Probability is Not Enough: Exploring and Counting Divergent Tokens for Reasoning Uncertainty Quantification in LLMs

Feiyang Li; Shengjing Liu; Qi Zhan; Sijie Cheng; Weiqing Wang; Hongwen Chen; Yuxuan Yang; Wen Wang

cs.AISep 29, 2026

Probability is Not Enough: Exploring and Counting Divergent Tokens for Reasoning Uncertainty Quantification in LLMs

Authors: Feiyang Li, Shengjing Liu, Qi Zhan, Sijie Cheng, Weiqing Wang, Hongwen Chen, Yuxuan Yang, Wen Wang, +2 more

Organizations: College of Computer Science and Software Engineering, Shenzhen University · Tsinghua University · Behavioral and Spatial AI Lab, Peking University & Tongji University

Abstract

As the chain-of-thought reasoning capabilities of large language models improve, evaluating and calibrating their reasoning confidence is becoming increasingly important for quantifying the uncertainty of their answers. Current methods for estimating the confidence of large language models are generally based on probabilities of selected key tokens, but the underlying mechanism remains unclear. Our pilot study finds that replacing selected token probabilities with coarse substitutes can also improve calibration, motivating us to further explore effective signals of model confidence. We introduce Divergent Token Confidence (DTC), a framework that estimates confidence by counting tokens at which two models strongly disagree during decoding. DTC identifies these divergent tokens using the Jensen-Shannon divergence between next-token distributions evaluated along the same reasoning trajectory. We find that their count is almost negatively associated with answer accuracy, thereby serving as a simple yet effective signal for uncertainty quantification. DTC supports both white-box and black-box evaluation using auxiliary models, without explicit training and affecting the generation process. Experiments across multiple model families and six mathematical benchmarks demonstrate improved calibration over probability-based and verbalized baselines. Under white-box evaluation, the count-only estimator achieves an average expected calibration error of 13.0%, compared with 32.7%-42.4% for standard full-sequence confidence methods. In black-box settings, it also improves calibration over the original verbalized scores. For example, mean expected calibration error falls from 32.1%-40.2% to 13.7%-16.3% on DeepSeek-V3.2. These findings provide new insights for improving reasoning uncertainty quantification in large language models. The code is released at https://github.com/szu-tera/DTC.git.

Figures & tables

Figure 1: Divergent-token selection and its relationship to path accuracy. (a) A generator produces a reasoning trajectory. (b) A token is selected when the JSD between two models’ next-token distributions exceeds $\theta$ . (c) Path accuracy decreases as the number of divergent tokens increases under single- and dual-auxiliary probing.

