\.

## Appendix NOverride and Best\-of\-N Analysis

Table 21:Override fire rates and accuracy decomposition by aggregator strength\.At strong aggregators, override rarely fires \(∼10%\{\\sim\}10\\%\) and accuracy is similar whether or not override is triggered\. At weak aggregators on code, override fires on the majority of problems \(5858–67%67\\%\) and is responsible for\+33\+33–4242pp of accuracy\. On QA, override never fires because the quality score is always 1\.0\.

#### Decomposition of the\+10\.5\+10\.5ppk=1→k=3k\{=\}1\\to k\{=\}3gain at agg=8b:

- •Genuinek×aggregatork\\times\\text\{aggregator\}interaction:\+2\.3\+2\.3pp
- •Override amplification \(cleanerk=3k\{=\}3clusters\):\+8\.2\+8\.2pp
- •Total:\+10\.5\+10\.5pp

#### Best\-of\-N ablation \(strong aggregator\)\.

On LCB\-Hard, SC\-MoA’s accuracy includes contributions from both aggregation synthesis \(Phase 4\) and score\-based override \(Phase 5\)\. Table[22](https://arxiv.org/html/2605.29116#A14.T22)isolates each component\.

Table 22:Decomposing SC\-MoA accuracy on LCB\-Hard \(n=171n\{=\}171,gpt\-oss\-120b\)\.At strong aggregators, the override fires on 11\.7% of problems \(20/171\) with 60% accuracy when it fires\. Aggregation alone \(65\.5%\) slightly*exceeds*the full pipeline \(63\.7%\)\. The override provides insurance against catastrophic aggregator failure \(the weak\-aggregator regime above, where it contributes\+33\+33–4242pp\) rather than a systematic accuracy boost at capable aggregators\.

## Appendix OOracle Ceiling Analysis

Table 23:Oracle ceiling: accuracy if the aggregator always selected the best proposal\. Gap = oracle−\-actual\.The aggregator leaves77–6363pp of correct answers on the table\. The gap grows withkk\(more candidates, same selection quality\) and is largest at weak proposers on QA \(\+62\.6\+62\.6pp at all\-8bk=3k\{=\}3\): the proposer pool contains the correct answer 94% of the time, but the system delivers only 31%\.

## Appendix PCluster Structure at Weak Proposers

At weak proposers \(llama\-3\.1\-8b\), the 5 SPUQ perturbations fragment into∼3\{\\sim\}3distinct answer clusters \(mean majority cluster size 2\.8/5, vs\. 4\.1/5 at strong proposers\)\.

Table 24:Pre\-refinement cluster shapes at all\-8b GPQAk=1k\{=\}1\.#### Unanimity reliability\.

At strong proposers, unanimous agreement \(C=1\.0C\{=\}1\.0\) is a strong signal:∼10%\{\\sim\}10\\%wrong rate\. At weak proposers, unanimity is a*negative*signal:∼43%\{\\sim\}43\\%wrong rate\. Unanimity at weak proposers reflects shared training\-data biases, not convergence to truth\.

Table 25:Unanimity wrong rate across configurations\.

## Appendix QSampling Protocol Comparison

Standard self\-consistency\[Wanget al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib15)\]draws diverse samples via temperature sampling \(T\>0T\>0\)\. Our baselines instead use greedy decoding \(T=0T\{=\}0\) with neutral variant tags \(§[J\.5](https://arxiv.org/html/2605.29116#A10.SS5)\) to create sample diversity\. To confirm that this protocol does not disadvantage the SC baseline, we run SC\(k=10k\{=\}10\) at four temperature settings and compare against our greedy\+tag variant\.

Table 26:Sampling protocol comparison: SC\(k=10k\{=\}10\) accuracy \(%\) under different diversity mechanisms\. On QA, greedy\+tag and temperature sampling are statistically indistinguishable \(McNemarp=0\.332p\{=\}0\.332for best temperature vs\. greedy\+tag\)\. On code, temperature sampling yields meaningfully higher SC accuracy than greedy\+tag \(67\.3% vs\. 57\.3% atT=0\.7T\{=\}0\.7\), reflecting the value of stochastic exploration for code generation\.On QA tasks, greedy decoding with a neutral variant tag produces diversity comparable to temperature sampling: all five conditions fall within a 2\.5 pp band \(70\.7–72\.2%\), and no pairwise McNemar test reaches significance\. SC\-MoA \(72\.7%\) slightly exceeds the strongest temperature SC \(72\.2% atT=0\.5T\{=\}0\.5\) by 0\.5 pp\.

On code tasks, temperature sampling provides substantially better diversity than greedy\+tag: accuracy peaks atT=0\.7T\{=\}0\.7\(67\.3%\) andT=0\.3T\{=\}0\.3\(65\.5%\), compared with 57\.3% for greedy\+tag\. Higher temperature \(T=1\.0T\{=\}1\.0, 59\.1%\) degrades accuracy, consistent with excessive token noise corrupting code syntax\. Token\-level stochasticity at moderate temperatures explores qualitatively different implementation strategies, while variant tags produce only minor prompt perturbations that often yield similar code\. SC\-MoA \(persona\-greedy\) remains competitive with the best temperature SC \(59\.6% vs\. 67\.3%, McNemarp=0\.711p\{=\}0\.711\)\.

