Mathematical Reasoning via Intervention-Based Time-Series Causal Discovery Using LLMs as Concept Mastery Simulators

Recent methods for improving LLM mathematical reasoning, whether through MCTS-based test-time search or causal graph-guided knowledge injection, cannot identify which concepts causally contribute to a correct answer, as the observed association may be spurious, driven by confounders such as problem difficulty. We propose CIKA (Causal Intervention for Knowledge Activation), a framework that uses the LLM itself as an interventional simulator: a prompt sets the concept state to ``mastered'' and the correctness change estimates the causal effect. We formalize this quantity as an Interventional Capability Probe (ICP), which diagnoses whether the LLM can use a given concept -- distinct from merely possessing knowledge. Because the intervention exogenously sets the concept state independently of problem difficulty, ICP separates confounding that observational methods cannot. On 67 screened problems, the ICP of the top-ranked concept (+0.219) is significantly larger than that of the negative control (+0.039; paired $t$-test, $p < 10^{-6}$, Cohen's $d = 0.86$), confirming that the probe discriminates causally relevant concepts from irrelevant ones. Analysis of 601 Omni-MATH problems further shows that solved problems have 6.1$\times$ higher ATE than unsolved ones (0.338 vs. 0.055), confirming that ICP is predictive of problem-solving success. With a 7B-parameter LLM whose weights are entirely frozen, CIKA achieves 69.7\% on the contamination-free Omni-MATH-Rule benchmark and 64.0\% overall, compared to 60.5\% for o1-mini, and 97.2\% on GSM8K, 46--50\% on AIME 2024--2026, and 46.2\% on MathArena. The Causal Knowledge Activation component contributes 33.8\% of correct answers on problems where the base model alone fails, demonstrating that the LLM already possessed but had not activated the requisite knowledge.