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KACE: Knowledge-Adaptive Context Engineering for Mathematical Reasoning

Context engineering can improve large language models without updating their weights, but mathematical reasoning exposes a key limitation: feedback accumulated in one growing prompt causes context bloat and limits the amount of learned guidance that can be used. Existing methods often conflate storage, what is learned across runs, with usage, what is included for a particular problem, and therefore inherit this prompt-size ceiling. We introduce Knowledge-Adaptive Context Engineering (KACE), which separates storage from usage through difficulty- and domain-based organization. Offline, a self-reflective learning loop distills training traces into an epistemic tree: a knowledge base of typed cards stratified by problem difficulty and epistemic domain. Each card is assigned to the difficulty-domain node corresponding to the failure from which it originated. At evaluation time, tiered self-consistency with per-tier agreement gates dynamically classifies each problem as easy, medium, or hard. Easy problems exit without retrieved cards, while harder problems retrieve only the matching branch of the tree. This tiered scheme matches or exceeds Best-of-N while using comparable compute, and it classifies problem difficulty with 78 percent pairwise concordance. The main empirical contribution is the construction and use of a difficulty- and domain-stratified knowledge base enabled by tiered self-consistency. On AIME 2025, KACE achieves 62.2 percent accuracy, a 10.4-point absolute gain over fixed Best-of-5 self-consistency at a comparable solver-call budget and a 5.6-point gain over the strongest learned-context baseline, Tiered + GEPA. We also observe consistent gains on MATH-HARD and the verifiable subset of OlymMATH.

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