Reasoning Models Acc

C_{\mathrm{NSL}}

C_{\mathrm{mean}}

Entropy Conf. BaseCal UQAC Verb. DTC (Ours) attn mean † prod lin MATH-500 Qwen2.5 - 7B 76.1 43.6 45.3 32.8 43.7 26.8 12.7 44.2 20.0 13.4 Qwen2.5 - 14B 80.0 43.0 44.9 38.9 43.0 26.8 14.2 34.7 16.2 0 9.9 Qwen2.5 - 32B 82.4 43.7 45.4 39.9 43.8 26.1 16.2 41.6 16.9 0 6.6 Qwen3 - 8B 83.9 38.1 41.5 31.8 36.7 37.1 37.8 47.8 0 7.9 15.6 Qwen3 - 14B 86.5 36.8 40.6 29.9 36.2 27.1 13.2 42.5 0 5.5 11.6 Qwen3 - 32B 83.7 36.1 40.2 29.4 — 38.8 38.6 37.0 0 6.1 11.0 Gemma3 - 12B 84.9 41.5 44.0 37.3 34.8 33.9 29.3 24.3 15.2 13.9 Gemma3 - 27B 89.2 42.3 44.5 38.4 36.0 16.0 0 7.0 33.4 15.0 0 7.1 AMC23 Qwen2.5 - 7B 53.6 43.6 45.3 31.7 44.0 24.7 0 8.2 42.5 15.7 0 6.7 Qwen2.5 - 14B 61.2 42.7 44.7 38.5 43.1 25.3 19.7 35.1 0 9.7 0 9.5 Qwen2.5 - 32B 66.4 43.3 45.1 39.5 44.0 24.9 22.7 38.3 0 9.8 10.8 Qwen3 - 8B 68.6 36.5 40.5 29.4 36.8 49.4 38.5 48.7 0 6.8 0 9.3 Qwen3 - 14B 74.1 34.6 39.2 26.9 35.9 27.2 10.5 35.3 12.3 11.8 Qwen3 - 32B 67.5 33.7 38.5 25.4 — 44.0 37.7 32.3 15.2 0 7.5 Gemma3 - 12B 66.8 40.8 43.6 36.4 35.3 34.5 24.0 25.9 12.7 10.9 Gemma3 - 27B 76.9 41.8 44.3 37.9 36.9 19.9 12.3 26.4 12.4 12.0 AIME24 Qwen2.5 - 7B 12.6 42.8 44.7 30.3 43.3 29.5 19.0 39.5 14.7 11.5 Qwen2.5 - 14B 13.8 41.8 44.0 37.5 42.3 25.1 15.6 31.4 13.1 16.9 Qwen2.5 - 32B 16.9 42.4 44.4 37.9 43.5 23.9 16.7 34.1 10.6 20.4 Qwen3 - 8B 27.7 34.9 39.3 26.5 36.4 40.3 42.9 45.8 14.2 14.9 Qwen3 - 14B 27.1 33.2 38.1 24.4 35.7 31.4 16.2 44.1 20.6 18.1 Qwen3 - 32B 28.3 32.0 37.4 23.7 — 44.2 37.6 25.7 21.6 16.4 Gemma3 - 12B 23.9 39.8 42.8 34.7 34.6 33.2 31.8 23.0 0 9.7 12.6 Gemma3 - 27B 29.0 40.7 43.5 36.2 36.1 29.6 29.4 21.1 0 8.1 24.3 AIME25 Qwen2.5 - 7B 0 9.1 42.4 44.4 31.0 43.1 26.6 16.2 37.3 11.2 20.5 Qwen2.5 - 14B 14.7 42.4 44.4 37.8 42.9 27.5 15.7 31.6 10.8 25.4 Qwen2.5 - 32B 12.2 42.7 44.7 38.3 43.8 30.6 18.4 33.0 0 8.7 28.3 Qwen3 - 8B 22.5 34.2 38.8 25.9 36.0 37.6 43.4 46.9 12.3 0 9.5 Qwen3 - 14B 27.5 32.6 37.8 24.0 35.3 30.8 19.7 53.0 17.1 0 5.3 Qwen3 - 32B 23.7 31.7 37.2 22.5 — 36.3 37.8 25.6 19.1 0 6.9 Gemma3 - 12B 18.8 40.3 43.2 35.5 34.5 32.1 29.4 15.9 16.7 0 7.8 Gemma3 - 27B 25.3 40.6 43.4 35.9 35.4 25.1 22.7 23.1 15.6 11.1 Average 48.0 39.3 42.4 32.7 39.0 30.8 23.6 35.0 13.2 13.0

PRM (ref.) 0 9.8 0 8.8 0 8.4 0 7.4 0 6.2 0 6.3 0 7.2 0 8.4 0 6.0 0 6.0 0 8.9 11.4 12.0 11.6 10.1 13.1 22.7 23.0 23.4 29.2 29.7 28.8 29.2 30.0 25.2 24.4 27.7 29.6 30.7 30.8 27.2 26.7 18.1

Table 1: Uncertainty quantification performance (ECE) in white-box settings. In each row, the best result is in bold and the second-best one is underlined (excluding PRM). †: UQAC variant by ours.

Figure 3: Calibration plots and probability histogram for Qwen2.5-14B on MATH-500. The $x$ -axis shows mean confidence within 20 probability bins. The calibration curve ( blue line with $\mu\pm\sigma$ ) displays actual accuracy per bin, while the gray shadow represents the probability proportion.

