Understanding Reasoning in LLMs through Strategic Information Allocation under Uncertainty

Jeonghye Kim; Xufang Luo; Minbeom Kim; Sangmook Lee; Dongsheng Li; Yuqing Yang

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Understanding Reasoning in LLMs through Strategic Information Allocation under Uncertainty

Jeonghye Kim, Xufang Luo, Minbeom Kim, Sangmook Lee, Dongsheng Li, Yuqing Yang

arXiv:2603.15500v2Mar 16, 2026cs.AIcs.LG

TL;DR

LLM self-correction can fail when reasoning drifts without producing an explicit contradiction, leaving the mechanism behind Aha moments unclear. The paper develops a token-level information-theoretic framework separating procedural advancement from epistemic verbalization, then shows that expressing uncertainty supports recovery and is readily learnable. It concludes that epistemic verbalization is a linguistic habit that helps explain and control self-correction.

Problem
Existing accounts do not explain how models recover from locally coherent but incorrect reasoning trajectories without explicit error triggers or external help.
Method
The paper separates procedural information from epistemic verbalization, defined as token-level externalization of uncertainty that makes latent assessments actionable for subsequent reasoning.
Results
A minimal doubt cue recovers around 15% of incorrect rollouts, and as few as 800 training samples can instill or suppress epistemic verbalization.
Takeaways & Limitations
Reasoning is reframed as strategic information allocation between procedural advancement and epistemic verbalization under uncertainty.
Takeaways & Limitations
The empirical analysis is mainly conducted on mathematical reasoning benchmarks, while extensions to other closed-world and open-world settings remain future work.

Abstract

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LLMs often exhibit Aha moments such as self-correction after tokens like "Wait," yet the underlying mechanism remains unclear. Standard LLMs collapse mainly through silent divergence, where trajectories drift from the correct answer yet remain locally coherent, so no explicit error triggers reactive self-correction. We introduce an information-theoretic framework that separates reasoning into procedural advancement and epistemic verbalization, the token-level externalization of uncertainty, and prove that sporadic verbalization restores convergence toward the correct answer even without explicit error triggers. Empirically, a minimal doubt cue recovers failed trajectories, and small-scale SFT suffices to instill or suppress this capability, suggesting that strong reasoning hinges less on an extraordinary inner mechanism than on the linguistic habit of externalizing uncertainty. Our framework recasts reasoning as strategic information allocation under uncertainty, offering a new lens for understanding and advancing LLM reasoning.

1 Introduction

The paper distinguishes reactive correction, which requires explicit error signals, from epistemic verbalization, which externalizes uncertainty and enables proactive self-checking. It argues that this distinction explains silent reasoning collapse and reframes reasoning as strategic information allocation under uncertainty.

Aha moments and tokens such as “Wait” are widely associated with self-correction, but their computational or informational role remains unclear.Prior work often groups Aha moments, reflection, self-correction, and specific tokens together, making their mechanisms difficult to disentangle.
Standard LLMs mainly correct reactively when explicit contradictions or failed checks surface, so latent errors can drive locally coherent reasoning into collapse.When an error remains hidden, no corrective trigger arises and the trajectory drifts away from the correct answer.
Large reasoning models additionally perform proactive correction by questioning prior steps without overt errors, although these signals can second-guess correct chains.In this setting, even an imprecise signal can be more useful than a precise signal that never fires.
Epistemic uncertainty verbalization is the token-level externalization of internal uncertainty, making otherwise latent assessments actionable for downstream self-correction.Because autoregressive generation conditions on preceding tokens, uncertainty that remains internal cannot directly influence subsequent generation.
A minimal doubt cue recovers around 15% of incorrect rollouts, while as few as 800 training samples can instill or suppress epistemic verbalization.These findings support interpreting externalized uncertainty as a learnable linguistic habit rather than an extraordinary capability.

2 Related Works

Prior work examines Aha-like behavior and reasoning trajectories but leaves unresolved how models recover from internally generated errors without external help. The paper addresses this gap with a token-level framework that identifies epistemic verbalization as the informational source for recovery and separates it from corrective actions.

