CASE: Causal Alignment and Structural Enforcement for Improving Chain-of-Thought Faithfulness

# Causal Alignment and Structural Enforcement for Improving Chain-of-Thought Faithfulness
Source: [https://arxiv.org/html/2607.18820](https://arxiv.org/html/2607.18820)
###### Abstract

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 chainZ→X→YZ\\rightarrow X\\rightarrow Y, withZZ,XX, andYYdenoting 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 athttps://github\.com/oddwang/CASE\.

## Introduction

Chain\-of\-thought \(CoT\) prompting generates intermediate reasoning steps before the final answer, which not only improves the performance of large language models \(LLMs\) but also provides a natural explanation for their decisions\(Weiet al\.[2022](https://arxiv.org/html/2607.18820#bib.bib1)\)\. By expressing reasoning steps in natural language, CoT reveals how the model derives its answer from the input\. However, despite its strong empirical performance, the faithfulness of CoT as an explanation remains under debate\(Turpinet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib2); Lanhamet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib4)\)\. A model may produce a plausible reasoning chain, yet its answer is driven by shortcuts that bypass the generated reasoning\.

To understand this issue,Baoet al\.\([2025](https://arxiv.org/html/2607.18820#bib.bib29)\)modeled CoT reasoning using three variables: the problem instructionZZ, the reasoning chainXX, and the final answerYY\. Through intervention\-based causal analysis\(Hagmayeret al\.[2007](https://arxiv.org/html/2607.18820#bib.bib30)\), they characterized faithful reasoning as the causal chainZ→X→YZ\\rightarrow X\\rightarrow Y, where the instruction affects the answer only through the CoT\. In contrast, unfaithful reasoning may involve a direct instruction\-to\-answer shortcutZ→YZ\\rightarrow Y, or a weak dependence of the answer on the reasoning chainX→YX\\rightarrow Y\(Baoet al\.[2025](https://arxiv.org/html/2607.18820#bib.bib29)\)\. This causal view suggests that improving CoT faithfulness requires two complementary goals: strengtheningX→YX\\rightarrow Yand suppressing the direct shortcutZ→YZ\\rightarrow Y\.

![Refer to caption](https://arxiv.org/html/2607.18820v1/motivation_fig.jpg)Figure 1:High\-level intuition of CASE\. CASE improves CoT faithfulness by strengthening the reasoning\-to\-answer pathX→YX\\rightarrow Yand weakening the direct instruction\-to\-answer shortcutZ→YZ\\rightarrow Y\. Training\-time alignment and inference\-time enforcement jointly guide the decision process toward the faithful causal chainZ→X→YZ\\rightarrow X\\rightarrow Y\.Some recent studies explore post\-hoc candidate selection, generating multiple CoTs and selecting one according to predefined faithfulness criteria\(Wanget al\.[2025](https://arxiv.org/html/2607.18820#bib.bib9); Jieet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib8); Liet al\.[2025a](https://arxiv.org/html/2607.18820#bib.bib10),[b](https://arxiv.org/html/2607.18820#bib.bib12)\)\. Although these methods can be efficient, they operate at the output level and do not change the model’s answer\-generation mechanism\. To modify this mechanism directly, a more principled strategy is to fine\-tune the LLM itself, using faithful CoT supervision\(Tanneruet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib27)\), condition extraction\(Linet al\.[2025](https://arxiv.org/html/2607.18820#bib.bib28)\), or dual\-model architectures\(Paulet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib31)\)\. Although these methods can improve reasoning quality, they do not explicitly enforce the faithful mechanism in which the final answer is derived from the CoT rather than directly from the instruction\. As a result, the model may still rely on theZ→YZ\\rightarrow Yshortcuts during answer generation\.

Motivated by the causal view, we propose CASE \(Causal Alignment and Structural Enforcement\), a framework for improving CoT faithfulness\. During training, CASE constructs counterfactual\-CoT, biased\-instruction, and empty\-instruction datasets, and applies selective\-loss fine\-tuning to strengthenX→YX\\rightarrow Ywhile reducing reliance onZ→YZ\\rightarrow Yshortcuts\. During inference, CASE applies an attention\-mask intervention that prevents answer tokens from directly attending to instruction tokens, thereby removing the explicit instruction\-to\-answer attention path while preserving the mediated CoT\-side path\. We further provide an information\-theoretic analysis showing that the training losses promote three complementary criteria: increasingI​\(X;Y\)I\(X;Y\), increasingI​\(X;Y∣Z\)I\(X;Y\\mid Z\), and reducingI​\(Y;Z∣X\)I\(Y;Z\\mid X\)\. Together with inference\-time masking, these components guide the decision process toward the faithful causal chainZ→X→YZ\\rightarrow X\\rightarrow Y, as illustrated in Figure[1](https://arxiv.org/html/2607.18820#Sx1.F1)\.

The main contributions of this paper are as follows:

- ∙\\bulletWe propose CASE, a framework that improves CoT faithfulness through training\-time causal alignment and inference\-time structural enforcement\.
- ∙\\bulletWe provide a theoretical analysis linking CASE’s training objectives to complementary information\-theoretic faithfulness criteria and showing how inference\-time masking removes the explicit instruction\-to\-answer attention path\.
- ∙\\bulletExperiments on four benchmark datasets using two non\-reasoning LLMs and one reasoning LLM demonstrate that CASE substantially improves CoT faithfulness, achieving more than 37% average improvement over the strongest baselines while maintaining competitive accuracy and strong cross\-dataset generalization\.

## Related Work

This section reviews prior work on evaluating and improving CoT faithfulness\.

### Evaluation of CoT Faithfulness

Existing research primarily evaluates CoT faithfulness by perturbing the input instruction\(Atanasovaet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib3); Turpinet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib2)\), the generated CoT\(Lanhamet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib4)\), or the model itself\(Tuteket al\.[2025](https://arxiv.org/html/2607.18820#bib.bib5)\), and then measuring whether the final answer changes\. Among these methods, Filler Tokens replaces the CoT with “…” tokens and evaluates whether the answer changes\(Lanhamet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib4); Zaman and Srivastava[2025](https://arxiv.org/html/2607.18820#bib.bib32)\)\. Early Answering truncates the CoT sentence by sentence and computes the area over the curve \(AOC\) of the probability of keeping the same answer at different truncation points, testing whether the model can answer before observing the full reasoning chain\(Lanhamet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib4)\)\. A recent systematic study byZaman and Srivastava \([2025](https://arxiv.org/html/2607.18820#bib.bib32)\)shows that Filler Tokens and Early Answering are among the most reliable metrics for CoT faithfulness\. Therefore, we adopt them as our main perturbation\-based evaluation metrics\.

We further include the controlled indirect effect \(CIE\) and controlled direct effect \(CDE\) metrics used byPaulet al\.\([2024](https://arxiv.org/html/2607.18820#bib.bib31)\)as complementary causal evaluations\. CIE replaces the CoT with a counterfactual CoT to test whether the answer depends on the reasoning chain\. CDE keeps the CoT fixed while replacing the original question with a counterfactual question\. If the answer still changes, this suggests a directZ→YZ\\rightarrow Yshortcut and lower faithfulness\. Together, CIE and CDE assess whether the answer is controlled by the CoT rather than directly by the instruction, providing a causal complement to perturbation\-based metrics\.

### Methods for Improving CoT Faithfulness

Recent studies have proposed various methods to improve CoT faithfulness\. Some methods perform post\-hoc candidate selection, generating multiple CoTs and selecting one through scoring or filtering\(Wanget al\.[2025](https://arxiv.org/html/2607.18820#bib.bib9); Jieet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib8); Liet al\.[2025a](https://arxiv.org/html/2607.18820#bib.bib10),[b](https://arxiv.org/html/2607.18820#bib.bib12)\)\. Although these methods can improve the selected output, they intervene only after generation and leave the answer\-generation mechanism unchanged\.

A more direct strategy is to fine\-tune the LLM itself\.Tanneruet al\.\([2024](https://arxiv.org/html/2607.18820#bib.bib27)\)fine\-tuned LLMs with faithful CoTs, but found that the improvement was limited\.Linet al\.\([2025](https://arxiv.org/html/2607.18820#bib.bib28)\)proposed FoCus, which extracts problem conditions, guides the model to use them during reasoning, and fine\-tunes the model on the resulting structured reasoning data\. However, these methods mainly improve reasoning quality and do not explicitly align answer generation with the faithful chainZ→X→YZ\\rightarrow X\\rightarrow Y\. Thus, the instruction may still provide a shortcut to the answer\.

![Refer to caption](https://arxiv.org/html/2607.18820v1/CASE_pipeline.jpg)Figure 2:Overview of the proposed CASE framework\.Paulet al\.\([2024](https://arxiv.org/html/2607.18820#bib.bib31)\)proposed FRODO, a dual\-model framework for improving CoT faithfulness\. FRODO trains an inference module with direct preference optimization \(DPO\)\(Rafailovet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib33)\)and an implicit causal reward, together with a reasoning module trained by a causal preference objective\. This design encourages the final answer to depend more on the generated CoT\. However, because the answer\-generation module can still access the original question, FRODO does not strictly remove the direct instruction\-to\-answer path\. Moreover, because FRODO assigns reasoning generation and answer prediction to two separate modules, it mainly improves pipeline\-level consistency rather than making a single LLM learn the desired dependency from its own CoT to its answer\. This separation also adds system complexity and computational cost\.

In contrast, CASE directly intervenes in the answer\-generation process of a single LLM\. It constructs counterfactual\-CoT, biased\-instruction, and empty\-instruction datasets with selective\-loss fine\-tuning, and applies an inference\-time attention mask to strengthenX→YX\\rightarrow Yand suppress the directZ→YZ\\rightarrow Ypath\. We therefore compare CASE with fine\-tuning\-based baselines that, like CASE, generate one CoT per instance\. Post\-hoc selection methods require multiple generations, operate at an orthogonal output\-selection level, and can be combined with CASE or any fine\-tuned model\.

## Methodology

This section first defines the CoT faithfulness, then presents CASE, and finally analyzes how its components support faithful answer generation\.

### Problem Formulation

Let𝒵\\mathcal\{Z\},𝒳\\mathcal\{X\}, and𝒴\\mathcal\{Y\}denote the spaces of instructions, reasoning chains, and final answers\. We useZ∈𝒵Z\\in\\mathcal\{Z\},X∈𝒳X\\in\\mathcal\{X\}, andY∈𝒴Y\\in\\mathcal\{Y\}for the corresponding random variables, andzz,xx, andyyfor concrete instances\.

Given an instructionzz, an autoregressive LLM parameterized byθ\\thetafirst generates a CoTxxand then predicts an answeryy\. This process can be factorized as

pθ​\(x,y∣z\)=pθ​\(x∣z\)​pθ​\(y∣z,x\),p\_\{\\theta\}\(x,y\\mid z\)=p\_\{\\theta\}\(x\\mid z\)\\,p\_\{\\theta\}\(y\\mid z,x\),wherepθ​\(x∣z\)p\_\{\\theta\}\(x\\mid z\)denotes the reasoning\-generation process, andpθ​\(y∣z,x\)p\_\{\\theta\}\(y\\mid z,x\)denotes the answer\-generation process conditioned on both the instruction and the CoT\.

