Reinforcement learning from verifiable rewards (RLVR) usually optimizes answer correctness, yet useful language-model behavior also requires high-quality reasoning and concise responses.
Existing multi-reward post-training methods typically scalarize rewards or combine specialists without explicitly protecting a reward priority order. This is problematic when trade-offs are asymmetric: conciseness, for example, should not improve at the cost of correctness.
We introduce Lexicographic Multi-Objective On-Policy Distillation (LMOPD), a multi-teacher method for integrating reward-specialized policies under explicit priorities.
For each student rollout, LMOPD selects the specialist for the first objective whose gate detects a deficiency, then locally projects its centered log-policy correction to remove components that oppose higher-priority specialists.
We evaluate 30B-A3B mixture-of-experts transformer models in two- and four-expert settings on three math benchmarks, measuring retained specialist gains.
With two experts, LMOPD's point estimates fully retain the accuracy and reasoning-quality gains while acquiring $46.9\%$ of the conciseness gain.
With four experts, it retains $\approx90\%$ of both the accuracy gain and reasoning-correctness gain, compared to only $\approx57\%$ by the next best evaluated baseline.
Matched four-expertablations show that lexicographic routing outperforms random routing and that projection further strengthens both top-priority capabilities.
Across both scales, LMOPD preserves the highest-priority capabilities more effectively than the existing baselines we evaluate, demonstrating the value of explicit priorities for specialist integration.