Uncertainty Localization in LLM Reasoning via Embedding Perturbations

Qihao Wen; Jiahao Wang; Yang Nan; Pengfei He; Ravi Tandon; Han Xu

cs.LGFeb 2, 2026

Uncertainty Localization in LLM Reasoning via Embedding Perturbations

Authors: Qihao Wen, Jiahao Wang, Yang Nan, Pengfei He, Ravi Tandon, Han Xu

Organizations: University of Arizona · Michigan State University

Abstract

Large Language Models (LLMs) have achieved significant breakthroughs across various domains, but they can still produce unreliable or misleading outputs. For responsible LLM applications, uncertainty quantification techniques are used to estimate a model's uncertainty about its outputs, indicating the likelihood that those outputs may be problematic. For LLM reasoning tasks, it is essential to estimate uncertainty not only in the final answer but also in the intermediate reasoning process, particularly to identify where uncertainty arises. Such information may enable more fine-grained and targeted interventions during inference. In this study, we investigate which metrics can effectively localize uncertain places within an LLM reasoning trajectory. Our study reveals that uncertain intermediate continuations are more likely to occur at tokens that are highly sensitive to perturbations in the embeddings of preceding tokens. In our experiments, we show that such perturbation-based metrics achieve stronger performance in localizing uncertain intermediate steps than baseline methods, including probability-based, sampling-based, and Bayesian-based approaches. Meanwhile, our proposed metrics also enjoy good simplicity and efficiency.

Figures & tables

Figure 1: A response of Llama in logical reasoning ( Suzgun et al., 2023 ) . The model first states that ‘Alice is now right midfielder’ and ‘Alice and Claire trade positions’ , it then incorrectly infers ‘Claire is now left midfielder.’ While, the correct inference should be ‘Claire is now right midfielder’. Our perturbation-based metrics flag the token ‘left’ with high uncertainty score. Note: the scores are min-max normalized in the figure.

Figure 2: The model’s alternative tokens at key positions, and the regenerated continuations for the example in Figure 1 . We provide additional positions & examples in Figure 7 in Appendix E .

	Math						Logic
	Llama		Qwen		Mistral		Llama		Qwen		Mistral
	10%	20%	10%	20%	10%	20%	10%	20%	10%	20%	10%	20%
NLL-max	0.19	0.30	0.21	0.38	0.16	0.28	0.21	0.41	0.27	0.50	0.15	0.32
Entropy-max	0.20	0.34	0.23	0.41	0.14	0.24	0.18	0.39	0.25	0.50	0.13	0.30
TokUR-max	0.27	0.48	0.26	0.45	0.28	0.44	0.27	0.42	0.29	0.47	0.25	0.45
NLL-avg	0.15	0.25	0.16	0.32	0.10	0.19	0.18	0.39	0.20	0.39	0.10	0.26

Table 1: Performance of each method in 300 responses with incorrect steps, The 95% conf. interval is

\pm 0.057

for 300 Bernoulli trials with probability around 0.5.

	GPQA
	gpt-oss-20b		Qwen3.5-27B
	10%	20%	10%	20%
NLL-max	0.21	0.30	0.23	0.38
Entropy-max	0.18	0.28	0.28	0.39
NLL-avg	0.13	0.31	0.13	0.28
Entropy-avg	0.15	0.28	0.15	0.30

Table 2: Performance on GPQA.

	Math				Logic
	Llama		Qwen		Llama		Qwen
	10%	20%	10%	20%	10%	20%	10%	20%
NLL-max	0.22	0.29	0.14	0.20	0.20	0.30	0.10	0.19
Entropy-max	0.23	0.30	0.14	0.19	0.22	0.30	0.09	0.18
TokUR-max	0.23	0.31	0.14	0.23	0.23	0.33	0.13	0.17
Rand. Pert.	0.27	0.31	0.21	0.27	0.24	0.31	0.17	0.19

Table 3: The max re-generation risk of flagged uncertain steps (95% conf. interval is

\pm 0.049

)

FactScore	Llama	Qwen	Mistral
NLL	0.610	0.638	0.604
Entropy	0.635	0.623	0.596
TokUR	0.706	0.670	0.674
CCP	0.660	0.659	0.687
TokenSAR	0.716	0.631	0.641
Graph Uncert.	0.778	0.878	0.887

Table 5: Hallucination Detection.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

	Math
	algebra	cnt. & prob.	number theory	precalculus	inter. algebra	prealgebra
Llama	5 / 5	4 / 5	5 / 5	4 / 5	5 / 5	5 / 5
Qwen	4 / 5	4 / 5	4 / 5	4 / 5	5 / 5	5 / 5
Mistral	4 / 5	5 / 5	4 / 5	5 / 5	5 / 5	5 / 5
	Logic
	logical deduction	tracking shuffled	web of lies	formal fallacies	navigate	temp. seq.

Appendix

Table 6: Human examination result for ‘first wrong step’ annotation in 180 responses.

Figure 4: Prompts used in the experiments. Title of each prompt is in bold, placeholders are marked as ‘[]’ with description within.

