Answer-Distribution Trajectories: A Stochastic-Dynamics View of LLM Reasoning
Mar Gonzàlez I Català, Haitz Sáez de Ocáriz Borde, Davide Murari, Carola-Bibiane Schönlieb, Pietro Liò, George Montañez
cs.AI · cs.CL · cs.IT · cs.LG
Quick verdict
LLM CoT seen through evolving answer distributions—finer than entropy/accuracy; framework sound, yet N=16 rollout estimates are noisy, no utility shown.
Summary
This preprint proposes analyzing chain-of-thought reasoning by tracking the model's full predictive distribution over candidate final answers as reasoning unfolds, rather than only the final answer or its entropy. At checkpoints every 16 tokens, the authors estimate the prefix-conditioned answer distribution by sampling 16 independent continuations, then summarize each trajectory along four dimensions: exploration, revision, motion, and commitment. Two short propositions show endpoint predictions and entropy profiles are non-injective summaries of these trajectories. Across sixteen open-weight models and four benchmarks (GSM8K, ARC, SVAMP, MATH algebra), traces with the same endpoint or similar entropy profiles instantiate different dynamics, and variance decomposition attributes the largest share of metric variation to individual traces rather than model-task identity. High-accuracy model-task pairs show narrow, early-committed dynamics, while shortest-CoT pairs show broad, switch-heavy dynamics; instruction tuning narrows support and hastens commitment, with weaker effects of scale and temperature. Limitations: the 16-continuation Monte Carlo estimates carry unquantified noise, the number of sampled traces per question is never stated, the intervention comparisons are descriptive and confounded, and no downstream utility is demonstrated.
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Assessment at a glance
5
risk flags identified
● high 0
● medium 4
● low 1
4 strengths · 5 weaknesses · 5 author questions
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Key claims under review
- Answer-distribution trajectories are a strictly finer representation of reasoning than endpoint predictions or entropy profiles (Props. 1-2, Cor. 1).
- Traces with identical endpoints or similar entropy profiles instantiate substantially different dominant-answer dynamics and success/failure mechanisms (Figs. 2-3).
- Trajectory-metric variation is largest within model-task conditions, not between them (Table 1).
- Different objectives favor different profiles: high-accuracy pairs show narrow, early-committed dynamics; shortest-CoT pairs show broad, switch-heavy dynamics (Table 2: 78.8% accuracy/240 tokens vs 18.6%/56 tokens).
- Instruction tuning, model scale, and decoding temperature shift dynamical profiles in measurably different directions (Figure 4).
Strengths
Rigorous finer-representation argument, matched by empirics
Propositions 1-2 and Corollary 1 (Sec. 3.2, proofs in App. C) establish formally that endpoint prediction and entropy trajectories are non-injective summaries; Figures 2-3 then show the collapse occurs in practice across real model pairs.
Exceptional reproducibility documentation
Appendix B gives exact construction recipes for every figure and table (checkpoint handling, RMSE binning at width 0.01, variance-component formulas, the six matched tuning/scale pairs, t-based CI construction with stated n), and Appendix D tabulates all licenses.
Breadth under one fixed protocol
Honest limitation section
Weaknesses
Monte Carlo estimator noise is never quantified
Underspecified experimental scale, no code release
Causal-flavored language on confounded descriptive comparisons
No utility demonstration or cheap-signal baseline
Theory thinner than the framing suggests
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Scope & Claims
This preprint (arXiv 2609.09030) introduces **answer-distribution trajectories**: sequences of prefix-conditioned predictive distributions over discrete final answers, estimated along a chain-of-thought trace. Three claims organize the paper: (1) trajectories are a *strictly finer* representation than endpoint accuracy or entropy profiles (Sec. 3.2, Props. 1-2, Cor. 1); (2) reasoning dynamics differ systematically across traces, models and tasks, profiled via exploration (effective support S_eff), revision (N_switch, N_return), motion (path length V, temporal concentration PR, directness R_direct), and commitment (t_commit) (Secs. 4.1, 5.2, Table 1); (3) instruction tuning, scale, and temperature reshape these profiles (Sec. 5.3, Fig. 4). A supplementary eight-way taxonomy of success/failure mechanisms (Sec. 4.2) orients trajectories to the gold answer.
The remaining sections cover Methodology Assessment, Statistical Evidence, Results & Reproducibility, Limitations & Risks, and Verdict & Recommendations.
- All 4 strengths & 5 weaknesses, analyzed in detail
- 5 risk flags with severity, section & reviewer questions
- Methods & statistical evidence assessment
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Frequently asked questions
What is an answer-distribution trajectory in LLM reasoning analysis?
It is the sequence of a model's predictive probability distributions over candidate final answers, estimated at checkpoints along a chain-of-thought trace. The paper samples 16 independent continuations from each reasoning prefix and counts parsed answers, so a trajectory records which hypotheses are supported and how probability mass moves as reasoning proceeds - information that the final answer alone, or the entropy curve alone, cannot recover.
How does this paper differ from entropy-based reasoning analysis?
Entropy profiles track how concentrated the model's uncertainty is over time, but entropy is invariant to which answers carry the probability. The paper proves (Proposition 2) that different answer distributions can share the same entropy trajectory, and shows empirically (Figure 3) that same-question traces with similar entropy profiles often disagree on the dominant answer and on success/failure mechanism.
Is code or data available for this paper?
All sixteen models and four benchmarks (GSM8K, ARC, SVAMP, MATH algebra) are public with licenses listed in Appendix D, and Appendix B describes how every figure and table was constructed. However, no code repository or released analysis pipeline is mentioned in the preprint text.
What did the paper find about instruction tuning, scale, and temperature?
Using matched model pairs, instruction tuning produced the clearest shift: narrower effective support, fewer hypothesis switches, less distributional movement, earlier commitment, and higher accuracy. Raising temperature from 0.2 to 1.0 (four models, GSM8K only) broadened support, increased movement, and delayed commitment, with model-dependent accuracy. Scale comparisons (six pairs) showed weaker, less uniform shifts in the same direction.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Figures 2 and 4 are image-only in the extraction; their plotted values (mechanism bar heights, per-pair standardized changes) cannot be verified from the text, so assessments of those figures rely on the prose and Appendix B descriptions.
- The unspecified M and per-dataset question counts were verified absent from the entire provided text including Appendix A - reported as a paper gap, not an extraction artifact.
- Minor extraction noise in the author block (e.g., 'Carola Bibiane-Schönlieb' name ordering) and some reference formatting; content unaffected.
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