###### Definition 1\(CoT faithfulness\)\.

A CoT\-based decision process is faithful if the instruction influences the final answer only through the generated CoT\. Formally, the generation process factorizes as

pθ​\(x,y∣z\)=pθ​\(x∣z\)​pθ​\(y∣x\),p\_\{\\theta\}\(x,y\\mid z\)=p\_\{\\theta\}\(x\\mid z\)\\,p\_\{\\theta\}\(y\\mid x\),or equivalently, the answer mechanism satisfies

pθ​\(y∣z,x\)=pθ​\(y∣x\)\.p\_\{\\theta\}\(y\\mid z,x\)=p\_\{\\theta\}\(y\\mid x\)\.

It implies the conditional independence condition

It corresponds to the faithful causal chainZ→X→YZ\\rightarrow X\\rightarrow Y: the instruction induces the reasoning chain, and the final answer is derived from the reasoning chain\(Baoet al\.[2025](https://arxiv.org/html/2607.18820#bib.bib29)\)\. Importantly, this does not mean thatZZdoes not influenceYY\. Instead, any influence ofZZonYYshould be mediated throughXX\. Here,XXis the visible natural\-language CoT, whereas answer generation accesses CoT\-side statesHXH\_\{X\}formed underZZ\. Instruction information retained inHXH\_\{X\}remains CoT\-mediated; attention masking encourages this structural path, while visible\-CoT perturbations evaluate whether the textualXXsupportsYY\.

An ideal way to encourage faithfulness is to train the model under the factorizationpθ​\(x∣z\)​pθ​\(y∣x\)p\_\{\\theta\}\(x\\mid z\)\\,p\_\{\\theta\}\(y\\mid x\), so that answer generation depends only on the CoT\. However, this is difficult in natural\-language reasoning: the final answeryyis supervised, whereas valid CoTsxxare not unique\. Forcingxxalone to determineyyfrom the start may produce degenerate or low\-quality CoTs, such as traces that simply reveal the answer\. Therefore, practical LLMs usually adopt the more flexible factorizationpθ​\(x∣z\)​pθ​\(y∣z,x\)p\_\{\\theta\}\(x\\mid z\)\\,p\_\{\\theta\}\(y\\mid z,x\), where the instruction helps generate and interpret the CoT, but also permits an unfaithful shortcutZ→YZ\\rightarrow Y\. CASE starts from this practical mechanism and regularizes answer generation toward the CoT\-mediated mechanism in Definition 1, preservingZ→XZ\\rightarrow Xwhile strengtheningX→YX\\rightarrow Yand suppressing the direct shortcutZ→YZ\\rightarrow Y\.

### CASE for CoT Faithfulness

Motivated by the above formulation, we propose CASE, a framework that combines training\-time causal alignment and inference\-time structural enforcement\. As shown in Figure[2](https://arxiv.org/html/2607.18820#Sx2.F2), CASE constructs mixed training datasets and applies selective\-loss supervised fine\-tuning \(SFT\) to strengthen the CoT\-to\-answer mechanism\. It then applies attention masking to block the direct instruction\-to\-answer path during answer generation\. The two stages are complementary: training\-time alignment learns a reliableX→YX\\rightarrow Ymechanism but does not remove the direct instruction\-to\-answer path, while inference\-time masking removes this path but relies on the learnedX→YX\\rightarrow Ymechanism to preserve answer quality\.

#### Training\-Time Causal Alignment

Given a standard SFT dataset

𝒟A=\{\(zi,xi,yi\)\}i=1n,\\mathcal\{D\}\_\{A\}=\\\{\(z\_\{i\},x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\},\(1\)each sample contains an instructionziz\_\{i\}, a CoTxix\_\{i\}, and a final answeryiy\_\{i\}\. In practice, we generate CoTs and answers using a strong LLM, and retain only instances where the generated answer matches the gold answer\. To make answer generation rely more on the CoT than on the instruction shortcut, CASE augments𝒟A\\mathcal\{D\}\_\{A\}with three auxiliary datasets:

𝒟B\\displaystyle\\mathcal\{D\}\_\{B\}=\{\(zi,xi′,yi′\)\}i=1n,\\displaystyle=\\\{\(z\_\{i\},x\_\{i\}^\{\\prime\},y\_\{i\}^\{\\prime\}\)\\\}\_\{i=1\}^\{n\},\(2\)𝒟C\\displaystyle\\mathcal\{D\}\_\{C\}=\{\(zi′,xi,yi\)\}i=1n,\\displaystyle=\\\{\(z\_\{i\}^\{\\prime\},x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\},\(3\)𝒟D\\displaystyle\\mathcal\{D\}\_\{D\}=\{\(∅,xi,yi\)\}i=1n\.\\displaystyle=\\\{\(\\emptyset,x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}\.\(4\)Here,𝒟B\\mathcal\{D\}\_\{B\}is the counterfactual CoT dataset, whereziz\_\{i\}is identical to that in𝒟A\\mathcal\{D\}\_\{A\}, while the answer is replaced by a stronger\-LLM\-generated counterfactual answeryi′y\_\{i\}^\{\\prime\}satisfyingyi′≠yiy\_\{i\}^\{\\prime\}\\neq y\_\{i\}, andxi′x\_\{i\}^\{\\prime\}is the corresponding reasoning chain that derivesyi′y\_\{i\}^\{\\prime\}\. This encourages answer prediction to respond to CoT changes under the same instruction\.𝒟C\\mathcal\{D\}\_\{C\}is the biased\-instruction dataset, where the instruction is modified intozi′z\_\{i\}^\{\\prime\}by adding misleading contextual information or an incorrect answer cue, while the CoT and answer remain unchanged\. This discourages answer changes caused only by misleading instruction cues\.𝒟D\\mathcal\{D\}\_\{D\}is the empty\-instruction dataset, where the instruction content is removed while the CoT and answer are preserved, encouraging answer prediction from the CoT alone\. We further conduct an automatic quality audit of the LLM\-generated CoT\-answer pairs in𝒟A\\mathcal\{D\}\_\{A\}and𝒟B\\mathcal\{D\}\_\{B\}, and the results in Appendix[A\.3](https://arxiv.org/html/2607.18820#A1.SS3)show that the generated data has high quality and consistency\.

##### Selective\-loss SFT\.

CASE uses selective\-loss because the four datasets serve different purposes\. The standard dataset𝒟A\\mathcal\{D\}\_\{A\}provides both CoT and answer supervision, while the auxiliary datasets𝒟B\\mathcal\{D\}\_\{B\},𝒟C\\mathcal\{D\}\_\{C\}, and𝒟D\\mathcal\{D\}\_\{D\}impose answer\-level intervention constraints\. Therefore, CASE supervises both CoT and answer tokens on𝒟A\\mathcal\{D\}\_\{A\}, but only answer tokens on the auxiliary datasets\.

For𝒟A\\mathcal\{D\}\_\{A\}, CASE optimizes both CoT and answer tokens:

ℒmain=𝔼𝒟A​\[−log⁡pθ​\(xi∣zi\)−log⁡pθ​\(yi∣zi,xi\)\]\.\\mathcal\{L\}\_\{\\mathrm\{main\}\}=\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{A\}\}\\left\[\-\\log p\_\{\\theta\}\(x\_\{i\}\\mid z\_\{i\}\)\-\\log p\_\{\\theta\}\(y\_\{i\}\\mid z\_\{i\},x\_\{i\}\)\\right\]\.\(5\)
For𝒟B\\mathcal\{D\}\_\{B\},𝒟C\\mathcal\{D\}\_\{C\}, and𝒟D\\mathcal\{D\}\_\{D\}, CASE computes answer\-token losses:

ℒX​\-int\\displaystyle\\mathcal\{L\}\_\{X\\text\{\-int\}\}=𝔼𝒟B​\[−log⁡pθ​\(yi′∣zi,xi′\)\],\\displaystyle=\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{B\}\}\\left\[\-\\log p\_\{\\theta\}\(y\_\{i\}^\{\\prime\}\\mid z\_\{i\},x\_\{i\}^\{\\prime\}\)\\right\],\(6\)ℒZ​\-int\\displaystyle\\mathcal\{L\}\_\{Z\\text\{\-int\}\}=𝔼𝒟C​\[−log⁡pθ​\(yi∣zi′,xi\)\],\\displaystyle=\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{C\}\}\\left\[\-\\log p\_\{\\theta\}\(y\_\{i\}\\mid z\_\{i\}^\{\\prime\},x\_\{i\}\)\\right\],\(7\)ℒZ​\-drop\\displaystyle\\mathcal\{L\}\_\{Z\\text\{\-drop\}\}=𝔼𝒟D​\[−log⁡pθ​\(yi∣∅,xi\)\]\.\\displaystyle=\\mathbb\{E\}\_\{\\mathcal\{D\}\_\{D\}\}\\left\[\-\\log p\_\{\\theta\}\(y\_\{i\}\\mid\\emptyset,x\_\{i\}\)\\right\]\.\(8\)
The full training objective is

ℒtrain=ℒmain\+ℒX​\-int\+ℒZ​\-int\+ℒZ​\-drop\.\\mathcal\{L\}\_\{\\mathrm\{train\}\}=\\mathcal\{L\}\_\{\\mathrm\{main\}\}\+\\mathcal\{L\}\_\{X\\text\{\-int\}\}\+\\mathcal\{L\}\_\{Z\\text\{\-int\}\}\+\\mathcal\{L\}\_\{Z\\text\{\-drop\}\}\.\(9\)This objective can be viewed as standard CoT SFT plus three answer\-level intervention losses\. Masking CoT\-token loss on the auxiliary datasets prevents the model from merely imitating constructed reasoning traces or being forced to generate counterfactual CoTs, while still teaching how the given CoT should determine the answer\.

#### Inference\-Time Structural Enforcement

Training\-time causal alignment encourages reliance on the CoT, but it does not structurally prevent answer tokens from accessing the instruction during inference\. Therefore, CASE further applies attention masking during answer generation\.

Given an input sequence consisting of instruction tokensZZ, CoT tokensXX, and answer tokensYY, standard causal self\-attention allows answer tokens to attend to bothZZandXX\. CASE modifies the attention mask only when generating answer tokens\. For every answer token positioni∈pos​\(Y\)i\\in\\mathrm\{pos\}\(Y\), instruction token positionj∈pos​\(Z\)j\\in\\mathrm\{pos\}\(Z\), layerℓ\\ell, and headhh, we setAi​jℓ,h=−∞A\_\{ij\}^\{\\ell,h\}=\-\\infty\. After softmax, these entries receive zero attention weight\. Thus, answer tokens cannot directly aggregate information from instruction tokens, but can still attend to the CoT and previously generated answer tokens\.

At the computation\-graph level, this masking removes the explicit directZ→YZ\\to Yattention path in the computation graph while preserving the mediated CoT\-side path\. Therefore, the CoT is still generated normally from the instruction, while answer generation proceeds through the CoT\-side states rather than directly through instruction\-token states\. This complements training\-time alignment: training teaches the model to use the CoT for answer prediction, and inference\-time masking prevents it from bypassing this learned mechanism by directly accessing instruction\-token states\.

