Reasoning in Two Dimensions: Quantifying Chain-of-Thought Faithfulness and Efficiency

Chain-of-thought (CoT) reasoning exposes intermediate steps, but a correct answer does not establish that an LLM relies on the dependence expressed by those steps or reasons efficiently. We operationalize CoT faithfulness and efficiency within a common framework. Specifically, we represent a CoT as a semantic reasoning graph and decompose the model's predictive dependence along two complementary axes: dependence explained by semantic support versus additional reasoning history, and dependence allocated to answer-reaching versus off-path reasoning. This decomposition yields global and step-level measures of unfaithfulness and inefficiency. Controlled interventions show that the proposed metrics respond to hidden dependence and induced off-path reasoning as intended. Across mathematical and multi-hop reasoning tasks, answer-level unfaithfulness is consistently associated with greater predictive uncertainty, whereas both unfaithfulness and inefficiency exhibit weak or inconsistent associations with answer correctness. We further derive theoretically grounded, graph-aware fine-tuning objectives that improve the targeted reasoning properties without directly optimizing the computationally expensive metrics or compromising answer accuracy. Results indicate that accuracy and brevity provide incomplete assessments of reasoning quality: reliable CoT evaluation should also consider whether the LLM's predictive dependence is faithfully represented and efficiently allocated.