	Math						Logic
	Llama		Qwen		Mistral		Llama		Qwen		Mistral
	10%	20%	10%	20%	10%	20%	10%	20%	10%	20%	10%	20%
NLL-max	0.18	0.29	0.21	0.39	0.15	0.28	0.21	0.39	0.28	0.51	0.17	0.33
Entropy-max	0.19	0.33	0.24	0.41	0.14	0.24	0.21	0.40	0.26	0.51	0.15	0.31
TokUR-max	0.27	0.49	0.26	0.44	0.28	0.44	0.27	0.42	0.29	0.45	0.26	0.45
NLL-avg	0.13	0.24	0.17	0.31	0.09	0.18	0.18	0.38	0.19	0.39	0.11	0.27

Appendix

Table 7: Performance of each method in 300 responses with incorrect final answer, The 95% conf. interval is

\approx\pm 0.057

for 300 Bernoulli trials (p=0.5).

	Math						Logic
	Llama		Qwen		Mistral		Llama		Qwen		Mistral
	10%	20%	10%	20%	10%	20%	10%	20%	10%	20%	10%	20%
NLL-max	0.20	0.33	0.21	0.42	0.17	0.32	0.20	0.39	0.26	0.47	0.14	0.32
Entropy-max	0.24	0.37	0.24	0.40	0.12	0.24	0.18	0.38	0.23	0.47	0.15	0.31
TokUR-max	0.25	0.46	0.27	0.45	0.28	0.45	0.26	0.40	0.29	0.49	0.23	0.45
NLL-avg	0.17	0.27	0.16	0.33	0.11	0.20	0.17	0.38	0.19	0.37	0.10	0.27

Appendix

Table 8: Results on MATH and BBH, following the same setting in Section 4.1 , with LLM temperature

T=0.5

Table 9: Experiment results using different system prompts.

Figure 5: Result under various hyper-parameters.

Figure 6: For the same setting as Figure 2 , we re-generate continuations conditioned on adversarial perturbed embeddings $\hat{\mathbf{H}}_{1:t-1}$ . We find that the continuations are also coherent and on-task, similar to Figure 2 . This shows that adversarial perturbation on the embedding also does not disrupt reasoning coherence.

Figure 7: Extra cases of regeneration from perturbed conditions.

Panel of Figure 8 — Figure 8: Comparison of un-normalized perturbation-based scores between uncertain responses (red) and certain responses (blue). Here certain responses indicate self-consistency $>0.7$ .

Panel of Figure 9 — Figure 9: Extra case studies on math reasoning task. Each uncertainty score is min-max normalized over the complete output to fit the figure, and only the critical segments are shown.

Panel of Figure 10 — Figure 10: Extra case studies on logical reasoning task. Each uncertainty score is min-max normalized over the complete output to fit the figure, and only the critical segments are shown.

Panel of Figure 11 — Figure 11: Case studies on logical reasoning task. Each uncertainty score is min-max normalized over the complete output to fit the figure, and only part of tokens are shown.

Figure 12: Correct but risky case. The LLM is performing a correct mathematical derivation from $y=-4-\frac{2x}{7}-\frac{6}{7}$ to $y=-\frac{2x}{7}-\frac{34}{7}$ . The token ‘34’ is flagged as uncertain though correct. Re-generation from this step reveals risk score of 0.6, where 0.5 of the risk is due to “ $y=-\frac{2x}{7}-\frac{32}{7}$ ” where model incorrectly generate ‘32’ instead of ‘34’ , and 0.1 is because model generate “ $y=-\frac{2x}{7}-\frac{38}{7}$ ”, both re-generations show the uncertainty of token ‘34’ .

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Large language models are informing decisions with ever-higher stakes. As the consequences of their errors grow, a central question becomes harder to ignore: how much can we trust an individual answer? Yet recognizing when to defer remains difficult because language models can present incorrect conclusions with fluent explanations and an authoritative tone. Uncertainty quantification seeks to address this disconnect by estimating the reliability of individual predictions. However, many existing methods require repeated generations or separately trained components, and their scalar estimates do not reveal where uncertainty arises or how it evolves during reasoning. Recent work has also shown that generation length can be strongly associated with uncertainty estimates and correctness, raising the question of how much of an estimator's predictive power comes from uncertainty-specific information rather than output length alone. Mechanistic interpretability offers a way to address these limitations by connecting human-interpretable concepts to intermediate model states. Building on this capability, we introduce the U-Space, a low-dimensional subspace that makes a model's evolving uncertainty measurable and interpretable. We identify semantic anchors for doubt and certainty, map their unembedding directions back into the residual space, and combine their contrasts into an orthogonal basis. The U-Lens projects each token state onto these basis vectors, yielding an interpretable token-level uncertainty map that can be inspected directly or aggregated into a scalar uncertainty score. Our approach requires no correctness labels, repeated generations, or training. Across reasoning benchmarks, its confidence score outperforms established baselines under both standard and length-controlled evaluation and transfers more reliably than supervised estimators. Code: https://github.com/s2labres/U-Space.

Abstract

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