FACT-E: Causality-Inspired Evaluation for Trustworthy Chain-of-Thought Reasoning

Yuxi Sun; Aoqi Zuo; Haotian Xie; Wei Gao; Mingming Gong; Jing Ma

cs.AIApr 12, 2026

FACT-E: Causality-Inspired Evaluation for Trustworthy Chain-of-Thought Reasoning

Authors: Yuxi Sun, Aoqi Zuo, Haotian Xie, Wei Gao, Mingming Gong, Jing Ma

Organizations: Hong Kong Baptist University · The University of Melbourne · Singapore Management University

Abstract

Chain-of-Thought (CoT) prompting has improved LLM reasoning, but models often generate explanations that appear coherent while containing unfaithful intermediate steps. Existing self-evaluation approaches are prone to inherent biases: the model may confidently endorse coherence even when the step-to-step implication is not valid, leading to unreliable faithfulness evaluation. We propose FACT-E, a causality-inspired framework for evaluating CoT quality. FACT-E uses controlled perturbations as an instrumental signal to separate genuine step-to-step dependence from bias-driven artifacts, producing more reliable faithfulness estimates (intra-chain faithfulness). To select trustworthy trajectories, FACT-E jointly considers intra-chain faithfulness and CoT-to-answer consistency, ensuring that selected chains are both faithful internally and supportive of the correct final answer. Experiments on GSM8K, MATH, and CommonsenseQA show that FACT-E improves reasoning-trajectory selection and yields stronger in-context learning exemplars. FACT-E also reliably detects flawed reasoning under noisy conditions, providing a robust metric for trustworthy LLM reasoning.

Figures & tables

Figure 1: Motivating example illustrating the limitation of LLM self-assessment on CoT evaluation. Two reasoning chains appear fluent and coherent, yet CoT 1 contains successive intermediate steps that are not logically necessary for subsequent reasoning. Conventional method (e.g., self-reflect) assigns similarly high quality scores to both chains, failing to detect this breakdown, whereas FACT-E evaluates the unfaithfulness in a chain and successfully identifies CoT 2 as more trustworthy.

Figure 2: Structural causal graphs for CoT quality estimation. (a) depicts the process of answering a query

Q

with CoT

\mathbf{S}

. (b) illustrates the traditional self-assessment approaches, where the LLM evaluates the faithfulness score

Y

between

\mathbf{S}_{\leq i}

and

\mathbf{S}_{>i}

, denoted by

X_{i}

, mediated by LLM’s self-evaluation

M

. The LLM’s internal bias

Z

is an unobserved confounder that affects all variables. (c) FACT-E introduces exogenous noise

E

as an instrumental variable to obtain a more reliable faithfulness evaluation for CoTs.

Question	Suppose $\sin D=0.7$ in the right triangle $DEF$ (where $\angle E=90^{\circ}$ and $EF=7$ ). What is the length of $DE$ ?
$(\mathbf{S}_{\leq i},\mathbf{S}_{>i})$	We are given that $\sin D=0.7$ and need to find the length of $DE$ . In right triangle $DEF$ , $EF=7$ is opposite to angle $D$ , and $DF$ is the hypotenuse. Thus, $0.7=\frac{7}{DF}$ , so $DF=10$ . By the Pythagorean theorem, $DE^{2}=DF^{2}-EF^{2}=100-49=51$ , hence $DE=\sqrt{51}$ .
$(\mathbf{S}_{\leq i},\mathbf{S}^{\prime}_{>i})$	We are given that $\sin D=0.7$ and need to find the length of $DE$ . In right triangle $DEF$ , $EF=7$ is opposite to angle $D$ , and $DF$ is the hypotenuse. Thus, $\sin D=\frac{EF}{DE}$ , so $DE=\frac{7}{0.7}=10$ .

