Privileged Information Does Not Guarantee Privileged Guidance: Auditing On-Policy Self-Distillation for Reasoning Models

arXiv:2611.16064v1 Announce Type: new Abstract: On-policy self-distillation (OPSD) trains a student on next-token targets from a teacher, a frozen copy of the student that additionally receives privileged information, such as a worked solution, hidden from the student. Whether student gains reflect that information remains unclear. Taking the worked solution as privileged information, we audit four stages across three Qwen3 scales on mathematics: Source, whether the teacher can use the solution; Encoding, whether the resulting token targets are useful; Transmission, whether the student learns from the targets; and Expression, whether the student applies the learned behavior without the solution. With the solution, the teacher solves hard problems and recovers from student errors, yet the teacher does not reliably recognize errors, and its token targets do not favor correct trajectories. Repeated training makes successes within the student's reach more frequent rather than unlocking new ones, and corrections the student learns to predict are not reliably expressed in free generation. To isolate the solution's contribution, we vary its availability separately from teacher and student thinking modes. The largest gains occur when a teacher in thinking mode supervises a student trained and evaluated without an explicit thinking trace, and problem-only teachers often match or exceed privileged-solution teachers: the gain tracks the teacher's thinking mode rather than the solution, a pattern we call reasoning arbitrage. Accuracy gains alone therefore do not establish transfer of reasoning from privileged solutions: bottlenecks lie after Source, across Encoding, Transmission, and Expression. Future evaluations must measure what the solution adds to otherwise identical training.