KACE: Knowledge-Adaptive Context Engineering for Mathematical Reasoning

Large language models can derive a plausible mathematical object yet still violate explicit requirements--for example, by omitting a modular reduction, returning a non-integer, or using the wrong encoded answer form. We introduce Constraint-First Reasoning (CFR), a training-free two-stage prompting protocol: Stage 1 extracts and summarizes constraints entailed by the problem, and Stage 2 solves while checking intermediate and final results against that summary. Routed-CFR activates the two-stage protocol only when a text-only regex router detects restrictive cues; otherwise it uses direct chain-of-thought (CoT). Across AIME, CMIMC, BRUMO, and AIMO_AMC, the method improves direct CoT on multiple backbones. We further report convention-controlled routing experiments, matched prompting baselines, problem-level paired tests, decoding robustness, constraint-quality audits, total-token accounting, and an OlympiadBench evaluation. These analyses position CFR as a targeted test-time intervention whose benefit depends on recoverable constraints and reliable Stage 1 extraction, rather than as a general-purpose replacement for mathematical reasoning.

The frontier of mathematics is defined by problems whose solutions are not yet known, yet it remains unclear whether language models can meaningfully engage with such problems without human intervention. A major obstacle is the lack of large-scale research-level math datasets. To this end, we introduce ResearchMath-14k, a set of $14{,}056$ problems curated from academic sources via a multi-agent pipeline, making it the largest collection of research-level mathematical problems to date. We further generate ResearchMath-Reasoning, $220$K teacher trajectories from two open models, where we observe recurring avoidance behaviors such as non-attempts and fabricated references. Interestingly, across eight open-weight models, newer generations produce $5.6\times$ more references and $5.0\times$ more fake references per trace. After agentic filtering of ResearchMath-Reasoning, fine-tuning Qwen3 models from 4B to 30B parameters improves over base models by $9.2$ points on average. This shows that filtered open-problem attempts can provide useful supervision even without fully correct reasoning traces. We make ResearchMath-14k publicly available for future works on research-level mathematical reasoning.

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.