Distilling Structure, Not Solutions: Structural Bottlenecks for On-Policy Self-Distillation

On-policy self-distillation (OPSD) aligns supervision with student-generated trajectories but retains an information asymmetry: the teacher receives a verified solution that is unavailable to the student at inference. We investigate whether restricting this privileged context to reasoning structure improves distillation. **Structural-Bottleneck OPSD (SB-OPSD)** replaces full solutions with compact, result-free skeletons that preserve ordered reasoning operations while excluding intermediate results and final answers. A frozen, skeleton-conditioned teacher supplies next-token targets at student-generated prefixes, while the student uses a question-only prompt throughout training and inference. Across Qwen3-1.7B, 4B, and 8B on three competition-mathematics benchmarks, SB-OPSD achieves the highest mean **Average@12** among the compared methods, improving over the respective base models by **5.70, 3.31, and 3.22 percentage points**. Structural controls favor compact, ordered, question-matched skeletons. Relative to full-solution OPSD, SB-OPSD also reduces teacher-context and output lengths and exhibits a smaller gap to its conditioned teacher during training. These results support compact reasoning structure as an effective form of privileged supervision.