### Q\.1Input\-Side vs\. Output\-Side Diversity

A natural question is whether input\-side diversity \(SPUQ perturbations\) fundamentally outperforms output\-side diversity \(temperature sampling\)\. To isolate the diversity source, we hold the total call budget fixed atN=5N\{=\}5proposals and compare three conditions, all using the same aggregation pipeline \(Phases 2–5\):

Table 27:Diversity source comparison, holding total proposals fixed atN=5N\{=\}5\. All conditions use the same aggregation pipeline \(Phases 2–5\)\. On GPQA, greedy decoding suffices: persona\-greedy and SPUQ\-greedy are statistically indistinguishable \(p=0\.480p\{=\}0\.480\), and temperature hurts\. On LCB\-Hard, input\-side and output\-side diversity are complementary: SPUQ\+\+T=0\.7T\{=\}0\.7yields the highest accuracy \(68\.4%\),\+4\.7\+4\.7pp over persona\-greedy\.QA\.SPUQ\-greedy \(73\.2%\) and persona\-greedy \(75\.3%\) are statistically indistinguishable \(McNemarp=0\.480p\{=\}0\.480,n=198n\{=\}198\): of the 18 problems where they disagree, personas win 11 and SPUQ wins 7\. Adding temperature to SPUQ does not help on QA: SPUQ\+\+T=0\.7T\{=\}0\.7\(72\.2%\) is 1\.0 pp below SPUQ\-greedy \(73\.2%\) and 3\.1 pp below persona\-greedy \(75\.3%\)\. Temperature\-only diversity \(T=0\.7T\{=\}0\.7, 69\.7%\) fares worse still, whileT=1\.0T\{=\}1\.0\(74\.7%\) partially recovers\. On QA, token\-level noise degrades the aggregator’s inputs without contributing useful diversity—greedy decoding suffices\.

Code\.On LCB\-Hard, temperature\-only diversity \(T=0\.7T\{=\}0\.7, single prompt\) yields 67\.3%—identical to SC\(T=0\.7T\{=\}0\.7\), indicating that the aggregation pipeline adds no value when all diversity comes from token\-level stochasticity\. Higher temperature \(T=1\.0T\{=\}1\.0\) degrades further to 62\.0%, below even persona\-greedy \(63\.7%\)\. Stacking SPUQ perturbations with moderate temperature \(T=0\.7T\{=\}0\.7\) recovers a clear gain: 68\.4%,\+4\.7\+4\.7pp over persona\-greedy and\+1\.1\+1\.1pp over SC\(T=0\.7T\{=\}0\.7\)\. Input\-side perturbation and output\-side stochasticity are*complementary*on code: SPUQ varies the problem framing while temperature varies the implementation strategy, together exploring a richer proposal space than either mechanism alone\.

## Appendix RToken\-Level Compute Accounting

To verify that SC\-MoA’s accuracy gains are not explained by disproportionate token expenditure, we report total tokens \(input\+\+output\) per method per problem\.

Table 28:Mean tokens per problem and accuracy\-per\-million\-tokens \(Acc/MT\) on GPQA\-Diamond and BBH\. SC\-MoA \(ungated\) achieves the highest accuracy at a fraction of the token cost of SC\(k=10k\{=\}10\)\.‡Token counts unavailable for the paraphrase\-based MoA variant \(N=4N\{=\}4, 9 calls\); see Appendix[K\.3](https://arxiv.org/html/2605.29116#A11.SS3)\.

On GPQA, SC\-MoA \(ungated\) uses3\.1×3\.1\{\\times\}fewer tokens than SC\(k=10k\{=\}10\) while achieving\+2\.0\+2\.0pp higher accuracy\. On BBH the ratio is2\.7×2\.7\{\\times\}fewer tokens at\+1\.6\+1\.6pp\. The efficiency comes from Phase 1 generating onlyN=5N\{=\}5base proposals plus a single aggregation call, whereas SC generates 10 independent samples\. The gated variant is more expensive because it also runs refinement and inner SC \(k=2k\{=\}2\)\. Across both benchmarks, SC\-MoA dominates every baseline on the accuracy\-per\-token frontier\.

## Appendix SContamination Analysis

We analyze potential training\-data contamination across all three benchmark families\.

#### BBH\.

We stratify the 27 BBH tasks by SC\-MoA’s improvement over SC\(k=10k\{=\}10\)\. Of 27 tasks, 8 show improvement \(\+1\+1to\+13\.5\+13\.5pp\), 3 show degradation \(−4\-4to−8\-8pp\), and 16 are tied \(14 of which are solved perfectly by both methods\)\. The improved tasks span diverse cognitive domains—state tracking \(*penguins\_in\_a\_table*,\+13\.5\+13\.5pp\), temporal reasoning \(*date\_understanding*,\+9\+9pp\), and linguistic parsing \(*disambiguation\_qa*,\+12\+12pp\)—with no concentration in tasks more likely to be contaminated\.

#### GPQA\-Diamond\.

GPQA was designed to be “Google\-proof”: domain experts achieve only∼\{\\sim\}65% accuracy, and the questions were validated against web search\[Reinet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib12)\]\. Stratifying by scientific domain: Chemistry \(n=93n\{=\}93\) improves by\+9\.7\+9\.7pp over SC, Physics \(n=86n\{=\}86\) by\+1\.2\+1\.2pp, and Biology \(n=20n\{=\}20\) degrades by−8\.2\-8\.2pp\. The Biology degradation is consistent with a small sample \(n=20n\{=\}20, not significant\) rather than domain\-specific contamination\.

