Reinforcement learning with verifiable rewards (RLVR) supervises mathematical reasoning through final-answer correctness, but provides little guidance on individual tokens.
On-policy distillation (OPD) supplies dense feedback on student-generated responses, yet teacher preference need not reflect correctness.
Recent hybrids combine OPD and verifier-derived advantages or reweight task credit using teacher ratios.
However, teacher guidance enters after verifier-based group normalization, and token reweighting need not preserve the total task credit assigned to each response.
We introduce Unified Entropy-Calibrated Credit Redistribution for GRPO (UECR-GRPO), which integrates verifier and teacher signals within a single GRPO-style update at both the response and token levels.
Path-Utility Unification (PUU) combines verifier reward and a teacher-to-anchor path log-ratio in a single KL-regularized objective. Its on-policy implementation uses a length-normalized teacher score and combines both rewards before group normalization and PPO clipping, allowing teacher evidence to influence the response ranking. Entropy-Calibrated Redistribution (ECR) then uses the signed teacher--old-policy token gap to redistribute the verifier-derived component. Full-vocabulary teacher entropy attenuates uncertain guidance, while a response-wise zero-sum projection preserves the total task credit and its token-wise sign before clipping.Across five mathematical reasoning benchmarks, UECR-GRPO achieves average \(\mathrm{Avg@12}\) accuracies of 17.21\% and 65.09\% with Qwen3-1.7B and Qwen3-4B students, respectively, exceeding the strongest baseline at each scale by 0.89 and 0.56 percentage points.