RL for LLM reasoning often plateaus early as its policy sharpens. Policy entropy is studied as both a diagnostic and a control signal, but increasing it alone does not reliably change the eventual plateau in our experiments. We analyze RL's implicit target policy using a sharpness factor $κ$ and effective counterpart $κ_{\mathrm{eff}}$ that describe how reward or advantage gaps translate into probability-ratio gaps. We introduce reward budgeting, a plug-in advantage construction assigning a fixed total reward budget to each prompt. For binary rewards and independent on-policy rollouts, its expected unclipped score-function update is the exact gradient of a finite-$G$ concave objective with diminishing returns; its advantage gap decreases as more rollouts become correct. For general rewards, a shifted log-mean interpretation holds asymptotically under the stated regularity conditions. On Qwen3-8B and Llama3.2-3B-Instruct, reward budgeting consistently improves RL baselines, with gains up to +12.50 on AMC23 and +10.00 on AIME24 and AIME25. These results support an update-dynamics view of premature convergence and substantial gains without explicitly preserving entropy. Code is available at https://github.com/Qwen-Applications/RewardBudget.

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