#### LCB\-Hard\.

All 171 problems havecontest\_date≥\\geqSeptember 2024, placing them after the training cutoff by construction\. We additionally verified that SC\-MoA does not preferentially solve older problems within this window: the date distribution of solved vs\. unsolved problems is indistinguishable \(Kolmogorov–SmirnovD=0\.114D\{=\}0\.114,p=0\.633p\{=\}0\.633\)\. Per\-month accuracy ranges from 28\.6% \(April 2025,n=7n\{=\}7\) to 77\.8% \(December 2024,n=18n\{=\}18\), with no monotonic trend\.

## Appendix TAggregation Value Analysis

On QA benchmarks, the score\-based override \(Phase 5\) never fires because the quality score is always 1\.0\. This isolates the aggregator’s contribution: any accuracy difference between majority\-vote\-only and SC\-MoA is attributable to Phase 4 aggregation\.

We decompose the aggregator’s effect on GPQA\-Diamond by comparing the pre\-aggregation majority answer against the post\-aggregation final answer for each problem:

Table 29:Aggregation value on GPQA\-Diamond \(n=198n\{=\}198\):2×22\\times 2contingency table comparing majority\-vote\-only vs\. aggregated answer\. McNemarχ2=2\.88\\chi^\{2\}\{=\}2\.88,p=0\.090p\{=\}0\.090\.Table 30:Aggregation value on BBH\-3 \(n=296n\{=\}296, pre\-refinement controlled setting\):2×22\\times 2contingency table comparing majority\-vote\-only vs\. synthesized answer\. McNemarχ2=5\.76\\chi^\{2\}\{=\}5\.76,p=0\.016p\{=\}0\.016\.Note:these are pre\-refinement numbers from the §[3](https://arxiv.org/html/2605.29116#S3)controlled setting \(synthesis vs\. voting on the same proposals\)\. After anchored refinement in the full pipeline, the harmful flips are eliminated \(Table[31](https://arxiv.org/html/2605.29116#A20.T31)reports 46 beneficial and 0 harmful on BBH\)\.

#### Synthesis advantage decomposition across benchmarks\.

Table[31](https://arxiv.org/html/2605.29116#A20.T31)extends the aggregation\-value analysis to four benchmarks using paraphrase perturbations, decomposing the net synthesis advantage into recovery \(Pw⋅RsP\_\{w\}\\cdot R\_\{s\}\) and corruption \(Pc⋅\(1−Fs\)P\_\{c\}\\cdot\(1\{\-\}F\_\{s\}\)\) terms per Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1)\(Figure[5](https://arxiv.org/html/2605.29116#S5.F5)b\)\. GPQA\-Diamond and LCB\-Hard use the SPUQ paraphrase variant \(N=4N\{=\}4,k=5k\{=\}5\); BBH and AIME useN=5N\{=\}5,k=2k\{=\}2\. Recovery dominates corruption on every benchmark, and all four McNemar tests are significant \(p<0\.05p\{<\}0\.05\), confirming that beneficial flips reliably outnumber harmful flips\.

Table 31:Synthesis advantage decomposition across four benchmarks \(paraphrase perturbation variants\)\. Config column specifies the operating point; the decisive subset excludes problems with degenerate answer\-only traces \(GPQA: 9 excluded, LCB: 7 excluded due to sandbox timeouts\)\. Recovery and corruption are expressed as percentage\-point contributions\. All McNemar tests are significant atp<0\.05p\{<\}0\.05\. MMLU\-ML \(n=112n\{=\}112\) is excluded because its small decisive set \(n<10n\{<\}10flips\) yields unreliable decomposition estimates\.
### T\.1Case Studies

We present a representative beneficial flip from GPQA\-Diamond to illustrate the aggregation mechanism\.

#### Example: GPQA problem 69 \(chemistry, point\-group symmetry\)\.

Five perturbations produce proposals:

- •Para 1\(k=2k\{=\}2, both wrong\): concludes C2v
- •Para 2\(k=2k\{=\}2, 1 wrong/1 correct\): first sample says D∞h, second says D∞h
- •Para 3\(k=2k\{=\}2, 1 correct/1 wrong\): first says B \(C2v\), second disagrees
- •Para 4\(k=2k\{=\}2, both wrong\): concludes D∞h\(answer D\)
- •Para 5\(k=2k\{=\}2, both wrong\): concludes C2v\(answer B\)

SC majority vote: 2/10 samples are correct→\\towrong\. Pre\-refinement consensus:C=0\.4C=0\.4\(split between B and D\)\. Three minorities adopt the emerging answer during refinement \(Cpost=0\.6C\_\{\\text\{post\}\}=0\.6\)\. The aggregator reads all traces—including the correct reasoning from Paras 2 and 3—and outputs the correct answerB\.

The mechanism is not best\-of\-N selection: the aggregator does not see correctness labels\. It synthesizes across reasoning traces, weighting arguments that identify the molecule as bent \(C2v\) over those that misidentify it as linear \(D∞h\)\.

Among the 16 beneficial flips in the SPUQ paraphrase variant \(Table[29](https://arxiv.org/html/2605.29116#A20.T29)\), the majority follow this pattern: a minority of perturbations contain the correct reasoning, and the aggregator recovers it by integrating evidence across traces\.

## Appendix UTrace Diversity Analysis

Figure[5](https://arxiv.org/html/2605.29116#S5.F5)\(a\) shows that trace\-level diversity \(DtD\_\{t\}\) predicts beneficial aggregation flips across four benchmarks \(Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1)\)\. Here we report the full measurement methodology and results\.

