KACE: Knowledge-Adaptive Context Engineering for Mathematical Reasoning

Abstract

Context-engineering methods improve a frozen language model by learning text from execution feedback and placing it in the prompt. When one learned prompt carries everything that was learned, irrelevant lessons compete with relevant ones, and the system can keep only what fits in that prompt. We present Knowledge-Adaptive Context Engineering (KACE), which decides per problem both how much inference to spend and which learned knowledge to show. A fixed tiered self-consistency schedule (2, 3, then 5 samples behind agreement gates) lets easy problems exit after two agreeing samples, and the tier at which a problem exits serves as its difficulty label. Offline, a teacher-guided learner distills failed training traces into short typed cards kept in a store partitioned by difficulty tier and domain, and each escalated tier reads only the cards of its (tier, domain) cell. On MATH-HARD, KACE reaches 77.3% against 72.7% for Best-of-5 self-consistency with fewer solver calls, and it scores above tiered ACE (70.0%) and tiered GEPA (72.0%). It also achieves the highest accuracy on AIME 2025 (61.1% against 53.3% for Best-of-5 and 57.8% for tiered GEPA) and on OlymMATH-EN (48.1%). A component analysis shows that the store adds 4.0 points on MATH-HARD and 3.7 on OlymMATH-EN over the tiered schedule alone, and that the (tier, domain) reading rule outperforms every alternative rule for choosing cards from the same store.