Table 1: Illustrative example of a perturbed rollout

\mathbf{S}^{\prime}_{>i}

generated from the same prefix

\mathbf{S}_{\leq i}

. The injected logical error leads to an incorrect continuation despite identical contextual and stylistic conditions.

Gpt-4o-mini				DeepSeek-V3
Method	Math-500	CommonsenseQA	GSM-8K	Math-500	CommonsenseQA	GSM-8K
CoT	78.69 (-4.253%)	83.00 (+0.029%)	92.40 (-0.154%)	85.52 (-4.587%)	83.80 (-1.671%)	96.00 (-0.243%)
Denoise	83.06 (+0.117%)	83.30 (+0.329%)	92.20 (-0.354%)	84.70 (-5.407%)	84.60 (-0.871%)	95.40 (-0.843%)
Polish	83.33 (+0.387%)	80.00 (-2.971%)	92.60 (+0.046%)	94.80 (+4.693%)	85.20 (-0.271%)	95.60 (-0.643%)
Refelect	80.60 (-2.343%)	83.20 (+0.229%)	92.10 (-0.454%)	87.43 (-2.677%)	85.80 (+0.329%)	95.80 (-0.443%)
Consistency	82.79 (-0.153%)	82.10 (-0.871%)	92.80 (+0.246%)	93.17 (+3.063%)	85.00 (-0.471%)	96.40 (+0.157%)

Table 2: Accuracy comparison (%) across three benchmarks for four LLMs. Values in parentheses denote the relative change with respect to the average performance of all methods for each model-dataset pair. Best results are shown in Bold , second-best results are underlined .

#	Answer	CoT (Chain of Thought)	Score
Question: Suppose $\sin D=0.7$ in the diagram below. What is $DE$ ? [asy] pair D,E,F; F = (0,0); D = (sqrt(51),7); E = (0,7); draw(D–E–F–D); draw(rightanglemark(D,E,F,15)); label(" $D$ ",D,NE); label(" $E$ ",E,NW); label(" $F$ ",F,SW); label(" $7$ ",(E+F)/2,W); [/asy]
1	$\sqrt{51}$ ✓	We are given that $\sin D=0.7$ and we need to find the length of $DE$ . In the right triangle $DEF$ , we know that $EF$ is the vertical side and has a length of $7$ . The sine of angle $D$ is defined as the ratio of the length of the opposite side to the hypotenuse. Therefore, we can express this relationship as: $\sin D=\frac{EF}{DF}$ . Substituting the known values, we have: $0.7=\frac{7}{DF}$ To find $DF$ , we can rearrange the equation: $DF=\frac{7}{0.7}$ . Calculating this gives: $DF=10$ Now, we can use the Pythagorean theorem to find $DE$ . In triangle $DEF$ : $DF^{2}=DE^{2}+EF^{2}$ Substituting the known values: $10^{2}=DE^{2}+7^{2}$ This simplifies to: $100=DE^{2}+49$ Now, isolating $DE^{2}$ : $DE^{2}=100-49=51$ Taking the square root gives: $DE=\sqrt{51}$ Thus, the length of $DE$ is $\sqrt{51}$ .	0.998
2	$10$ ✗	We are given that $\sin D=0.7$ and we need to find the length of $DE$ . In the right triangle $DEF$ , we know that $EF$ is the opposite side to angle $D$ and $DE$ is the hypotenuse. The sine function is defined as the ratio of the length of the opposite side to the hypotenuse. Therefore, we can express this relationship as: $\sin D=\frac{EF}{DE}$ From the diagram, we see that $EF=7$ . Substituting the known values into the sine equation gives us: $0.7=\frac{7}{DE}$ To find $DE$ , we can rearrange this equation: $DE=\frac{7}{0.7}$ Calculating this gives: $DE=\frac{7}{0.7}=10$ Thus, the length of $DE$ is $10$ .	0
Question: What is $1^{(2^{235423523})}$ ?
1	$1$ ✓	To solve the expression $1^{(2^{235423523})}$ , we need to understand the properties of exponents. The base of the exponent is 1. According to the rules of exponents, any number raised to any power is equal to 1, as long as the base is 1. Therefore, regardless of the value of the exponent, $1$ raised to any power will always equal $1$ .	0.999
2	$1$ ✓	The expression $1^{(2^{235423523})}$ involves raising 1 to the power of $2^{235423523}$ . The key property of exponentiation to note here is that any number raised to the power of 0 is 1 , and any number raised to any power (including very large numbers) is still 1 if the base is 1. Therefore, regardless of the exponent, since the base is 1, the result will always be 1.	0.596

