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Replication or Corroboration? An Axiomatic Framework for Evidential Dependence in Multi-Agent Debate

Hoang-Loc Cao, Truong Thanh Hung Nguyen, Phuc Truong Loc Nguyen, Vo Thanh Khang Nguyen, Phuc Ho, Hung Nguyen, Duc Pham, Long Tran-Thanh, Hung Cao

arXiv:2610.36151v1cs.MAcs.AIcs.LO

TL;DR

Matching reports can reflect repeated evidence or distinct evidential roots, a distinction that report counts and textual agreement alone do not preserve. The paper formalizes replication and corroboration with a supplied root partition, derives representation and response guarantees, and audits judges under matched interventions. The audits show the predicted separation between replication and corroboration, while the guarantees remain conditional on supplied dependence information.

  • Problem

    Matching reports may repeat one evidential root or represent distinct roots, while report counts and textual agreement do not reveal that distinction.

    Source: §1 · p. 1

  • Method

    The paper formalizes a judgment game and introduces a lineage quotient that merges renderings of one root while preserving distinct roots.

    Source: §1 · p. 3, §5 · p. 5

  • Results

    Audits over 28,224 direct-star profiles, 2,052 multi-layer DAGs, and 277 AVeriTeC-derived profiles confirm the predicted separation achieved by the lineage quotient.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The lineage quotient is the coarsest representation sufficient for deterministic judges invariant to replication and root-label renaming on an orbit-connected domain.

    Source: §9 · p. 9

    The guarantees take the root partition as supplied and do not establish provenance discovery or fact-checking accuracy.

    Source: §8 · p. 8

Abstract

from arXiv · show

In multi-agent debates, matching reports can arise for two different reasons: agents may repeat the same evidence (replication), or they may provide separate evidential roots that support the same claim (corroboration). A judge must distinguish these two cases. Counting report occurrences gives extra weight to relays, paraphrases, alias resubmissions, and excerpt splits, while merging all matching content avoids this only by also removing genuine corroboration. We formalize this tension as a minimal judgment game in which acquiring new evidence is costly, but replication is cheap. We show that root replication invariance (RRI) is exactly threshold-uniform replication immunity. Under RRI, replication-only deviations cannot change the verdict, and positive-cost replication is strictly dominated, whereas occurrence-sensitive and content-keyed judges admit profitable deviations. We also establish an information boundary: if a judge's observation interface cannot distinguish a replica from a matched fresh root, no judge can both ignore the replica and respond strictly to the fresh root. To preserve this distinction, we introduce the lineage quotient, which removes wording, identity, occurrence multiplicity, and arbitrary root names while keeping distinct roots separate. On orbit-connected domains, it is the coarsest representation sufficient for every deterministic judge invariant to replication and root-label renaming. We further lift this construction to quantitative bipolar argumentation frameworks (QBAFs), characterize strict fresh-root response, prove its propagation along sign-consistent paths under DF-QuAD, and derive a tight closed-form response modulus. Exhaustive audits over 28,224 direct-star profiles, 2,052 multi-layer DAGs, with matched interventions on 277 AVeriTeC-derived profiles, show the predicted separation between replication and corroboration under a supplied partition.

1 Introduction

Matching reports may represent either repeated evidence or distinct evidential roots, so judges need to distinguish replication from corroboration. The paper develops a dependence-aware representation and formal guarantees for doing so, while treating the root partition as supplied input.

Trust Boundary

  • The guarantees assume a supplied root partition; they do not establish provenance discovery or fact-checking accuracy.The partitioning and authentication of roots remain upstream tasks.Source: §1 · p. 2

Replication Immunity and Information Requirements

  • RRI is equivalent to threshold-uniform replication immunity; under RRI, replication-only deviations cannot change verdicts and costly replication is strictly dominated.Occurrence aggregation, plurality voting, and a content-keyed quotient admit profitable deviations.Source: §1 · p. 3
  • A judge cannot ignore a replica yet respond strictly to a fresh root if its observation interface represents them identically.This applies to any interface, including an LLM prompt.Source: §1 · p. 3

A Dependence-Aware Representation

  • The lineage quotient removes wording, identity, and occurrence multiplicity while merging renderings of one root and preserving distinct roots.On a fixed orbit-connected domain, it is the coarsest representation sufficient for deterministic judges satisfying RRI and LNI.Source: §1 · p. 3