Method	AIME24			AIME25			HMMT25			HMMT26
Method	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$
Default CoT	72.2	–	–	60.9	–	–	46.1	–	–	50.9	–	–
$\rightarrow$ + PRM	–	39.0	88.2	–	39.5	80.4	–	41.1	60.4	–	38.1	75.1
$\rightarrow$ + $\mathrm{DTC}_{\mathrm{lin}}$	–	11.2	74.6	–	0 9.4	78.7	–	12.7	67.9	–	12.5	72.7
$\rightarrow$ + $\mathrm{DTC}_{\mathrm{prod}}$	–	24.6	79.4	–	23.9	82.1	–	29.1	75.0	–	26.4	76.5
Verb. Conf.	67.4	40.8	85.5	55.3	39.6	86.8	38.3	40.3	78.2	45.0	40.1	84.3

Table 2: Uncertainty quantification performance with DeepSeek-V3.2 in black-box settings.

Panel of Figure 4 — Figure 4: Sensitivity of $\mathrm{DTC}_{\mathrm{lin}}$ to $\theta$ and auxiliary size, single-auxiliary Qwen2.5 on AMC23. We search $\theta$ in steps of $0.05$ and report the $\theta$ (a) with the lowest ECE (b) and its AUROC (c).

Figure 5:

\mathrm{DTC}_{\mathrm{lin}}

sensitivity to

\theta

with auxiliary fixed at 1.5B and generating models

\in\{7,14,32\}

B (solid: AMC23; dashed: AIME24). The shaded band marks the white-box operating point

\theta{=}0.50

Method	AIME24		AIME25		HMMT25		HMMT26
Method	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$
Verb. Conf.	40.8	85.5	39.6	86.8	40.3	78.2	40.1	84.3
w/ DTC	0 6.3	85.8	0 3.7	87.7	10.5	77.8	0 6.2	84.2
Verb. TopK	32.0	88.4	32.6	85.7	31.6	79.3	32.0	83.6
w/ DTC	17.0	90.3	14.6	87.3	18.8	79.1	15.7	83.6
Verb. PD	32.6	86.2	33.2	84.9	31.5	74.3	32.3	83.1

Table 3: Verbalized vs. w/ DTC on DeepSeek-V3.2.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

	White-box	Black-box
Benchmarks	MATH-500 ( Lightman et al., 2024 ) AMC23 ( Balunovic et al., 2025 ) AIME24 ( Balunovic et al., 2025 ) AIME25 ( Balunovic et al., 2025 )	AIME24 ( Balunovic et al., 2025 ) AIME25 ( Balunovic et al., 2025 ) HMMT25 ( Balunovic et al., 2025 ) HMMT26 ( Balunovic et al., 2025 )
Generators	Qwen2.5-7B-Instruct ( Qwen Team, 2024 ) Qwen2.5-14B-Instruct ( Qwen Team, 2024 ) Qwen2.5-32B-Instruct ( Qwen Team, 2024 ) Qwen3-8B ( Yang et al., 2025 ) Qwen3-14B ( Yang et al., 2025 ) Qwen3-32B ( Yang et al., 2025 ) Gemma3-12B-IT ( Gemma Team, 2025 ) Gemma3-27B-IT ( Gemma Team, 2025 )	DeepSeek-V3.2 ( DeepSeek-AI, 2025 ) Qwen3-30B-A3B-Instruct-2507 ( Yang et al., 2025 ) Qwen3-4B-Instruct-2507 ( Yang et al., 2025 )
Auxiliaries	Qwen2.5-1.5B-Instruct ( Qwen Team, 2024 ) Qwen3-1.7B ( Yang et al., 2025 ) Gemma3-4B-IT ( Gemma Team, 2025 )	Qwen2.5-7B-Instruct ( Qwen Team, 2024 ) Qwen2.5-1.5B-Instruct ( Qwen Team, 2024 )
Sample number per question	$8$ on MATH-500 $64$ on AMC23 $64$ on AIME24 $64$ on AIME25	$32$ for DeepSeek-V3.2 $64$ for Qwen3-30B-A3B-Instruct-2507 $64$ for Qwen3-4B-Instruct-2507

Appendix

Table 4: Benchmarks, models, and sample counts per question. The three black-box generators share one auxiliary pair.

Panel of Figure 6 — Figure 6: Prompt for Verbalized confidence. [Question] is replaced by the problem.

Panel of Figure 7 — Figure 7: Prompt for Verbalized TopK with $k{=}2$ . [Question] is replaced by the problem.

Panel of Figure 8 — Figure 8: Prompt for Verbalized probability distribution. [Question] is replaced by the problem.