#### Methodology\.

For each problem, we extract the first reasoning trace from each perturbation and computeDtD\_\{t\}as the mean pairwise cosine distance in TF\-IDF space \(5000 features, English stop words removed\)\. Problems where all traces are answer\-only \(degenerateDt=0D\_\{t\}\{=\}0\) are excluded from the distributional analysis \(24 problems total: 7 in GPQA, 17 in BBH\)\. GPQA\-Diamond and LCB\-Hard use SPUQ paraphrases \(N=4N\{=\}4,k=5k\{=\}5\); BBH \(date understanding, disambiguation QA, penguins in a table\) and AIME 2022–24 useN=5N\{=\}5,k=2k\{=\}2\.

#### Results\.

Table[32](https://arxiv.org/html/2605.29116#A21.T32)reportsDtD\_\{t\}stratified by aggregation flip type across all four benchmarks\.

Table 32:Trace diversity \(D¯t\\bar\{D\}\_\{t\}\) by aggregation outcome across four benchmarks\. Beneficial flips consistently show higher trace diversity than non\-flips\. Degenerate zero\-diversity traces excluded\.Welchtt\-tests \(beneficial vs\. non\-flip\): GPQA\-Diamondt=2\.70t\{=\}2\.70,p=0\.015p\{=\}0\.015; BBHt=2\.05t\{=\}2\.05,p=0\.045p\{=\}0\.045; AIMEt=1\.99t\{=\}1\.99,p=0\.057p\{=\}0\.057; LCB\-Hardt=1\.80t\{=\}1\.80,p=0\.081p\{=\}0\.081\. The effect is significant atp<0\.05p\{<\}0\.05on GPQA\-Diamond and BBH, and directionally consistent on AIME and LCB\-Hard\. Point\-biserial correlations betweenDtD\_\{t\}and beneficial flip occurrence:r=0\.20r\{=\}0\.20\(p=0\.007p\{=\}0\.007\) on GPQA,r=0\.13r\{=\}0\.13\(p=0\.037p\{=\}0\.037\) on BBH,r=0\.22r\{=\}0\.22\(p=0\.042p\{=\}0\.042\) on AIME,r=0\.15r\{=\}0\.15\(p=0\.055p\{=\}0\.055\) on LCB\-Hard\.

#### Harmful flips\.

Harmful flips occur only on GPQA\-Diamond \(n=4n\{=\}4\) and LCB\-Hard \(n=5n\{=\}5\); BBH and AIME have zero harmful flips\. On both benchmarks where harmful flips exist, theirDtD\_\{t\}is comparable to beneficial flips \(0\.5210\.521and0\.4850\.485, respectively\)\. This is consistent with Proposition[1](https://arxiv.org/html/2605.29116#Thmtheorem1): high trace diversity increases*both*flip types, but the net advantage is positive because synthesis fidelityFsF\_\{s\}is high \(Table[31](https://arxiv.org/html/2605.29116#A20.T31)\)\.

## Appendix VEmbedding\-Based Diversity Validation

The TF\-IDF representation used in Appendix[U](https://arxiv.org/html/2605.29116#A21)captures lexical variation but fits a small per\-problem vocabulary \(4 documents per problem atN=4N\{=\}4\), raising the question of whether theDtD\_\{t\}–flip association reflects genuine structural differences in reasoning or merely surface rewording\. We validate the finding under dense pre\-trained embeddings that encode semantic structure independently of per\-problem term frequencies\.

#### Methodology\.

We embed all reasoning traces using Amazon Titan Text Embeddings v2 \(1024 dimensions, L2\-normalized\), a pre\-trained dense encoder whose representation is fixed across problems—unlike TF\-IDF, which fits a separate vocabulary per problem on onlyNNdocuments\. We computeDtD\_\{t\}as1−𝐯i⊤​𝐯j1\-\\mathbf\{v\}\_\{i\}^\{\\top\}\\mathbf\{v\}\_\{j\}over all perturbation pairs, identically to the TF\-IDF pipeline but replacing the sparse representation\. The analysis uses the mechanistic configuration \(N=4N\{=\}4,k=5k\{=\}5, GPQA\-Diamond, 198 problems, 3,390 unique traces\)\. TF\-IDF excludes 8 degenerate problems with answer\-only traces \(n=190n\{=\}190\); embeddings retain all 198\.

#### Results\.

Table[33](https://arxiv.org/html/2605.29116#A22.T33)reportsDtD\_\{t\}under both representations side by side on the same dataset\.

Table 33:Trace diversity \(D¯t\\bar\{D\}\_\{t\}\) under TF\-IDF vs\. dense embeddings on GPQA\-Diamond \(N=4N\{=\}4,k=5k\{=\}5\)\. Both representations yield large, significant effects \(Cohen’sd≈0\.92d\\approx 0\.92\)\.Welchtt\-tests: TF\-IDFt=3\.44t\{=\}3\.44,p=0\.0007p\{=\}0\.0007; embeddingt=3\.53t\{=\}3\.53,p=0\.0005p\{=\}0\.0005\. Point\-biserial correlations are nearly identical:r=0\.246r\{=\}0\.246under both \(p<0\.001p<0\.001\)\.

