# PICO

Verifiable-reward reinforcement learning hits a learning cliff on problems beyond the model's reach: when a rollout group is entirely wrong, group-relative advantage collapses to zero and the update carries no signal. PICO triggers distribution-constrained joint decoding only on all-failure groups, intersecting the teacher's next-token candidates with the student's while a dynamic cutoff keeps the sampling distribution inside a finite divergence budget, yielding trajectories that pass verification and stay close to the current policy. On Qwen3-4B-Base and Qwen3-8B-Base, PICO averages 32.0 and 35.4 across AIME25, AIME26, HMMT26, MATH500, GPQA-Diamond and HLE, against 27.9 and 31.8 for GRPO.

Verifiable-reward reinforcement learning hits a learning cliff on problems beyond the model's reach: when a rollout group is entirely wrong, group-relative advantage collapses to zero and the update carries no signal. PICO triggers distribution-constrained joint decoding only on all-failure groups, intersecting the teacher's next-token candidates with the student's while a dynamic cutoff keeps the sampling distribution inside a finite divergence budget, yielding trajectories that pass verification and stay close to the current policy. On Qwen3-4B-Base and Qwen3-8B-Base, PICO averages 32.0 and 35.4 across AIME25, AIME26, HMMT26, MATH500, GPQA-Diamond and HLE, against 27.9 and 31.8 for GRPO.