A Privileged Teacher That Only Scores Cannot Search: Critique-Conditioned On-Policy Self-Distillation for Reasoning

On-policy self-distillation promises to replace the sparse outcome signal of verifiable-reward RL with dense token-level credit: a teacher conditioned on the reference solution scores the student's own rollout, and the resulting distributional shift is distilled back into the policy. We investigate whether this scoring signal actually contains corrective credit. Using gold error spans in student rollouts, we test the two signals consumed by self-distillation objectives, the full-distribution KL and the token log-ratio, for two properties: whether they localize errors and whether they push probability away from erroneous student tokens. They exhibit neither of these properties nor can they reliably distinguish correct from incorrect rollouts within the same problem. We argue that this failure is structural: locating an error in a reasoning trace is itself a search problem, while a scoring teacher is never explicitly asked to perform that search. The same model, with the same privileged information, localizes errors substantially better when allowed to generate a critique. This forms the motivation for \emph{critique-conditioned self-distillation}: first generate a critique of the current rollout, then condition the scoring teacher on that critique. The resulting signal localizes errors, is corrective within error spans, and shifts probability toward tokens that initiate backtracking such as ``Wait'', ``But'', and ``Actually''. We further show that the critique is useful both at test time, where resuming a wrong rollout with the critique in context converts math problems the model cannot solve by sampling, and in cold-start GRPO, where it provides an advantage signal on prompts for which verifiable rewards alone provide none.