#### Interpretation\.

Dense embeddings compress the absoluteDtD\_\{t\}scale \(the overall mean drops from0\.5290\.529to0\.3300\.330\) because semantically similar traces cluster more tightly in embedding space\. Despite this compression, the beneficial\-flip separation is preserved with essentially identical effect size, indicating that the signal is carried by genuine reasoning\-strategy variation rather than lexical noise\.

The most informative discrepancy is in harmful flips: under TF\-IDF, harmful flips show moderate diversity \(D¯t=0\.558\\bar\{D\}\_\{t\}\{=\}0\.558, above the no\-flip baseline of0\.5140\.514\), but under embeddings they collapse to baseline \(D¯t=0\.310\\bar\{D\}\_\{t\}\{=\}0\.310vs\. no\-flip0\.3160\.316\)\. This indicates that the TF\-IDF diversity in harmful cases is driven by surface rewording—different words expressing the same flawed reasoning—rather than the structural reasoning divergence that enables recovery\. The embedding representation strips away this superficial variation, cleanly separating the beneficial signal from noise\.

## Appendix WControlled Perturbation vs\. Free\-Form Paraphrasing

SC\-MoA’s default perturbation strategy \(Phase 0\) uses free\-form paraphrasing: a single LLM call generatesNNunconstrained meaning\-preserving rephrasings \(instantiated via SPUQ\[Gaoet al\.,[2024](https://arxiv.org/html/2605.29116#bib.bib42)\]\)\. An alternative is*controlled perturbation*, where each paraphrase applies exactly one named linguistic operation drawn from a fixed taxonomy\. This section reports a head\-to\-head comparison on GPQA\-Diamond\.

We adapt the controlled\-perturbation taxonomy from Polyjuice\[Wuet al\.,[2021](https://arxiv.org/html/2605.29116#bib.bib47)\]and Tailor\[Rosset al\.,[2022](https://arxiv.org/html/2605.29116#bib.bib48)\]to an LLM\-based generator, defining five meaning\-preserving operation types \(lexical, syntactic, discourse, register, compression\)\. Each ofN=5N\{=\}5agents receives a paraphrase generated under a different type; the same protected\-token validation pipeline applies\.

#### Experimental setup\.

We evaluate on a 50\-problem subset of GPQA\-Diamond usinggpt\-oss\-120bfor both paraphrase generation and SC\-MoA proposing/aggregation \(N=5N\{=\}5,k=2k\{=\}2, always\-aggregate\)\. Both conditions use identical model, temperature, and caching infrastructure; the only difference is the perturbation generator\. Paraphrases are pre\-generated and cached before the SC\-MoA run to avoid rate\-limit interference\.

#### Results\.

Table[34](https://arxiv.org/html/2605.29116#A23.T34)summarizes the comparison\.

Table 34:Head\-to\-head on GPQA\-Diamond \(n=50n\{=\}50,gpt\-oss\-120b,N=5N\{=\}5,k=2k\{=\}2\)\. Both methods use identical SC\-MoA infrastructure; the only variable is the perturbation generator\.The\+2\.0\+2\.0pp difference is within the margin expected fromn=50n\{=\}50\(McNemar’s test: 4 vs\. 3 discordant pairs,p=1\.0p\{=\}1\.0\)\. Both methods produce nearly identical consensus profiles \(C¯pre≈0\.80\\bar\{C\}\_\{\\text\{pre\}\}\\approx 0\.80,C¯post≈0\.96\\bar\{C\}\_\{\\text\{post\}\}\\approx 0\.96\) and use the same number of LLM calls\.

#### Power analysis\.

Then=50n\{=\}50sample yields 80% power to detect a≥12\{\\geq\}12pp effect \(two\-sided McNemar,α=0\.05\\alpha\{=\}0\.05\)\. The observed 2 pp difference is well within the noise floor\. We interpret this as absence of evidence for a large perturbation\-content effect, not as evidence of absence\. A properly powered comparison \(n≥400n\\geq 400\) would require additional compute budget\.

#### Per\-type validation\.

Controlled perturbations achieve 97\.2% overall validation \(243/250\), compared to 98\.0% for free\-form SPUQ \(245/250\)\. Lexical and syntactic perturbations preserve all protected tokens \(100%\); compression is hardest \(92%\), as tightening prose occasionally drops a unit or number\.

We retain SPUQ as the default because it requires one LLM call vs\. five, reducing Phase 0 cost by∼5×\{\\sim\}5\{\\times\}\.

### W\.1Lightweightk=1k\{=\}1variant

Table 35:k=1k\{=\}1variant\(N=4N\{=\}4, SPUQ paraphrases,gpt\-oss\-120b\)\. SC\-MoA atk=1k\{=\}1uses∼5\.2\{\\sim\}5\.2LLM calls and matches or exceeds SC\(k=10k\{=\}10\) on 3 of 4 benchmarks\.Note:this table usesN=4N\{=\}4; Table[1](https://arxiv.org/html/2605.29116#S5.T1)usesN=5N\{=\}5,k=2k\{=\}2—numbers are not directly comparable\.Atk=1k\{=\}1, SC\-MoA uses∼5\.2\{\\sim\}5\.2calls \(no inner SC\) and exceeds SC\(k=10k\{=\}10\) on GPQA \(\+5\.0\+5\.0pp\) at roughly half the compute \(Figure[3](https://arxiv.org/html/2605.29116#S5.F3); discussion in §[5](https://arxiv.org/html/2605.29116#S5.SS0.SSS0.Px3)\)\.

