Despite being powerful, reinforcement learning with verifiable rewards (RLVR) induces extremely long COT, making it computationally expensive. Since RLVR per-step cost is dominated by long-context rollout generation, sparse attention offers a promising way to accelerate dense rollout. However, sparse rollouts require a delicate stability-efficiency tradeoff: overly aggressive sparsity causes collapse, while overly lenient sparsity gives insufficient speedup. In this work, we study this tradeoff through sparse-to-dense actor-policy mismatch. We first observe that sparse rollout collapse is not driven by uniform degradation across tokens: most sparse tokens align perfectly with dense even under aggressive sparsity. Motivated by this, we hypothesize that sparse rollout training remains stable if the lower tail of per-token actor-policy mismatch stays above a critical threshold throughout the trajectory. We introduce a dynamic sparsity schedule that keeps this tail statistic constant during generation and validate our hypothesis. Across Qwen3 thinking-family models, keeping the tail mismatch statistic near a consistent threshold generally enables stable training. We then use a cost model to find the sparsity schedule for maximum speedup under this mismatch threshold, achieving 2.2x, 2.4x, and 2.0x rollout speedups when training Qwen3-1.7B, Qwen3-4B, and Qwen3-8B. Empirically, we show the thresholds generalize to a larger model (Qwen3-14B) and another RL domain (coding). Finally, our analysis naturally motivates DistillSparse: lightweight LoRA-based distillation on sparse rollout lets more aggressive sparsity reach the same sparse-to-dense mismatch threshold, yielding higher speedup.