Training large language models for complex reasoning tasks relies on two complementary signals: sparse, outcome-grounded reinforcement learning (RL) for correctness, and dense, token-level distillation for guidance. However, complementarity in supervision does not guarantee compatibility in optimization; imitation-focused distillation can conflict with the outcome-driven RL objective, and existing hybrids lack a principled resolution.

We propose GIFT (Gradient-Informed FilTering for Distillation-Augmented Reasoning RL), a novel method that directly addresses the allocation problem of which teacher updates to apply. GIFT performs a token-level geometric analysis, decomposing the distillation gradient into a component parallel to the RL gradient, which measures first-order conflict and provides a safety filter, and a component orthogonal to it, which quantifies novel directional guidance and provides a measure of value. This decomposition yields a principled scoring rule to rank token-level guidance.

We combine this with a progressive coverage annealing schedule that shrinks the budget of tokens receiving distillation, ensuring that the most conflicting updates are discarded first while the most complementary ones are retained at full strength. This facilitates a smooth transition from dense, geometrically-filtered guidance to a pure, outcome-grounded objective at convergence.

Extensive experiments on mathematical and logical reasoning benchmarks show that GIFT consistently outperforms strong baselines across multiple model scales, achieving relative gains of up to 9.5% on difficult mathematical reasoning tasks. Code: https://github.com/lilifree/GIFT