Reasoning Models Acc

C_{\mathrm{NSL}}

C_{\mathrm{mean}}

Entropy Conf. BaseCal UQAC Verb. DTC (Ours) attn mean † prod lin MATH-500 Qwen2.5 - 7B 76.1 67.0 66.5 61.4 58.5 59.2 62.0 74.7 80.4 83.8 Qwen2.5 - 14B 80.0 65.7 65.0 65.0 56.0 56.6 64.9 74.8 81.9 85.0 Qwen2.5 - 32B 82.4 66.7 65.8 65.5 50.6 59.7 65.0 83.8 82.7 85.8 Qwen3 - 8B 83.9 75.0 73.4 72.4 52.3 59.1 61.6 57.9 79.3 73.9 Qwen3 - 14B 86.5 74.3 72.8 72.0 53.2 66.1 75.6 61.8 79.4 75.6 Qwen3 - 32B 83.7 73.6 71.6 70.9 — 58.8 63.1 63.3 80.6 78.9 Gemma3 - 12B 84.9 61.9 60.3 62.7 40.9 61.9 58.1 84.1 81.9 84.0 Gemma3 - 27B 89.2 63.6 61.8 63.3 38.9 76.5 81.7 87.2 85.5 87.9 AMC23 Qwen2.5 - 7B 53.6 72.4 71.8 67.9 66.6 63.9 68.1 67.1 81.3 81.2 Qwen2.5 - 14B 61.2 66.9 66.4 67.7 62.3 57.1 68.5 73.0 79.2 80.3 Qwen2.5 - 32B 66.4 67.1 66.3 69.8 60.8 62.2 68.6 73.7 77.6 78.0 Qwen3 - 8B 68.6 73.8 72.0 71.6 56.2 41.1 47.7 57.8 78.7 76.0 Qwen3 - 14B 74.1 73.5 71.5 72.6 56.6 66.1 72.9 68.8 79.2 75.3 Qwen3 - 32B 67.5 75.3 73.4 74.4 — 55.1 57.2 67.7 80.3 78.6 Gemma3 - 12B 66.8 69.7 68.6 70.0 54.2 61.2 61.9 78.7 79.9 78.7 Gemma3 - 27B 76.9 68.5 67.1 68.5 47.8 72.7 76.4 85.6 82.6 80.4 AIME24 Qwen2.5 - 7B 12.6 79.0 78.0 74.4 75.2 55.1 61.5 79.0 84.4 75.6 Qwen2.5 - 14B 13.8 71.3 71.0 75.2 65.5 61.4 70.1 77.8 76.5 74.2 Qwen2.5 - 32B 16.9 75.6 75.2 79.3 72.2 67.5 73.5 84.1 80.0 70.4 Qwen3 - 8B 27.7 71.3 69.9 72.1 58.4 53.5 56.4 65.5 74.1 65.7 Qwen3 - 14B 27.1 70.8 69.4 70.6 61.4 64.9 72.6 63.9 74.3 67.0 Qwen3 - 32B 28.3 71.2 69.9 73.4 — 52.1 52.5 74.7 76.2 69.6 Gemma3 - 12B 23.9 81.0 80.1 79.8 71.1 64.8 60.9 74.8 81.8 66.4 Gemma3 - 27B 29.0 78.5 78.1 78.3 68.2 66.4 65.1 87.8 80.2 64.0 AIME25 Qwen2.5 - 7B 0 9.1 79.3 78.9 77.8 78.7 67.7 71.1 69.6 76.5 64.3 Qwen2.5 - 14B 14.7 75.1 74.7 75.9 71.8 68.5 75.5 74.4 75.9 62.2 Qwen2.5 - 32B 12.2 79.0 79.1 79.4 75.4 52.4 77.5 72.1 76.7 60.1 Qwen3 - 8B 22.5 78.9 77.4 77.2 67.9 55.1 56.2 63.3 82.4 70.9 Qwen3 - 14B 27.5 82.5 81.3 82.8 73.8 67.7 84.1 52.1 89.1 81.8 Qwen3 - 32B 23.7 80.3 78.8 81.3 — 61.2 61.7 74.8 86.8 79.9 Gemma3 - 12B 18.8 84.9 84.1 84.4 69.0 64.8 66.7 85.7 88.9 76.3 Gemma3 - 27B 25.3 86.9 85.8 86.4 67.2 70.9 74.8 91.3 89.8 79.0 Average 48.0 73.8 72.7 73.2 61.8 61.6 66.7 73.5 80.8 75.3