A simpler alternative is to remove the instruction before answer generation and let the model answer based solely on the generated CoT\. However, this replaces the original instruction\-conditioned CoT\-side states with CoT\-only states\. Attention masking instead preserves these states while blocking direct instruction\-to\-answer access, and Appendix[A\.5](https://arxiv.org/html/2607.18820#A1.SS5)shows that it achieves better accuracy and faithfulness\.

### Theoretical Analysis

This subsection analyzes how CASE promotes the faithful causal chain in Definition 1 from an information\-level perspective\. All proofs are provided in Appendix[A\.1](https://arxiv.org/html/2607.18820#A1.SS1)\. Here,IIdenotes mutual information, andHHdenotes entropy\. To strengthen the CoT\-mediated path and suppress the residual instruction\-to\-answer shortcut, we consider three complementary information\-level criteria:

max⁡I​\(X;Y\),\\displaystyle\\max I\(X;Y\),max⁡I​\(X;Y∣Z\),\\displaystyle\\max I\(X;Y\\mid Z\),min⁡I​\(Y;Z∣X\)\.\\displaystyle\\min I\(Y;Z\\mid X\)\.The first criterion requires the CoT to be predictive of the answer\. The second requires the answer to remain dependent on the CoT even under the same instruction\. These two criteria strengthenX→YX\\rightarrow Y\. The third suppresses the residual direct shortcut from instruction to answer after conditioning on the CoT\. Since these mutual\-information quantities are intractable for high\-dimensional natural\-language sequences, CASE uses practical surrogate losses and an inference\-time structural mask\.

###### Proposition 1\(CoT\-only likelihood lower boundsI​\(X;Y\)I\(X;Y\)\)\.

Under the data distribution over\(X,Y\)\(X,Y\), letrϕ​\(y∣x\)r\_\{\\phi\}\(y\\mid x\)be any variational CoT\-conditioned answer predictor\. Then

I​\(X;Y\)≥H​\(Y\)\+𝔼x,y​\[log⁡rϕ​\(y∣x\)\]\.I\(X;Y\)\\geq H\(Y\)\+\\mathbb\{E\}\_\{x,y\}\[\\log r\_\{\\phi\}\(y\\mid x\)\]\.

Aspθ​\(y∣∅,x\)p\_\{\\theta\}\(y\\mid\\emptyset,x\)instantiatesrϕ​\(y∣x\)r\_\{\\phi\}\(y\\mid x\), minimizingℒZ​\-drop\\mathcal\{L\}\_\{Z\\text\{\-drop\}\}in Eq\. \([8](https://arxiv.org/html/2607.18820#Sx3.E8)\) is equivalent to maximizing the lower bound ofI​\(X;Y\)I\(X;Y\)\.

###### Proposition 2\(Counterfactual\-CoT augmented likelihood lower boundsIP~​\(X;Y∣Z\)I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)\)\.

LetP~\\widetilde\{P\}denote the augmented data distribution induced by the original and counterfactual\-CoT triplets,

D~X​\-​int=\{\(zi,xi,yi\),\(zi,xi′,yi′\)\}i=1n,yi′≠yi\.\\widetilde\{D\}\_\{X\\text\{\-\}\\mathrm\{int\}\}=\\\{\(z\_\{i\},x\_\{i\},y\_\{i\}\),\(z\_\{i\},x^\{\\prime\}\_\{i\},y^\{\\prime\}\_\{i\}\)\\\}\_\{i=1\}^\{n\},\\quad y^\{\\prime\}\_\{i\}\\neq y\_\{i\}\.For any variational answer predictorrϕ​\(y∣z,x\)r\_\{\\phi\}\(y\\mid z,x\), we have

IP~​\(X;Y∣Z\)≥HP~​\(Y∣Z\)\+𝔼P~​\[log⁡rϕ​\(Y∣Z,X\)\]\.I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)\\geq H\_\{\\widetilde\{P\}\}\(Y\\mid Z\)\+\\mathbb\{E\}\_\{\\widetilde\{P\}\}\\left\[\\log r\_\{\\phi\}\(Y\\mid Z,X\)\\right\]\.

SinceHP~​\(Y∣Z\)H\_\{\\widetilde\{P\}\}\(Y\\mid Z\)is constant under the augmented distribution, minimizing the answer\-token negative log\-likelihood is equivalent to maximizing the lower bound ofIP~​\(X;Y∣Z\)I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)\. The augmented distribution is crucial because it associates the same instructionziz\_\{i\}with multiple CoT\-answer pairs, namely\(zi,xi,yi\)\(z\_\{i\},x\_\{i\},y\_\{i\}\)and\(zi,xi′,yi′\)\(z\_\{i\},x\_\{i\}^\{\\prime\},y\_\{i\}^\{\\prime\}\)\. Consequently, a predictor that ignoresXXcannot correctly model both conditional answer distributions under the same instruction\. In CASE, the answer\-token part ofℒmain\\mathcal\{L\}\_\{\\mathrm\{main\}\}in Eq\. \([5](https://arxiv.org/html/2607.18820#Sx3.E5)\) trains with the original pairs, whileℒX​\-int\\mathcal\{L\}\_\{X\\text\{\-int\}\}in Eq\. \([6](https://arxiv.org/html/2607.18820#Sx3.E6)\) trains with the counterfactual\-CoT pairs\. Together, they encourage the model to distinguish different reasoning chains under the same instruction, thereby increasing the conditional dependence ofYYonXXgivenZZ\.

###### Proposition 3\(Pairwise instruction\-discrepancy upper bound forI​\(Y;Z∣X\)I\(Y;Z\\mid X\)\)\.

LetpXp\_\{X\}denote the marginal distribution of reasoning chains,pZ\(⋅∣x\)p\_\{Z\}\(\\cdot\\mid x\)the conditional distribution of instructions givenxx, andpY\(⋅∣x,z\)p\_\{Y\}\(\\cdot\\mid x,z\)the conditional answer distribution\. Then

I​\(Y;Z∣X\)\\displaystyle I\(Y;Z\\mid X\)≤𝔼x∼pX,z∼pZ\(⋅∣x\),z′∼pZ\(⋅∣x\)\[KL\(pY\(⋅∣x,z\)\\displaystyle\\leq\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}x\\sim p\_\{X\},\\\\ z\\sim p\_\{Z\}\(\\cdot\\mid x\),\\\\ z^\{\\prime\}\\sim p\_\{Z\}\(\\cdot\\mid x\)\\end\{subarray\}\}\\Big\[\\mathrm\{KL\}\\Big\(p\_\{Y\}\(\\cdot\\mid x,z\)∥pY\(⋅∣x,z′\)\)\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\\|\\,p\_\{Y\}\(\\cdot\\mid x,z^\{\\prime\}\)\\Big\)\\Big\]\.\(10\)

Consequently, reducing the discrepancy between answer distributions induced by different instructions under the same reasoning chain reduces an upper bound on the residual dependence ofYYonZZconditioned onXX\. However, the true conditional answer distributions are unknown, and the pairwise KL divergence cannot be optimized directly\. CASE uses likelihood maximization as a tractable pointwise surrogate\. Specifically, the answer\-token term inℒmain\\mathcal\{L\}\_\{\\mathrm\{main\}\}in Eq\. \([5](https://arxiv.org/html/2607.18820#Sx3.E5)\) trainspθ​\(yi∣zi,xi\)p\_\{\\theta\}\(y\_\{i\}\\mid z\_\{i\},x\_\{i\}\)to assign high likelihood toyiy\_\{i\}under the original instructionziz\_\{i\}, whileℒZ​\-int\\mathcal\{L\}\_\{Z\\text\{\-int\}\}in Eq\. \([7](https://arxiv.org/html/2607.18820#Sx3.E7)\) trainspθ​\(yi∣zi′,xi\)p\_\{\\theta\}\(y\_\{i\}\\mid z\_\{i\}^\{\\prime\},x\_\{i\}\)to assign high likelihood to the same answeryiy\_\{i\}under an intervened instructionzi′z\_\{i\}^\{\\prime\}with the reasoning chain fixed\. Together, they encourage the same reasoning chain to support the same answer under both the original and intervened instructions, serving as a likelihood\-based surrogate for reducing the pairwise instruction discrepancy in Eq\. \([10](https://arxiv.org/html/2607.18820#Sx3.E10)\)\.

###### Proposition 4\(Attention masking removes the explicit instruction\-to\-answer attention path\)\.

LetpθAMp\_\{\\theta\}^\{\\mathrm\{AM\}\}denote the model distribution under inference\-time attention masking, and letHXH\_\{X\}denote the CoT\-side computational states available to answer generation, i\.e\., the hidden or key\-value states at CoT token positions formed during autoregressive CoT generation\. If, for every answer token positioni∈pos​\(Y\)i\\in\\mathrm\{pos\}\(Y\), instruction token positionj∈pos​\(Z\)j\\in\\mathrm\{pos\}\(Z\), layerℓ\\ell, and headhh, the masked pre\-softmax attention logit is set toAi​jℓ,h=−∞A\_\{ij\}^\{\\ell,h\}=\-\\infty, then answer tokens cannot directly attend to instruction\-token states\. Consequently, once the CoT\-side statehXh\_\{X\}is fixed, answer generation has no direct computation\-graph dependence on the instruction\-token states, in the explicit computation\-graph sense,

pθAM​\(y∣z,hX\)=pθAM​\(y∣hX\),p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\\mid z,h\_\{X\}\)=p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\\mid h\_\{X\}\),wherehXh\_\{X\}is a realized value ofHXH\_\{X\}\.

This is a structural computation\-graph statement: attention masking removes the explicitZ→YZ\\rightarrow Yattention path, but does not remove instruction information already encoded in the CoT\-side mediatorHXH\_\{X\}\. Thus, Proposition 4 supports the inference\-time stage: answer tokens cannot directly attend to instruction\-token states and must use the CoT\-mediated path\.

Together, these propositions show how CASE combines training\-time alignment and inference\-time masking\. The training losses provide causal alignment: they increase CoT\-only answer predictiveness, strengthen answer dependence on CoT changes, and reduce answer variation under instruction changes\. The attention mask then provides a structural constraint that prevents answer tokens from directly attending to instruction tokens\. Thus, CASE learns a reliable CoT\-to\-answer mechanism and encourages inference to use this mechanism, pushing the model toward a CoT\-mediated computation consistent with the faithful mechanism in Definition 1\.

## Experimental Studies

In this section, we conduct empirical evaluations to answer the following research questions \(RQs\):

- ∙\\bulletRQ1:How effectively does CASE improve CoT faithfulness compared with state\-of\-the\-art \(SOTA\) baselines?
- ∙\\bulletRQ2:How well does CASE maintain CoT faithfulness improvements under cross\-dataset transfer evaluation?
- ∙\\bulletRQ3:How do training\-time causal alignment and inference\-time structural enforcement individually contribute to improving CoT faithfulness?