Quantitative Response Guarantees

  • In QBAFs, strict fresh-root response propagates to the topic along paths with consistent attack-edge parity, and DF-QuAD has a tight closed-form response modulus.The modulus quantifies how much corroboration can move the score as the aggregate approaches saturation.Source: §1 · p. 3

Replication–Corroboration Stress Test

  • Audits confirm the predicted separation between replication and corroboration across 28,224 direct-star profiles, 2,052 multi-layer DAGs, and 277 AVeriTeC-derived profiles.The audits expose complementary failures of occurrence-sensitive and content-collapsing baselines.Source: §1 · p. 3

2 Related Work

Prior work studies debate, argumentation, dependence, and strategic reporting, but typically does not combine a supplied evidential-root partition with a representation-level analysis of replication and corroboration.

Multi-Agent Debate and Post-hoc Judgment

  • Multi-agent debate work examines scalable oversight and collaborative reasoning, while LLM judges show systematic biases and may miss nonlinear argument structures.A cited judge represents agent-indexed opinions but no relations between reports, allowing report weight to be inflated through added identities.Source: §2 · p. 3

QBAF Semantics and Duplicate Handling

  • Gradual QBAF semantics and duplicate-handling approaches study repeated or near-copy information, but can suppress every near-copy rather than preserve corroboration from distinct roots.This paper instead builds the graph from a supplied root partition, allowing matching reports from distinct roots to reinforce one another.Source: §2 · p. 3

Provenance and Dependent Evidence

  • Provenance and dependence research models source dependence or its effects under given probabilistic models; this paper asks what distinctions a representation must retain when dependence is supplied as a partition.Partial or graded dependence is left to the discussion of limitations.Source: §2 · p. 3

Strategic Multiplicity

  • The replication guarantee is narrower than false-name-proofness: it covers replication of already represented evidence under authenticated root labels, not forged labels or modified content.Strategic reporting and multiple-identity models provide related context.Source: §2 · p. 3

3 Model and the Judgment Game

The model treats reports as occurrences associated with content, role, strength, and a represented evidential root. It then defines replication actions and matched interventions to compare replicas with otherwise identical fresh-root reports.

3.1 Occurrences, Roots, and Units

  • An occurrence records displayed identity, surface text, normalized content, root label, target claim, argumentative sign, and strength; the aggregation unit is the root together with its role.Paraphrases can differ in surface text while sharing normalized content, and matching content can come from either the same or distinct roots.Source: §3.1 · pp. 3–4
  • A root is fresh when it is not already represented, and its strength is the maximum strength among occurrences of its root–role unit.Root integrity assigns all renderings of one represented root the same label; agents can relay, paraphrase, resubmit under an alias, or split excerpts without acquiring evidence or increasing root strength.Source: §3.1 · p. 4, §3.2 · p. 4

3.2 Agent Actions and Minimal Judgment Game

  • The judgment game makes fresh-root acquisition more costly than replication and defines agents’ utilities through verdict preferences and action costs.Replication deviations add or remove actions in the replication class; a matched replica and fresh extension agree on every specified report field except the supplied dependence annotation.Source: §3.2 · p. 4, §3.3 · p. 4

3.3 Matched Interventions

  • Matched interventions compare a root-replica extension with a fresh-root extension that differs only in the supplied dependence annotation.Eligibility is fixed before score comparison and may use freshness, role, strength, structural well-formedness, and a non-saturation test, but not the score inequality being tested.Source: §3.3 · p. 4

4 Principles and the Information Boundary

The section defines replication, naming, and fresh-evidence response principles, then establishes when they can coexist. It characterizes replication immunity, an information boundary, and the strategic consequences of satisfying or violating the replication requirement.