PRM (ref.) 95.4 94.3 94.4 90.9 89.8 91.3 93.6 93.2 90.8 90.5 87.2 90.6 89.5 89.0 86.6 91.1 96.2 89.7 89.6 89.0 88.6 91.7 93.8 90.8 88.6 88.3 83.8 87.5 87.5 88.1 89.6 91.1 90.4

Appendix

Table 5: Uncertainty quantification performance (AUROC,

\uparrow

higher is better) in white-box settings. In each row, the best result is in bold and the second-best one is underlined (excluding PRM). †: UQAC variant by ours.

Method	AIME24			AIME25			HMMT25			HMMT26
Method	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$
Default CoT	61.4	–	–	45.6	–	–	30.3	–	–	34.7	–	–
$\rightarrow$ + PRM	–	36.0	82.9	–	36.6	80.3	–	37.8	53.3	–	33.4	78.0
$\rightarrow$ + $\mathrm{DTC}_{\mathrm{lin}}$	–	11.3	72.9	–	0 9.0	81.5	–	13.1	69.6	–	13.0	74.6
$\rightarrow$ + $\mathrm{DTC}_{\mathrm{prod}}$	–	16.3	75.8	–	10.7	84.8	–	15.4	78.0	–	14.3	79.0
Verb. Conf.	61.8	45.6	80.8	46.6	45.9	85.6	27.3	45.6	81.9	33.0	45.1	79.8

Appendix

Table 6: Uncertainty quantification performance with Qwen3-4B in black-box settings.

Method	AIME24			AIME25			HMMT25			HMMT26
Method	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	Acc $\uparrow$	ECE $\downarrow$	AUC $\uparrow$
Default CoT	73.4	–	–	59.8	–	–	42.4	–	–	43.5	–	–
$\rightarrow$ + PRM	–	37.5	86.0	–	37.8	78.4	–	38.7	53.6	–	34.6	77.4
$\rightarrow$ + $\mathrm{DTC}_{\mathrm{lin}}$	–	15.9	70.2	–	12.3	75.8	–	13.1	72.1	–	0 9.6	77.8
$\rightarrow$ + $\mathrm{DTC}_{\mathrm{prod}}$	–	18.2	75.1	–	16.6	79.6	–	19.5	78.7	–	17.1	80.5
Verb. Conf.	75.7	44.5	83.9	60.6	44.3	80.8	41.8	42.9	85.3	43.9	43.9	79.2

Appendix

Table 7: Uncertainty quantification performance with Qwen3-30B-A3B in black-box settings.

Method	AIME24		AIME25		HMMT25		HMMT26
Method	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$
Verb. Conf.	44.5	83.9	44.3	80.8	42.9	85.3	43.9	79.2
w/ DTC	17.8	87.0	15.7	83.8	13.0	87.9	17.4	85.9
Verb. TopK	40.2	88.2	40.6	88.7	41.2	85.4	41.6	83.0
w/ DTC	11.1	83.0	10.4	84.9	0 5.4	86.5	0 6.1	86.4
Verb. PD	33.5	75.5	34.1	69.6	31.1	77.3	34.9	72.3

Appendix

Table 8: Verbalized vs. w/ DTC on Qwen3-30B-A3B.

Method	AIME24		AIME25		HMMT25		HMMT26
Method	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$	ECE $\downarrow$	AUC $\uparrow$
Verb. Conf.	45.6	80.8	45.9	85.6	45.6	81.9	45.1	79.8
w/ DTC	30.8	82.0	29.4	87.0	31.0	81.6	31.1	81.2
Verb. TopK	47.0	68.3	47.0	75.3	46.8	74.0	45.5	77.3
w/ DTC	33.7	68.6	32.8	75.5	34.3	73.8	30.9	76.9
Verb. PD	46.9	69.9	48.1	73.7	43.9	75.5	42.9	70.9

Appendix

Table 9: Verbalized vs. w/ DTC on Qwen3-4B.