### Experimental Setup

This subsection summarizes the models, datasets, baselines, metrics, and implementation details used in our experiments\.

#### Models

The models employed include two non\-reasoning models, Llama\-3\.1\-8B\(Meta AI[2024](https://arxiv.org/html/2607.18820#bib.bib17)\)and Qwen3\-8B\(Yanget al\.[2025](https://arxiv.org/html/2607.18820#bib.bib34)\), as well as one reasoning model, DeepSeek\-R1\-Distill\-Qwen\-7B\(Guoet al\.[2025](https://arxiv.org/html/2607.18820#bib.bib35)\)\.

#### Datasets

We evaluate on four benchmarks: ARC\-Easy, ARC\-Challenge\(Clarket al\.[2018](https://arxiv.org/html/2607.18820#bib.bib14)\), StrategyQA\(Gevaet al\.[2021](https://arxiv.org/html/2607.18820#bib.bib15)\), and LogiQA\(Liuet al\.[2020](https://arxiv.org/html/2607.18820#bib.bib36)\)\. All methods are fine\-tuned on the training set and evaluated on 500 randomly sampled test questions\. For cross\-dataset transfer, we fine\-tune on ARC\-Challenge and evaluate on OpenBookQA\(Mihaylovet al\.[2018](https://arxiv.org/html/2607.18820#bib.bib37)\)to test transfer within science question answering \(QA\), and fine\-tune on StrategyQA and evaluate on ARC\-Easy to test a larger domain and format shift\.

#### Baselines

We compared CASE with an untuned Base LLM and three fine\-tuning\-based methods for improving CoT faithfulness:D​UcDU^\{c\}\(Tanneruet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib27)\), FRODO\(Paulet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib31)\), and FoCus\(Linet al\.[2025](https://arxiv.org/html/2607.18820#bib.bib28)\)\.

#### Evaluation Metrics

Besides the accuracy \(Acc\) metric, we employed four CoT faithfulness metrics: Filler Tokens \(FT\), Early Answering \(EA\)\(Lanhamet al\.[2023](https://arxiv.org/html/2607.18820#bib.bib4)\), CIE, and CDE\(Paulet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib31)\)\. All metrics range from 0 to 1\. Higher FT, EA, and CIE indicate higher faithfulness, while lower CDE indicates higher faithfulness\. Detailed definitions and computation procedures are provided in Appendix[A\.4](https://arxiv.org/html/2607.18820#A1.SS4)\.

Table 1:Main comparison across three models and four datasets\. Best and second\-best results are highlighted in bold and underlined, respectively\. CASE is shaded\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.Since these metrics capture CoT faithfulness from different perspectives, we adopt the geometric mean \(G\-mean\)\(Derringer[1994](https://arxiv.org/html/2607.18820#bib.bib38)\), which is commonly used to aggregate multiple objectives, to obtain an overall faithfulness score\. After converting all metrics into the higher\-is\-better form, we define

G\-meanF​a​i​t​h=F​T×E​A×C​I​E×\(1−C​D​E\)4\.\\text\{G\-mean\}\_\{Faith\}=\\sqrt\[4\]\{FT\\times EA\\times CIE\\times\(1\-CDE\)\}\.

#### Implementation Details

For fair comparisons, all methods use the same LoRA configuration, few\-shot prompts, and decoding settings\. CoT demonstrations and counterfactual CoTs for CASE and the baselines are generated by DeepSeek\-V3\.2\(Liuet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib18)\)\. Detailed checkpoints, hyperparameters, and data construction are provided in Appendix[A\.2](https://arxiv.org/html/2607.18820#A1.SS2), while all prompt templates are given in Appendix[A\.11](https://arxiv.org/html/2607.18820#A1.SS11)\.

For CASE, biased instructions are constructed by adding an incorrect\-answer cue to the question\. For multiple\-choice tasks, empty\-instruction training and inference\-time masking remove only the question content while preserving answer options, allowing the model to map the reasoning process to the option space\. For true/false tasks, we mask most question tokens while retaining the target proposition needed for the True/False decision\. Further details, including the biased\-instruction construction rule and mask settings, are provided in Appendix[A\.2](https://arxiv.org/html/2607.18820#A1.SS2)\.

### Experimental Results

This section answers the three RQs through experiments\.

#### Main Results \(RQ1\)

Table[1](https://arxiv.org/html/2607.18820#Sx4.T1)presents the main results across three models and four datasets\. We compare CASE with the Base LLM and three fine\-tuning baselines using both task accuracy and CoT faithfulness metrics\. Overall, CASE consistently improves CoT faithfulness across models and task types while maintaining competitive accuracy\.

For overall faithfulness, CASE achieves the bestG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}in all 12 model–dataset settings\. Compared with the strongest baseline in each setting, CASE achieves an average relativeG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}improvement of 37% across the 12 model–dataset settings\. Under a stricter single\-LLM comparison that excludes FRODO, whose gains rely on a decoupled multi\-module pipeline, the relative improvement further increases to 65%\. These results show that CASE provides stronger and more stable CoT faithfulness\. Meanwhile, CASE achieves an average accuracy of 74\.88%, comparable to the 74\.80% of the per\-setting strongest baselines, showing that its faithfulness gains do not reduce average task accuracy\.

The faithfulness gains are also reflected in individual metrics\. Across all faithfulness metrics and model–dataset settings, CASE ranks first in 67% of the cases and second in 21%\. The improvements are especially clear on Filler Tokens and Early Answering, suggesting stronger dependence on the generated CoT\. CASE also reduces CDE across most settings, indicating weaker dependence on direct instruction\-to\-answer\. Appendix[A\.7](https://arxiv.org/html/2607.18820#A1.SS7)further shows that CASE has inference runtime comparable to other single\-model methods and is about3\.0×3\.0\\timesfaster than the dual\-model FRODO pipeline\.

Appendix[A\.10](https://arxiv.org/html/2607.18820#A1.SS10)provides a qualitative analysis of two examples to supplement the quantitative results, showing that baselines\(Tanneruet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib27); Paulet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib31); Linet al\.[2025](https://arxiv.org/html/2607.18820#bib.bib28)\)may still rely on directZ→YZ\\rightarrow Yshortcuts or produce answers inconsistent with their CoTs, whereas CASE more faithfully follows theZ→X→YZ\\rightarrow X\\rightarrow Ychain\.

#### Cross\-Dataset Generalization \(RQ2\)

Table[2](https://arxiv.org/html/2607.18820#Sx4.T2)reports cross\-dataset generalization results on Llama\-3\.1\-8B under two transfer settings\. ARC\-Challenge→\\rightarrowOpenBookQA tests transfer within science\-oriented multiple\-choice QA, while StrategyQA→\\rightarrowARC\-Easy tests a larger shift from Boolean commonsense QA to multiple\-choice science QA\. Thus, the two settings evaluate transfer under both related and larger dataset shifts\. Full results on the other two models are provided in Appendix[A\.8](https://arxiv.org/html/2607.18820#A1.SS8)\.

Table 2:Cross\-dataset generalization on Llama\-3\.1\-8B under two transfer settings\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.Existing baselines show limited faithfulness transfer\. Although they improve individual metrics in some settings, their overall cross\-dataset performance remains unstable\. Across the six transfer model–dataset settings, the averageG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}scores ofD​UcDU^\{c\}, FRODO, and FoCus are 0\.310, 0\.365, and 0\.328, respectively, none exceeding the Base LLM average of 0\.366\. Thus, their faithfulness improvements do not transfer reliably across datasets\.

In contrast, CASE achieves the bestG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}in all six settings, with an average score of 0\.650\. Averaging the per\-setting relative improvements, CASE outperforms the Base LLM by 102% and the strongest competing baseline in each setting by 61%\. It also achieves the best EA and CDE in all settings and the best or second\-best FT and CIE in each setting\. These results indicate that CASE yields more transferable faithfulness improvements rather than gains limited to the source fine\-tuning dataset\.

Regarding task performance, most methods achieve accuracy close to the Base LLM, and no method consistently performs best\. CASE likewise maintains competitive accuracy, indicating that its cross\-dataset faithfulness gains do not require a substantial loss in task performance\.

#### Ablation Study \(RQ3\)

Table[3](https://arxiv.org/html/2607.18820#Sx4.T3)presents the ablation results of CASE on Llama\-3\.1\-8B across two ARC benchmarks\. We compare full CASE with the Base LLM and two single\-stage variants: training\-time causal alignment \(TCA\) only and inference\-time structural enforcement \(ISE\) only\. Due to space limits, results on the other two datasets and the other two models are provided in Appendix[A\.9](https://arxiv.org/html/2607.18820#A1.SS9)\.

Table 3:Ablation study of CASE on Llama\-3\.1\-8B across two ARC benchmarks\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.The results show that the two stages contribute in complementary ways\. TCA mainly improves causal faithfulness metrics, consistently increasing CIE and reducing CDE across all ablations, suggesting weaker direct instruction\-to\-answer dependence\. However, its gains on FT and EA are limited\. By contrast, ISE substantially improves FT and EA, indicating that attention masking makes the final answer rely more on the generated CoT, though it may slightly affect accuracy\.

Combining both stages, CASE achieves the bestG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}across all ablation settings\. These results confirm that TCA makes the CoT\-to\-answer mechanism more reliable, while ISE prevents the model from bypassing this mechanism during inference\. Appendix[A\.9](https://arxiv.org/html/2607.18820#A1.SS9)further shows that this trend holds across additional datasets and models\.

## Conclusion

This paper studies CoT faithfulness from a causal perspective\. We argue that standard SFT\(Tanneruet al\.[2024](https://arxiv.org/html/2607.18820#bib.bib27)\)does not explicitly encourage the faithful CoT\-to\-answer mechanism and can still allow instruction\-to\-answer shortcuts\. To address this issue, we propose CASE, a framework that combines training\-time causal alignment with inference\-time structural enforcement\. By constructing counterfactual\-CoT, biased\-instruction, and empty\-instruction datasets in fine\-tuning and blocking direct instruction\-to\-answer attention during inference, CASE pushes answer generation toward the faithful chainZ→X→YZ\\rightarrow X\\rightarrow Y\. Experiments show that CASE consistently improves CoT faithfulness, achieves better cross\-dataset generalization, and preserves task accuracy\.

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## Appendix AAppendix

### A\.1Proofs for Theoretical Analysis

We provide the proofs for the propositions in the main text\. Throughout this appendix, all expectations are taken under the corresponding data\- or model\-induced distributions, and all distributions are assumed to have compatible supports so that the KL divergences are finite\. For autoregressive answer generation, conditional probabilities such aspθ​\(y∣z,x\)p\_\{\\theta\}\(y\\mid z,x\)denote sequence\-level probabilities, with dependence on previous answer tokens omitted for notation\. In Proposition 4,HXH\_\{X\}denotes the CoT\-side mediator available to answer generation, including generated CoT tokens and contextual hidden states used by answer\-generation attention across layers\.