## Appendix XAdditional Ablations and Stability

### X\.1Trace Ablation: Full Results

We ablate the trace content visible to the aggregator, holding Phases 1–3 constant \(same SPUQ\-paraphrased proposals, same refinement, same post\-refinement clusters\)\. Only the Phase 4 aggregation call differs\. Eight conditions test distinct hypotheses about what information drives the aggregator’s beneficial corrections \(Table[36](https://arxiv.org/html/2605.29116#A24.T36)\)\. “Ben\.” and “Harm\.” count beneficial and harmful flips vs\. majority vote\.

Table 36:Trace ablation on GPQA\-Diamond\(n=198n\{=\}198, SPUQ paraphrases,N=5N\{=\}5,k=2k\{=\}2\)\. Beneficial and harmful flips are measured against majority vote\.p∗p^\{\*\}: Holm–Bonferroni\-corrected McNemarpp\-value \(7 comparisons against majority vote\)\.Table[36](https://arxiv.org/html/2605.29116#A24.T36)reports all conditions; we highlight the consensus\-stratified findings below\.

#### Consensus\-stratified analysis\.

The effect concentrates in the strong\-consensus stratum \(23 problems, 12% of total\): full\-trace accuracy jumps from 21\.7% \(majority vote\) to 47\.8%, and minority\-only traces reach 47\.8% as well—more than doubling majority\-vote accuracy on the hardest disputed problems\. On unanimous problems \(175/198, 88%\), all conditions perform within±1\.7\{\\pm\}1\.7pp of majority vote, as expected: when no minority traces exist, the aggregator has no complementary reasoning to integrate\.

#### Degenerate cases\.

In condition \(d\), 175/198 problems \(88\.4%\) have unanimous post\-refinement consensus, leaving no genuine minority traces\. For these, the aggregator receives a synthetic message noting unanimity; it flips away from the unanimous answer on only 2/175 problems \(1\.1%\), confirming that the aggregator respects strong consensus even without the evolution narrative\.

### X\.2Information Ladder: Compute vs\. Evidence

The trace ablation \(§[X\.1](https://arxiv.org/html/2605.29116#A24.SS1)\) establishes that answer\-only and majority\-only conditions collapse to MV, but cannot distinguish whether the aggregator’s gain comes from*re\-solving with additional compute*or from*synthesizing complementary trace evidence*\. We construct an information ladder that progressively reveals proposer information while holding Phases 1–3 constant \(Table[37](https://arxiv.org/html/2605.29116#A24.T37)\)\.

#### Conditions\.

Eight conditions span zero to full information\.Q\-only \(agg\):the aggregator receives only the problem under its standard system prompt \(mentioning “specialist proposals”\), with zero proposals—a pure re\-solve baseline\.Q\-only \(CoT\):identical, but using the proposer’s chain\-of\-thought prompt, controlling for whether the aggregator prompt biases re\-solving\.Answers bare:an unnumbered list of extracted answer labels \(e\.g\., “A, A, A, B, C”\) with no metadata—testing whether the raw vote distribution helps beyond re\-solving\.Answers \+ metadata:theanswer\_onlycondition from §[X\.1](https://arxiv.org/html/2605.29116#A24.SS1), including persona names, quality, intra\-agreement, cluster labels, and evolution narrative\.Traces bare:full proposal text \(1,800 char cap\) with generic headers, no persona names, quality, clusters, or evolution\.Traces \+ partial meta:theno\_metadatacondition\.Traces−\-evolution:theno\_evolutioncondition\.Full:the unmodified Phase 4 call\.

Table 37:Information ladder on GPQA\-Diamond\(n=198n\{=\}198, SPUQ paraphrases,N=5N\{=\}5,k=2k\{=\}2\)\. Conditions ordered by increasing information available to the aggregator\. Ben\./Harm\. = beneficial/harmful flips vs\. majority vote\.pp: McNemar, Holm–Bonferroni corrected \(7 comparisons vs\. MV\)\.ConditionAcc\. \(%\)Ben\.Harm\.RatioppMajority vote69\.7––––re\-solveQ\-only \(agg prompt\)55\.610380\.26<<0\.001Q\-only \(CoT prompt\)63\.110230\.430\.035votesAnswers bare72\.2942\.250\.267Answers \+ metadata69\.7661\.001\.000tracesTraces bare72\.71152\.200\.210Traces \+ partial meta72\.7933\.000\.146Traces−\-evolution74\.21125\.500\.023Full \(all info\)72\.2832\.670\.227
#### Value decomposition\.

\+2\.5​pp⏟full−MV=−14\.1​pp⏟re\-solve\+\+16\.7​pp⏟trace evidence\\displaystyle\\underbrace\{\+2\.5\\text\{\\,pp\}\}\_\{\\text\{full\}\-\\text\{MV\}\}\\;=\\;\\underbrace\{\-14\.1\\text\{\\,pp\}\}\_\{\\text\{re\-solve\}\}\\;\+\\;\\underbrace\{\+16\.7\\text\{\\,pp\}\}\_\{\\text\{trace evidence\}\}The re\-solve component is strongly negative: the aggregator destroys accuracy when forced to reason from scratch\. The entire\+16\.7\+16\.7pp trace\-evidence component must compensate for the re\-solve penalty to produce the net\+2\.5\+2\.5pp gain\.

#### The CoT prompt control\.

Q\-only \(CoT\) scores 63\.1%—\+7\.5\+7\.5pp above Q\-only \(agg\) but still−6\.6\-6\.6pp below MV\. The aggregator system prompt, which references “specialist proposals” and “minority positions,” confuses the model when no proposals are present\. Neither re\-solve variant approaches MV\.