4 Principles and the Information Boundary

  • RRI requires a root replica that adds no unit or strength to leave the score unchanged; LNI makes consistent renaming of root labels score-neutral.FLR requires eligible fresh evidence to move the score in the correct direction, while qFLR lower-bounds that movement.Source: §4 · p. 4

4.1 Replication-Proofness Characterizes RRI

  • RRI is equivalent to unchanged verdicts at every threshold under a root-replica extension, or along any replication path.Stability at one fixed threshold is weaker: a relay can change the score while leaving a verdict unchanged at that threshold.Source: §4.1 · p. 5

4.2 Information Necessity

  • If an interface maps a matched replica and fresh extension to the same observation, no judge using that interface can satisfy both RRI and strict positive fresh-root response on that witness.The same boundary applies to a lineage-blind interface, including an LLM prompt that represents the cases identically.Source: §4.2 · p. 5

4.3 Strategic Consequences

  • With RRI, replication-only deviations cannot change verdicts, and positive-cost replication is strictly dominated.Deleting k replication actions raises utility by exactly k times their replication cost; at zero cost, replication is payoff-equivalent to abstaining.Source: §4.3 · p. 5
  • Occurrence-sensitive voting and DF-QuAD can reward a relay, while a content-keyed quotient can reward an excerpt split; the lineage quotient is neutral to both.The content-keyed quotient absorbs the relay but distinguishes the excerpt’s new content key, so it satisfies CRI while violating RRI.Source: §4.3 · p. 5

5 The Lineage Quotient

The lineage quotient treats repeated renderings of one root–role as a single unit while preserving distinct roots. On orbit-connected domains, it is the coarsest representation through which every deterministic judge satisfying RRI and LNI can score profiles.

5 The Lineage Quotient

  • The lineage quotient merges renderings of the same root–role, while keeping a genuinely fresh root as a new unit.It removes wording, identity, and occurrence multiplicity while retaining the root-linked structure.Source: §5 · p. 5
  • Content-only deduplication can merge matching reports from distinct roots, preventing strict response to matched fresh evidence.For such a fresh-root extension, the content quotient is unchanged and any judge based on it violates strict FLR.Source: §5 · p. 6
  • The coarseness result requires orbit-connectedness; without it, RRI can be vacuous between profiles with equal quotients.A single judge may need less information; on direct stars, LQ-DF-QuAD needs only the base score and both aggregates.Source: §5 · p. 6
  • On an orbit-connected domain, a deterministic judge satisfies RRI and LNI exactly when its score factors through the lineage quotient.The quotient is both sufficient and, up to recoding, the coarsest representation supporting every such judge.Source: §5 · p. 6

6 Lifting the Lineage Quotient to QBAFs

The paper lifts lineage-based replication invariance to QBAFs by quotienting root–role replicas before applying a semantics. For DF-QuAD, fresh-root response holds directly and propagates through sign-consistent paths, with a tight lower bound on its magnitude.

6 Lifting the Lineage Quotient to QBAFs

  • The generic lifting quotients equivalent occurrence nodes before evaluation, satisfying structural RRI and LNI while matching the original semantics on singleton classes.Equivalent occurrences must have matching signed neighborhoods after quotienting; classes that violate compatibility are malformed.Source: §6 · p. 6
  • For direct topic leaves, strict fresh-root response holds exactly when the corresponding supporter or attacker aggregate strictly increases.A maximum aggregator can fail this condition when a new input does not exceed the current maximum.Source: §6 · p. 6

6.1 Propagation and a Quantitative Modulus

  • Under DF-QuAD, a fresh root at an intermediate node strictly changes the topic score when all paths to it have the same attack-edge parity.The change has sign ε0ε; each attack edge flips the sign, and opposite-parity paths can cancel.Source: §6.1 · p. 7
  • A fresh direct supporter changes the score by at least t(1 − v_s) min{v_0, 1 − v_0}, and the bound is tight.The corresponding attacker bound is t(1 − v_a) min{v_0, 1 − v_0}; the guaranteed effect shrinks as the same-side aggregate saturates.Source: §6.1 · p. 7
  • Replica extensions leave LQ[σ] and threshold decisions unchanged, while eligible fresh contributions move the score strictly in their sign, directly or along sign-consistent paths.This graph-level result is stated with the scope limits of Section 8.Source: §6.1 · p. 7

7 Experiments and Results

Matched audits test whether judges ignore repeated evidence while responding to fresh roots, using structural profiles and AVeriTeC-derived profiles under supplied lineage. Across these settings, the lineage quotient separates replication from corroboration, unlike occurrence-sensitive and content-collapsing baselines.