###### Proposition 1\(CoT\-only likelihood lower boundsI​\(X;Y\)I\(X;Y\)\)\.

Under the data distribution over\(X,Y\)\(X,Y\), letrϕ​\(y∣x\)r\_\{\\phi\}\(y\\mid x\)be any variational CoT\-conditioned answer predictor\. Then

I​\(X;Y\)≥H​\(Y\)\+𝔼x,y​\[log⁡rϕ​\(y∣x\)\]\.I\(X;Y\)\\geq H\(Y\)\+\\mathbb\{E\}\_\{x,y\}\[\\log r\_\{\\phi\}\(y\\mid x\)\]\.\(A\.1\)

###### Proof\.

By definition,I​\(X;Y\)=H​\(Y\)−H​\(Y∣X\)I\(X;Y\)=H\(Y\)\-H\(Y\\mid X\)\. For any variational predictorrϕ​\(y∣x\)r\_\{\\phi\}\(y\\mid x\), the conditional entropy satisfies

H​\(Y∣X\)=𝔼x​\[−∑yp​\(y∣x\)​log⁡p​\(y∣x\)\]\.H\(Y\\mid X\)=\\mathbb\{E\}\_\{x\}\\left\[\-\\sum\_\{y\}p\(y\\mid x\)\\log p\(y\\mid x\)\\right\]\.Using the non\-negativity of KL divergence,

𝔼x\[KL\(p\(⋅∣x\)∥rϕ\(⋅∣x\)\)\]≥0,\\mathbb\{E\}\_\{x\}\\left\[\\mathrm\{KL\}\\left\(p\(\\cdot\\mid x\)\\,\\\|\\,r\_\{\\phi\}\(\\cdot\\mid x\)\\right\)\\right\]\\geq 0,we have

𝔼x​\[−∑yp​\(y∣x\)​log⁡p​\(y∣x\)\]\\displaystyle\\mathbb\{E\}\_\{x\}\\left\[\-\\sum\_\{y\}p\(y\\mid x\)\\log p\(y\\mid x\)\\right\]≤𝔼x​\[−∑yp​\(y∣x\)​log⁡rϕ​\(y∣x\)\]\.\\displaystyle\\leq\\mathbb\{E\}\_\{x\}\\left\[\-\\sum\_\{y\}p\(y\\mid x\)\\log r\_\{\\phi\}\(y\\mid x\)\\right\]\.Therefore,

H​\(Y∣X\)≤𝔼x,y​\[−log⁡rϕ​\(y∣x\)\],H\(Y\\mid X\)\\leq\\mathbb\{E\}\_\{x,y\}\\left\[\-\\log r\_\{\\phi\}\(y\\mid x\)\\right\],which implies

−H​\(Y∣X\)≥𝔼x,y​\[log⁡rϕ​\(y∣x\)\]\.\-H\(Y\\mid X\)\\geq\\mathbb\{E\}\_\{x,y\}\\left\[\\log r\_\{\\phi\}\(y\\mid x\)\\right\]\.Substituting this inequality intoI​\(X;Y\)=H​\(Y\)−H​\(Y∣X\)I\(X;Y\)=H\(Y\)\-H\(Y\\mid X\)gives Eq\. \([A\.1](https://arxiv.org/html/2607.18820#A1.E1)\)\. ∎

###### Proposition 2\(Counterfactual\-CoT likelihood lower boundsIP~​\(X;Y∣Z\)I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)\)\.

LetP~\\widetilde\{P\}denote the augmented CoT\-intervention distribution induced by the original and counterfactual\-CoT pairs,

D~X​\-​int=\{\(zi,xi,yi\),\(zi,xi′,yi′\)\}i=1n,yi′≠yi\.\\widetilde\{D\}\_\{X\\text\{\-\}\\mathrm\{int\}\}=\\\{\(z\_\{i\},x\_\{i\},y\_\{i\}\),\(z\_\{i\},x^\{\\prime\}\_\{i\},y^\{\\prime\}\_\{i\}\)\\\}\_\{i=1\}^\{n\},\\quad y^\{\\prime\}\_\{i\}\\neq y\_\{i\}\.For any variational answer predictorrϕ​\(y∣z,x\)r\_\{\\phi\}\(y\\mid z,x\), we have

IP~​\(X;Y∣Z\)≥HP~​\(Y∣Z\)\+𝔼P~​\[log⁡rϕ​\(Y∣Z,X\)\]\.I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)\\geq H\_\{\\widetilde\{P\}\}\(Y\\mid Z\)\+\\mathbb\{E\}\_\{\\widetilde\{P\}\}\\left\[\\log r\_\{\\phi\}\(Y\\mid Z,X\)\\right\]\.\(A\.2\)

###### Proof\.

Under the augmented distributionP~\\widetilde\{P\},

IP~​\(X;Y∣Z\)=HP~​\(Y∣Z\)−HP~​\(Y∣Z,X\)\.I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)=H\_\{\\widetilde\{P\}\}\(Y\\mid Z\)\-H\_\{\\widetilde\{P\}\}\(Y\\mid Z,X\)\.For any variational predictorrϕ​\(y∣z,x\)r\_\{\\phi\}\(y\\mid z,x\), the conditional cross\-entropy decomposes as

𝔼P~\\displaystyle\\mathbb\{E\}\_\{\\widetilde\{P\}\}\[−log⁡rϕ​\(Y∣Z,X\)\]=HP~​\(Y∣Z,X\)\\displaystyle\\left\[\-\\log r\_\{\\phi\}\(Y\\mid Z,X\)\\right\]=H\_\{\\widetilde\{P\}\}\(Y\\mid Z,X\)\+𝔼P~\(z,x\)\[KL\(P~\(⋅∣z,x\)∥rϕ\(⋅∣z,x\)\)\]\.\\displaystyle\+\\mathbb\{E\}\_\{\\widetilde\{P\}\_\{\(z,x\)\}\}\\left\[\\mathrm\{KL\}\\left\(\\widetilde\{P\}\(\\cdot\\mid z,x\)\\,\\\|\\,r\_\{\\phi\}\(\\cdot\\mid z,x\)\\,\\right\)\\right\]\.Since the KL term is non\-negative,

HP~​\(Y∣Z,X\)≤𝔼P~​\[−log⁡rϕ​\(Y∣Z,X\)\]\.H\_\{\\widetilde\{P\}\}\(Y\\mid Z,X\)\\leq\\mathbb\{E\}\_\{\\widetilde\{P\}\}\\left\[\-\\log r\_\{\\phi\}\(Y\\mid Z,X\)\\right\]\.Therefore,

−HP~​\(Y∣Z,X\)≥𝔼P~​\[log⁡rϕ​\(Y∣Z,X\)\]\.\-H\_\{\\widetilde\{P\}\}\(Y\\mid Z,X\)\\geq\\mathbb\{E\}\_\{\\widetilde\{P\}\}\\left\[\\log r\_\{\\phi\}\(Y\\mid Z,X\)\\right\]\.Substituting this inequality intoIP~​\(X;Y∣Z\)=HP~​\(Y∣Z\)−HP~​\(Y∣Z,X\)I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)=H\_\{\\widetilde\{P\}\}\(Y\\mid Z\)\-H\_\{\\widetilde\{P\}\}\(Y\\mid Z,X\)gives Eq\. \([A\.2](https://arxiv.org/html/2607.18820#A1.E2)\)\.

∎

The inequality above is a generic variational lower bound forIP~​\(X;Y∣Z\)I\_\{\\widetilde\{P\}\}\(X;Y\\mid Z\)\. The counterfactual\-CoT augmentation is important because it changes the distributionP~\\widetilde\{P\}: the same instruction can be paired with different CoTs and different answers\. Therefore, optimizing the likelihood term underP~\\widetilde\{P\}cannot be satisfied by a predictor that ignoresXXand relies only onZZ\. This makes the bound useful as a surrogate for strengthening theX→YX\\rightarrow Ydependence under fixedZZ\.

###### Proposition 3\(Pairwise instruction\-discrepancy upper bound forI​\(Y;Z∣X\)I\(Y;Z\\mid X\)\)\.

LetpXp\_\{X\}denote the marginal distribution of reasoning chains,pZ\(⋅∣x\)p\_\{Z\}\(\\cdot\\mid x\)the conditional distribution of instructions givenxx, andpY\(⋅∣x,z\)p\_\{Y\}\(\\cdot\\mid x,z\)the conditional answer distribution\. Then

I​\(Y;Z∣X\)\\displaystyle I\(Y;Z\\mid X\)≤𝔼x∼pX,z∼pZ\(⋅∣x\),z′∼pZ\(⋅∣x\)\[KL\(pY\(⋅∣x,z\)\\displaystyle\\leq\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}x\\sim p\_\{X\},\\\\ z\\sim p\_\{Z\}\(\\cdot\\mid x\),\\\\ z^\{\\prime\}\\sim p\_\{Z\}\(\\cdot\\mid x\)\\end\{subarray\}\}\\Big\[\\mathrm\{KL\}\\Big\(p\_\{Y\}\(\\cdot\\mid x,z\)∥pY\(⋅∣x,z′\)\)\]\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\\|\\,p\_\{Y\}\(\\cdot\\mid x,z^\{\\prime\}\)\\Big\)\\Big\]\.\(A\.3\)

###### Proof\.

For a fixed reasoning chainxx, the conditional mutual information can be written as

I\(Y;Z\\displaystyle I\(Y;Z∣X=x\)\\displaystyle\\mid X=x\)=𝔼z∼pZ\(⋅∣x\)\[KL\(pY\(⋅∣x,z\)∥pY\(⋅∣x\)\)\],\\displaystyle=\\mathbb\{E\}\_\{z\\sim p\_\{Z\}\(\\cdot\\mid x\)\}\\left\[\\mathrm\{KL\}\\left\(p\_\{Y\}\(\\cdot\\mid x,z\)\\\|p\_\{Y\}\(\\cdot\\mid x\)\\right\)\\right\],wherepY\(⋅∣x\)=𝔼z′∼pZ\(⋅∣x\)\[pY\(⋅∣x,z′\)\]p\_\{Y\}\(\\cdot\\mid x\)=\\mathbb\{E\}\_\{z^\{\\prime\}\\sim p\_\{Z\}\(\\cdot\\mid x\)\}\\left\[p\_\{Y\}\(\\cdot\\mid x,z^\{\\prime\}\)\\right\]\. By the convexity of the KL divergence, we have

KL\(pY\(⋅∣x,z\)∥pY\(⋅∣x\)\)\\displaystyle\\mathrm\{KL\}\\left\(p\_\{Y\}\(\\cdot\\mid x,z\)\\,\\\|\\,p\_\{Y\}\(\\cdot\\mid x\)\\right\)=KL\(pY\(⋅∣x,z\)∥𝔼z′∼pZ\(⋅∣x\)\[pY\(⋅∣x,z′\)\]\)\\displaystyle=\\mathrm\{KL\}\\left\(p\_\{Y\}\(\\cdot\\mid x,z\)\\;\\middle\\\|\\;\\mathbb\{E\}\_\{z^\{\\prime\}\\sim p\_\{Z\}\(\\cdot\\mid x\)\}\\left\[p\_\{Y\}\(\\cdot\\mid x,z^\{\\prime\}\)\\right\]\\right\)≤𝔼z′∼pZ\(⋅∣x\)\[KL\(pY\(⋅∣x,z\)∥pY\(⋅∣x,z′\)\)\]\.\\displaystyle\\leq\\mathbb\{E\}\_\{z^\{\\prime\}\\sim p\_\{Z\}\(\\cdot\\mid x\)\}\\left\[\\mathrm\{KL\}\\left\(p\_\{Y\}\(\\cdot\\mid x,z\)\\,\\\|\\,p\_\{Y\}\(\\cdot\\mid x,z^\{\\prime\}\)\\right\)\\right\]\.Taking expectation overz∼pZ\(⋅∣x\)z\\sim p\_\{Z\}\(\\cdot\\mid x\)gives