#### Vote distribution provides weak signal\.

Answers bare \(72\.2%\) matches full\-trace accuracy and exceeds MV, suggesting the raw vote distribution provides some signal\. Adding structured metadata to answer labels*degrades*accuracy back to MV \(69\.7%, ratio 1\.00\)—metadata without supporting reasoning introduces noise that overrides the implicit vote signal\.

#### Traces are the active ingredient\.

Traces bare \(72\.7%\) and traces with partial metadata \(72\.7%\) both exceed MV, with beneficial\-to\-harmful ratios of 2\.20 and 3\.00\. Removing the evolution narrative while keeping full metadata yields the highest accuracy \(74\.2%, ratio 5\.50,p=0\.023p\{=\}0\.023\)—consistent with the finding that evolution narrative introduces mild noise\.

#### Consensus\-stratified analysis\.

On unanimous problems \(175/198, 88%\), all trace\-level conditions cluster within±0\.6\{\\pm\}0\.6pp of MV \(76\.0%\)\. The re\-solve conditions are the exception: Q\-only \(agg\) drops to 58\.9% \(−17\.1\-17\.1pp\), showing that re\-solving corrupts even easy problems\.

On contested problems \(23/198, 12%\), MV scores 21\.7%; Q\-only \(agg\) reaches 30\.4%; but traces bare jumps to 47\.8% and traces−\-evolution reaches 60\.9%\. The aggregator nearly triples contested\-problem accuracy when given reasoning traces, confirming complementary evidence from diverse proposers is the mechanism\.

Table 38:Consensus\-stratified information ladderon GPQA\-Diamond\. Unanimous: post\-refinement consensus≥80%\\geq 80\\%; contested:<80%<80\\%\.
#### Adjacent\-pair McNemar tests\.

Four pre\-registered adjacent contrasts along the ladder \(Table[39](https://arxiv.org/html/2605.29116#A24.T39)\), corrected with Holm–Bonferroni \(m=4m\{=\}4\)\.

Table 39:Adjacent\-pair McNemar testsalong the information ladder\.Only the first contrast is significant: the jump from zero proposer information to vote distribution accounts for essentially all recoverable accuracy\. Subsequent steps do not produce individually significant gains against their immediate predecessor, though trace\-level conditions collectively dominate answer\-level conditions in flip ratio \(2\.20–5\.50 vs\. 1\.00–2\.25\)\.

![Refer to caption](https://arxiv.org/html/2605.29116v1/x8.png)Figure 10:Information ladderon GPQA\-Diamond \(n=198n\{=\}198\)\. Accuracy with Wilson 95% CIs, ordered by increasing information\. The dashed line marks majority vote \(69\.7%\)\. Re\-solve conditions \(red\) fall well below MV; trace\-level conditions \(green\) cluster above it\.![Refer to caption](https://arxiv.org/html/2605.29116v1/x9.png)Figure 11:Consensus\-stratified information ladder\.On unanimous problems \(left,n=175n\{=\}175\), all conditions cluster near MV except re\-solve, which collapses\. On contested problems \(right,n=23n\{=\}23\), trace conditions nearly triple MV accuracy\.![Refer to caption](https://arxiv.org/html/2605.29116v1/x10.png)Figure 12:Beneficial\-to\-harmful flip ratioacross the information ladder\. Re\-solve conditions \(<1×<1\{\\times\}\) are corruption\-dominated; trace conditions \(22–5\.5×5\.5\{\\times\}\) are recovery\-dominated\.

### X\.3Selection Baselines: Full Results

We compare SC\-MoA’s synthesis aggregation against two selection baselines that choose the best*existing*proposal rather than generating a new one, using the same Phase 1 proposals \(N=5N\{=\}5,k=2k\{=\}2\)\.

#### Dawid–Skene EM\[Dawid and Skene,[1979](https://arxiv.org/html/2605.29116#bib.bib2)\]\.

Multinomial Dawid–Skene estimates a per\-agent confusion matrixπj​\[ctrue,cobs\]\\pi\_\{j\}\[c\_\{\\text\{true\}\},c\_\{\\text\{obs\}\}\]via EM across all 198 GPQA problems, then selects the answer with highest reliability\-weighted posterior per problem\. This adds 0 extra LLM calls—it is a pure post\-hoc reweighting of the existing agent votes\.

#### Pairwise LLM judge\[Zhenget al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib1)\]\.

For each problem, all\(52\)=10\\binom\{5\}\{2\}\{=\}10proposal pairs are presented to the same model \(gpt\-oss\-120b\) with a judge prompt requesting a binary “A” or “B” verdict\. Borda count aggregates pairwise wins; the proposal with the highest Borda score is selected\. This adds 10 LLM calls per problem\.

Table 40:Selection vs\. synthesis on GPQA\-Diamond\(n=198n\{=\}198\)\. McNemar tests are against SC\-MoA synthesis\.SC\-MoA synthesis significantly outperforms all voting and selection baselines atp<0\.03p<0\.03\. The pairwise judge matches accuracy but at nearly2×2\{\\times\}compute \(21 vs\. 11\.5 calls per problem\), making SC\-MoA Pareto\-dominant on the accuracy–compute frontier\.

