Belief-Trajectory Energy: Measuring the Path to a Prediction

Jiahao Ying; Wei Tang; Boxian Ai; Yaoning Wang; Haotian Chen; Wenhe Sun; Caijun Xu; Haozhan Cai

cs.CLOct 4, 2026

Belief-Trajectory Energy: Measuring the Path to a Prediction

Authors: Jiahao Ying, Wei Tang, Boxian Ai, Yaoning Wang, Haotian Chen, Wenhe Sun, Caijun Xu, Haozhan Cai, +2 more

Organizations: Fudan University · University of Science and Technology of China · Shanghai Innovation Institute

Abstract

Large language models (LLMs) progressively revise their predictions across Transformer layers, yet we typically observe only the final output, discarding the trajectory through which it is formed. We introduce Belief-Trajectory Energy(BTE), a model-grounded measure that characterizes an input through the layerwise predictive revisions it induces in a model. By mapping intermediate states into a shared predictive space, BTE provides a principled measure of belief change that can be summarized as either a scalar or a structured depth profile. Theoretically, we show that local BTE corresponds to predictive revision under the Fisher-Rao geometry, while the sequence of revisions captures information beyond the initial-to-final belief change. Empirically, scalar BTE provides a model-relative signal of difficulty across diverse reasoning tasks, while richer BTE representations support human-LLM review detection and fine-grained generator attribution, reaching up to $0.998$ macro-AUROC and $95.6\%$ eight-way attribution accuracy. Further analysis shows that BTE develops throughout pretraining and is selectively reshaped by targeted training, demonstrating that the resulting measurement reflects what the scoring model has learned. Together, our results establish belief trajectories as a principled model-grounded signal and suggest a broader perspective in which learned models can themselves serve as instruments for characterizing the data they process. More demonstrations can be found at https://yingjiahao14.github.io/BTE-web/.

Figures & tables

Figure 1 : Overview of Belief-Trajectory Energy (BTE). BTE quantifies the layerwise predictive revisions induced by an input and aggregates them into a depth profile. The profile can be further summarized into a scalar for training-free data difficulty measurement, and used directly to distinguish human-LLM reviews and attribute responses to generators.

Figure 2 : Response-disruption control. Layer-wise BTE change relative to normal responses.

Figure 3 : Level-wise change in $\bar{e}_{M}$ relative to the easiest level, within task (top row, SpaRTUN) and across tasks (GSM8K $\to$ MATH-500 $\to$ OlympiadBench, StepGame $\to$ SpaRTUN). Solid: reference solution scored given its problem, five scorers ordered red $\to$ teal by strength; dashed: problem statement only (Llama). Bars: 95% bootstrap CI (Llama).

Figure 4 : Depth profiles of generated reviews relative to human reviews of the same papers, per layer, under the Llama-3.1-8B (left) and Qwen-3.5-9B (right) scorers.

Figure 6 : Familiarity and the depth profile. A : OLMo-2-7B stage-1 checkpoints from 0.5T to the 4T final model. B : mean profile before and after fine-tuning on 1k, 5k and 9.7k MATH solutions; shading gives the GSM8K drop at each dose and the dashed line the MATH-500 drop (right axis). C : relative change per layer for the three training arms, with the GSM8K drop per arm shaded and the MATH-500 drop dashed as in A. Grey band: layers 12-19.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Solution source	$\bm{n}$	$\bm{\rho}$ [95% CI]
Official references	150	0.44 [0.29, 0.57]
GPT-5.5 rewrites	150	0.75 [0.68, 0.81]

Appendix

Table 5: Writer control for cross-task mathematical difficulty. Spearman correlation between middle-layer BTE and the benchmark ordering GSM8K

\rightarrow

MATH-500

\rightarrow

OlympiadBench, using either official reference solutions or GPT-5.5 rewrites under a shared instruction.

Figure 7 : Review-only depth profiles of generated reviews relative to human reviews under five scorers, with layer index normalised to $[0,1]$ . Subtitles give macro-AUROC (full 32D / shape-only) in settings A / B / C. Solid: legacy generators; dashed: newer generators.

	A		B		C
Test conference-year	Llama	Qwen	Llama	Qwen	Llama	Qwen
ICLR 2022	0.8129	0.9314	0.8939	0.9997	0.9601	1.0000
ICLR 2023	0.7556	0.9357	0.9441	0.9999	0.9827	0.9998
ICLR 2024	0.7723	0.8729	0.8966	0.9865	0.9616	0.9932
NeurIPS 2021	0.7430	0.9282	0.9444	1.0000	0.9822	1.0000
NeurIPS 2022	0.7450	0.9231	0.9389	1.0000	0.9872	0.9999

Appendix

Table 9: AUROC by test conference-year, averaged over the seven generators (Original human reviews, full 32D). Settings A-C and scorers as in Table 1 .