7 Experiments and Results

  • The audits pair repetition and matched fresh-root interventions to measure score changes, verdict changes, signed fresh-root gains, and clean-score preservation.The AVeriTeC audit uses source-linked evidence from Supported and Refuted claims, with a supplied controlled root per original report.Source: §7 · p. 7, §7.3 · p. 8

7.1 A Toy Debate

  • In the toy debate, a replica shifts occurrence DF-QuAD from .5 to .42 and reverses the verdict, while the lineage quotient stays at .5; matched fresh reports move its score to .42 or .58.The fresh-root changes equal the .08 response bound; Max-QuAD and the content quotient ignore both fresh reports.Source: §7.1 · pp. 7–8, Figure 2 · p. 7

7.2 Exhaustive Structural Audit

  • Among tested judges, only the lineage quotient meets the joint target: CD, RSS8, RFR8, and FASR are zero, while FLG = .0298 and MinG = .0013.The exhaustive audit enumerates 2,052 chain, fork, and diamond DAGs with strengths in {0.2, 0.5, 0.8}. Lineage lifting has zero maximum clean deviation and replication shift, with minimum signed fresh-root gain 1.28 × 10−4.Source: §7.2 · p. 8, Table 2 · p. 8

7.3 Real-profile Audit on AVeriTeC

  • Across 277 AVeriTeC-derived profiles, only the lineage quotient meets the joint target: zero CD, RSS8, RFR8, and FASR, with positive gains for all 711 fresh contributions.Mean signed gain is .0088 (95% CI [.0071, .0107]).Source: §7.3 · p. 8, Table 3 · p. 8

8 Discussion

The lineage partition is supplied rather than inferred, and the framework leaves dependence inference and authentication upstream. Its guarantees are bounded by assumptions and modeling choices, including unmodeled graded dependence and extensions beyond DF-QuAD.

8 Discussion

  • The lineage partition is an input, not an inferred output, and distinct roots do not certify causal or statistical independence.Building and authenticating the partition, including deciding how documents decompose into roots, is upstream work.Source: §8 · p. 8
  • A partition cannot express graded dependence, and realistic sensitivity to false splits and merges remains open.In the direct same-sign case, false splits inflate aggregation while false merges suppress it.Source: §8 · p. 8
  • Replication resistance under authenticated roots does not guarantee dominant-strategy truthfulness or false-name-proofness.The analysis excludes forged identities, strategic misreports, strength revelation, and genuinely new evidence; extending propagation beyond DF-QuAD remains open.Source: §8 · pp. 8–9

9 Conclusion

The paper distinguishes replication from corroboration by requiring representations to separate further renderings of a root from distinct roots. It develops formal guarantees, a lineage quotient, QBAF results, and audits within a supplied partition.

9 Conclusion

  • Under RRI, replication cannot change the verdict and is strictly dominated when costly, while occurrence-sensitive and content-keyed judges admit profitable deviations.No interface that represents a replica and matched fresh root identically can both ignore the replica and respond strictly to the fresh root.Source: §9 · p. 9
  • The lineage quotient is the coarsest representation sufficient for deterministic RRI–LNI judges on orbit-connected domains.The paper also lifts the construction to QBAFs and derives a tight DF-QuAD response modulus for fresh-root effects along sign-consistent paths.Source: §9 · p. 9
  • Exhaustive and AVeriTeC-derived audits confirm the predicted separation under a supplied partition, while the result rules out interfaces that represent replicas and fresh roots identically from being both replica-neutral and strictly fresh-responsive.Source: §9 · p. 9

Appendices

The supplement supplies omitted proofs, QBAF and DF-QuAD derivations, and experimental details. It also establishes the root aggregate forced by absorption and completes the lineage-quotient equivalence proof.

Appendices

  • The supplement provides deferred proofs, QBAF and DF-QuAD derivations, and experimental construction, preprocessing, uncertainty, and sensitivity details.Its sections fill in omitted profile-level representation proofs, technical derivations, and audit details.Source: Appendices · p. 10

A.1 Why Absorption Fixes the Root Aggregate

  • The singleton and absorption conditions force the within-root aggregate to be the maximum, and maximum itself satisfies both conditions.The arithmetic mean can let a weaker rendering lower the aggregate; the maximum rule prevents that effect.Source: §A.1 · p. 10

A.2 Quotient Completeness

  • Profiles with equal lineage quotients can be reduced to one maximal-strength representative per root–role unit and transformed into one another through permitted steps.The transformation can replace representatives that differ in displayed identity, wording, or canonical proposition; deleting a maximum-strength occurrence is not a permitted reversal.Source: §A.2 · p. 10

A.3 Factorization and Minimal Sufficiency

The appendix establishes the lineage quotient’s minimal sufficiency and details conditions for replication admissibility, acyclicity, and strict fresh-root response, including a cancellation boundary.