I\(Y;Z∣X\\displaystyle I\(Y;Z\\mid X=x\)\\displaystyle=x\)≤𝔼z∼pZ\(⋅∣x\)𝔼z′∼pZ\(⋅∣x\)\[KL\(pY\(⋅∣x,z\)\\displaystyle\\leq\\mathbb\{E\}\_\{z\\sim p\_\{Z\}\(\\cdot\\mid x\)\}\\mathbb\{E\}\_\{z^\{\\prime\}\\sim p\_\{Z\}\(\\cdot\\mid x\)\}\\Bigg\[\\mathrm\{KL\}\\Big\(p\_\{Y\}\(\\cdot\\mid x,z\)∥pY\(⋅∣x,z′\)\)\]\.\\displaystyle\\qquad\\qquad\\;\\\|\\;p\_\{Y\}\(\\cdot\\mid x,z^\{\\prime\}\)\\Big\)\\Bigg\]\.Finally, averaging both sides overx∼pXx\\sim p\_\{X\}yields Eq\. \([A\.3](https://arxiv.org/html/2607.18820#A1.E3)\)\. ∎

###### Proposition 4\(Attention masking removes the explicit instruction\-to\-answer attention path\)\.

LetpθAMp\_\{\\theta\}^\{\\mathrm\{AM\}\}denote the model distribution under inference\-time attention masking, and letHXH\_\{X\}denote the CoT\-side computational states available to answer generation, i\.e\., the hidden or key\-value states at CoT token positions formed during autoregressive CoT generation\. If, for every answer token positioni∈pos​\(Y\)i\\in\\mathrm\{pos\}\(Y\), instruction token positionj∈pos​\(Z\)j\\in\\mathrm\{pos\}\(Z\), layerℓ\\ell, and headhh, the masked pre\-softmax attention logit is set toAi​jℓ,h=−∞A\_\{ij\}^\{\\ell,h\}=\-\\infty, then answer tokens cannot directly attend to instruction\-token states\. Consequently, once the CoT\-side statehXh\_\{X\}is fixed, answer generation has no direct computation\-graph dependence on the instruction\-token states, in the explicit computation\-graph sense,

pθAM​\(y∣z,hX\)=pθAM​\(y∣hX\),p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\\mid z,h\_\{X\}\)=p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\\mid h\_\{X\}\),\(A\.4\)wherehXh\_\{X\}is a realized value ofHXH\_\{X\}\.

###### Proof\.

LetSZS\_\{Z\},SXS\_\{X\}, andSYS\_\{Y\}be the token\-position sets of the instruction, CoT, and answer, respectively\. LetC​\(i\)C\(i\)denote the causal\-visible positions for an answer positioni∈SYi\\in S\_\{Y\}\. At layerℓ\\elland headhh,

αi​jℓ,h=exp⁡\(Ai​jℓ,h\)∑k∈C​\(i\)exp⁡\(Ai​kℓ,h\),\\alpha\_\{ij\}^\{\\ell,h\}=\\frac\{\\exp\(A\_\{ij\}^\{\\ell,h\}\)\}\{\\sum\_\{k\\in C\(i\)\}\\exp\(A\_\{ik\}^\{\\ell,h\}\)\},whereAi​jℓ,hA\_\{ij\}^\{\\ell,h\}is the masked pre\-softmax attention logit\. The CASE mask sets

Ai​jℓ,h=−∞,∀i∈SY,j∈SZ,ℓ,h,A\_\{ij\}^\{\\ell,h\}=\-\\infty,\\quad\\forall i\\in S\_\{Y\},\\ j\\in S\_\{Z\},\\ell,h,and hence

αi​jℓ,h=0,∀i∈SY,j∈SZ,ℓ,h\.\\alpha\_\{ij\}^\{\\ell,h\}=0,\\quad\\forall i\\in S\_\{Y\},\\ j\\in S\_\{Z\},\\ell,h\.Thus, the attention output at answer positioniicontains no direct contribution from instruction positions:

oiℓ,h=∑k∈C​\(i\)\\SZαi​kℓ,h​WVℓ,h​hkℓ−1\.o\_\{i\}^\{\\ell,h\}=\\sum\_\{k\\in C\(i\)\\backslash S\_\{Z\}\}\\alpha\_\{ik\}^\{\\ell,h\}W\_\{V\}^\{\\ell,h\}h\_\{k\}^\{\\ell\-1\}\.Residual connections and feed\-forward blocks introduce no new cross\-token edges\. Therefore, conditioning on the CoT\-side mediatorhXh\_\{X\}, answer\-token states can depend on the instruction only throughhXh\_\{X\}and previous answer tokens\. By induction over layers, for each answer tokenyty\_\{t\},

pθAM​\(yt∣z,hX,y<t\)=pθAM​\(yt∣hX,y<t\)\.p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\_\{t\}\\mid z,h\_\{X\},y\_\{<t\}\)=p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\_\{t\}\\mid h\_\{X\},y\_\{<t\}\)\.Using the autoregressive factorization, we obtain

pθAM​\(y∣z,hX\)\\displaystyle p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\\mid z,h\_\{X\}\)=∏tpθAM​\(yt∣z,hX,y<t\)\\displaystyle=\\prod\_\{t\}p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\_\{t\}\\mid z,h\_\{X\},y\_\{<t\}\)=∏tpθAM​\(yt∣hX,y<t\)\\displaystyle=\\prod\_\{t\}p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\_\{t\}\\mid h\_\{X\},y\_\{<t\}\)=pθAM​\(y∣hX\)\.\\displaystyle=p\_\{\\theta\}^\{\\mathrm\{AM\}\}\(y\\mid h\_\{X\}\)\.This proves Eq\. \([A\.4](https://arxiv.org/html/2607.18820#A1.E4)\)\. ∎

### A\.2Detailed Experiment Settings

Table[S\-I](https://arxiv.org/html/2607.18820#A1.T1)lists the model checkpoints used in our experiments\.

Table S\-I:The LLMs used in the experiments and their corresponding checkpoints\.We use a fixed random seed of 42 for data sampling, model training, and evaluation\. For all fine\-tuning experiments, we use LoRA with rankr=64r=64and scaling factorα=128\\alpha=128\. For CASE, we use AdamW for 5 epochs with a learning rate of5×10−55\\times 10^\{\-5\}, weight decay of 0\.01, warmup ratio of 0\.1, epsilon of10−810^\{\-8\}, and maximum gradient norm of 0\.5\. The maximum sequence length, per\-device batch size, and gradient\-accumulation steps are set to 1024, 2, and 4, respectively\. During inference, all methods use the same few\-shot prompts, with temperature and top−p\-pset to 0\.7 and 0\.9, respectively\. The remaining hyperparameters of baseline methods follow their original implementations\.

For CASE, we use DeepSeek\-V3\.2 to generate CoT demonstrations and counterfactual CoTs\. To construct the biased\-instruction dataset𝒟C\\mathcal\{D\}\_\{C\}, we append the sentence “An MIT professor believes the answer is\{chosen\}\.” to the end of the question, where\{chosen\}is randomly selected from the incorrect answer candidates\. For the empty\-instruction dataset𝒟D\\mathcal\{D\}\_\{D\}, we remove only the question content while preserving the answer options for multiple\-choice tasks\. Similarly, during inference, attention masking blocks only the question content while retaining the option information\. For true/false tasks, we mask only a proportionβ\\betaof the question tokens to preserve the answer format\. We setβ=80%\\beta=80\\%, and analyze the impact of differentβ\\betavalues in Appendix[A\.6](https://arxiv.org/html/2607.18820#A1.SS6)\. Appendix[A\.11](https://arxiv.org/html/2607.18820#A1.SS11)provides all prompt templates used in our experiments\.

### A\.3Quality Audit of Generated Training Data

Since CASE uses LLM\-generated CoTs for training, we conduct an automatic quality audit for the generated training data\. The audit mainly covers𝒟A\\mathcal\{D\}\_\{A\}and𝒟B\\mathcal\{D\}\_\{B\}, because they contain the newly generated CoT\-answer and counterfactual CoT\-answer pairs produced by DeepSeek\-V3\.2\. The other two datasets,𝒟C\\mathcal\{D\}\_\{C\}and𝒟D\\mathcal\{D\}\_\{D\}, are constructed by modifying or removing the instruction while reusing the CoT\-answer pairs from𝒟A\\mathcal\{D\}\_\{A\}\. Thus, auditing𝒟A\\mathcal\{D\}\_\{A\}and𝒟B\\mathcal\{D\}\_\{B\}checks the quality of the LLM\-generated supervision used by CASE\. For each benchmark, we randomly sampleNNgenerated cases from each audited dataset, whereN=500N=500in our experiments\. The goal of this audit is not to prove that every generated CoT is uniquely correct, but to check whether the generated data satisfies the consistency properties required by our training objective\. To avoid self\-verification, we use GPT\-5\.5\(OpenAI[2026](https://arxiv.org/html/2607.18820#bib.bib39)\)as an independent LLM verifier, which is different from DeepSeek\-V3\.2 used for data generation and from the models fine\-tuned in our experiments\.

We use three automatic metrics\. Let𝒴i\\mathcal\{Y\}\_\{i\}denote the candidate answer set for instanceii, and letyigoldy\_\{i\}^\{\\mathrm\{gold\}\}denote the gold answer\. For𝒟A\\mathcal\{D\}\_\{A\},yiy\_\{i\}denotes the generated answer, while for𝒟B\\mathcal\{D\}\_\{B\},yi′y\_\{i\}^\{\\prime\}denotes the generated counterfactual answer\. We first compute the answer validity rate \(AVR\), which checks whether the generated answer satisfies the required answer constraint:

A​V​R𝒟A=1N​∑i=1N𝟏​\[yi=yigold\],AVR\_\{\\mathcal\{D\}\_\{A\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\\\!\\left\[y\_\{i\}=y\_\{i\}^\{\\mathrm\{gold\}\}\\right\],A​V​R𝒟B=1N​∑i=1N𝟏​\[yi′∈𝒴i∧yi′≠yi\]\.AVR\_\{\\mathcal\{D\}\_\{B\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\\\!\\left\[y\_\{i\}^\{\\prime\}\\in\\mathcal\{Y\}\_\{i\}\\ \\land\\ y\_\{i\}^\{\\prime\}\\neq y\_\{i\}\\right\]\.
Second, we compute the CoT\-answer consistency rate \(CAC\), which checks whether the generated answer can be inferred from the question and the generated CoT\. LetVans​\(z,x\)V\_\{\\mathrm\{ans\}\}\(z,x\)denote the answer predicted by the independent LLM verifier when given the questionzzand the CoTxx\. We define

C​A​C𝒟A=1N​∑i=1N𝟏​\[Vans​\(zi,xi\)=yi\],CAC\_\{\\mathcal\{D\}\_\{A\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\\\!\\left\[V\_\{\\mathrm\{ans\}\}\(z\_\{i\},x\_\{i\}\)=y\_\{i\}\\right\],C​A​C𝒟B=1N​∑i=1N𝟏​\[Vans​\(zi,xi′\)=yi′\]\.CAC\_\{\\mathcal\{D\}\_\{B\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\\\!\\left\[V\_\{\\mathrm\{ans\}\}\(z\_\{i\},x\_\{i\}^\{\\prime\}\)=y\_\{i\}^\{\\prime\}\\right\]\.This metric evaluates whether the generated CoT, together with the question context, leads the verifier to the generated answer\.