Test conference-year	GPT-4o	Gem-1.5	Clau-3.5	GPT-5.5	GPT-5.6	Gem-3.1	Opus-5
Llama-3.1-8B, setting A
ICLR 2022	0.999	0.999	0.999	0.937	0.860	0.792	0.105
ICLR 2023	1.000	0.999	0.998	0.689	0.498	0.958	0.147
ICLR 2024	0.990	0.968	0.939	0.788	0.738	0.895	0.088
NeurIPS 2021	1.000	0.998	0.999	0.649	0.493	0.943	0.120
NeurIPS 2022	1.000	0.998	0.996	0.639	0.510	0.934	0.139

Appendix

Table 10: Setting A (three legacy training generators, ICLR 2021 papers) by test conference-year and generator, full 32D, Original human reviews. The first three generators are seen in training.

Generator	$\bm{W}$	$\bm{T}$	$\bm{H}$	$\bm{P_{g}}$	95% CI
GPT-4o	23	6	31	0.433	$[0.300,0.575]$
Gemini 1.5 Pro	38	2	20	0.650	$[0.508,0.783]$
Claude 3.5 Sonnet	29	3	28	0.508	$[0.367,0.650]$
GPT-5.5	49	6	5	0.867	$[0.775,0.942]$
GPT-5.6	51	5	4	0.892	$[0.800,0.967]$
Gemini 3.1 Pro	28	2	30	0.483	$[0.333,0.633]$

Appendix

Table 11: Human preference results on the primary Overall dimension.

W

T

, and

H

denote model wins, materially similar judgments, and human wins. Each row contains 60 judgments over the same 30 papers, with two evaluators per paper. Scores above

0.5

favor the generated review. Confidence intervals use a paper-clustered bootstrap.

Figure 8 : Attribution features. Each layer column of the token $\times$ layer BTE matrix (left) is summarised over tokens (middle) and the resulting per-layer statistics are concatenated into 32D-352D feature vectors (right).

		Llama-3.1-8B-Instruction			Qwen-3.5-9B
Per-layer features	Dim.	A	B	C	A	B	C
Token mean	32	72.1	82.9	86.4	82.9	89.5	91.3
+ std, P10/P50/P90	160	77.9	88.5	91.8	87.1	93.1	94.4
+ means in 4 position bins	288	76.4	86.2	92.7	87.0	93.2	95.0
+ AC, top-10% mean	352	77.1	87.1	93.2	86.6	94.0	95.6
Position bins only	128	70.5	82.1	88.6	79.6	90.1	92.6

Appendix

Table 13: Eight-way source accuracy (%) on generated reviews under settings A-C.

	Llama-3.1-8B-Instruct			Qwen-3.5-9B
Per-layer features	A	B	C	A	B	C
Token mean	79.1 / 2.3	74.2 / 0.2	85.6 / 1.2	93.9 / 0.0	97.1 / 0.1	98.6 / 0.3
+ std, P10/P50/P90	85.1 / 0.7	94.8 / 0.5	93.3 / 0.8	94.1 / 0.0	97.8 / 0.1	98.3 / 0.1
+ means in 4 position bins	86.7 / 0.8	94.6 / 0.5	93.6 / 0.5	94.9 / 0.1	98.2 / 0.1	98.0 / 0.0
+ AC, top-10% mean	85.6 / 0.7	95.4 / 0.3	94.1 / 0.5	94.4 / 0.0	98.1 / 0.0	98.0 / 0.1
Position bins only	84.3 / 2.3	89.2 / 0.5	90.2 / 1.0	94.9 / 0.1	97.4 / 0.2	98.7 / 0.2

Appendix

Table 14: Human-versus-machine behaviour of the same eight-way classifiers as Table 13 : human recall (%) / share of generated reviews assigned to the Human class (%).

Figure 9 : Held-out reviews of the eight sources under the Llama scorer, for the four feature sets of Table 2 . Each panel is a nonlinear (t-SNE) map of the reviews’ scores on the seven LDA discriminant axes fitted on calibration papers; contours enclose 90% and 50% of each source’s estimated density, dots mark centroids. Map distances are not calibrated: only cluster overlap is meaningful. Richer token statistics separate Human and the three legacy generators from the newer ones; GPT-5.5 and GPT-5.6 remain overlapped and account for most residual errors.