Table 4: Question-answer evaluation results. Flawed portions of the CoT are highlighted in red . Correct and incorrect answers are marked with ✓ and ✗ , respectively.

Figure 3: Performance (%) on noisy-rationale benchmarks. Results are shown for Qwen3 and DeepSeek-V3 on NoRa-Commonsense, and Gpt-4o-mini on NoRa-Math/Commonsense, under in-context learning with different numbers of noisy rationale demonstrations.

Figure 4: Performance (%) of lightweight FACT-E (a) and standard FACT-E (b) varying the number of sampling trials (

N

) on three benchmarks.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Method	# LLM inference requests
Polish	7
Consistency	3
Reflect	6
Denoise	7
FACT-E ${}_{\text{lightweight}}$	7
FACT-E ${}_{\text{standard}}$	$3\cdot\ell(c)+1$

Appendix

Table 5: Number of LLM inference requests per query under

N=3

and a single perturbation type per inspected split point. Here

\ell(c)

denotes the number of reasoning steps in the CoT. The exact count may vary depending on prompt batching in implementation.

Noise Type	Description (MATH Task)	Description (Commonsense)
Operation Error	Modification of a specific mathematical operation or procedural step.	Modification of a specific logical step or operative element within the reasoning.
Conceptual Swap	Substitution of distinct mathematical or logical concepts.	Substitution of semantically or logically related entities, properties, or concepts.
Misgeneralization	Erroneous extrapolation from a specific concept to an invalid general rule.	Improper extension of a specific concept or heuristic to an invalid or broader context.
Reordered Logic	Permutation of the sequential order of reasoning steps.	Permutation of the sequential or causal order of reasoning steps.
Contradiction	Introduction of an inconsistency with established premises or prior conclusions.	Introduction of an assertion that conflicts with established facts, premises, or prior logic.

Appendix

Table 6: Unified Definitions of Noise Types across MATH and Commonsense Tasks.

Figure 5: Ablation study of FACT-E.