Panel of Figure 9 — Figure 9: Black-box sensitivity of $\mathrm{DTC}_{\mathrm{lin}}$ to $\theta$ and auxiliary size. We search $\theta$ in steps of $0.05$ and report the $\theta$ (a) with the lowest ECE (b) and its AUROC (c). Cells without a finished probe are left blank.

Figure 10:

\mathrm{DTC}_{\mathrm{lin}}

sensitivity to

\theta

with

{\mathcal{A}}^{\prime\prime}

fixed at 1.5B (DeepSeek-V3.2 generating; solid: AIME25; dashed: HMMT25). The shaded band marks the black-box operating point

\theta{=}0.70

Figure 11: Divergent-token count and ratio against path accuracy, single-auxiliary Qwen2.5-7B on MATH-500. Each curve is one auxiliary at the threshold used in Figure 1 (c).

Figure 12: Divergence choice for divergent-token selection. Each column is one disagreement score (JSD, forward KL, reverse KL) for

\mathrm{DTC}_{\mathrm{lin}}

with auxiliary fixed at 1.5B and generating models

\in\{7,14,32\}

B (solid: AMC23; dashed: AIME24). The top row shows ECE (%) and the bottom row AUROC (%) versus

\theta

Figure 13: Divergence choice for black-box divergent-token selection. Each column is one disagreement score (JSD, forward KL, reverse KL) for

\mathrm{DTC}_{\mathrm{lin}}

with auxiliaries fixed at Qwen2.5-7B and Qwen2.5-1.5B and generators Qwen3-4B, Qwen3-30B-A3B, and DeepSeek-V3.2 (solid: AIME25; dashed: HMMT25). The top row shows ECE (%) and the bottom row AUROC (%) versus

\theta

Figure 14: Saturation length

n

for

\mathrm{DTC}_{\mathrm{lin}}

. The top row shows ECE (%) and the bottom row AUROC (%) versus

\theta

at step

0.01

, averaged over MATH-500, AMC23, AIME24, and AIME25.

Figure 15: Saturation length

n

for black-box

\mathrm{DTC}_{\mathrm{lin}}

. Each column is one generator with Qwen2.5 auxiliaries

7

B and

1.5

B. The top row shows ECE (%) and the bottom row AUROC (%) versus

\theta

at step

0.01

, averaged over AIME24, AIME25, and HMMT25.

Figure 16: Offset

k

for

\mathrm{DTC}_{\mathrm{prod}}

. Each column is the smallest white-box generator in its family (Qwen2.5-7B, Qwen3-8B, Gemma3-12B) with the family auxiliary. ECE (%) is shown versus

\theta

at step

0.01

, averaged over MATH-500, AMC23, AIME24, and AIME25.

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Reliable uncertainty communication is critical to the trustworthiness of LLMs, yet faithful calibration (FC)--the alignment between models' intrinsic and (linguistically) expressed confidence--is a persistent failure mode. This challenge is key for large reasoning models (LRMs), whose extended reasoning traces are often interpreted by users as evidence of deliberation, competence, and confidence. Despite the importance of FC and wide usage of LRMs, the extent to which LRMs can faithfully express their confidence remains poorly understood. Moreover, the prevailing paradigm to measure FC does not generalize well to the long chain-of-thought outputs generated by LRMs, which tend to lack clear step boundaries, involve inconsistent step structure, and encode complex conditional dependencies throughout the trace--complicating estimation of intrinsic confidence. To address this challenge, we introduce a novel framework to systematically quantify FC of LRMs. Our framework analyzes linguistic decisiveness relative to three sources of internal uncertainty, based on token probabilities, hidden states, and sampled response consistency. We also devise a prefix-conditioned sampling approach to control for conditional and structural variation across traces. Applying our framework to a diverse suite of leading models, datasets, and prompts, we find that faithful confidence expression is a significant challenge for LRMs. Reasoning behaviors do not automatically translate to improved FC, and prompt interventions for non-reasoning models do not improve faithfulness in the reasoning setting. Different confidence estimators further produce divergent assessments of the same traces, revealing fragility in prior evaluation methodologies. Taken together, our work establishes FC as a distinct reliability and alignment target for LRMs, particularly as such systems are increasingly deployed in high-stakes contexts.

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