Auditing the groups with the fewest weak passes first drove reward hacking in GRPO to near zero while using 28 percent of the labels a full audit needs

Reinforcement learning with verifiable rewards trains on cheap checks such as unit tests, which a policy can learn to game, while a stronger check is too expensive to run on every rollout. The paper works out how to spend a limited strong-verifier budget in GRPO, separating which weakly passing rollouts to audit from what reward a caught hack should receive. At a fixed policy state with centered advantages the expected update depends on the design only through each rollout's marginal audit probability and correction, reflecting a caught hack's reward about the failure reward negates the advantages of an all-hack group, and auditing whole groups with the fewest weak passes first, called low-pass budgeted audits, minimizes the advantage mass left on hacks. Tested in actual LLM training, the sign of that mass on a frozen panel predicted whether hacking would be suppressed in 33 of 39 runs, including 13 of 16 preregistered ones. In LeetCode and Countdown-Code, low-pass budgeted audits drive hacking to near zero, while random audits with the same penalty and budget do not and use more labels. Two unbiased corrections also brought hacking down but ended 6.1-11.8 and 8.0-11.5 points lower in held-out accuracy than the low-pass method at the same budget, while a full audit stayed within about two points of it in three seeds using 28% of its labels. The limitation: the prediction held in 33 of 39 runs rather than all of them, and the held-out accuracy cost was measured on two training environments.