#	Answer	CoT (Chain of Thought)	Score
Question: Suppose $\sin D=0.7$ in the diagram below. What is $DE$ ? [asy] pair D,E,F; F = (0,0); D = (sqrt(51),7); E = (0,7); draw(D–E–F–D); draw(rightanglemark(D,E,F,15)); label(" $D$ ",D,NE); label(" $E$ ",E,NW); label(" $F$ ",F,SW); label(" $7$ ",(E+F)/2,W); [/asy]
1	$6.86$ ✗	We are given that $\sin D=0.7$ and we need to find the length of $DE$ . In the right triangle $DEF$ , we know that $DE$ is the side opposite angle $D$ and $EF$ is the side adjacent to angle $D$ . The length of $EF$ is given as $7$ . Using the definition of sine, we have: $\sin D=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{DE}{DF}$ . We can also express $DF$ in terms of $DE$ and $EF$ using the Pythagorean theorem: $DF=\sqrt{DE^{2}+EF^{2}}$ . Substituting $EF=7$ into the equation gives: $DF=\sqrt{DE^{2}+7^{2}}=\sqrt{DE^{2}+49}$ . Now substituting this into the sine equation: $0.7=\frac{DE}{\sqrt{DE^{2}+49}}$ . To eliminate the square root, we can square both sides: $0.7^{2}=\frac{DE^{2}}{DE^{2}+49}$ . Calculating $0.7^{2}$ : $0.49=\frac{DE^{2}}{DE^{2}+49}$ . Cross-multiplying gives: $0.49(DE^{2}+49)=DE^{2}$ . Expanding this: $0.49DE^{2}+24.01=DE^{2}$ . Rearranging terms: $DE^{2}-0.49DE^{2}=24.01$ . This simplifies to: $0.51DE^{2}=24.01$ . Dividing both sides by $0.51$ : $DE^{2}=\frac{24.01}{0.51}\approx 47.0$ . Taking the square root to find $DE$ : $DE\approx\sqrt{47.0}\approx 6.86$ .	0.338
Question: $\overline{BC}$ is parallel to the segment through $A$ , and $AB=BC$ . What is the number of degrees represented by $x$ ? [asy] draw((0,0)–(10,0)); draw((0,3)–(10,3)); draw((2,3)–(8,0)); draw((2,3)–(4,0)); label(" $A$ ",(2,3),N); label(" $B$ ",(4,0),S); label(" $C$ ",(8,0),S); label(" $124^{\circ}$ ",(2,3),SW); label(" $x^{\circ}$ ",(4.5,3),S); [/asy]
1	$124^{\circ}$ ✗	We start by analyzing the given information. We have two parallel lines, one at the top (where point A is located) and one at the bottom (where points B and C are located). The angle at point A is given as $124^{\circ}$ . Since $\overline{BC}$ is parallel to the line through $A$ , the angle at $A$ and the angle $x$ at point $B$ are corresponding angles. According to the properties of parallel lines, corresponding angles are equal. Therefore, we can set up the equation: $x=124^{\circ}$ . Thus, the value of $x$ is directly equal to the angle at point $A$ .	0.398
2	$28^{\circ}$ ✓	In the given problem, we have two parallel lines represented by the segments through points $A$ and $B$ and $C$ . Since $\overline{BC}$ is parallel to the segment through $A$ , we can use the properties of alternate interior angles. The angle at point $A$ is given as $124^{\circ}$ . Since $AB$ is equal to $BC$ , triangle $ABC$ is isosceles with $AB=BC$ . Therefore, the angles at points $B$ and $C$ must be equal. Let the angle at point $B$ be $x$ . The angle at point $C$ will also be $x$ . The sum of the angles in triangle $ABC$ must equal $180^{\circ}$ . Thus, we can set up the equation: $124^{\circ}+x+x=180^{\circ}$ This simplifies to: $124^{\circ}+2x=180^{\circ}$ Subtracting $124^{\circ}$ from both sides gives: $2x=180^{\circ}-124^{\circ}=56^{\circ}$ Dividing both sides by $2$ yields: $x=\frac{56^{\circ}}{2}=28^{\circ}$ . Thus, the value of $x$ is $28^{\circ}$ .	0.798
3	$56^{\circ}$ ✗	In the given problem, we have two parallel lines, and we know that $\overline{BC}$ is parallel to the segment through $A$ . The angle at point $A$ is given as $124^{\circ}$ . Since $AB$ is equal to $BC$ , triangle $ABC$ is isosceles with $AB=BC$ . The angle $x$ is the angle at point $B$ which is opposite to the angle at point $A$ . By the properties of parallel lines and the angles formed, we can use the fact that the angle at $A$ and the angle at $B$ are supplementary because they are on the same side of the transversal line $AC$ . Thus, we can set up the equation: $x+124^{\circ}=180^{\circ}$ Now, we can solve for $x$ : $x=180^{\circ}-124^{\circ}=56^{\circ}$ Therefore, the value of $x$ is $56^{\circ}$ .	0.177

Appendix

Table 7: Question and Answer Evaluation Results, the flawed part of CoT is highlighted in red . The correct and wrong answers are shown in ✓ and ✗ , respectively.

