Privileged Likelihood Is Not Automatically Value: Three Checks for Token Credit in On-Policy Self-Distillation

On-policy self-distillation aims to improve upon reinforcement learning from verifiable rewards (RLVR) by providing token-level scores derived from privileged information, such as reference solutions or critic feedback. These scores are treated as estimates of token-level action values, yet they answer a fundamentally different question: how the model's prediction changes when its input context is enriched, rather than how the expected outcome changes when a token is changed. We examine this gap along three dimensions: (i) whether the token-level score tracks task success; (ii) whether feedback generated from the same rollout causes the score to reflect agreement with its own description, and whether using feedback from other rollouts in the group mitigates this self-referential effect; and (iii) what behavior the resulting training objective actually reinforces. In experiments on AIME 2025, the implemented score distinguishes correct from incorrect rollouts at approximately chance level (AUC=0.505); using feedback from a different rollout does not consistently improve this discrimination; and all training configurations achieve only 24.2-33.9% Avg@4, compared with 64.2% for outcome-only GRPO. Moreover, the highest-entropy token decile accounts for 57-71% of the total absolute token-advantage mass, despite the score being least informative about reasoning quality in this regime. By contrast, similar experiments on SciKnowEval Biology improves held-out Avg@8 by 28.0%, while its corresponding trajectory scores achieve AUCs of 0.81-0.92. Together, these results suggest that dense credit assignment through distillation can be effective when its likelihood-based scores are empirically validated as meaningful proxies for outcome-relevant credit. When this alignment does not hold, however, the resulting supervision can fail to generalize and may substantially underperform outcome-based RL.