A.3 Factorization and Minimal Sufficiency

  • On orbit-connected domains, RRI- and LNI-invariant judges factor through the lineage quotient, which is coarsest for supporting the full deterministic judge class.Source: §A.3 · pp. 10–11

A.4 Admissibility of Replication-only Deviations

  • Replication-only deviations preserve which non-replication actions are admissible, because replication adds no roots or root-role units and leaves root strengths unchanged.Source: §A.4 · p. 11

B.1 Acyclicity after Lineage Quotienting

  • Lineage stratification guarantees an acyclic quotient, but is not necessary; quotienting can create cycles even when the raw graph is acyclic.Source: §B.1 · p. 11

B.2 Full Case Analysis for Proposition 4

  • Every eligible direct supporter strictly increases the score, while the attacker result is symmetric.Source: §B.2 · p. 11

B.3 Full Induction for Theorem 6

  • Under DF-QuAD, adding a fresh leaf changes the topic strictly with parity-determined sign when all leaf-to-topic paths share attack-edge parity.Source: §B.3 · p. 12

B.4 Why Sign Consistency cannot be Removed

  • With opposite-parity paths, both intermediate scores can rise while equal topic support and attack aggregates leave the topic score unchanged.Source: §B.4 · p. 12

B.5 Full Derivation of the Tight Response Modulus

The appendix derives a tight direct-star response bound, calculates the effect of lineage split and merge errors, and specifies structural and AVeriTeC audit construction, sensitivity, and scope. The audits use supplied lineage; they do not test provenance discovery.

B.5 Full Derivation of the Tight Response Modulus

  • 0.00128 is the analytic minimum response modulus on the stated grid, attained at the example configuration of topic strength 0.8, supporter strengths (0.2, 0.8, 0.8), attacker strength 0.2, and intervention strength 0.2.The derivation also establishes a lower bound of 0.0064 for t(1 − v_s) and a lower bound of 0.2 for min{v_0, 1 − v_0}.Source: §B.5 · p. 12

B.6 Two Additional Boundary Calculations

  • 1 − (1 − t)^m > t: falsely splitting one same-sign root into m roots inflates its aggregate, while merging m distinct equal-strength roots suppresses it.The calculation is limited to direct same-sign contributions; it does not quantify mixed-sign or multi-layer graph errors.Source: §B.6 · p. 13

C.1 Structural Audits

  • The 2,052 enumerated multi-layer DAGs received 2,700 replica-position tests and 4,104 fresh direct-root interventions; the appendix records construction details needed for reproduction.The enumerated families are chains, forks, and diamonds; intermediate-node fresh-root interventions are excluded.Source: §C.1 · p. 13, C Additional Experimental Specification · p. 13

C.2 AVeriTeC Profile Construction

  • The AVeriTeC audit contains 277 profiles—127 Supported and 150 Refuted—with 711 reports after stance assignment and independent human cross-checking.A frozen NLI model assigns support, attack, or discard; retained report strengths use predicted class probabilities clipped to [0.05, 0.95].Source: §C.2 · p. 13

C.3 Controlled Lineage, Canonicalization and Uncertainty

  • AVeriTeC reports receive controlled roots, and matched replica and fresh-root interventions preserve judged fields while changing whether the root is retained or new.Claim-clustered bootstrap intervals use 10,000 resamples stratified by label; MinG is the observed minimum over evaluated report positions.Source: §C.3 · pp. 13–14

C.4 Sensitivity Analyses

  • Across sensitivity conditions, controlled replicas remain exactly neutral and every evaluated fresh root has a positive signed gain.The analysis varies NLI thresholds to 0.50 and 0.70 and also tests fixed report strength 0.5; it does not evaluate the NLI model as a fact checker.Source: §C.4 · p. 14

C.5 Reproduction and Scope Notes

  • The AVeriTeC audit uses a constructed root partition, only 47 profiles with both support and attack reports, and matched interventions that keep canonical content fixed.Thus the experiments do not empirically distinguish content-keyed from root–role lineage quotients under excerpt splitting; that distinction is established analytically in the main paper.Source: §C.5 · p. 14

D AI-Assisted Preparation Disclosure

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