Third, we compute the LLM\-judge support rate \(SR\), which checks whether the reasoning chain supports the predicted answer\. LetVsup​\(z,x,y\)∈\{Yes,No\}V\_\{\\mathrm\{sup\}\}\(z,x,y\)\\in\\\{\\mathrm\{Yes\},\\mathrm\{No\}\\\}denote the LLM verifier’s judgment on whether CoTxxsupports answeryyfor questionzz\. We define

S​R𝒟A=1N​∑i=1N𝟏​\[Vsup​\(zi,xi,yi\)=Yes\],SR\_\{\\mathcal\{D\}\_\{A\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\\\!\\left\[V\_\{\\mathrm\{sup\}\}\(z\_\{i\},x\_\{i\},y\_\{i\}\)=\\mathrm\{Yes\}\\right\],S​R𝒟B=1N​∑i=1N𝟏​\[Vsup​\(zi,xi′,yi′\)=Yes\]\.SR\_\{\\mathcal\{D\}\_\{B\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{1\}\\\!\\left\[V\_\{\\mathrm\{sup\}\}\(z\_\{i\},x\_\{i\}^\{\\prime\},y\_\{i\}^\{\\prime\}\)=\\mathrm\{Yes\}\\right\]\.The prompt template for SR is provided in Table[S\-XVII](https://arxiv.org/html/2607.18820#A1.T17)in Appendix[A\.11](https://arxiv.org/html/2607.18820#A1.SS11)\.

Table[S\-II](https://arxiv.org/html/2607.18820#A1.T2)reports the audit results across four datasets\.

Table S\-II:Automatic quality audit of generated training data\.The results show that the generated training data has high automatic quality across all datasets\. Both𝒟A\\mathcal\{D\}\_\{A\}and𝒟B\\mathcal\{D\}\_\{B\}achieve perfect answer validity, indicating that the generated answers are valid under their corresponding data\-construction requirements\. The CAC scores are also consistently high, showing that the generated CoTs usually lead to the intended answers\. In addition, the SR scores remain above 0\.986 across all settings, suggesting that the verifier judges most generated reasoning chains as supporting their target answers\. These results provide scalable evidence that the generated CoT data is reliable for training CASE, especially for learning the intended CoT\-to\-answer dependency\.

### A\.4Evaluation Metric Details

We describe the four faithfulness metrics used in our experiments\. All metrics are computed overMMevaluation samples \(whereM=500M=500in this paper\)\. Higher values indicate higher CoT faithfulness for FT, EA, and CIE, while lower CDE indicates higher faithfulness\.

##### Filler Tokens \(FT\)\.

Filler Tokens measures whether the answer depends on the generated CoT\. For each question, we first record the original predictiony^i\\hat\{y\}\_\{i\}using the generated CoTxix\_\{i\}\. We then replace all CoT tokens with the filler string “…” and obtain a new predictiony^iFT\\hat\{y\}\_\{i\}^\{\\mathrm\{FT\}\}\. The metric is the proportion of samples whose answers change:

FT=1M​∑i=1M𝟏​\[y^iFT≠y^i\]\.\\mathrm\{FT\}=\\frac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\mathbf\{1\}\[\\hat\{y\}\_\{i\}^\{\\mathrm\{FT\}\}\\neq\\hat\{y\}\_\{i\}\]\.A higher FT score indicates stronger reliance on the CoT\.

##### Early Answering \(EA\)\.

Early Answering tests whether the model can predict the final answer before observing the full CoT\. For sampleii, we split the CoT intoNiN\_\{i\}sentences and construct prefixesri\(0\),ri\(1\),…,ri\(Ni\)r\_\{i\}^\{\(0\)\},r\_\{i\}^\{\(1\)\},\\ldots,r\_\{i\}^\{\(N\_\{i\}\)\}, whereri\(0\)=∅r\_\{i\}^\{\(0\)\}=\\emptysetandri\(k\)r\_\{i\}^\{\(k\)\}contains the firstkksentences\. Lety^i\\hat\{y\}\_\{i\}be the prediction using the full CoT, andy^i\(k\)\\hat\{y\}\_\{i\}^\{\(k\)\}be the prediction using prefixri\(k\)r\_\{i\}^\{\(k\)\}\. We define the consistency sequence as

ci\(k\)=𝟏​\[y^i\(k\)=y^i\]\.c\_\{i\}^\{\(k\)\}=\\mathbf\{1\}\[\\hat\{y\}\_\{i\}^\{\(k\)\}=\\hat\{y\}\_\{i\}\]\.We compute the normalized area under this consistency curve, denoted byAUC​\(ci\)\\mathrm\{AUC\}\(c\_\{i\}\), and defineAOCi=1−AUC​\(ci\)\\mathrm\{AOC\}\_\{i\}=1\-\\mathrm\{AUC\}\(c\_\{i\}\)\. The final metric is

EA=1M​∑i=1MAOCi\.\\mathrm\{EA\}=\\frac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\mathrm\{AOC\}\_\{i\}\.A higher EA score means that the model is less able to answer early, indicating stronger dependence on the complete CoT\.

##### Controlled Indirect Effect \(CIE\)\.

CIE measures whether changing the CoT changes the final answer while keeping the question fixed\. For each questionziz\_\{i\}and the generated CoTxix\_\{i\}, we use DeepSeek\-V3\.2 to generate a counterfactual CoTxi′x\_\{i\}^\{\\prime\}, with prompt templates provided in Appendix[A\.11](https://arxiv.org/html/2607.18820#A1.SS11)\. We compare the original predictiony^i\\hat\{y\}\_\{i\}based on\(zi,xi\)\(z\_\{i\},x\_\{i\}\)with the predictiony^iCIE\\hat\{y\}\_\{i\}^\{\\mathrm\{CIE\}\}based on\(zi,xi′\)\(z\_\{i\},x\_\{i\}^\{\\prime\}\)\. The metric is

CIE=1M​∑i=1M𝟏​\[y^iCIE≠y^i\]\.\\mathrm\{CIE\}=\\frac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\mathbf\{1\}\[\\hat\{y\}\_\{i\}^\{\\mathrm\{CIE\}\}\\neq\\hat\{y\}\_\{i\}\]\.A higher CIE score indicates that the answer is more sensitive to changes in the reasoning chain\.

##### Controlled Direct Effect \(CDE\)\.

CDE measures whether changing the question still changes the answer when the CoT is fixed\. For each sample, we use DeepSeek\-V3\.2 to generate a counterfactual questionzi′z\_\{i\}^\{\\prime\}, with prompt templates provided in Appendix[A\.11](https://arxiv.org/html/2607.18820#A1.SS11)\. We compare the original predictiony^i\\hat\{y\}\_\{i\}based on\(zi,xi\)\(z\_\{i\},x\_\{i\}\)with the predictiony^iCDE\\hat\{y\}\_\{i\}^\{\\mathrm\{CDE\}\}based on\(zi′,xi\)\(z\_\{i\}^\{\\prime\},x\_\{i\}\)\. The metric is

CDE=1M​∑i=1M𝟏​\[y^iCDE≠y^i\]\.\\mathrm\{CDE\}=\\frac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\mathbf\{1\}\[\\hat\{y\}\_\{i\}^\{\\mathrm\{CDE\}\}\\neq\\hat\{y\}\_\{i\}\]\.A lower CDE score indicates weaker direct instruction\-to\-answer dependence after conditioning on the CoT\.

### A\.5Attention Masking vs\. CoT\-only Decoding

A direct alternative to attention masking is to first generate the CoT from the instruction, then remove the instruction and generate the answer from the CoT alone\. We call this strategy CoT\-only decoding \(COD\)\. To isolate the effect of the inference\-time strategy, we keep the same CASE training procedure and replace only the attention\-masking stage with COD\. Table[S\-III](https://arxiv.org/html/2607.18820#A1.T3)compares CASE with its COD variant on Llama\-3\.1\-8B across two ARC benchmarks\.

Table S\-III:Comparison between CASE and its CoT\-only decoding variant on Llama\-3\.1\-8B across two ARC benchmarks\. Best results are highlighted in bold\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.The results show that directly deleting the instruction is less effective than attention masking\. Although CASE w/ COD obtains slightly higher CIE, CASE performs better in task accuracy and most faithfulness metrics, including FT, EA, CDE, and the overall G\-M score\. These results do not imply that the visible CoT is insufficient as an explanation\. Rather, COD changes both the prompt and the original instruction\-conditioned CoT\-side states, whereas attention masking isolates direct\-path removal while preserving these states\. This leads to better accuracy and more balanced faithfulness\.

### A\.6Sensitivity Analysis of Masking Ratioβ\\beta

For true/false tasks, fully masking the question can hide the target proposition to be judged\. As a result, even with correct reasoning, the model may not know whether to answer True or False\. Therefore, we introduce a masking ratioβ\\beta, which controls the proportion of question tokens masked during inference\. To study its effect, we varyβ\\betafrom 0% to 100% on StrategyQA using Llama\-3\.1\-8B, whereβ=0%\\beta=0\\%means no question masking andβ=100%\\beta=100\\%means masking the entire question content\. The results are shown in Table[S\-IV](https://arxiv.org/html/2607.18820#A1.T4)\.

Table S\-IV:Sensitivity analysis of the masking ratioβ\\betaon StrategyQA with Llama\-3\.1\-8B\. Best and second\-best results are highlighted in bold and underlined, respectively\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.Asβ\\betaincreases, FT and EA improve steadily, indicating that stronger masking forces the model to rely more on the generated CoT when producing the final answer\. However, overly strong masking degrades the intervention\-based metrics, because removing too much question information makes it harder to maintain the correct true/false decision under interventions\. Accuracy also improves with moderate masking but drops when the question is fully masked\. Overall,β\\betavalues between60%60\\%and80%80\\%provide the best trade\-off between faithfulness and accuracy\. Althoughβ=70%\\beta=70\\%gives the highest G\-M score,β=80%\\beta=80\\%achieves the best accuracy and similar G\-M while further improving FT and EA\. Therefore, we setβ=80%\\beta=80\\%in our main experiments\.

