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arxiv: 2606.17312 · v1 · pith:YQBTNIMZnew · submitted 2026-06-15 · 💻 cs.AI

Quantifying Consistency in LLM Logical Reasoning via Structural Uncertainty

Baishali Chaudhury , Mengdie Flora Wang , Hyunji Hayley Park , Rahul Ghosh , Sungmin Hong , Jae Oh Woo This is my paper

Pith reviewed 2026-06-27 03:14 UTC · model grok-4.3

classification 💻 cs.AI
keywords structural uncertaintyLLM reasoning consistencyself-preference rankingsBradley-Terry modelanswer dispersionlogical reasoningreliability estimationentropy components
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The pith

Structural uncertainty from self-preference rankings complements answer dispersion to identify unreliable LLM reasoning on logic and math tasks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes structural uncertainty as a way to quantify consistency in how large language models reason. Instead of only looking at how much the final answers vary across samples, it also checks whether the model can stably rank its own candidate reasoning paths through repeated pairwise comparisons. This ranking stability is turned into entropy measures of instability across different trials and ambiguity within each trial. When added to standard dispersion measures, it helps flag unreliable answers more effectively on deductive reasoning benchmarks, but the ranking signal becomes uninformative on simple fact retrieval. The two entropy parts show opposite ties to actual correctness.

Core claim

Structural uncertainty is derived from the stability of rankings obtained by aggregating an LLM's pairwise self-preferences among multiple sampled solutions using Bradley-Terry modeling and PageRank. This yields two components—across-trial ranking instability that correlates negatively with accuracy, and within-trial candidate ambiguity that correlates positively—providing a signal complementary to answer dispersion specifically for logical and mathematical reasoning tasks while collapsing to uniformity on factual retrieval.

What carries the argument

Structural uncertainty, computed as entropy decompositions of ranking distributions from self-preference judgments.

Load-bearing premise

The model's pairwise self-preferences among its own outputs form a stable ranking signal that reflects genuine reasoning consistency rather than its own judgment biases.

What would settle it

An experiment showing that on logical reasoning tasks the combination of structural uncertainty and answer dispersion does not improve detection of incorrect answers compared to dispersion alone would falsify the main claim.

Figures

Figures reproduced from arXiv: 2606.17312 by Baishali Chaudhury, Hyunji Hayley Park, Jae Oh Woo, Mengdie Flora Wang, Rahul Ghosh, Sungmin Hong.

Figure 1
Figure 1. Figure 1: An illustrative example contrasting dispersion-based and structural consistency evaluation. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the consistency-aware reasoning evaluation framework. Given a query, we (1) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Across-trial vs. within-trial structural uncertainty: two contrasting examples. Sce￾nario I (left): The model produces a confident, concentrated ranking within each trial (low within￾trial entropy H[π (m) ]), but the dominant candidate changes across trials as different spanning trees are sampled—indicating substantial across-trial instability in the induced ranking distribution (StructUacross ↑, StructUwi… view at source ↗
Figure 4
Figure 4. Figure 4: Sel-AUC lift of STRUCTU+SELF-CONSU over SELF-CONSU. ∆Sel-AUC for five models × eight benchmarks. Teal = hybrid wins; coral = Self-ConsU dominates [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spearman correlation (ρ) between the two uncertainty components and accuracy. Each cell reports Spearman rank correlation between per-question uncertainty and per-question accuracy (fraction correct among N=5 samples). Blue indicates negative correlation; red indicates positive correlation. Stars denote statistical significance (∗p < 0.05, ∗∗p < 0.01, ∗∗∗p < 0.001). Regime Analysis: Reasoning vs. Retrieval… view at source ↗
Figure 6
Figure 6. Figure 6: Regime analysis: reasoning consistency vs. retrieval collapse. Claude 4.5 Sonnet on Math-Synth (reasoning) and HotpotQA (retrieval), conditioned on correctness (BT + PageRank). (a) STRUCTUacross: Math-Synth shows correctness separation; HotpotQA concentrates near zero for both. (b) STRUCTUwithin: HotpotQA clusters at maximum entropy (log 5 ≈ 1.61, dotted line). (c) Joint space: reasoning tasks separate alo… view at source ↗
Figure 7
Figure 7. Figure 7: Ablation study on AIME benchmarks: effect of number of sampled responses showing [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ablation study on AIME benchmarks: effect of number of trials showing performance [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Sensitivity of Sel-AUC to inverse regularization strength C (BT+PageRank, Math￾Synth). Each panel shows one uncertainty component across all five models as a function of C ∈ {0.1, 0.5, 1, 3, 5, 10}, where larger C corresponds to weaker regularization. (a) Across-trial (STRUCTUacross): models with stronger preference signal (DeepSeek R1, GPT-OSS 20B, Amazon Nova Premier) show the largest degradation at low … view at source ↗
Figure 10
Figure 10. Figure 10: Structural collapse on factual retrieval across four additional models. Rows corre￾spond to Amazon Nova Premier, DeepSeek R1, GPT-OSS 20B, and Qwen 3 32B. Each row shows the across-trial uncertainty distribution (left), within-trial uncertainty distribution (center), and joint across-trial–within-trial scatter (right) on Math-Synth and HotpotQA, conditioned on correctness (Bradley–Terry + PageRank). The “… view at source ↗
read the original abstract

