Reliable Chain-of-Thought via Prefix Consistency

arxiv:2605.07654

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Reasoning and the "physics" of language models · 2026-05-08

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Contents

• Reliable Chain-of-Thought via Prefix Consistency
• Abstract
• 1 Introduction
• 2 Preliminary
• 3 Prefix Consistency
• 3.1 Prefix Consistency as a Reliability Signal
• 3.2 Prefix-Consistency-Weighted Majority Voting (Algorithm 1 )
• Hyperparameters.
• 4 Experiments
• Hyperparameters of prefix consistency.
• 4.1 Prefix Consistency as a Correctness Predictor
• 4.2 Weighted Majority Voting Results
• Weighting variants.
• Cost-equivalent evaluation.
• 4.3 Token Efficiency
• Comparison with adaptive-stopping baselines.
• Remark on the cost accounting.
• 4.4 How Problem Difficulty Affects the Discrimination Gap D D
• 5 Conclusion
• Appendix A Related Work
• A.1 Test-Time Scaling
• Sequential and parallel scaling.
• Majority voting.
• Confidence-based weighting.
• Prefix continuation as a per-sample signal.
• A.2 Chain-of-Thought
• Internal answer determination.
• Error propagation.
• Reasoning robustness.
• Overthinking and length scaling.
• Appendix B Limitations and Future Work
• Appendix C Asymptotic Analysis
• C.1 Notation and Useful Tools
• Proposition 1 (Population objective) .
• Proof.
• Proposition 2 (Exact decomposition of Φ w \Phi_{w} ) .
• Proof.
• C.2 Asymptotic Convergence of PC-WMV
• Assumption 1 (Self-reproduction dominance) .
• Interpretation.
• Assumption 2 (Pooled-mass dominance) .
• Interpretation.
• Theorem 2 (Asymptotic convergence of PC-WMV) .
• Proof.
• C.3 Proof of Theorem 1
• The two assumptions coincide.
• The exact margin.
• Proof.
• C.4 Empirical Verification of Assumptions
• Appendix D Additional Experiments
• D.1 ROC Curves for Correctness Predictors
• D.2 Cost–Accuracy Curves
• D.3 Full Baseline Comparison
• D.4 Pool Coverage vs. Reweighting
• Oracle upper bounds.
• Per-problem marginal correctness.
• D.5 Sensitivity to τ \tau and K K
• D.6 Robustness to Judge Choice
• PC’s AUROC ¯ \overline{\mathrm{AUROC}} advantage is preserved on every cell.
• WMV cost–accuracy curves under both judges nearly overlap.
• Pool-level shifts are concentrated on FrontierScience-Olympiad.
• D.7 Reliability Signals vs. Pass@1
• D.7.1 GLM Panels and Slopes
• GLM fit kernel.
• Inference protocol.
• D.7.2 Robustness under Per-Benchmark Pooling
• D.7.3 Robustness under Alternative Estimators
• Trial-pooled binned estimator (Figure 13 ).
• LOWESS smoother (Figure 14 ).
• D.7.4 Baseline Confidence Signals vs. Pass@1
• Appendix E Baseline Implementation Details
• Glossary of baseline labels.
• Deep Think with Confidence (DeepConf) [ Fu et al., 2026 ] .
• Confidence-Informed Self-Consistency (CISC) [ Taubenfeld et al., 2025 ] .
• Adaptive Consistency (AC) [ Aggarwal et al., 2023 ] .
• Early-Stopping Self-Consistency (ESC) [ Li et al., 2024 ] .
• Cost accounting for AC and ESC.
• SubthoughtReasoner [ Hammoud et al., 2025 ] .
• Prefix consistency (ours).
• Appendix F Examples of Prefix Consistency
• Appendix G Evaluation Protocol
• G.1 Setup
• Benchmarks.
• Pre-generated pool and cost vectors.
• Hyperparameter defaults.
• Confidence intervals (CIs) and bootstrap conventions.
• G.2 AUROC ¯ \overline{\mathrm{AUROC}} for the Correctness-Predictor Evaluation (Section 4.1 )
• Definition of AUROC ¯ \overline{\mathrm{AUROC}} .
• AUROC computation.
• G.3 Cost-Accuracy Evaluation (Sections 4.2 and 4.3 )
• Sample-with-replacement trials.
• Dense token-budget grid.
• Standard MV plateau and Pass@1.
• Monotone envelope and log-budget interpolation.
• Parametric bootstrap for ratio CIs.
• G.4 Reproduction-Rate GLM (Section 4.4 )
• Logistic GLM and cluster bootstrap.
• Appendix H Answer Extraction and Equivalence Judging
• H.1 Answer Extraction
• Boxed-answer parses.
• Verbal score handling.
• Verbal 0–100 parser fragility.
• H.2 Equivalence Judging
• Judge configuration.
• FrontierScience-Olympiad.
• HMMT Feb 2026, Brumo 2025.
• Appendix I Reproducibility Statement
• Code and data.
• Inference and sampling.
• Generation prompts.
• Instructions for reporting errors