Intervention-Certified Amplification: A Validity Gate for $σ$-Weighted Post-Training

Reinforcement learning post-training drives reasoning and agentic capabilities in modern AI systems, yet a growing body of work shows that it is most effective when used to fine-tune an already capable base model. We question whether existing pipelines yield models that are most suitable for reinforcement learning. Building on prior work highlighting the role of coverage and pass@K as predictors of post-RL performance, we design a simple modification to supervised fine-tuning, TailSFT, which filters out already fit sequences during training, thereby focusing learning on under-modeled regions, or the tail, of the data distribution. We justify and validate the design choices in TailSFT, particularly the specific filtering criteria, through a combination of controlled experiments and theoretical analysis. On OLMo-3 7B, TailSFT often improves pass@16 performance on math and coding evaluations, with gains up to 17% absolute, while incurring minimal computational overhead. These higher-coverage checkpoints consistently translate to up to 4% absolute pass@1 gains in subsequent GRPO runs, demonstrating that TailSFT checkpoints serve as better initializations for RL. We further introduce a lightweight diagnostic for identifying settings where TailSFT is most likely to help. More broadly, our results motivate a principled, stage-aware approach to model development, in which intermediate checkpoints are judged by how effectively they support subsequent training.

Post-training large language models usually applies a single training recipe to all samples, even though the model's own rollouts reveal different sample-level learning states. We propose Self-Routing, a behavior-conditioned post-training framework that uses rollout correctness and confidence to decide how each sample should be optimized. Depending on its behavior state, a sample is routed to GRPO, on-policy self-distillation, regularization, or skipping, allowing training to adapt without external teachers, extra annotations, or additional sampling. Experiments on mathematical reasoning across Qwen3 and Qwen3.5 backbones show that Self-Routing consistently improves over uniform GRPO, uniform OPSD, fixed mixtures, and simpler routing baselines. Further analyses show that the routing distribution changes over training and reduces unnecessary updates on low-signal or already stable samples.

Modern LLM post-training composes supervised fine-tuning (SFT), reinforcement learning with verifiable rewards (RLVR), and on-policy distillation (OPD) into multi-stage pipelines, yet these stages are typically designed and evaluated in isolation. We show that this composition is consequential: a stage that improves the current model can make the next stage less effective. Through controlled experiments with Qwen3 models on math and science reasoning, we first characterize OPD across nine student-teacher pairs spanning 2x to 53x parameter ratios and show that OPD effectiveness depends on student-teacher compatibility rather than teacher scale alone. The surrounding stages of OPD reshape this compatibility in three ways: (1) A brief SFT warm-up improves subsequent OPD, while an RLVR-strengthened student regresses under distillation from the same teacher. (2) Adapting the teacher with RLVR raises downstream OPD accuracy in proportion to the capability it adds. Following these two interventions, we find that combining teacher adaptation and student warm-up alone raise average OPD accuracy from 29.2% to 43.8% (50% relative improvement) after the same number of distillation steps, with additional preparatory training. (3) At comparable accuracy, OPD leaves a stronger initialization for downstream RLVR than SFT, with a gap that widens as RL compute scales. Our results suggest that each post-training stage should be chosen not only for the capability it adds, but for the learning interface it creates for the next stage.