Figure 6: Analysis of different levels of MATH-500.

	5 Examples		10 Examples		15 Examples
Method	DeepSeek-V3	Qwen3-14B	DeepSeek-V3	Qwen3-14B	DeepSeek-V3	Qwen3-14B
Standard CoT	92.35	90.16	92.35	91.53	92.62	92.62
Denoise	92.62	85.79	94.54	89.34	92.08	90.16
Polish	94.81	92.08	92.62	89.07	91.80	90.98
Reflect	93.44	88.25	94.54	87.98	90.44	90.71
Consistency	90.98	91.26	93.17	90.98	93.00	89.89

Appendix

Table 8: Performance comparison of DeepSeek-V3 and Qwen3-14B on MATH-500 across different numbers of prompting examples.

Method	Prompt Template
zero shot	Please answer the following question without any explanation.

	Please format your response as follows:

	Answer: Final numeric answer

Appendix

Table 9: Baseline prompts

Module	Prompt Template
$\mathcal{C}_{\mathbf{S}}$	Question: {question}
	CoT: {cot}
	Answer: {ground_truth}

	Task: Determine whether the provided Chain of Thought (CoT) logically deduces the correct answer for the given question. Respond with "True" if the reasoning leads to the answer, or "False" if it does not.
The counterfactual chain generation	You are given a math question and its corresponding reasoning chain.

Appendix

Table 10: Prompts used in our method.

Method	Stage	Prompt Content and Some examples of demo
Consistency	—	Base Prompt: Think step by step to solve the following question. First provide the reasoning process (CoT), then give the numeric final answer.
		Format: CoT: Step-by-step reasoning Answer: Final numeric answer
		Example: {Example}
Reflect	1	(Same as the Base Prompt above)
	2	Reflection Prompt: Based on the Chain-of-Thought (COT) reasoning and the answer you just provided, please reconsider the following question. Confirm the correctness of your prior answer, and then answer it again, also using the Chain-of-Thought (COT) format followed by the final answer.
Denoise	1	(Same as the Base Prompt above)

Appendix

Table 11: The configuration of baseline methods (for MATH-500). Consistency samples the base prompt

N

times to reach a consensus, while Reflect and Polish utilize a sequential two-stage process for self-correction and refinement, respectively. Denoise incorporates a token recovery task, where the masking procedure is a programmatic implementation-level operation rather than a textual prompt.

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CoT Faithfulness Faithfulness of Language Model Explanations Causal Reasoning in Language Models Causal Interventions in Language Models LLM Reasoning

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Chain-of-thought (CoT) reasoning is widely used to improve both the performance and interpretability of large language models (LLMs), yet the generated reasoning may not faithfully support the final answer. We study this problem from a causal perspective, where a faithful CoT process should follow the chain

Z\rightarrow X\rightarrow Y

, with

Z

X

, and

Y

denoting the instruction, reasoning chain, and final answer, respectively. In this process, the instruction should affect the answer only through the reasoning chain. However, conventional autoregressive LLMs condition answer generation on both the instruction and the CoT, which still allows a direct instruction-to-answer shortcut. To address this issue, we propose CASE, a framework that combines training-time causal alignment and inference-time structural enforcement. During training, CASE builds counterfactual-CoT, biased-instruction, and empty-instruction datasets, and applies selective-loss fine-tuning to strengthen CoT-to-answer dependence while suppressing instruction shortcuts. During inference, CASE masks direct attention from instruction tokens to answer tokens, preventing the model from bypassing the generated CoT. We provide an information-theoretic analysis showing how these components promote faithful chains. Experiments on three models and four benchmarks show that CASE achieves a 37% average per-setting relative improvement in overall CoT faithfulness over the strongest baselines, exhibits stronger cross-dataset faithfulness transfer, and maintains competitive average accuracy. Code is available at https://github.com/oddwang/CASE.

Abstract

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