RL for LLM reasoning often reaches an early performance plateau as the policy distribution becomes increasingly sharp. Recent work has studied policy entropy as both a diagnostic and a control signal for this behavior. In our experiments, however, increasing entropy alone does not reliably change the eventual plateau. We analyze RL through its implicit target policy and derive a sharpness factor $κ$, together with an effective counterpart $κ_{\mathrm{eff}}$, that describes how strongly reward or advantage gaps are translated into probability-ratio gaps. Motivated by this view, we introduce reward budgeting, a plug-in advantage construction that assigns a fixed total reward budget to each prompt. For binary rewards and independent on-policy rollouts, its unclipped score-function estimator follows the gradient of an exact finite-$G$ concave objective with diminishing returns, while its advantage gap decreases as more rollouts become correct. For general rewards, a shifted log-mean interpretation holds asymptotically under the stated regularity conditions. These results motivate, rather than establish exact objective equivalence for, the clipped implementations. On Qwen3-8B and Llama3.2-3B-Instruct, reward budgeting consistently improves a range of RL baselines, with gains up to +12.50 on AMC23 and +10.00 on AIME24 and AIME25. Overall, our results support an update-dynamics view of premature convergence and show that strong gains can be obtained without explicitly preserving entropy.

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