Studies report that “Wait” markers correlate weakly with performance gains and that apparent self-reflection can degenerate into repetition.
Research also finds that models may correct externally supplied errors while failing to repair the same errors in their own outputs.These findings leave open whether Aha markers are unreliable, corrective policies are missing, or both.
Aggregate information-theoretic accounts describe information flow across reasoning trajectories but treat every token as procedural, leaving self-recovery after drift unexplained.
The paper identifies epistemic verbalization as the informational source that lets models regain traction after procedural missteps and proves that sporadic occurrences can restore convergence.It also separates this informational role from control actions such as self-correction.

3 A Self-Conditioning Framework for LLM Reasoning

The framework models reasoning as self-conditioning over generated tokens, separating procedural execution from informational signals that enable recovery. Empirical analyses show that standard LLMs commonly undergo silent collapse, whereas proactive correction in LRMs surfaces hidden errors despite imperfect precision.

3.1 Reasoning as Self-Conditioning: In closed-world inference, generated tokens are the model’s only evidence, and reasoning advances by self-conditioning its belief over the target variable.The objective is to reduce the target entropy H(Y | s_T) without introducing external observations.
3.2 Procedural Reasoning and Its Collapse: Procedural reasoning executes explicit computations, symbolic manipulations, variable instantiations, and learned subroutines across task-level states.The execution operator maps one task-level state to the next while autoregressively generating the trace.
3.2 Procedural Reasoning and Its Collapse: Silent divergence preserves step-by-step surface structure while the model’s predictive distribution drifts from the correct answer without overt error.Because no explicit local mistake surfaces, reactive correction has no trigger to act on.
3.3 How Models Escape (or Fail to Escape) Collapse: LRMs show 22–35% proactive corrections among self-corrections, while standard LLMs self-corrected in at most 35 of 4,800 generations, or under 1%.Proactive corrections question prior steps without overt errors; standard LLM corrections are overwhelmingly reactive.
3.4 Proactive Correction in LRMs: Proactive signals average 24.4% precision across LRMs, yet their noisy checks can expose hidden errors that precise reactive mechanisms never trigger on.DeepSeek-R1 distillations report 37.3% and 23.9% precision versus 20.5% and 15.8% for Qwen3, with no improvement from scale.

4 Epistemic Verbalization

Epistemic verbalization externalizes uncertainty about a reasoning trajectory, making latent assessments actionable for subsequent generation. Interventions show that even minimal doubt cues can help failed trajectories recover.

Epistemic verbalization externalizes uncertainty about a model’s trajectory, turning latent assessments into conditionable tokens that later reasoning can act on.The paper distinguishes this informational channel from procedural execution and self-correction.
The intervention truncates incorrect rollouts at several relative positions and compares uncertainty or revisit cues against NONE re-sampling controls.The recovery rate is the fraction of originally incorrect rollouts whose continuation reaches the correct answer.
Even DOUBT cues that express uncertainty without identifying an error suffice to support recovery across benchmarks and injection settings.This indicates that pinpointing the failure is unnecessary for the cue to provide useful signal.
The study uses surface tokens as practical indicators of diverse uncertainty expressions, adopting nine terms after measuring their co-occurrence in traces from four models.The most frequent candidate was “wait” at 73.0%, followed by “maybe” at 32.9%.

5 A Unified Framework: Reasoning as Strategic Information Allocation

The framework separates reasoning into procedural advancement and epistemic verbalization, then distinguishes both informational processes from the control actions that use them. It proves that sporadic epistemic verbalization can restore convergence even when overt-error triggers are absent.

Reasoning operates along procedural and epistemic informational axes, combining trajectory advancement with uncertainty verbalization before control actions act on that information.The framework presents reasoning as strategic allocation of information under uncertainty.
Epistemic verbalization provides information about trajectory reliability independently of whether an overt error has surfaced.Under the framework’s assumption, an epistemic token reduces uncertainty whenever uncertainty remains above threshold η.
Procedural information accumulates only when overt errors surface, so silent divergence stalls information acquisition when the trigger probability pE is approximately zero.The trajectory may remain locally coherent while its belief about the correct answer drifts.
If epistemic tokens occur with non-zero probability ρ under high uncertainty, expected uncertainty converges to zero regardless of procedural trigger probability pE.The formal proposition states E[H(Y | ˜S_t)] → 0 as t → ∞.
Self-correction is a control action, whereas epistemic verbalization is an informational mechanism that exposes uncertainty for the policy to use.This distinction separates the signal from the corrective behavior it enables.