		Original			Rewrite
Setting	Scorer	Length	Raw 32D	Residual 32D	Length	Raw 32D	Residual 32D
A	Llama	0.5383	0.7416	0.7489	0.5349	0.7197	0.7233
A	Qwen	0.5344	0.8911	0.8848	0.5308	0.8640	0.8574
B	Llama	0.5123	0.9094	0.9167	0.5119	0.9046	0.9127
B	Qwen	0.5344	0.9940	0.9933	0.5308	0.9934	0.9925
C	Llama	0.5383	0.9680	0.9695	0.5349	0.9654	0.9680

Appendix

Table 15: Response-length ablation for review detection. Length is a class-balanced logistic detector using only

\log(1+n)

; Residual 32D removes the train-fitted linear length component from every BTE coordinate. Values are macro-AUROC over the same conference-year

\times

generator as Table 1 .

Scorer	$\bm{\rho(E,n)}$	$\bm{\rho(E,d)}$	$\bm{\rho(E,d\mid n)}$	$\bm{\rho(E,d\mid n,m)}$
Llama-3.1-8B-Instruct	0.576	0.444	0.212	0.151
OpenMath2-Llama3.1-8B	0.431	0.391	0.219	0.191
DeepSeek-R1-Distill-Llama-8B	0.106	0.143	0.104	0.082
Nemotron-Nano-8B-v1	0.219	0.164	0.061	$-0.016$

Appendix

Table 16: Length sensitivity on MATH-500.

E

is mean Total BTE over layers 12-19,

d

is difficulty level,

n

is response-token count, and

m

is problem-token count. The response-length-adjusted association is significant for the first three scorers (

p\leq 0.021

), but not Nemotron (

p=0.173

); after controlling both lengths, only Llama and OpenMath2 remain significant (

p<0.001

Checkpoint	Step	Tokens	L0-3	L12-19	L28-31	$\bm{\rho}$	95% CI
Gaussian init (same architecture)	-	0	0.130	0.013	0.007	$-0.01$	$[-0.10,\,+0.07]$
stage1-step150	150	1B	0.123	0.006	0.002	$-0.14$	$[-0.23,\,-0.05]$
stage1-step1000	1,000	5B	0.139	0.012	0.007	$-0.15$	$[-0.24,\,-0.07]$
stage1-step5000	5,000	21B	0.230	0.031	0.015	$-0.12$	$[-0.20,\,-0.03]$
stage1-step12000	12,000	51B	0.289	0.044	0.020	$-0.14$	$[-0.23,\,-0.06]$
stage1-step24000	24,000	101B	0.344	0.058	0.021	$-0.15$	$[-0.24,\,-0.06]$

Appendix

Table 17: OLMo-2-7B checkpoints (HuggingFace branches of allenai/OLMo-2-1124-7B ) with mean Total-BTE on MATH-500 solutions in three layer bands and the Spearman correlation

\rho

between middle-layer BTE and MATH level (paper-level bootstrap CI). The middle block is the set shown in Fig. 6 C; Llama-3.1-8B is listed for reference.

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Large language models (LLMs) generate not only reasoning text, but also token-level confidence trajectories that record how uncertainty evolves during inference. Whether these trajectories are relevant to reasoning correctness remains unclear. Here we show that confidence trajectories encode a content-agnostic confidence geometry associated with trace-level final-answer correctness. Using only token-level confidence values, without access to the input question, reasoning text, hidden states, or external verifiers, we find that low-dimensional representations of confidence trajectories separate correct from incorrect reasoning traces. Across GSM8K, MATH, and MMLU, this geometric separation is quantitatively linked to downstream predictability: stronger clustering of correct and incorrect traces, measured by the Davies--Bouldin index, consistently corresponds to higher correctness-discrimination AUC. We further show that correctness-related information is enriched in the tail of reasoning, suggesting that late-stage confidence dynamics carry key correctness signals. We propose NeuralConf, a lightweight estimator that learns from confidence trajectories for correctness evaluation. Under a fixed trace budget, NeuralConf-derived scores improve confidence-weighted answer aggregation over majority voting, tail confidence, and other static baselines. These results reveal that LLMs expose trace-intrinsic statistical signals of correctness through their own confidence dynamics, offering a route to improve inference using information already present within generation.

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