Dawid–Skene slightly*underperforms*naive majority vote \(68\.7%68\.7\\%vs\.69\.2%69\.2\\%\), consistent with the known limitation that EM\-based annotator models require substantially more annotators thanN=5N\{=\}5to estimate reliable confusion matrices\[Dawid and Skene,[1979](https://arxiv.org/html/2605.29116#bib.bib2)\]\. The learned weights add noise rather than signal at this scale\.

The pairwise judge result is informative: it shows that the correct answer*is present*among the proposals∼75%\{\\sim\}75\\%of the time—the challenge is extracting it\. Selection via\(52\)\\binom\{5\}\{2\}comparisons and synthesis via a single aggregation call reach the same accuracy, but synthesis does so at half the cost\. This confirms that the aggregator’s advantage is efficiency, not access to information that selection methods lack\.

### X\.4Stability Under Input Perturbation

All experiments use greedy decoding with content\-hash caching, so re\-running produces bitwise\-identical results\. We quantify stability to the*choice*of input perturbation—a stronger test than seed variation, since it varies the actual problem text\.

#### Bootstrap accuracy CIs\.

10,000 bootstrap resamples over the 198 GPQA\-Diamond problems yield 95% CIs for each method \(Table[40](https://arxiv.org/html/2605.29116#A24.T40)\)\. SC\-MoA’s CI\[69\.2%,81\.3%\]\[69\.2\\%,81\.3\\%\]excludes all voting baselines’ point estimates, consistent with the McNemar tests\.

#### Cross\-perturbation agreement\.

The SPUQ paraphrase\-mode run generates 5 independent perturbations per problem\. Each perturbation produces a best\-of\-kkproposal whose extracted answer may differ\. Across 198 problems, unanimous agreement \(all 5 paraphrases yield the same answer\) occurs on 56\.4% of problems, with 50\.9% all\-correct\. The 56\.4% unanimity rate is a lower bound on stability: after aggregation, the pipeline absorbs remaining disagreement into a single synthesized answer\.

## Appendix YArchitectural Comparison of Aggregation Strategies

Figure[13](https://arxiv.org/html/2605.29116#A25.F13)compares the architectural structure of six multi\-agent aggregation strategies\. Self\-consistency\[Wanget al\.,[2023](https://arxiv.org/html/2605.29116#bib.bib15)\]sampleskkresponses from a single prompt and takes a majority vote over extracted answers\. Mixture of Agents\[Wanget al\.,[2025a](https://arxiv.org/html/2605.29116#bib.bib39)\]usesNNheterogeneous proposers followed by a layered debate and aggregation stage\. Self\-MoA\[Yuksekgonulet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib49)\]applies MoA’s layered aggregation tokksamples from a single prompt, combining Self\-consistency’s single\-model setting with MoA’s synthesis mechanism\. TextGrad\[Yuksekgonulet al\.,[2025](https://arxiv.org/html/2605.29116#bib.bib49)\]iteratively refines a single solution through evaluate–gradient–update cycles\. GoA\[Yunet al\.,[2026](https://arxiv.org/html/2605.29116#bib.bib41)\]structures agent interactions as a graph with meta\-LLM\-controlled node/edge sampling, message passing, and graph pooling\.

SC\-MoA \(rightmost column, highlighted\) implements the three design constraints from §[4](https://arxiv.org/html/2605.29116#S4): perturbation diversity, anchored refinement, and universal synthesis\. The detailed pipeline is shown in Figure[2](https://arxiv.org/html/2605.29116#S4.F2)\.

Self\-ConsistencyMoASelf\-MoATextGradGoASC\-MoA \(ours\)qqSingle promptr1r\_\{1\}rkr\_\{k\}⋯\\cdotsMajority votea^\\hat\{a\}qqa1a\_\{1\}aNa\_\{N\}⋯\\cdotsp1p\_\{1\}pNp\_\{N\}⋯\\cdotsLayer 2: debateAggregatea^\\hat\{a\}qqSingle promptr1r\_\{1\}rkr\_\{k\}⋯\\cdotsAggregatea^\\hat\{a\}qqGenerateEvaluateText gradientUpdatea^\\hat\{a\}loopqqMeta\-LLM\(node sampling\)a1a\_\{1\}a2a\_\{2\}a3a\_\{3\}Edge sampling\(score graph\)Message passingGraph poolinga^\\hat\{a\}qqx1x\_\{1\}xNx\_\{N\}⋯\\cdotsNNperturbationsr1r\_\{1\}rkr\_\{k\}⋯\\cdotsr1r\_\{1\}rkr\_\{k\}⋯\\cdots⋯\\cdotskksamples eachp1p\_\{1\}pNp\_\{N\}⋯\\cdotsInner SCMajority vote:\{p1,…,pN\}\\\{p\_\{1\},\\ldots,p\_\{N\}\\\}Refine minorities w\.r\.t\.a∗a^\{\*\}Synthesize allNNproposalsa^\\hat\{a\}Phase 2Phase 3Phase 4majority frozen

Figure 13:Architectural comparison of six multi\-agent aggregation strategies\.Self\-Consistency votes over samples from a single prompt\. MoA layers heterogeneous proposers with debate\. Self\-MoA applies MoA\-style aggregation to single\-model samples\. TextGrad iteratively refines via text gradients\. GoA structures agent interaction as a scored graph\. SC\-MoA \(highlighted, rightmost\) uniquely combines perturbation\-based diversity, anchored refinement with the majority answer frozen, and universal trace\-level synthesis with verification—never gating on consensus\.