6 Experiments

Experiments show that suppressing epistemic verbalization harms reasoning, while small-scale distillation can rapidly improve performance when student models are distributionally aligned with epistemic tokens. The effects are attributed to a learnable reasoning habit rather than mere verbosity.

Masking epistemic tokens in DeepSeek-R1-Distill-Qwen-14B/32B causes performance drops around 10%, although alternative verbalizations prevent complete collapse.The result is reported for comparisons between standard inference and inference with epistemic tokens suppressed.
Training on 800 traces with epistemic verbalization suppressed consistently degrades performance, cutting accuracy by more than half in some cases despite correct training answers.The controlled SFT experiment is designed to reduce circumvention through alternative uncertainty expressions.
LIMO contains many epistemic tokens, including an average of 77 occurrences of “Wait” per response.The dataset was gathered from several reasoning models and contains only 800 examples.
On Qwen2.5-7B and Qwen3-8B/14B-Base, LIMO training improves AIME24 pass@1 by up to 2.6x using only 800 samples.Other models trained on the same dataset show substantial degradation despite similar initial accuracy and model size.
Successful distillation requires epistemic tokens to remain within the student model’s support; poorly aligned models place them outside support and perform poorly.In successful models, these tokens are nevertheless low-probability and high-entropy relative to other tokens.
When a base model can absorb the source model’s epistemic verbalization, performance improves rapidly with a small dataset regardless of model size.The paper links distillation effectiveness to pre-existing distributional alignment on epistemic tokens.

7 Conclusion

The paper frames effective reasoning as strategic allocation between procedural information and epistemic verbalization under uncertainty. It argues that externalized uncertainty supports information acquisition and self-correction and can be learned or suppressed with small datasets.

Externalizing uncertainty enables continued information acquisition and self-correction while reframing reasoning as allocation across procedural and epistemic axes.The framework also separates these informational roles from their associated control actions.
A minimal doubt cue can recover failed trajectories, while as few as 800 training samples can instill or suppress epistemic verbalization.The conclusion characterizes the capability as a learnable linguistic habit rather than an intrinsic model capability.

Limitations

The analysis is mainly limited to mathematical reasoning benchmarks and uses practical proxies for uncertainty expressions. Extensions to open-world settings and broader linguistic characterization remain future work, while automated classification may introduce annotation noise.

The empirical analysis mainly covers mathematical reasoning benchmarks with objectively verifiable correctness.
Open-world extensions, including tool-augmented and interactive agents, are left for future work.External observations may independently surface latent errors and reduce reliance on epistemic verbalization.
The nine epistemic tokens are practical proxies that do not cover the full range of uncertainty expressions.A broader linguistic analysis of these proxies may be warranted.
GPT-5-based classification of reactive and proactive correction and reasoning collapse may introduce annotation noise.

A Proof of Proposition 5.3

The proposition formalizes convergence when epistemic tokens occur with a positive minimum probability before uncertainty falls below a threshold. Its bound is independent of the procedural trigger probability, and suitable parameters imply expected uncertainty vanishes over time.

If epistemic tokens occur with probability at least ρ whenever H_t−1 exceeds η, the proposition establishes a bound regardless of procedural trigger probability pE.
If suitable pairs (ρ(η), δ(η)) exist for every η > 0, expected uncertainty satisfies E[H_t] → 0 as t → ∞.
The proof models epistemic-token generation with an indicator V_t and tracks entropy decrease Δ_t = H_t−1 − H_t.
The resulting bound depends on ρ and δ, not on pE, after telescoping until the stopping time τ_η.