Large language models can arrive at the same answer through reasoning paths that are unstable, contradictory, or difficult to rank consistently -- a failure mode especially prevalent in multi-step deductive reasoning. Existing methods assess reliability primarily through output dispersion -- measuring how much sampled answers differ -- but this discards a complementary signal: whether the model can consistently rank competing reasoning candidates. We propose structural uncertainty, a consistency-aware framework derived from the stability of self-preference-induced rankings over sampled reasoning solutions. Given a query, we generate multiple candidate solutions and ask the model to judge pairwise preferences among its own outputs. We aggregate self-preferences into ranking distributions via Bradley-Terry modeling with PageRank, and decompose the signal into two entropy-based components: across-trial ranking instability and within-trial candidate ambiguity. Across five LLMs and eight benchmarks, structural signals provide information complementary to answer dispersion: on logical and mathematical reasoning tasks, the combination improves identification of unreliable instances, while on factual retrieval the structural signal collapses toward uniformity, diagnosing a regime boundary where reasoning-level consistency evaluation is uninformative. The two components relate differently to accuracy: within-trial ambiguity correlates positively with correctness -- consistent with settings where multiple plausible solution paths remain competitive -- while across-trial instability correlates negatively, signaling unreliable reasoning. Structural uncertainty is best understood not as a universal confidence estimator, but as a regime-sensitive evaluator of logical reasoning consistency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that structural uncertainty—derived from aggregating model self-preferences over sampled reasoning solutions via Bradley-Terry modeling and PageRank, then decomposing into across-trial ranking instability (negatively correlated with accuracy) and within-trial candidate ambiguity (positively correlated)—supplies information complementary to answer dispersion. This complementarity improves detection of unreliable instances on logical and mathematical tasks but collapses toward uniformity on factual retrieval, across five LLMs and eight benchmarks; the framework is positioned as a regime-sensitive evaluator of logical reasoning consistency rather than a universal confidence measure.

Significance. If the structural signal is shown to track reasoning consistency independent of elicitation artifacts, the work supplies a useful decomposition and task-regime distinction for LLM reliability assessment. The reported differential correlations and the breadth of models/benchmarks constitute a concrete empirical contribution that could be strengthened by targeted validation.

major comments (2)
  1. [Abstract] Abstract and framework description: the central claim that self-preference rankings yield entropy components tracking genuine reasoning instability (rather than LLM-as-judge artifacts such as position or length bias) is load-bearing for both the complementarity result and the logical-vs-factual regime boundary, yet the manuscript does not report controls for order randomization, length normalization, or consistency under prompt perturbation.
  2. [Results on task regimes] Task-regime results: the reported improvement from combining structural uncertainty with dispersion on logical/math tasks (and collapse on factual retrieval) depends on the pairwise tournament matrix being driven by logical content; without explicit checks that the Bradley-Terry + PageRank aggregation is robust to known self-preference biases, the task-type distinction risks being an artifact of the elicitation procedure.
minor comments (2)
  1. [Methods] Specify the precise entropy formulas used for the across-trial and within-trial components and how they are normalized across varying numbers of samples.
  2. [Experimental setup] Clarify the exact set of eight benchmarks, their categorization into logical/math vs. factual, and any selection criteria.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback emphasizing the need to validate the structural uncertainty framework against potential elicitation artifacts. We address the major comments point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and framework description: the central claim that self-preference rankings yield entropy components tracking genuine reasoning instability (rather than LLM-as-judge artifacts such as position or length bias) is load-bearing for both the complementarity result and the logical-vs-factual regime boundary, yet the manuscript does not report controls for order randomization, length normalization, or consistency under prompt perturbation.

    Authors: We agree that the manuscript does not include explicit controls for order randomization, length normalization, or prompt perturbation consistency, and that such controls are necessary to more rigorously support the claim that the entropy components track reasoning instability rather than judge artifacts. The differential task-regime results provide indirect evidence, but direct validation is warranted. In the revised version we will add these controls as additional experiments and report the outcomes. revision: yes

  2. Referee: [Results on task regimes] Task-regime results: the reported improvement from combining structural uncertainty with dispersion on logical/math tasks (and collapse on factual retrieval) depends on the pairwise tournament matrix being driven by logical content; without explicit checks that the Bradley-Terry + PageRank aggregation is robust to known self-preference biases, the task-type distinction risks being an artifact of the elicitation procedure.

    Authors: We acknowledge that the manuscript lacks explicit robustness checks against known self-preference biases (position, length, etc.) in the Bradley-Terry + PageRank aggregation, and that this leaves open the possibility that the logical-vs-factual distinction is partly elicitation-driven. While the observed complementarity on logical tasks versus uniformity on factual retrieval is consistent with content-driven behavior, targeted bias-mitigation experiments are needed. We will incorporate such checks in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard models to self-generated data without reduction to inputs

full rationale

The paper's chain generates candidate solutions, elicits pairwise self-preferences, aggregates them via Bradley-Terry + PageRank into ranking distributions, and decomposes the resulting entropy into across-trial instability and within-trial ambiguity. These steps use externally standard ranking techniques on the model's own outputs; the central claim of complementarity to answer dispersion (and task-regime differences) is presented as an empirical observation across benchmarks rather than a quantity forced by the equations themselves. No self-citations, fitted parameters renamed as predictions, or ansatzes imported via prior work appear as load-bearing in the provided abstract or derivation description. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard ranking models without introducing new fitted parameters or invented entities beyond the modeling framework itself; no ad-hoc constants or new physical assumptions are described.

axioms (2)
  • standard math Bradley-Terry model produces valid preference probabilities from pairwise comparisons
    Invoked when aggregating self-preferences into ranking distributions
  • standard math PageRank applied to preference graph yields stable ranking distributions
    Used to convert pairwise judgments into distributions for entropy calculation

pith-pipeline@v0.9.1-grok · 5785 in / 1315 out tokens · 39125 ms · 2026-06-27T03:14:48.157355+00:00 · methodology

discussion (0)

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Reference graph

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