Confidence Trap in CoT Reasoning: How Entropy Exposes LLM's "Confident Errors"

This study focuses on the "confident error" phenomenon in Chain-of-Thought (CoT) reasoning of large language models (LLMs). By analyzing step-by-step entropy values and algebraic consistency of Qwen2.5-1.5B during polynomial equation solving, it was found that models often exhibit low entropy (high confidence) when algebraic operations violate mathematical rules. This reveals the limitations of relying on confidence to judge the correctness of reasoning and proposes directions for detection and improvement.

The CoT reasoning ability of LLMs improves accuracy by decomposing complex tasks, but there is a core problem: models often generate seemingly reasonable but incorrect reasoning steps ("confident errors"), which may lead to serious consequences especially in precise scenarios like mathematical reasoning, and are hard for users to detect.

Core research question: Can step-by-step entropy predict violations of algebraic consistency?

• Step-by-step entropy: Reflects the model's confidence in each reasoning step (low entropy = high certainty, high entropy = hesitation);
• PACS score: Quantifies the consistency of algebraic operations in polynomial solving (checks equation balance, correctness of transformations, etc.).

Core finding: Models often exhibit low entropy (high confidence) when making mistakes. Phenomenon: Low entropy ≠ correctness; algebraic errors do not always correspond to high entropy regions—models are unusually certain about some wrong paths. Reasons: Bias in training data (familiarity with error patterns), limitations of pattern matching (non-symbolic reasoning), cumulative effect of CoT (early errors lead to consistent progression in subsequent steps).

• Experimental design: The test set covers quadratic, cubic, and quartic polynomials (to verify reasoning ability across different complexities);
• Measurement methods: Record step-by-step entropy (Shannon entropy), PACS scores, and the correlation between the two;
• Model selection: Qwen2.5-1.5B (moderate scale, open-source, good mathematical ability).

• Model evaluation: Challenges the assumption that "confidence = correctness";
• Error detection: Requires external validators (e.g., PACS), multi-step consistency checks, and adversarial testing;
• Training improvements: Uncertainty calibration, error awareness training, and hybrid symbolic-neural validation.

Limitations: Verified only on Qwen2.5-1.5B, focused on the polynomial domain, and used a single entropy metric; Future directions: Cross-model validation (GPT/Llama, etc.), expansion to multiple domains (code/logical reasoning), improvement of detection metrics, and real-time error intervention.

The study reveals that LLMs can produce algebraically inconsistent reasoning with high confidence. Practical applications require: not blindly trusting confidence, developing domain-specific validation mechanisms, and establishing multi-layered reliability checks. Understanding and resolving "confident errors" is a core challenge in building trustworthy LLM applications.