B World-Bayesian Reasoning with External Observations

World-Bayesian reasoning adds external observations that can directly reduce uncertainty and surface latent errors, weakening but not eliminating the role of epistemic verbalization. Token-level entropy alone may not reliably distinguish productive from incorrect reasoning.

World-Bayesian setup: World-Bayesian reasoning extends the framework to agents that receive external observations during inference.The state incorporates actions and observations, with observations providing exogenous information about the target.
External information gain: External-observation information gain is non-negative and can resolve ambiguities even when procedural self-conditioning adds no information.A single informative tool call or environmental observation may suffice.
Reduced dependence: Richer external observation channels can surface latent errors directly, narrowing the performance gap between models with and without epistemic verbalization.
Residual role: Epistemic verbalization retains a monitoring role by helping trigger decisions to seek tools, clarification, or experiments.
Limits of internal uncertainty: Token-level entropy captures local next-token confidence rather than uncertainty about the target variable Y.Figure 9 shows similar entropy decreases in correct and incorrect solutions, so entropy alone may not suffice as a corrective signal.

D.1 Reasoning Collapse Analysis

The appendix describes a sampled-trace pipeline that uses GPT-5 to identify reasoning collapse and classify self-correction triggers. Across Qwen models, collapse comprises most incorrect responses, while the analysis also operationalizes proactive doubt and epistemic-token characterization.

Sampling and collapse detection: GPT-5 judges incorrect traces for collapse, its dominant type, onset sentence, and justification; collapse rate is the fraction flagged as collapsed.
Aggregate findings: Collapse accounts for 54–62% of incorrect responses across Qwen models, with 58.4% overall.Scaling lowers total error rates while leaving collapse’s share among remaining errors roughly unchanged.
Trigger classification: Self-correction cases are classified as reactive when triggered by explicit evidence and proactive when triggered by suspicion without overt error.
Proactive-signal analysis: Proactive-pattern verbalizations are checked for whether the model was actually on a wrong path at the doubt point.The analysis samples 80 responses per model across four LRMs for this verification.
Epistemic-token characterization: GPT-5 characterizes epistemic tokens by extracting minimal trigger phrases, assigning labels, and describing their epistemic function.
Training setup: Distillation experiments use LLaMA-Factory under the default LIMO configuration with four B200 GPUs.
Decoding settings: Sampling temperature affects models differently, and Top-P 0.8 substantially improves DeepSeek-Distill performance.

F Epistemic Verbalization Produces Information Gain

The section argues that epistemic verbalization, rather than thinking tokens alone, produces information gain that supports recovery from incorrect reasoning trajectories. It further relates verbalization frequency to model uncertainty and shows that the behavior can vary with training data and survive lexical suppression through substitutions.

On AIME24 #7, both models initially fail, but only Qwen3-8B-SFT recovers through self-correction while Qwen3-8B-Base remains incorrect.The SFT model sustains elevated MI during evaluative expressions, while the base model’s MI collapses near zero after divergence.
Mutual information rises most reliably during utterances that externalize uncertainty, not during thinking tokens alone.Tokens such as “Hmm” can occur without an MI increase, whereas evaluative expressions coincide with elevated MI.
Epistemic verbalization makes latent uncertainty actionable by externalizing the model’s epistemic state for conditioning and reuse during inference.The framework treats this as an informational axis distinct from procedural advancement.
75% more “Wait” and 235% more “Perhaps” occurrences appear in the 1.5B than the 14B model on AIME24, while harder benchmarks elicit more epistemic tokens.The reported pattern suggests that verbalization frequency tracks uncertainty associated with problem difficulty and model capacity.
Smaller Qwen3-Base LIMO models also verbalize more uncertainty as difficulty increases, but training-data distributions shift which tokens dominate.These models rely mainly on “Wait,” unlike DeepSeek-R1-Distill-Qwen models, where “Perhaps” rises more strongly with uncertainty.
Around 10% performance drops after suppressing nine epistemic tokens, yet models bypass the mask with equivalent expressions or paragraph breaks.The underlying doubt-and-verify pattern can persist even when its lexical surface is rerouted.