### A\.7Inference Runtime Analysis

To evaluate inference efficiency, we measure the average runtime per question on ARC\-Easy using Llama\-3\.1\-8B\. All experiments are conducted on a single NVIDIA A100 GPU with 40GB of memory under the same evaluation setting\. The results are reported in Table[S\-V](https://arxiv.org/html/2607.18820#A1.T5)\.

Table S\-V:Inference runtime comparison on Llama\-3\.1\-8B and ARC\-Easy\. Best and second\-best results are highlighted in bold and underlined, respectively\. CASE is shaded\.The results show that CASE has a comparable inference cost to other single\-model methods\. Its runtime is close toD​UcDU^\{c\}and FoCus, while being only moderately slower than the untuned Base LLM due to the inference\-time attention masking\. In contrast, FRODO requires much longer inference time because it relies on a dual\-model pipeline\. Specifically, CASE is about3\.0×3\.0\\timesfaster than FRODO in this setting\. These results suggest that CASE improves CoT faithfulness without introducing substantial inference overhead compared with other single\-model approaches\.

### A\.8Additional Cross\-Dataset Generalization Results

Due to space limits, the main text reports cross\-dataset generalization results on Llama\-3\.1\-8B\. Table[S\-VI](https://arxiv.org/html/2607.18820#A1.T6)provides the remaining results on Qwen3\-8B and DeepSeek\-R1\-Distill\-Qwen\-7B under the same two transfer settings: ARC\-Challenge→\\rightarrowOpenBookQA and StrategyQA→\\rightarrowARC\-Easy\. The former evaluates transfer within science\-oriented multiple\-choice QA, while the latter evaluates a larger shift from Boolean commonsense QA to multiple\-choice science QA\.

Table S\-VI:Supplementary cross\-dataset generalization results on Qwen3\-8B and DeepSeek\-R1\-Distill\-Qwen\-7B under two transfer settings\. Best and second\-best results are highlighted in bold and underlined, respectively\. CASE is shaded\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.The results are consistent with the findings in the main text\. CASE achieves the bestG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}in all four supplementary settings, while maintaining accuracy close to the Base LLM or the strongest baselines\. It also consistently improves EA and reduces CDE, suggesting stronger reliance on the generated CoT and weaker direct instruction\-to\-answer dependence\. Together with Table[2](https://arxiv.org/html/2607.18820#Sx4.T2), these results show that CASE yields more transferable faithfulness improvements under cross\-dataset evaluation\.

### A\.9Additional Ablation Results

We provide the remaining ablation results not included in the main paper, including Llama\-3\.1\-8B on StrategyQA and LogiQA, as well as all results on Qwen3\-8B and DeepSeek\-R1\-Distill\-Qwen\-7B, in Tables[S\-VII](https://arxiv.org/html/2607.18820#A1.T7)–[S\-IX](https://arxiv.org/html/2607.18820#A1.T9)\. The results are consistent with the observations in the main paper\. TCA mainly improves causal faithfulness metrics such as CIE and CDE, while ISE brings larger gains on FT and EA by encouraging stronger reliance on the generated CoT during answer generation\. The full CASE framework achieves the bestG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}in all additional model–dataset settings, confirming that the complementary effect of TCA and ISE is consistent across datasets and models, rather than being specific to the ARC benchmarks or Llama\-3\.1\-8B\.

Table S\-VII:Ablation study of CASE on Llama\-3\.1\-8B across StrategyQA and LogiQA benchmarks\. CASE is shaded\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.Table S\-VIII:Ablation study of CASE on Qwen3\-8B across four reasoning benchmarks\. CASE is shaded\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.Table S\-IX:Ablation study of CASE on DeepSeek\-R1\-Distill\-Qwen\-7B across four reasoning benchmarks\. CASE is shaded\. G\-M denotesG\-meanF​a​i​t​h\\text\{G\-mean\}\_\{Faith\}\.
### A\.10Qualitative Analysis

To complement the quantitative results, we provide two qualitative examples from ARC\-Challenge using Llama\-3\.1\-8B\. Tables[S\-X](https://arxiv.org/html/2607.18820#A1.T10)and[S\-XI](https://arxiv.org/html/2607.18820#A1.T11)compare the CoTs and final answers generated by different methods\. These examples illustrate two common failure modes of existing baselines: directly relying on instruction\-level cues when choosing the answer, and producing a final answer that is not well supported by the generated CoT\. In contrast, CASE produces answers that are more consistent with the faithful chainZ→X→YZ\\rightarrow X\\rightarrow Y, where the instruction first induces the reasoning process, and the final answer is then derived from that reasoning\.

Question:Which change in Earth’s surface is most directly related to the water cycle?Options:\(A\) deposition of sediments \(B\) uplifting of a mountain \(C\) formation of ocean trenches \(D\) movement of tectonic platesCorrect answer:\(A\)MethodCoTAnswerBase LLMThe water cycle involves the movement of water from the Earth to the atmosphere and back to the Earth\. This is closely related to the weathering and erosion processes which are most directly related to the movement of tectonic plates\.✗\(D\)D​UcDU^\{c\}The water cycle involves the movement of water between Earth and atmosphere, and processes like evaporation, condensation, and precipitation\. Among the options, the movement of tectonic plates can create landscapes \(like mountains, valleys, and lakes\) that are filled and shaped by water from the cycle, making it the most directly related\.✗\(D\)FRODOThe most directly related to the water cycle is actually evaporation and precipitation \(not listed\)\. But from the options provided, the water cycle involves the movement of water between Earth and atmosphere, so it includes processes like evaporation and precipitation\. From the options D is the answer\.✗\(D\)FoCusThe water cycle involves the movement of water on Earth’s surface, which is directly related to the movement of tectonic plates as they can cause changes in the Earth’s surface, such as creating new land or altering existing landscapes, which in turn affect the water cycle\.✗\(D\)CASEThe water cycle involves processes like evaporation, condensation, and precipitation, which are driven by Earth’s heat and gravity\. While movement of tectonic plates can shape the landscape, the water cycle directly influences weathering and erosion, which in turn can move sediments and change the surface\. However, the most direct relation is the continuous movement and transformation of water across Earth’s surface, which is part of the water cycle itself\.Among the options, deposition of sediments is a result of erosion and weathering, which are both influenced by the water cycle\. The other choices are geological processes not directly driven by the water cycle in the same way\.✓\(A\)

Table S\-X:Qualitative comparison on Case 1 from ARC\-Challenge using Llama\-3\.1\-8B\.Question:Students are observing two fossils in a lab\. They are told by the teacher that the fossils were found in different rock layers\. What can the students most likely determine from just observing the fossils?Options:\(A\) how each organism died \(B\) the exact age of the fossils \(C\) when the fossils were discovered \(D\) the type of environment in which the organisms livedCorrect answer:\(D\)MethodCoTAnswerBase LLMThe fossils found in different rock layers would likely indicate that the organisms lived at different times, as rock layers are typically formed in chronological order\.✗\(B\)D​UcDU^\{c\}Fossils in different rock layers are layered in a specific order \(stratigraphy\)\. By observing the fossils, students can infer the likely environments in which the organisms lived, because certain fossils are associated with specific ecological settings \(e\.g\., marine vs\. terrestrial\)\.✓\(D\)FRODONot directly from just the fossils, but from the layers they are in: Fossils in lower layers are likely older than those in higher layers \(due to layering\)\. However, more accurately they can determine the likely time of which organism was older than the other\.✗\(B\)FoCusThe students can determine the relative time period in which each organism lived, as fossils in lower rock layers are typically older than those in higher layers\. This is based on the principle of superposition in geology\. But more specifically it’s the relative age of the fossils, not the exact age\.✗\(B\)CASEFossils found in different rock layers are typically layered in chronological order\. By observing the fossils, students can infer the likely time period and environment in which each organism lived, because similar fossils tend to appear in similar environments across layers\.✓\(D\)

Table S\-XI:Qualitative comparison on Case 2 from ARC\-Challenge using Llama\-3\.1\-8B\.In Case 1, the question asks which surface change is most directly related to the water cycle\. Most baselines mention the water cycle in their CoTs, but then incorrectly choose “movement of tectonic plates\.” This suggests that their final decisions are influenced by a surface\-level shortcut from the instruction and options, such as associating “Earth’s surface” with tectonic processes, rather than by the reasoning chain itself\. FRODO even notes that evaporation and precipitation are the most relevant water\-cycle processes, but still selects option D\. These outputs show a clear mismatch between reasoning and answer, indicating that the generated CoT does not fully control the final prediction\. By contrast, CASE explicitly connects the water cycle to weathering, erosion, and sediment deposition, and then selects the correct answer\. This behavior is more consistent with a faithfulZ→X→YZ\\rightarrow X\\rightarrow Yprocess\.

Case 2 shows another form of shortcut behavior\. The question asks what students can determine by observing fossils, but several baselines focus on the phrase “different rock layers” and directly infer fossil age\. Base LLM, FRODO, and FoCus therefore choose option B, even though the option asks for the exact age, which cannot be determined from observation alone\. FoCus is especially illustrative: its CoT states that the exact age cannot be determined, but its final answer is still B\. This indicates that the final answer is not fully derived from the stated reasoning\. In contrast, CASE identifies that fossil features can support an inference about the organisms’ living environment and selects option D\. These cases suggest that CASE better aligns answer generation with the generated CoT, which is consistent with its design: training\-time causal alignment strengthens the CoT\-to\-answer dependency, while inference\-time attention masking blocks the direct instruction\-to\-answer shortcut\.

### A\.11Prompt Templates

This section provides the prompt templates used in our experiments\. Table[S\-XII](https://arxiv.org/html/2607.18820#A1.T12)shows the prompt used to generate CoT demonstrations for constructing𝒟A\\mathcal\{D\}\_\{A\}\. Table[S\-XIII](https://arxiv.org/html/2607.18820#A1.T13)shows the shared prompt used to generate counterfactual CoTs and answers for constructing𝒟B\\mathcal\{D\}\_\{B\}and evaluating the CIE metric\. Table[S\-XIV](https://arxiv.org/html/2607.18820#A1.T14)shows the prompt used to generate counterfactual questions for evaluating the CDE metric\. Table[S\-XV](https://arxiv.org/html/2607.18820#A1.T15)shows the few\-shot inference prompt used by all methods during evaluation\. Tables[S\-XVI](https://arxiv.org/html/2607.18820#A1.T16)and[S\-XVII](https://arxiv.org/html/2607.18820#A1.T17)show the prompt used for the CAC and SR metrics\.

Table S\-XII:Prompt used to generate CoT demonstrations for constructing𝒟A\\mathcal\{D\}\_\{A\}\.Table S\-XIII:Prompt used to generate counterfactual CoTs and answers for constructing𝒟B\\mathcal\{D\}\_\{B\}and evaluating the CIE metric\.Table S\-XIV:Prompt used to generate counterfactual questions for the CDE metric\.Table S\-XV:Few\-shot inference prompt used by all methods during evaluation\.Table S\-XVI:Prompt used for computing the CoT\-Answer Consistency Evaluation \(CAC\)\.Table S\-XVII:Prompt used for computing the LLM\-judge support rate \(SR\)\.